Download Three problems from quantum optics

Institute of Theoretical Physics and Astrophysics
Faculty of Science, Masaryk University
Brno, Czech Republic
Three problems from quantum optics
(habilitation thesis)
Tomáš Tyc
Brno 2005
2
Contents
1 Introduction
1.1 Quantum state sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Fermion coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Some important terms and concepts of quantum optics
2.1 Field operators and quadratures . . . . . . . . . . . . . .
2.2 Linear mode transformation . . . . . . . . . . . . . . . . .
2.3 Coherent states of light . . . . . . . . . . . . . . . . . . .
2.4 POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
5
7
7
.
.
.
.
9
9
10
12
13
.
.
.
.
.
.
.
.
.
.
.
.
14
15
15
16
17
18
19
20
20
21
22
23
23
3 Quantum state sharing
3.1 Continuous-variable quantum state sharing in the Schrödinger picture
3.1.1 Encoding the quantum secret . . . . . . . . . . . . . . . . . . .
3.1.2 Extraction of the quantum secret . . . . . . . . . . . . . . . . .
3.1.3 Example: the (2,3) threshold scheme . . . . . . . . . . . . . . .
3.1.4 Optimizing the secret extraction . . . . . . . . . . . . . . . . .
3.1.5 Finite squeezing in dealer’s encoding procedure . . . . . . . . .
3.2 Heisenberg picture of continuous-variable quantum state sharing . . .
3.2.1 Encoding the secret . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Extraction of the secret state by players 1 and 2 . . . . . . . .
3.2.3 Extraction of the secret state by players 1 and 3 . . . . . . . .
3.3 Experimental realization of the (2, 3) threshold scheme . . . . . . . . .
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
4 Homodyne detection
4.1 Homodyne detection as a phase-sensitive method . . . .
4.2 Why homodyne detection measures the field quadrature
4.3 POVM calculation using the SU(2) Wigner functions . .
4.4 POVM calculation using the Glauber-Sudarshan
P -representation . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Properties of the series expressing the probability
4.5 Strong local oscillator . . . . . . . . . . . . . . . . . . .
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
. . .
j
Pm
. . .
. . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
29
30
32
32
5 Fermion coherent states
5.1 Introduction . . . . . . . . . . . . . . . . . . .
5.2 The options for introducing coherent states of
5.3 Fermion analogy of the boson coherent state .
5.4 Properties of fermion correlators . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
34
34
34
36
37
3
. . .
light
. . .
. . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
25
. . . . . . . . . . . . . . . . . 25
. . . . . . . . . . . . . . . . . 26
. . . . . . . . . . . . . . . . . 27
.
.
.
.
.
.
.
.
5.5
5.4.1 Correlators of chaotic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4
Chapter 1
Introduction
This thesis is concerned with three problems from the field of quantum optics. In their choice I
was not motivated by attempting to explain quantum optics systematically but I rather talk about
problems I was working on in the past five years. The thesis is based on a set of articles that have
been published in international physical journals and are attached at the end of the thesis.
The first topic, quantum secret sharing, deals with protection of quantum information that is
physically realized by a quantum-optical system. The second topic, theory of homodyne detection, is
concerned with the description of one of the most important detection methods in quantum optics.
The third topic, fermion coherent states, deals with generalization to fermion fields of coherent states,
one of the key concepts of quantum optics. Each of these topics is described briefly below and in
detail in a separated chapter.
I have tried to write this thesis clearly so that a physicist not specialized in quantum optics can
understand it, and also that it can be of some use to a non-physicist. To some extent the thesis
re-tells the papers it is based on; at the same time, I have tried to include all the important results
and explain the steps that led to them so that a reader does not have to look into the papers too
often. This is also the reason why I have included Chapter 2 that explains some important terms
and concepts of quantum optics that are used in this thesis.
As I said, all the topics this thesis deals with are directly connected to quantum optics. This area
of physics is, as is clear from its name, the quantum theory of light. In many situations light behaves
as a wave governed by the laws of classical physics but sooner or later one comes across a situation
where the classical description is completely unsatisfactory and the quantum nature of light presents
itself in the full extent. It is enough just to sit down by a fireplace and think what is the color
of the light emitted by the glowing coals. Classical physics would give us a completely wrong answer
in the form of the “ultraviolet catastrophe” [1] while quantum optics allows to find the spectral
composition of the emitted light in a full agreement with the observation. And of course, quantum
optics offers much more. The consequences of the quantum nature of light are vast and many of
them are very practical. We just remind of the laser, which is a source of light commonly used
for precise measurements, communication, reading information media, for medical therapy etc., and
which can work thanks to the quantum properties of light. Also, quantum optics enables realization
of various cryptographic protocols, the security of which is guaranteed by the very laws of nature and
not e.g. by just computational difficulty. Last but not least, it is in quantum optics where the laws
of quantum physics often appear in a crystalline pure form and so it enables us to deeper understand
the rules that the world around us is governed by.
1.1
Quantum state sharing
At the present time, the importance of quantum optics for practical implementation of quantum
information protocols is growing as quantum states of light belong to the best carriers of quantum
5
Three problems from quantum optics
information [2]. Moreover, the experimental effort for realizing certain quantum-information protocols such as quantum teleportation has been most successful in quantum optics [3, 4, 5]. Quantum
information theory is a fast-developing interdisciplinary field that offers options that would otherwise be impossible or very difficult [6]. For example, quantum cryptography provides nowadays
a practically usable method for an unconditionally secure information transfer without the risk of
eavesdropping [7, 8]. At the same time, processing of quantum information in quantum computers
enables solving problems that would take an incomparably longer time on a classical computer (e.g.
billions of years compared to a few minutes) [9, 10], and simulating quantum systems that is highly
ineffective on a classical computer [11]. There are several algorithms that have been proposed for
quantum computers that are designed for solving very specific problems such as large number factorization or search in a database [12]. However, these algorithms have a relatively limited use and so
new algorithms that would exploit the full potential of quantum computers are still to be discovered.
Similarly, quantum computers themselves are waiting for their practical realization.
Quantum information differs significantly from its counterpart, the classical information. The
basic unit of quantum information is a quantum bit (qubit). A qubit can have, just as a classical
bit, the values 0 and 1, but it can also be in a so-called superposition of these two values. The
superposition is a way of simultaneous existence of the two options that a human has no direct
experience of, which makes it hard to imagine. A qubit is realized practically by a two-level quantum
system, e.g. the spin state of an electron, photon polarization or even by the options “there is
a photon in mode k” and “there is no photon in mode k”. In principle, one can perform similar logical
operations with qubits as with classical bits. However, it is not possible to copy (or clone) them,
which is a fundamental difference compared to classical information that can be copied arbitrarily.
The impossibility of copying quantum information is an important consequence of linearity of the
laws of quantum physics and it became known as the no-cloning theorem [13]. It is also connected
to the fact that it is not possible to read quantum information completely even if one possesses the
system carrying it; there is always some information that escapes, no matter what measurements one
performs on the system [14].
With the expansion of quantum information theory, there is a growing interest for its storage,
transfer and protection against misuse. More specifically, quantum teleportation enables to transfer quantum information between stations where it cannot be be sent physically (i.e., that are not
connected by a so-called quantum channel) [3, 5]. On the other hand, for protecting quantum information one can use the protocol of quantum state sharing that enables the access to the information
only based on collaboration between several participants; without such collaboration, the access is
denied completely.
In the last few years I have been working on the theory of quantum state sharing. With co-workers
I have achieved several results, the most important of which was proposing optical quantum state
sharing scheme and its experimental realization at the Australian National University in Canberra.
Theory of quantum state sharing and the experimental realization is discussed in Chapter 3. It
is based on the following articles:
(1) Tomáš Tyc and Barry C. Sanders, ”How to share a continuous-variable quantum secret by optical
interferometry”, Physical Review A 65, 042310 (2002)
(2) Tomáš Tyc, David Rowe and Barry Sanders, ”Efficient sharing of a continuous-variable quantum
secret”, Journal of Physics A 36, 7625 (2003)
(3) Andrew M. Lance, Thomas Symul, Warwick P. Bowen, Tomáš Tyc, Barry C. Sanders and Ping
Koy Lam, ”Continuous variable (2,3) threshold quantum secret sharing schemes”, New Journal
of Physics 5, 4 (2003)
(4) Andrew M. Lance, Thomas Symul, Warwick P. Bowen, Barry C. Sanders, Tomáš Tyc, Timothy
C. Ralph and Ping K. Lam, ”Continuous Variable Quantum State Sharing via Quantum Disentanglement”, Physical Review A 71, 33814 (2005).
6
Chapter 1. Introduction
1.2
Homodyne detection
Modern quantum optics is, to a large extent, an experimental discipline for which the precise measurement of a quantum state of light is of a key importance. One of the essential detection methods is
the homodyne detection based on interference of the measured light beam with a beam of well-known
properties (the so-called local oscillator). Homodyne detection is a phase-sensitive method and it
enables a direct measurement of quadratures, basic quantities used for describing the quantized electromagnetic field. Many important quantum-optical experiments are literally based on homodyne
detection that has become a standard experimental tool. Homodyne detection is the ultimate detection method in experiments with squeezed light, in quantum teleportation and cryptography with
so-called continuous variables and in many other situations.
Therefore it is surprising that until recently the full quantum description of homodyne detection was missing, especially the knowledge of POVM (its meaning will be explained in Chapter 2,
Sec. 2.4). The theory of homodyne detection was based on indirect calculations employing characteristic functions and quadrature moments of the electromagnetic field [15, 16, 17, 18, 19, 20], but
no direct derivation of the probability distribution of homodyne detector output was known. When
I discussed this problem with colleagues at a conference in Vienna in 2000, we decided to work on
finding the POVM for homodyne detection and directly calculating the corresponding probability
distribution. In the following two years we have managed to find the POVM including correction
terms that enables to describe the detection even in the non-ideal conditions of a weak local oscillator.
This result is important both for theoretical understanding of homodyne detection and for practical
application in which a weak reference field has to be used.
Theory of homodyne detection that I have developed in collaboration with Barry Sanders is
explained in Chapter 4 and it was published in the paper
(5) Tomáš Tyc and Barry C. Sanders, ”Operational formulation of homodyne detection”, Journal
of Physics A 37, 7341 (2004).
1.3
Fermion coherent states
For describing the quantized electromagnetic field, quantum optics uses an extended mathematical
formalism that is sometimes very elegant. One of the most important representations of quantum
states and operators is provided by coherent states of light that possess many useful physical and
mathematical properties [21]. These states exhibit high coherence, they are close to classical states
of light, do not change their character when subject to a linear mode transformation at a beam
splitter etc. Thanks to the so-called overcompleteness of the set of coherent states they can be
employed even for describing situations in which coherent states themselves do not take part, and
simplify calculations significantly.
Coherent states became best known in their connection with the electromagnetic field. Photons,
the quanta of this field, belong to bosons, the group of particles that tend to gather in the same
quantum state. This property is a consequence of their quantum indistinguishability – it is not
possible, even in principle, to distinguish two particles of the same sort. For the other type of particles,
the fermions, quantum indistinguishability has just the opposite effect: it is not possible to find more
than one fermion in the same state. This is expressed by the Pauli exclusion principle, one of the most
fundamental statements in quantum physics. Thanks to the common properties of all bosons, one can
easily extend coherent states to arbitrary boson fields. This raises a natural question: is it possible
to generalize the concept of coherent states also to fermion fields, for example electrons or neutrons?
Such a generalization is indeed possible and was performed several decades ago [22, 23, 24] based on
the so-called Grassmann numbers. However, the analogy with boson coherent states is just partial and
not direct. The largest problem is that the algebra of non-commuting Grassmann numbers provides
coherent states without any physical interpretation. Therefore I was thinking about introducing
7
Three problems from quantum optics
fermion coherent states in a direct analogy to the boson case without employing the Grassmann
numbers. When discussing this with my colleagues at Macquarie University in Sydney, we attacked
this problem and have achieved several results. We have shown that the desired generalization is
not possible and that the Grassmann variables are probably the only possibility how to introduce
fermion coherent states meaningfully. A side effect of our effort was deriving several theorems that
are valid for fermion correlation functions and that have no analogy for boson fields.
The problem of fermion coherent states is discussed in Chapter 5 that is based on the paper
(6) Tomáš Tyc and Barry C. Sanders, “Investigating complex fermion coherent states”, at the present
time in the review process in New Journal of Physics.
8
Chapter 2
Some important terms and concepts
of quantum optics
In this chapter we remind of some important terms that will be used in this thesis. We do not intend
to provide a complete introduction but rather mention some basic quantities and relations between
them so that a reader who is not trained in the area of quantum optics could read the following
chapters without having to look often into a quantum optics textbook.
2.1
Field operators and quadratures
Basic quantities used for describing the quantized electromagnetic field are creation and annihilation
operators ↠, â that are generally called field operators. The creation and annihilation operator raises
and lowers, respectively, the number of quanta (photons) in the field. If |ni denotes the state with n
photons (nth Fock state), then it holds
â|0i = 0,
√
â|ni = n |n − 1i
√
↠|ni = n + 1 |n + 1i
(n = 1, 2, 3, . . . )
(2.1)
(n = 0, 1, 2, . . . )
The creation and annihilation operators satisfy the commutation relations
[âi , â†j ] = δij 1̂,
[âi , âj ] = 0,
[â†i , â†j ] = 0,
(2.2)
where the indexes i, j distinguish individual modes, that is, the ways of the possible field oscillations.
It is well-known that the electromagnetic field is equivalent to a system of harmonic oscillators
and a single mode corresponds to a single harmonic oscillator. We will not show here this equivalence
as it is explained in most textbooks of quantum optics (see e.g. [21]). For a given mode of the field
one can define dimensionless position and momentum operators of the corresponding oscillator as
x̂ =
â + â†
√ ,
2
p̂ =
â − â†
√ .
i 2
(2.3)
The operators x̂ and p̂ satisfy the canonical commutation relation [x̂, p̂] = i 1 that follows from the
relations (2.2). The quantities x̂ and p̂ are called quadratures of the field and they are fundamental for
describing quantum-optical phenomena with so-called continuous variables. The word “continuous”
is related to the fact that the spectrum of the quadratures is continuous in contrast to the discrete
spectrum of the photon-number operator n̂ = ↠â. Instead of x̂ and p̂, scaled quadratures X̂ ± are
1
We take the Planck constant ~ to be equal to unity; otherwise the commutator would be equal to i~
9
Three problems from quantum optics
often used that are defined as X̂ + =
a general quadrature
√
√
2 x̂ and X̂ − = 2 p̂. Along with x̂ and p̂ one can also define
âe−iϕ + ↠eiϕ
√
(2.4)
2
as a linear combination of x̂ and p̂ that can be interpreted as a rotated position in the phase space
of the given mode.
Continuous-variable quantum information protocols correspond to analog systems in the classical
information theory, while discrete-variable protocols correspond to digital systems. As the digital
technology has some clear advantages over the analog technology, discrete variables are often preferred
also in quantum information theory. However, continuous variables have other advantages such as
possibility of manipulating with quantum information by linear optical elements.
For the mathematical description of mode transformation one can employ the continuous basis
of the Hilbert space, namely the basis of the eigenstates |xi of the quadrature operator x̂ such that
x̂ϕ = x̂ cos ϕ + p̂ sin ϕ =
x̂|xi = x|xi.
(2.5)
Some operations with the states |xi may be problematic from the rigorous mathematical point of view
as these states cannot be properly normalized, which is the case of some calculations in Chapter 3.
However, it turns out that by a relatively simple way, namely introducing the so-called Gelfand
triplet [25, 26] one can make the explained theory mathematically rigorous.
2.2
Linear mode transformation
It is well-known that physical quantities that characterize a quantum system can change in time.
The time evolution of a quantum system can be described in two different ways, namely using the
Schrödinger and Heisenberg pictures. The Schrödinger description views the operators corresponding
to physical quantities as fixed and the evolution is given by changing the quantum state in the Hilbert
space. On the other hand, the Heisenberg description considers the quantum state to be fixed and
the evolution is ascribed to the operators of the physical quantities. Both approaches are equivalent,
and none of them should be regarded as “more correct”.
The time evolution of a given mode of the electromagnetic field proceeds spontaneously due to the
nonzero energy of the mode and due to its interaction with various optical elements. The spontaneous
evolution is of little interest as it is given by a uniform phase change; we will concentrate on the
evolution caused by the optical elements. A beam splitter, phase shifter and squeezer are typical
such elements. They have one or two input modes and the same number of output modes that can
be considered as the transformed input modes. In the Schrödinger picture the elements transform
the quantum state while in the Heisenberg picture they transform the field operators describing the
modes.
An important class of mode transformations is formed by linear canonical (symplectic) transformations for which the output quadratures can be expressed as linear combinations of the input
quadratures. We consider here a special case only, namely the so-called point transformations, for
which the positions and momenta transform separately and do not mix. Such as general transformation of m modes can be expressed in the Heisenberg picture as
x̂0i
=
m
X
Tij x̂j ,
p̂0i
=
j=1
m
X
Sij p̂j
(i = 1, . . . , m) ,
(2.6)
j=1
where the matrices T and S satisfy S = (T −1 )T . The corresponding transformation in the Schrödinger
picture changes the eigenstate of the quadratures x̂1 , . . . , x̂m according to
¯m
¯m
+ ¯m
+
+
¯X
¯X
¯X
p
¯
¯
¯
T1k xk ¯
|x1 i1 |x2 i2 · · · |xm im → | det T | ¯
T2k xk · · · ¯
Tmk xk
.
(2.7)
¯
¯
¯
k=1
1 k=1
10
2
k=1
m
Chapter 2. Some important terms and concepts of quantum optics
p
The indexes at the kets | . . . i label the modes of the field and the factor | det T | ensures the correct
normalization of the states or, in other words, the unitarity of the transformation (2.7).
As can be seen by comparing Equations (2.6) and (2.7), the eigenvalues in the Schrödinger picture
transform in the same way as the quadratures in the Heisenberg picture. This must be so because the
eigenvalues are measurable quantities and both pictures have to provide an equivalent description.
There is a special class of so-called passive transformations among (2.6) and (2.7) for which the
matrices T and S are orthogonal. These transformations can be realized experimentally by passive
optical elements only, i.e., linear mode couplers (usually beam splitters) and phase shifters. On the
other hand, realizing a non-orthogonal transformation requires employing active elements such as
optical parametric oscillators and it is much more challenging experimentally.
The simplest example of passive mode transformation is a phase shift that does not, however,
belong to the the point transformations as it mixes position and momentum, and therefore we will
not consider it here. Another example is mixing two modes on a beam splitter (e.g. a half-silvered
glass) that can be expressed as
µ 0 ¶ µ
¶µ
¶
x̂1
cos θ/2 − sin θ/2
x̂1
=
.
(2.8)
x̂02
sin θ/2
cos θ/2
x̂2
For a symmetric beam splitter with both transmissivity and reflectivity equal to 50%, it holds θ = π/2
and therefore
x̂1 + x̂2
x̂1 − x̂2
,
x̂02 = √
.
(2.9)
x̂01 = √
2
2
The simplest example of the active transformation is a single-mode squeezing operation:
x̂
,
p̂0 = sp̂ ,
(2.10)
s
where s is the squeezing factor. For |s| > 1, the operation (2.10) squeezes the quadrature x̂ and
for |s| < 1 it squeezes p̂. In practice the squeezing operation is realized e.g. by a degenerate downconversion in an optical parametric oscillator (OPA) pumped by a beam of double frequency. With
some probability amplitude a pump photon in the nonlinear crystal realizing OPA can change into
a pair of photons of the mode being transformed, and the opposite process is also possible. If one
transforms the vacuum, that is, the state with the wavefunction
1 −x2 /2
ψvac (x) = hx|vaci = √
e
(2.11)
4
π
x̂0 =
by the single-mode squeezer, then the output state in the Schrödinger picture will be
p
|s| −s2 x2 /2
ψs (x) = hx|si = √
e
.
4
π
(2.12)
Clearly, this state differs from the vacuum (2.11) for s 6= ±1 and hence its expansion in the Fock
basis hn|si must have nonzero coefficients for some n > 0. Therefore the state |si contains photons
that were added by the transformation (2.10), which is where the name “active” comes from. Is is
not hard to show that hn|si 6= 0 for n even and hn|si = 0 for n odd. This is related to the realization
of the squeezing transformation mentioned above – photons emerge in pairs and if there was no
photon in the field initially, there can only be an even number of them after the squeezing operation.
One can show [27] that an arbitrary matrix T from Eq. (2.6) can be decomposed as T = O 2 DO1 ,
where the matrices O1 and O2 are orthogonal and the matrix D = diag (d1 , . . . , dm ) is diagonal.
Therefore the transformation (2.6) or (2.7) can be realized in three steps (see Figure 2.1): the first
and last steps are passive transformations corresponding to the matrices O 1 and O2 , respectively. The
middle step consists of m single-mode squeezing operations corresponding to the diagonal elements
of the matrix D and scaling the quadrature x̂i and p̂i by the factor di and 1/di , respectively. Thus the
number of active elements needed for realizing an arbitrary symplectic transformation of m modes
does not exceed the number of modes.
11
Three problems from quantum optics
1
2
3
S
S
S
PI
m
PI
1’
2’
3’
m’
S
Figure 2.1: Decomposition of a general symplectic transformation of m modes: first the modes are combined
in a passive interferometer (PI), then each mode undergoes a squeezing transformation (S) individually and
finally the modes are combined in another passive interferometer.
2.3
Coherent states of light
Coherent states play an important role in quantum optics for their numerous useful physical and
mathematical properties. First of all, they are states that are closest to classical states of light and
that exhibit large coherence. Coherent states have the minimum product of uncertainties of the
quadratures x̂ and p̂; in both x-, and p-representations they are Gaussian wavepackets. Another
useful property of coherent states is their elegant transformation on a linear mode coupler. Coherent states also have interesting mathematical properties that enable constructing representations
very useful for describing quantized electromagnetic field. One of them is the Glauber-Sudarshan
P -representation [28, 29, 30] that will be discussed in a moment. Thanks to their physical and mathematical properties, coherent states are useful for various optical measurements, as local oscillators
for homodyne detection, for pumping down-converters and squeezers, as testing states for quantum
teleportation, quantum state sharing etc.
With respect to what we just said about the importance of coherent states, it may be surprising
that they have not yet been realized at optical frequencies as was emphasized by K. Mølmer [31]
and B. C. Sanders and T. Rudolph [32]. Hence, coherent states are a “convenient fiction” rather
than a physical reality. For example, laser light is not in a coherent state as one often hears in the
community of quantum opticians, but rather in a mixture of coherent states with equal amplitude
and with the phase distributed uniformly over the interval h0, 2π) [21]. However, when describing an
experiment with a laser source using coherent states instead of their mixtures, one does not make
a serious mistake; the result expected by the theory is the same in both cases because the measured
beam and the reference beam (local oscillator) are derived from the same source and hence have a
fixed relative phase. This way, most experiments that one should, strictly speaking, describe using
mixtures of coherent states can equivalently be described using pure coherent states.
Coherent states can be defined by several equivalent ways that will be discussed in Chapter 5;
here we mention just the most common definition. Coherent state is the eigenstate of the annihilation
operator â, i.e., the state satisfying â|αi = α|αi for some complex number α. This definition yields
immediately the expansion of the coherent state in the Fock basis:
|αi = e
−|α|2 /2
∞
X
αn
√ |ni .
n!
n=0
(2.13)
The photon number distribution in the coherent state |αi is Poissonian with both the mean and
variance equal to |α|2 .
It is an important property of coherent states that they provide the following decomposition of the
unit operator:
Z
1
1̂ =
|βihβ| d2 β ,
(2.14)
π
where the integration runs over the whole complex plane. At the same time, no two coherent states
2
are orthogonal as |hα|βi|2 = e−|α−β| . These two properties have an interesting consequence – any
12
Chapter 2. Some important terms and concepts of quantum optics
state |ψi from the Hilbert space can be expressed as a superposition of coherent states by an infinite
number of ways. To show this, assume that |ψi itself is a coherent state. Then
Z
|αi = δ 2 (β − α) |βi d2 β
(2.15)
certainly holds, where δ 2 (β) = δ(Re β) δ(Im β) is the two-dimensional Dirac delta-function. At the
same time, using the unit operator expansion (2.14) one arrives at
Z
Z
1
1
∗
2
2
2
|αi =
|βihβ|αi d β =
eβ α−(|β| +|α| )/2 |βi d2 β.
(2.16)
π
π
Equations (2.15) and (2.16) give two different and valid expansions of the state |αi in terms of coherent
states.
Similarly, a general density matrix ρ̂ of the mode can be expressed in many different ways as
follows,
Z Z
ρ̂ =
ρ(β, γ)|βihγ| d2 β d2 γ.
(2.17)
There is so much freedom in the choice of the function ρ(β, γ) that it enables something seemingly
impossible: one can choose it such that ρ(β, γ) = 0 for β 6= γ, that is, the density matrix can be
expressed in terms of coherent states in a diagonal way:
Z
ρ̂ = P (β)|βihβ| d2 β.
(2.18)
P (β) is the so-called Glauber-Sudarshan P -function; it has some unusual properties and for many
states it is a distribution rather than an ordinary function. This is quite natural with respect to
the very strong requirement of diagonality of ρ(β, γ). For a coherent state ρ̂ = |αihα| one has
1 −|β|2 /N
P (β) = δ 2 (β − α), for a thermal state with the mean photon number N it is P (β) = πN
e
and
for a Fock state |ni the P -function is proportional to the nth derivative of the Dirac delta-function
δ 2 (β). The P -function forms the basis for mathematical description of homodyne detection as will
be discussed in Chapter 4.
2.4
POVM
POVM (positive operator-valued measure) is a set of positive-semidefinite Hermitian operators Π̂i
that characterizes completely a given quantum-mechanical measurement. If ρ̂ is the density matrix
of the system, the probability of the ith measurement
output is given by pi = Tr (ρ̂Π̂i ). The operators
P
Π̂i satisfy the unit operator decomposition i Π̂i = 1̂.
The POVM is a generalization of a projective quantum-mechanical measurement. A general
measurement can be performed by adding an ancilla system in a known state to the measured
system and making a projective measurement on the composite system [33].
13
Chapter 3
Quantum state sharing
Quantum state sharing is an important quantum-information protocol. Its goal is to protect a quantum information (called quantum secret) that is distributed among a group of parties (called players)
against its misuse by unauthorized groups of players and to enable the access to this information
to other, authorized groups. Initially, the dealer who owns the quantum information in the form
of a quantum state of a given system encodes this state into an entangled state of n quantum systems and distributes these systems to the individual players. The encoding is performed in such a
way that for any authorized group of players there exists a unitary operation that the players can
apply to their systems (called shares) and in this way obtain one system in the same state as was
the original secret. This is called secret reconstruction or extraction. On the other hand, the density
matrix of the systems of the players from any unauthorized group is independent of the secret and
hence the unauthorized groups cannot get any information about the secret, no matter what operations they perform with their shares. At first sight, it might seem odd that such a protocol can exist
at all.
It is important to note that the quantum secret may be in a mixed state and it can even be
entangled with another quantum system. In this case, the entanglement is recovered after the secret
extraction. This way it is possible to share e.g. just a component of a quantum state of a larger
system.
The method of the secret encoding is a public information and it is closely related to the so-called
access structure, which is the set of all authorized groups of players that should be able to extract
the secret. The access structure cannot be arbitrary but it must satisfy certain conditions. An
obvious rule is that when adding a player to an authorized group, it remains authorized. Another
condition says that there cannot exist two separated (disjoint) authorized groups of players. If this
were possible, one could create two copies of the extracted secret from a single original secret state,
and this way effectively clone a quantum state. However, cloning quantum states is impossible, as
has been shown by W. K. Wootters and W. H. Zurek [13] (the no-cloning theorem). The condition
does not apply to classical secret sharing, which the classical analogy of quantum state sharing; the
secret in this case is a classical information that can be copied or cloned arbitrarily.
Among quantum state sharing schemes there is an important class of the so-called self-dual access
structures with the following property: for every division of all players into two groups, exactly one
group is authorized. It turns out that any access structure that is not self-dual can be derived from
some self-dual one by discarding one or more shares [34]. Therefore exploring only self-dual structures
is sufficient for describing quantum state sharing. Another important class of access structures are
the so-called threshold schemes for which it is only the number of players in the group that determines
whether the group is authorized or not. For the (k, n) threshold scheme there are n players in total
and any k of them are authorized to extract the secret. It can be seen easily that self-dual structures
are those for which n = 2k − 1 holds; in the following we will consider these structures only.
Quantum state sharing can be implemented is quantum systems described by both discrete and
14
Chapter 3. Quantum state sharing
(a)
(b)
(c)
Figure 3.1: Three examples of access structures; only the minimal authorized sets are shown. The access
structure in (a) is not allowed in quantum state sharing as two disjoint groups of players can access the secret;
however, it is allowed in classical secret sharing; the access structure in (b) is allowed also in the quantum
case, and (c) shows the access structure of the (2, 3) threshold scheme in which any two players can access the
secret.
continuous variables. In discrete variables where the secret is realized as qubits (or more generally
qudits), the encoding can effectively be designed by employing properties of matrices over finite
number fields, and the theory of quantum state sharing is well developed [35, 36, 34]. The theory of quantum state sharing with continuous variables was developed by me, B. C. Sanders and
D. J. Rowe at Macquarie University in Sydney [37, 38]. Later we have, together with co-workers
at Australian National University in Canberra, proposed [39] and realized successfully [40, 41] an
experiment that demonstrated quantum state sharing for the first time. The proposed scheme was
designed for optical implementation and the quantum systems carrying the secret and the shares
were realized as modes of the electromagnetic field. The fundamental quantities used for describing
the quantum system are the field quadratures that have been discussed in Sec. 2.1.
Originally, we have formulated the theory of continuous-variable quantum state sharing in the
Schrödinger picture [37, 38] in analogy to the discrete-variable case. However, later the Heisenberg
picture was preferred [39, 40, 41] (to compare both pictures, see Sec. 2.2). In the following section we
will describe the Schrödinger approach and in Sec. 3.2 the Heisenberg approach to continuous-variable
quantum state sharing.
3.1
Continuous-variable quantum state sharing in the Schrödinger
picture
In the following we explain continuous-variable quantum state sharing in the Schrödinger picture on
the example of the (k, 2k − 1) threshold scheme that has total 2k − 1 players and any k of them can
extract the secret. Generalization of the protocol to an arbitrary access structure is straightforward.
3.1.1
Encoding the quantum secret
The first step in the protocol is the encoding of the quantum secret into an entangled state of 2k − 1
modes of the field and distributing these modes to the players. The initial state of the dealer is
formed by 2k − 1 modes of the electromagnetic field: the first of them is the quantum secret
Z
|ψi =
ψ(x) |xi dx
(3.1)
R
and the remaining 2k − 2 are squeezed vacuum states. Half of them, that is k − 1, are squeezed in the
quadrature p̂, so they are the states from Eq. (2.12) with s < 1, and the other half are squeezed in
the quadrature x̂ so they are the states |si with s > 1. In order for the secret extraction to be perfect,
the squeezing must be infinite, which corresponds to the limits s → 0 and s → ∞, respectively. In
this case one can express both states as
Z
|xi dx and |0i .
(3.2)
R
15
Three problems from quantum optics
In the following we will assume this ideal case of infinite squeezing. The more realistic situation
of finite squeezing will be discussed in sections 3.1.5 and 3.2 that talks about the (2, 3) threshold
scheme and its experimental realization.
In the ideal situation of infinite squeezing, the initial state of the dealer is
Z
|Φ0 i =
ψ(x1 ) |x1 i1 |x2 i2 · · · |xk ik |0ik+1 · · · |0i2k−1 dk x ,
(3.3)
Rk
and the indexes of the kets mark modes of the field. The dealer then applies a particular symplectic
transformation (see Eq. (2.7)) to the state |Φ0 i to create the following entangled state:
Z
ψ(x1 ) |L1 (x)i1 |L2 (x)i2 · · · |L2k−1 (x)i2k−1 dk x .
(3.4)
|Φi =
Rk
Here x denotes the set of variables x1 , x2 , . . . , xk and Li (x) with i = 1, . . . , 2k − 1 are linear combinations of the variables x1 , x2 , . . . , xk that satisfy a certain condition that ensures that any k players can
extract the secret. The condition is that any k elements from the 2k-element set {x 1 , L1 , . . . , L2k−1 }
must be linearly independent. By a proper choice of L1 (x), . . . , L2k−1 (x) one can ensure that the
transformation |Φ0 i → |Φi is orthogonal. This means that the dealer does not need active operations
for encoding the quantum secret but only for creating the initial squeezed states.
3.1.2
Extraction of the quantum secret
Next we show how a group of k players can extract the quantum secret. Without loss of generality
we can assume that the first k players collaborate, in the opposite case we can relabel the players.
When thinking about the linear combinations L1 (x), . . . , L2k−1 (x) of the variables x1 , . . . , xk , it
is useful to view these objects as vectors in a k-dimensional vector space V with the basis vectors
x1 , . . . , xk . This makes our considerations much clearer. It then follows from our assumptions
about x1 , L1 (x), . . . , L2k−1 (x) from last section that the vectors L1 , L2 , . . . , Lk as well as the vectors
x1 , Lk+1 , . . . , L2k−1 are linearly independent. At the same time, in both groups there are k vectors,
which is the same number as the dimension of the vector space V . Therefore there must exist
a non-singular matrix T such that

 

x1
L1
 L2   Lk+1 

 

(3.5)
T . =

..

 ..  
.
L2k−1
Lk
holds. The existence of the matrix T implies the existence of a unitary operator Û (T ) acting on the
modes 1, 2, . . . , k as follows:
p
Û (T )|L1 i1 |L2 i2 · · · |Lk ik = | det T | |x1 i1 |Lk+1 i2 · · · |L2k−1 ik .
(3.6)
Now, if players 1, 2, . . . , k apply the operation Û to their shares, the total state of all shares will
be
Û |Φi = J
=J
p
p
| det T |
Z
Rk
ψ(x1 ) |x1 i1 |Lk+1 i2 · · · |L2k−1 ik |Lk+1 ik+1 · · · |L2k−1 i2k−1 dx1 dLk+1 · · · dL2k−1
| det T | |ψi1 |Θi2,k+1 |Θi3,k+2 · · · |Θik,2k−1 .
(3.7)
In the integral in Eq. (3.7) we changed the integration variables x1 , . . . , xk to x1 , Lk+1 , . . . , L2k−1 , we
have denoted the Jacobian of this transformation by J and have defined a two-mode state
Z
|xii |xij dx .
(3.8)
|Θiij ≡
R
16
Chapter 3. Quantum state sharing
Eq. (3.7) shows that the first player’s share is left in the state |ψi, so the secret is extracted in its
original form in mode 1. The shares of players 2, 3, . . . , k form strongly entangled pairs |Θi ij with
the shares of players k + 1, . . . , 2k − 1 who did not participate in the extraction process.
The state |Θiij is the EPR (Einstein-Podolsky-Rosen) state that A. Einstein and co-workers used
in 1935 in their famous paper [42] attacking completeness of quantum mechanics. As can be seen
from Eq. (3.8), in the EPR state the quadratures x̂i , x̂j are perfectly correlated; at the same time,
the quadratures p̂i , p̂j are perfectly anticorrelated. The EPR state |Θiij plays an important role in
many continuous-variable quantum-information protocols, e.g. in quantum teleportation [3, 4, 5].
It still remains to show that unauthorized groups cannot obtain any information about the quantum secret. For this it is enough to know that the access structure of the (k, 2k − 1) threshold scheme
is self-dual, so the complement of any unauthorized group is a group that can extract the secret
perfectly. This itself denies any information leakage to the unauthorized group as such a leakage
would prevent the authorized group from perfect extraction of the secret. This is because any information about a quantum state that escapes to the environment changes the state of the system. This
is a general property of quantum states and it forms the basis for important quantum-information
protocols, in particular of quantum key distribution [7]. To show that a group of k − 1 players cannot
get any information about the secret, one can also calculate the trace of the total state of all shares
|ΦihΦ| over the shares of the remaining k shares. It is not hard to show that the resulting density
matrix is independent of the secret |ψi, so it cannot provide any information about it.
3.1.3
Example: the (2,3) threshold scheme
In this section we illustrate the quantum state sharing protocol on the example of the (2, 3) threshold
scheme in which there are three players in total and any two of them can obtain the quantum secret
by collaboration. This scheme is important in that it has been realized experimentally, which will be
discussed in Sec. 3.2 in detail.
The initial state of the dealer |Φ0 i consists of the quantum secret |ψi and two states squeezed
infinitely in the quadratures p̂ and x̂, respectively:
Z
ψ(x1 ) |x1 i1 |x2 i2 |0i3 dx1 dx2 .
(3.9)
|Φ0 i =
R2
The dealer chooses the following linear combinations L1 , L2 , L3 according to Eq. (3.4):
x1
x2
L1 = √ + ,
2
2
x1
x2
L2 = √ − ,
2
2
x2
L3 = √
2
and employing a passive transformation (2.7) with the orthogonal matrix
 1

1
1
√

T =
2
√1
2
0
2
− 21
√1
2
(3.10)
2
− 21 

− √12
he encodes |Φ0 i into the three-share entangled state
¯
À ¯
À ¯
À
Z
¯ x1
¯ x2
¯ x1
x
x
2
2
¯√ −
¯√
|Φi =
dx1 dx2 .
ψ(x1 ) ¯¯ √ +
2 1¯ 2
2 2¯ 2 3
2
R2
(3.11)
(3.12)
The dealer then distributes the shares to the players.
Players 1 and 2 can extract the secret via a passive transformation (2.7) with the matrix
¶
µ
1
1
1
.
(3.13)
T12 = √
2 1 −1
17
Three problems from quantum optics
The resulting state
¯
À ¯
À
¯ x2
¯ x2
¯√
ψ(x1 )|x1 i1 ¯¯ √
=
dx1 dx2
(3.14)
2 2¯ 2 3
R2
clearly contains the quantum secret |ψi in mode 1.
Players 1 and 3 can extract the secret via an active transformation (2.7) with the matrix
µ √
¶
2 −1
√
T13 =
,
(3.15)
1 − 2
Z
|Φ012 i
which yields the state
|Φ013 i
¯
À ¯
À
¯ x1
¯ x1
x2
x2
¯
¯
√
√
−
=
ψ(x1 ) |x1 i1 ¯
dx1 dx2
−
2 2¯ 2
2 3
2
R2
Z
ψ(x1 ) |x1 i1 |L2 i2 |L2 i3 dx1 dL2
=2
Z
R2
(3.16)
(3.17)
and hence the secret is again reconstructed in mode 1.
The secret extraction from components 2 and 3 is almost identical to the extraction from components 1 and 3, so we will not discuss it.
3.1.4
Optimizing the secret extraction
On order to realize the k-mode transformation (3.6) for extracting the quantum secret, the collaborating players have to employ k active optical elements (squeezers) in general (see Sec. 2.2). However,
it would be highly desirable to reduce the number of active elements in some way because of their
high experimental cost and difficulty. In out work [38] we have shown that the transformation (3.6) is
not the only one that enables the secret extraction, and by optimizing the extraction procedure one
can reduce the number of active elements down to two, independent of the number of collaborating
players k.
To understand this, we return to Eqs. (3.6) and (3.7) and note that even though the variable x 1
is still present in the linear combinations Lk+1 , . . . , L2k−1 , by changing the integration variables to
x1 , Lk+1 , . . . , L2k−1 it was possible formally eliminate it. In this way the quantum secret in the first
mode has been disentangled from all the other modes. The same would be achieved, however, if the
matrix T from Eq. (3.5) was replaced by a matrix T 0 that satisfies

 

x1
L1
 L2   M2 (Lk+1 , . . . , L2k−1 ) 

 

(3.18)
T0  .  = 
,
..

 ..  
.
Lk
Mk (Lk+1 , . . . , L2k−1 )
where M2 , . . . , Mk are linear combinations of the vectors Lk+1 , . . . , L2k−1 . Also in this case the first
mode is disentangled from all the other modes and yields the quantum secret. The only difference is
that the modes of the collaborating players 2, . . . , k would no more form EPR pairs with the modes
of the non-collaborating players but rather a more complicated entangled state.
Equation (3.18) provides a large freedom thanks to the possibility of choosing the linear combinations Mi (the only condition is that the matrix T 0 is non-singular). To minimize the number
of squeezing elements, we tried to find the T 0 to be close to some orthogonal matrix. We have shown
that the matrix T 0 can be found in the form


α β 0 ... 0
 0 γ 0 ... 0 




0
T = 0 0
(3.19)
O.
 . .

 .. ..

Ik−2
0 0
18
Chapter 3. Quantum state sharing
Here O is an orthogonal matrix, Ik−2 is the unit matrix of dimension k − 2, the numbers α and β
are determined by vectors L1 , . . . , L2k−1 , and γ is a free parameter. Hence, one can decompose the
extraction transformation
into
µ
¶ a passive operation corresponding to the matrix O and a two-mode
α β
transformation R ≡
(as the remaining k − 2 modes are no more transformed). For the two0 γ
mode transformation R2 one needs just two squeezing elements and it can be decomposed according
to Sec. 2.2 as R = O2 DO1 Altogether, the transformation T 0 can be realized in three steps (see
Fig. 3.2): first comes the passive operation O1 O followed by two single-mode squeezing operations
corresponding to D and the last step is another passive operation O2 . Hence no more than two active
operations are needed to extract the quantum secret. Furthermore, by choosing γ one can minimize
the overall squeezing cost of the two squeezers.
1
2
3
PI
S
S
PI
T
k
Figure 3.2: The optimum extraction of the quantum secret: the k modes are first combined in a passive
interferometer, then the first two of them are squeezed individually and finally the two modes are combined
in a passive interferometer. One of the outputs is then the extracted secret T.
3.1.5
Finite squeezing in dealer’s encoding procedure
Until now we have assumed that the dealer uses 2k − 2 infinitely squeezed states for his encoding
procedure (see Sec. 3.1.1). In practice this is not possible, however, as the mean number of photons
and mean energy are infinite in an infinitely squeezed state. Nowadays one can achieve the squeezing
factor of the order of several units only, so an important question arises of how the protocol will
work if the squeezing employed by the dealer is finite. In this case the secret will not be extracted
perfectly but will be degraded increasingly with decreasing amount of squeezing used by the dealer.
At the same time, the protection of the secret against unauthorized groups will no more be perfect,
and some information can escape to them.
In order to quantify the quality of the secret extraction, it is useful to express the density matrix
ρ̂out of the extracted secret with the help of the density matrix ρ̂ of the original secret. In our
work [38] we have derived the following relation between the two matrices in the x-representation:
· 2
¸Z
· 2 2¸
s
u (x − x0 )2
s y
0
0
exp −
ρ(x − y, x − y) exp − 2 dy .
ρout (x, x ) = √
(3.20)
2
4s
v
πv
R
Here s denotes the squeezing parameter of the squeezed vacuum states employed by the dealer and
u, v are parameters depending on the dealer encoding operation and the choice of the collaborating
players. The factor in front of the integral in Eq. (3.20) reduces the magnitude of the non-diagonal
elements of the density matrix and the integral itself convolutes the secret density matrix with
a Gaussian, which both degrades the secret. It would be even more advantageous to express this
degradation in terms of the Wigner function W (x, p) that provides description equivalent to that
of the density matrix, but symmetrical with respect to quadratures x̂ and p̂. It turns out that the
Wigner function of the extracted secret Wout (x, p) is a two-dimensional convolution of the original
Wigner function with a Gaussian function of x, p.
As the parameters u and v differ in general for different groups of collaborating players, the
degradation of the extracted secret differs as well. It this sense, the protocol is “unjust” as some
19
Three problems from quantum optics
Figure 3.3: The encoding of the secret in the (2, 3) threshold scheme: the dealer first creates squeezed ancilla
states by squeezing the vacuum states in two optical parametric oscillators (OPA), and combines them on
a symmetric beam splitter. One of the outputs is then combined with the quantum secret state on another
beam splitter. This way the dealer obtains three shares that he distributes to the players.
groups can extract the secret better than other ones1 . I believe that for a large number of players
one cannot design a “just” protocol because the dealer does not have enough parameters to vary in
order to satisfy the large number of conditions requiring equal secret degradation for all authorized
groups of players.
3.2
Heisenberg picture of continuous-variable quantum state sharing
During the period of theoretical preparation of the quantum state sharing experiment at the Australian National University it turned out that the Heisenberg picture is more advantageous in some
aspects than the Schrödinger picture for describing the protocol, in particular for the easier treatment of finitely squeezed states in the dealer’s process. In this section we describe the (2, 3) threshold
scheme in the Heisenberg picture that has been realized experimentally. Formally, the transition from
the Schrödinger to Heisenberg pictures is simple and it is explained in the basic course of quantum
mechanics. However, in a particular calculation it may not be trivial and we will not perform this
transition here, but rather describe quantum state sharing in the Heisenberg picture directly.
When using the Heisenberg picture, one has to consider both quadratures x̂ and p̂ of the transformed modes. This is in contrast with the Schrödinger picture where we used the x-representation
and did not consider the momenta at all as the wavefunction provided a complete information about
a pure quantum state.
3.2.1
Encoding the secret
As has been said in Sec. 3.1.3, the dealer owns initially the quantum secret (we will label its quadratures by the index S) and two ancilla squeezed states, one squeezed in the quadrature p̂ and the
other one squeezed in x̂. We will label the quadratures of these squeezed states by the indexes sqz1
and sqz2. Due to the squeezing, the
√ uncertainties of p̂√sqz1 and x̂sqz2 are lower than would be for the
vacuum state |0i, so ∆psqz1 < 1/ 2 and ∆xsqz2 < 1/ 2 hold. The dealer encodes the secret by the
1
For example, in the (2, 3) threshold scheme discussed in Sec. 3.1.3 players 1 and 2 can still extract the secret
perfectly even if the dealer employs finite squeezing, while players 1 and 3 or 2 and 3 cannot.
20
Chapter 3. Quantum state sharing
!"
01 325476
$#"
0 4%B24B=
')(
01 32548
051 32548
>@?@A 0 =BC=4
01 3254:9
051 3254:9
&%"
>@?@A 0 =BD=4
01 325476
01 32548
01 3254:9
*
')(
*
>@?@A 0 E= D=4
051 325476
&+"
T I FUTWVYXQZ
0 4;%<254<=
=NM GCOPFHG Q+ 4 05R IQ4<4;S<4;2
051 325476
')(
051 32548
0 4;%<254<=
051 3254:9
-/.
',(
FHG +JIJKLD= G 2
0 4%B24B=
Figure 3.4: The extraction of the secret by the authorized groups {1, 2} a {1, 3} in the (2, 3) threshold scheme.
(a) Players 1 and 2 simply combine their shares on a beam splitter [BS]; players 1 and 3 (or similarly 2 and 3)
have several options: they can employ (b) a two-mode squeezer to transform their shares, (c) a combination
of two beam splitters and two single-mode squeezers, or (d) a non-symmetric beam splitter combined with
a homodyne detector [HD] and an electro-optical modulator.
transformation (2.6) with the matrix (3.11) to obtain the shares with the following quadratures:
√
√
x̂1 = x̂S / 2 + (x̂sqz1 + x̂sqz2 )/2,
p̂1 = p̂S / 2 + (p̂sqz1 + p̂sqz2 )/2
√
√
x̂2 = x̂S / 2 − (x̂sqz1 + x̂sqz2 )/2,
p̂2 = p̂S / 2 − (p̂sqz1 + p̂sqz2 )/2
(3.21)
√
√
p̂3 = (p̂sqz1 − p̂sqz2 )/ 2.
x̂3 = (x̂sqz1 − x̂sqz2 )/ 2,
This transformation can be realized passively in two steps (see Fig. 3.3). First, the squeezed ancillas
are combined on a symmetric beam splitter, thus forming an approximate EPR pair (see Eq. (3.8)),
that is, a pair of entangled beams with correlated quadratures x̂ and anticorrelated quadratures p̂.
One of the beam splitter outputs is then combined with the secret state on another symmetric beam
splitter whose outputs yield the first two shares; the last share is the second beam of the EPR pair.
The three shares are then distributed to the players.
3.2.2
Extraction of the secret state by players 1 and 2
If players 1 and 2 wish to extract the secret, they simply combine their shares on a 1:1 beam splitter
(see Fig. 3.4 (a)). This way, the dealer’s and players’ operations effectively form a Mach-Zehnder
interferometer whose output replicates the input and thus yields the quantum secret. The quadratures
of the beam splitter outputs are
√
√
x̂1out = (x̂1 + x̂2 )/ 2 = x̂S , p̂1out = (p̂1 + p̂2 )/ 2 = p̂S ,
(3.22)
√
√
√
x̂2out = (x̂1 − x̂2 )/ 2 = (x̂sqz1 + x̂sqz2 )/ 2, p̂2out = (p̂sqz1 + p̂sqz2 )/ 2.
21
Three problems from quantum optics
Eqs. (3.22) show that the quadratures of the first output are identical to the quadratures of the
original secret, so the secret is extracted. It is also important that the quadratures x̂ sqz1,2 a p̂sqz1,2
are not contained in the output quadratures x̂1out , p̂1out and hence players 1 and 2 can extract the
secret state with an arbitrary precision, independent of the amount of squeezing employed by the
dealer.
3.2.3
Extraction of the secret state by players 1 and 3
Extraction of the secret by players 1 and 3 is more complicated than for players 1 and 2 because
of the asymmetry of shares 1 and 3 with respect to the content of the anti-squeezed quadratures x̂ sqz1
a p̂sqz2 in the quadratures x̂1,3 a p̂1,3 (see Eqs. (3.21)). These anti-squeezed quadratures have to be
eliminated, which can be achieved in several ways.
Ideally, players 1 and 3 perform the two-mode active operation
√
√
x̂1out = 2 x̂1 − x̂3 ,
p̂1out = 2 p̂1 + p̂3 ,
√
√
x̂2out = −x̂1 + 2 x̂2 ,
p̂2out = p̂1 + 2 p̂2 ,
(3.23)
which yields the following quadratures of the first output:
√
√
x̂1out = x̂S + 2 x̂sqz2 ,
p̂1out = p̂S + 2 p̂sqz1 .
(3.24)
If the squeezing of the quadratures p̂sqz1 and x̂sqz2 is infinite, the state of all shares is an eigenstate
of these quadratures with the eigenvalue zero. Then one can omit p̂sqz1 and x̂sqz2 in Eqs. (3.24),
which yields x̂1out = x̂S and p̂1out = p̂S . This means that for infinite squeezing of the quadratures
p̂sqz1 and x̂sqz2 , the secret is exactly replicated at the first output. However, for finite squeezing the
extraction is not perfect and the quantum noise (uncertainty) of the quadratures p̂ sqz1 and x̂sqz2 is
transfered to the extracted secret.
In principle, the transformation (3.23) could be achieved directly by employing a two-mode
squeezer realized by a phase insensitive amplifier [43] (see Fig. 3.4 (b)), which is, however, infeasible experimentally. Another option is to use a pair of symmetric beam splitters and two singlemode squeezers realized by phase-sensitive parametric amplifiers (see Fig. 3.4 (c)), which is also very
challenging experimentally. Therefore we have proposed an alternative extraction method that is
experimentally feasible; it employs linear optical elements, homodyne detection and electro-optical
modulation.
Secret extraction via electro-optical modulation
The electro-optical modulation method for extracting the quantum secret has the major advantage
of being feasible experimentally. However, the disadvantage is that even for infinite squeezing in the
dealer setup, the secret is not extracted in its original form but is subject to a unitary squeezing
transformation.
In this scheme, shares 1 and 3 are first interfered on a beam splitter with transmissivity 2/3 and
reflectivity 1/3 (see Fig. 3.4 (d)). The quadrature x̂ of one output is then measured via homodyne
detection (see Chapter 4) and the detected signal is imparted onto the x̂ quadrature of the second
beam splitter output via an electro-optic modulator. The beam splitter reflectivity and other parameters are chosen such that the anti-squeezed quadratures x̂sqz1 and p̂sqz2 of the dealer’s ancilla states
cancel in the output. The quadratures of the output beam are then
x̂out =
√
√
3 (x̂S + 2 x̂sqz2 ),
√
1
p̂out = √ (p̂S + 2 p̂sqz1 ).
3
(3.25)
For infinite squeezing in the dealer procedure, these equations become x̂ out = 31/2 x̂S and p̂out =
3−1/2 p̂S , which means that this method reconstructs the secret up to a squeezing transformation with
22
Chapter 3. Quantum state sharing
√
the factor s = 1/ 3 (see Eq. (2.10)). To obtain the secret in its original form, it would be necessary
to invert this squeezing transformation, which would require additional quantum resources. On the
other hand, from the perspective of quantum information theory the reconstructed secret contains all
the information that was in the original secret as both the secrets differ by a unitary transformation.
From this point of view, the electro-optical modulation method can be considered as an adequate
method for the quantum secret extraction.
3.3
Experimental realization of the (2, 3) threshold scheme
Shortly after the idea of continuous-variable quantum state sharing arose, we started to discuss
a possible experimental realization of this protocol with our colleagues from the Australian National
University in Canberra. The university has an excellent experimental background in continuousvariable quantum information and several important results [44] have been achieved there. We
decided to work on the (2, 3) threshold scheme that is not trivial and is feasible at the same time,
and we have chosen the electro-optical modulation method for the secret extraction by players 1 and
3 (see Sec. 3.2.3). Two years later the scheme was realized successfully [41, 40], which was the first
realization of quantum state sharing.
The light source for the experimental setup is a Nd:YAG laser at 1064 nm wavelength that pumps
a second harmonic generator based on a non-linear crystal MgO : LiNbO 3 . The resulting frequencydoubled light is used to pump two MgO : LiNbO3 optical parametric amplifiers that produce two
beams squeezed 4.5±0.2 dB below the vacuum noise limit. The squeezed beams are mixed on a beam
splitter to produce a pair of approximate EPR beams. The quantum secret state is represented by
a coherent state at the sideband separated by 6.12 MHz from the carrier wave. The secret is mixed
with one beam of the EPR entangled pair, which yields the first two shares, and the last share is the
remaining beam of the EPR pair (see Fig. 3.3). To increase the security of the scheme, additional
Gaussian noise is added onto the three shares using electro-optic modulation techniques. This noise
does not degrade the secret extracted by the authorized groups while it reduces the information that
can escape to adversary players if the dealer uses finite squeezing.
For quantifying the quality of the extraction, we defined the extraction fidelity (overlap) for a pure
secret state |ψi as F = hψ|ρ̂out |ψi [45], where ρ̂out is the density matrix of the extracted secret. If
F = 1, the secret is perfectly extracted. We also used criterions of added noise and signal transfer.
It was relatively easy to extract the secret from shares 1 and 2 by simply combining them at the
beam splitter with the phase set properly (See Fig. 3.4 (a)). The best fidelity achieved for players 1
and 2 was F{1,2} = 0.95 ± 0.05, which is a value fairly close to unity and a very good result.
To extract the secret from shares 1 and 3, we employed the electro-optical modulator method
(See Fig. 3.4 (d)). In order to determine the overlap of the extracted and original secrets, an
a posteriori symplectic transform was applied to the extracted state. The calculated fidelity was up
to F{1,3} = 0.62 ± 0.02. If the quantum secret had been shared classically, i.e., without squeezing in
the dealer procedure, the highest achievable fidelity would have been 1/2. As the fidelity achieved in
the experiment exceeded this value, the quantum nature of our protocol was proved.
3.4
Conclusion
With my Australian colleagues from Macquarie University in Sydney I have introduced a general
protocol for sharing quantum states in continuous variables and optimized it with respect to the
number of active (squeezing) operations. We have shown that for extracting the quantum secret by
any number of collaborating players, only two active elements are needed. Further, together with the
colleagues from Australian National University in Canberra I have proposed the experimental realization of the (2, 3) threshold scheme and the experiment was later completed successfully. This way,
23
Three problems from quantum optics
the collection of practically feasible quantum-information protocols has been increased by another element. Even though quantum state sharing is nowadays interesting mainly from the theoretical point
of view, it will probably become an important tool for protecting data in future quantum-information
technologies.
24
Chapter 4
Homodyne detection
Homodyne detection is an important detection method in modern quantum-optical experiments [46,
47] that is used especially when working with continuous variables. It is based on interference of the
detected field with a coherent beam of the local oscillator and measuring the intensity difference
of the resulting beams. At proper conditions, homodyne detection effectively measures the field
quadratures and thus it is a phase-sensitive method.
The standard theory of homodyne detection [15, 16, 17, 18, 19, 20] is developed based on several
approaches mostly using characteristic functions and quasiprobabilistic distributions in the phase
space. The standard theory clearly showed the connection between the field quadrature and the
quantities directly measured by a homodyne detector; however, it did not provide a complete description of homodyne detection, in particular the explicit derivation of the corresponding POVM
(see Sec. 2.4). I was attracted by this problem in 2000 and after discussions with Barry C. Sanders
I started working on it. During my stay at Macquarie University in Sydney I managed to find the
POVM of homodyne detection by two different methods. Both of these methods are based on a direct calculation of the probability distribution of the detection outcome and they differ by the way
of calculation. We will explain both methods in Sections 4.3 and 4.4 of this chapter but first we
introduce homodyne detection in mode detail and say a few words about the idea of the standard
description of homodyne detection.
4.1
Homodyne detection as a phase-sensitive method
There are situations, especially in modern quantum optics, where one needs to measure the intensity
E of the electric field associated with a certain electromagnetic wave. At low frequencies, it is possible
to detect E directly (e.g. from the force that the field acts on electrons in an antenna) and so one
can determine both the amplitude and phase of the field. However, for a number of reasons such
a direct measurement is not possible at optical frequencies because of the impossibility of processing
an electronic signal of an optical frequency, problems with the reference time etc. Moreover, at
optical frequencies the quantum nature of light presents itself significantly, i.e., the fact that light
interacts with matter in the form of quanta (photons). A detector of light has to absorb a quantum
of energy in order to report a detection event and so the most common method of light detection is
based on absorption of photons at photodetectors. As the operators describing such photodetection
are diagonal in the Fock basis, photodetection alone cannot provide information about the phase
of the field but only about its intensity. Indeed, it follows from the uncertainty relations that if one
knows the number of photons in the field, its phase is completely unknown.
However, the phase information can be accessed by interfering the measured field with a reference
field with known properties, which is typically done in holographic imaging. Homodyne detection
is based on the same principle – the measured field (called signal field) is interfered with a local
oscillator beam on a beam splitter (usually a half-silvered mirror). The resulting two output modes
25
Three problems from quantum optics
are then subject to a photodetection that measures the photon numbers in the ideal case. If the
local oscillator is in a coherent state with a large amplitude (for the exact condition see Sec. 4.5),
then the photon number difference at the two outputs is closely related to field quadrature operator
x̂ϕ (see Eq. (2.4)) with ϕ being the phase of the local oscillator. More precisely, the probability
ˆ approaches the distribution of the quadrature
distribution of the scaled photon number difference ∆
ˆ one can thus measure the field quadrature just as if one measured the position
x̂ϕ . By detecting ∆
of the harmonic oscillator that represents the mode of the field. Optical homodyne detection has
been taken over from electronics where it is quite common – a homodyne detector can be found
almost in every radio or television receiver.
4.2
Why homodyne detection measures the field quadrature
It is not trivial to show the connection between the photon number difference at a homodyne detector
with the field quadrature. Now we explain one way how one can see this connection, which is no
rigorous proof, however. The standard description of homodyne detection is based on this idea but
it is much more elaborated.
We will consider here the balanced homodyne detection that uses a symmetric beam splitter with
transmissivity and reflectivity equal to 50%. The annihilation operators of the output modes â 01 , â02
of a symmetrical beam splitter are connected with the input modes operators â 1 , â2 by the relations
â01 =
â1 − â2
√
,
2
â02 =
â1 + â2
√
.
2
(4.1)
ˆ at the two beam splitter outputs can then be expressed
The photon number difference operator ∆
in terms of the input operators as
ˆ = n̂01 − n̂02 = â01 † â01 − â02 † â02 = −↠â2 − ↠â1 .
∆
1
2
(4.2)
Now, assume the first input mode to be in a coherent state | − Aeiϕ i with A > 0 (the minus sign is
ˆ is then
convenient for further calculations). The expectation value of the operator ∆
√
ˆ = Ae−iϕ hâ2 i + Aeiϕ h↠i = 2 A hx̂ϕ i.
(4.3)
h∆i
2
ˆ is, up to a multiplicative factor, equal to the expectation value of the quadrature x̂ ϕ .
Hence h∆i
ˆ
Similarly, one can calculate the second moment of the operator ∆:
¡
¢
ˆ 2 i = A2 e−2iϕ hâ22 i + 2h↠â2 i + e2iϕ h↠2 i + 1 + h↠â2 i = 2A2 hx̂2ϕ i + h↠â2 i.
h∆
(4.4)
2
2
2
2
√
ˆ 2 i = h( 2 A x̂ϕ )2 i.
This equation shows that for a large amplitude A, it holds approximately h∆
Expressing higher moments as well, one can show that for A → ∞,
√
ˆ n i → h( 2 A x̂ϕ )n i.
h∆
(4.5)
√
ˆ and 2 A x̂ϕ are the same for A → ∞, then also
holds. Now, if all the moments of quantities ∆
√
ˆ
2A we
their statistical distributions should be the same. This means that by measuring X̂ϕ ≡ ∆/
ˆ has
effectively measure the quadrature x̂ϕ . One could make an objection here that the quantity ∆
a discrete spectrum while x̂ϕ has a continuous spectrum so the two quantities cannot have the same
ˆ is unity)
probability distribution. However, for A → ∞ the step of X̂ϕ goes to zero (as the step of ∆
so the spectra of both quantities become practically equal in the limit of large A.
The method we have just explained shows connection between the field quadrature and the photon
number difference at the beam splitter outputs, but it is not fully sufficient for describing homodyne
detection. For a more precise description one would have to show how the difference of the n th
26
Chapter 4. Homodyne detection
moments of X̂ϕ and x̂ϕ depends on the amplitude A as n grows and in what sense the convergence (4.5)
occurs. Moreover, in practice is is not possible to increase the local oscillator amplitude arbitrarily,
so for practical use of homodyne detection it is important to know the connection between X̂ϕ and x̂ϕ
for finite A. Most importantly, the explained method does not allow to find the POVM of homodyne
detection that would show the direct correspondence of the probability of finding a given photon
number difference ∆ and the probability for the quadrature to have a given value x.
4.3
POVM calculation using the SU(2) Wigner functions
(a)
j−m
ψ
α
j+m’
(b)
j+m
j−m’
Figure 4.1: Balanced homodyne detection scheme: (a) the input state |ψi is mixed with a local oscillator in
coherent state |αi, and photodetection is performed at the two output ports; (b) the probability amplitude
of finding j ± m photons at the beam splitter outputs provided there were j ± m 0 photons at the inputs is
given by the Wigner function djmm0 .
The first method of finding the POVM of homodyne detection that we have developed [48] works
with the photon number (Fock) basis and it can only be used for large amplitudes A of the local
oscillator. The matrix elements of a beam splitter in this basis are given by so-called Wigner SU(2)
functions that were originally defined in the angular momentum theory [49]. The Hilbert space H F
of a pair of modes of the electromagnetic field is isomorphic with the Hilbert space H J of a quantum
system described by the operators Jˆx , Jˆy , Jˆz satisfying the usual angular momentum commutation
relations [Jˆi , Jˆj ] = iεijk Jk 1 . The operators Jˆx , Jˆy , Jˆz are related to the field operators â1,2 , â†1,2 of the
pair of the modes by the Schwinger boson representation [50]
1
Jˆx = (â†1 â2 + â†2 â1 ),
2
i
Jˆy = − (â†1 â2 − â†2 â1 ),
2
1
Jˆz = (â†1 â1 − â†2 â2 )
2
(4.6)
and the commutation relations mentioned above follow from the commutation relations of the field
operators. The basis of the Hilbert space HJ is given by the states {|jmi} with 2j = 0, 1, 2, . . . and
m = −j, −j + 1, . . . , j that are the eigenstates of Jˆ2 and Jˆz with the eigenvalues j(j + 1) and m,
respectively. In the state |jmi there are j + m photons in the first mode and j − m in the second
mode. The beam splitter operator is given by
ˆ
B̂(θ) = e−iθJy ,
(4.7)
where for the symmetric beam splitter θ = π/2 holds. The state |jmi is transformed on the beam
splitter as
X j
B̂(θ)|jmi =
dm0 m (θ) |jm0 i,
(4.8)
m0
ˆ
and djm0 m (θ) = hjm0 |e−iθJy |jmi are the SU(2) Wigner functions, that is, the matrix elements of the
beam splitter transformation (4.7) in the basis |jmi.
1
εijk denotes the Levi-Civita symbol that is equal to 1 and −1 for ijk an even and odd permutation of the numbers
1,2,3, respectively, and equal to zero if some of the numbers i, j, k coincide
27
Three problems from quantum optics
If the signal state is |ψi and the amplitude of the local oscillator is −A (we set the phase ϕ to
zero for simplicity), then the beam splitter input state is | − Ai|ψi. We can express this state in
the basis {|jmi} and by applying the transformation (4.8) to it we obtain the output state in the
j
same basis. The probability amplitude Mm
of finding j + m and j − m photons at the beam splitter
outputs is then equal to the coefficient at the state |jmi in this decomposition, that is,
j
Mm
= hjm|B̂(π/2)| − Ai1 |ψi2 = e−A
2 /2
2j
X
(−A)2j−n j
dm,j−n (π/2) ,
ψn p
(2j − n)!
n=0
(4.9)
with ψn being the nth coefficient in the Fock basis expansion of the signal state |ψi. The corresponding
j
j
probability Pm
is equal to the square of the modulus of the amplitude Mm
.
j
The key step in the calculation of the the probability Pm
is using the asymptotic form of the
Wigner functions for large j that was derived in [51]:
p
djm,j−n (π/2) ≈ (−1)n j −1/4 un (m/ j).
(4.10)
Here un (x) = hx|ni denotes the x-representation of the nth stationary state of the harmonic oscillator
with the Hamiltonian Ĥ = (x̂2 + p̂2 ) ω/2, i.e.,
2
e−x /2
√
un (x) = √
Hn (x),
4
π 2n n!
(4.11)
and Hn (x) is the Hermite polynomial.
j
Using various expansions we have arrived at the following result for the probability P m
that holds
for large A:
2 2
2
e−(2j−A ) /2A
j
√
Pm
=
hx|ρ̂|xi.
(4.12)
π A2
√
Here |xi denotes the eigenstate of the quadrature x̂ with the eigenvalue x = m/ j, and ρ̂ = |ψihψ|
is the density matrix of the signal state. The corresponding POVM is then
2 2
Π̂jm =
2
e−(2j−A ) /2A
√
|xihx|.
π A2
(4.13)
The projection operator |xihx| is of a key importance here. As the POVM is proportional to it
means that homodyne detection really measures the field quadrature x̂. The factor in front of |xihx|
j
in Eq. (4.13) is connected with the normalization of the POVM and the fact that the probability P m
is related not only to the photon number difference but also to the photon number sum. Eq. (4.13)
shows that the total photon number 2j has the Gaussian distribution with both the mean value and
dispersion equal to A2 . This is not surprising as the Poissonian distribution of the photon number in
the local oscillator converges to such Gaussian distribution for A → ∞; even though there are some
photons from the signal state among the 2j photons total, they do not influence the distribution
of 2j much as there is a negligible minority of them for very large A.
The probability Pm of finding the photon number difference 2m at the beam splitter outputs
j
regardless of the photon number sum 2j is equal to the sum of Pm
over all possible j:
Pm =
∞
X
j
Pm
(4.14)
j=|m|,|m|+1,...
In the limit of large A this sum can be evaluated by replacing the summation by integration. Also,
for large A the width of the distribution
of 2j has a very small relative width, so the eigenvalue
√
√
x = m/ j can be replaced by x = 2 m/A. The result is then
1
hx|ρ̂|xi.
Pm = √
2A
28
(4.15)
Chapter 4. Homodyne detection
√
√
√
The factor 1/ 2A in Eq. (4.15) is connected by the Jacobian 2 m/A of the map m → x = 2 m/A
and with the fact that m changes in half-integer steps. The probability Pm is normalized properly
as for large A, Eq. (4.15) yields
Z
X
(4.16)
Pm = hx|ρ̂|xi dx = Tr ρ̂ = 1.
m
R
We also mention the situation of a general phase of the local oscillator. If the local oscillator is
in a coherent state | − Aeiϕ i, then hx| and |xi in Eqs. (4.12), (4.13) and (4.15) have to be replaced by
which are the left- and right-eigenstates of the quadrature x̂ϕ , respectively, with the
ϕ hx| and |xiϕ ,√
eigenvalue m/ j. Hence, by setting the phase of the local oscillator one can choose what quadrature
will be measured by the homodyne detector.
In this way, we have shown for the first time by a direct calculation that the POVM of homodyne
detection is proportional to the projector |xihx|. However, this method did not allow us to find
correction terms to Eqs. (4.12) and (4.13) for small amplitudes A, the main reason being the absence
of correction terms in Eq. (4.10). This problem can be overcome by using a different method that
works with coherent states instead of Fock states.
4.4
POVM calculation using the Glauber-Sudarshan
P -representation
The advantage of working in the coherent state basis is the extremely simple description of the
beam splitter transformation in this basis. This optical element transforms a pair of coherent states
|α1 i, |α2 i into another pair of coherent states as follows:
B̂(θ)|α1 i ⊗ |α2 i = |α1 cos
θ
θ
θ
θ
− α2 sin i ⊗ |α1 sin + α2 cos i,
2
2
2
2
(4.17)
and θ = π/2 for a symmetric beam splitter. The transformation of the coherent state amplitudes is
the same as the corresponding transformation of annihilation operators would be in the Heisenberg
picture.
Let the (generally mixed) signal state ρ̂ be represented by the Glauber-Sudarshan P -function
P (β) (see Eq. (2.18)) and let the local oscillator coherent state be again | − Ai. The beam splitter
input state is then
Z
ρ̂in = | − Aih−A| ⊗
and the output state is
Z
ρ̂out = P (β)
P (β) |βihβ| d2 β
(4.18)
¯
¯ ¯
¯
À ¿
À ¿
¯ −A − β
−A − β ¯¯ ¯¯ −A + β
−A + β ¯¯ 2
¯ √
√
√
√
⊗
d β.
¯
2
2 ¯ ¯
2
2 ¯
1
2
(4.19)
j
The probability Pm
of finding j + m and j − m photons at the beam splitter outputs is then
j
Pm
= 2 hj − m| 1 hj + m|ρ̂out |j + mi1 |j − mi2
(4.20)
and we have used Eq. (2.13) for its calculation. To be able to simplify the resulting expressions
³
´j±m
β
containing powers such as 1 ± A
, we have used the expansion
"
n
(1 + x) = exp[n ln(1 + x)] = exp n
29
∞
X
(−1)k−1 xk
k=1
k
#
,
(4.21)
Three problems from quantum optics
that is based on the Taylor expansion of the logarithm and is of key importance for the calculation.
The radius of convergence of the series in the exponent in Eq. (4.21) is equal to unity. Therefore
we had to make sure that the expansion was not used for |x| ≥ 1 and hence for |β| ≥ A. However,
when performing the integral in Eq. (4.19), the variable β runs over the whole complex plane and
the expansion can therefore be used only if
P (β) = 0 for |β| ≥ A
(4.22)
holds. We will come back to this condition later.
With the help of Eq. (4.21) and after some algebra we have arrived at the following expression
j
for the probability Pm
:
j
Pm
·
µ
√ −2j −A2 4j 2m2 /A2 ½
π2
e
A e
2j − A2 2
=
Tr ρ̂ : |xihx| exp −
{â + (↠)2 }
(j + m)! (j − m)!
2A2
¶¸ ¾
∞
∞
X
X
1
â2k−1 + (↠)2k−1
1 â2k + (↠)2k
+ 2m
−j
: , (4.23)
2k − 1
k
A2k−1
A2k
k=2
k=2
√
2m
. The normal-ordering symbol : : should be
where the eigenvalue of the quadrature is x = A
understood such as all the creation operators stand to the left of the projector |xihx| and all the
annihilation operators stand to the right of it, that is,
: |xihx|âr (↠)s : = (↠)s |xihx|âr .
(4.24)
The exponential in Eq. (4.23) can be expanded as a series with an increasing number of the creation
j
and annihilation operators, which yields the probability Pm
in the form of the following series:
j
Pm
=
√ −2j −A2 4j 2m2 /A2 ½
2j − A2
π2
e
A e
hx|ρ̂|xi −
[hx|â2 ρ̂|xi + hx|ρ̂(↠)2 |xi]
(j + m)! (j − m)!
2A2
¾
2m
3
† 3
+
[hx|â ρ̂|xi + hx|ρ̂(â ) |xi] + . . . , (4.25)
3A3
and the corresponding POVM is
Π̂jm
=
√
2
π 2−2j e−A A4j e2m
(j + m)! (j − m)!
2 /A2
½
2j − A2
|xihx| −
[|xihx|â2 + (↠)2 |xihx|]
2A2
¾
2m
3
† 3
[|xihx|â + (â ) |xihx|] + . . . . (4.26)
+
3A3
It is not hard to show that for large A the first term in the parentheses in Eq. 4.25 dominates. As the
corresponding POVM is proportional to the projector |xihx|, homodyne detection clearly measures
the field quadrature for a strong local oscillator. Furthermore, for large A the fraction before the
parentheses in Eq. (4.25) approaches the fraction in Eq. (4.12), which makes the results (4.12)
and (4.25) in the limit A → ∞ equal. However, Eq. (4.25) yields, in contrast to Eq. (4.12), correction
terms that express how much the real homodyne detection POVM differs from the ideal quadrature
POVM and that can be used for describing homodyne detection with a weak local oscillator. Also
these correction terms were derived for the first time.
4.4.1
Properties of the series expressing the probability Pmj
j
Before we analyze the result (4.25) for the probability Pm
, let us first return to the validity condition
of the calculation. As we have mentioned, the use of expansion (4.21) is the key step in the derivation
30
Chapter 4. Homodyne detection
j
of Pm
and it is allowed only if the condition (4.22) holds. In other words, the support of the P function of the signal state has to lie inside the circle with the radius A, which is clearly a very strong
condition. We have shown that this condition is satisfied by the class of so-called z-regular states
(with z < A) that we defined as states whose coefficients in the Fock basis decrease asymptotically
at least as quickly as the coefficients of the coherent state |zi. Examples of z–regular states include
coherent states with the amplitude smaller than z, all Fock states and superpositions or mixtures
of a finite number of such states. At the same time, many states that are typically subject to
homodyne detection do not satisfy the criterion (4.22), e.g. squeezed states or thermal states with
a non-zero mean number of photons. This might be a serious problem as one always comes across
some thermal noise in real experiments and our calculation would hence not be useful for describing
such experiments. However, we will show now that this problem can be avoided due to the interesting
properties of the P -representation and one can use the series (4.25) also for the states that do not
satisfy the condition (4.22).
First consider the situation when the density matrix ρ̂ has just a finite number of terms in the
Fock basis, for example for the signal state in a superposition of a finite number of Fock states.
Such a state is z-regular for any z > 0 and the calculation is hence correct. Moreover, there exists
a number N such that the density matrix satisfies ρmn ≡ hm|ρ̂|ni = 0 for all m > N, n > N . Now, in
the terms in the parentheses in Eq. (4.25) all the annihilation operators are to the left from ρ̂ and all
the creation operators are to the right of ρ̂. It then follows that the series has only a finite number
of elements as the number of the field operators increases gradually.
Next consider the situation when the P -function of the state ρ̂ does not satisfy the condition (4.22),
for example for a squeezed or thermal state. We will show that even in this case one can use the
POVM by the procedure of the state truncation. We define the state ρ̂ [N ] for a given N ∈ N as
follows:
(¡P
¢−1
N
ρmn
for m ≤ N, n ≤ N
[N ]
i=1 ρii
ρmn =
(4.27)
0
else
This definition truncates the state ρ̂ in the Fock basis and normalizes the resulting state. Clearly,
j
(ρ̂[N ] ) calculated for the
for N → ∞ the state ρ̂[N ] converges to ρ̂. Therefore the probability Pm
j
state ρ̂[N ] converges to the probability Pm (ρ̂) for the state ρ̂. Hence, for any ρ̂ one can find N0 large
j
j
enough such that the probabilities Pm
(ρ̂) and Pm
(ρ̂[N0 ] ) differ at an arbitrarily small level. Then
j
j
the probability Pm (ρ̂) is approximated by a finite series for Pm
(ρ̂[N0 ] ) to a very high precision, even
j
though the series expressing Pm (ρ̂) itself possibly diverges. From the practical point of view this
means that the the series (4.25) can be used for evaluating the POVM even for the state for which
it diverges.
These properties of the series (4.25) may seem quite odd and they are connected with the following
properties of the P -function. The P -function of the Fock state |ni is equal to zero for all β 6= 0 2 .
The same applies to the P -function of an arbitrary truncated state. Of course, for a general (nontruncated) state P (β) can be nonzero also for β 6= 0. If we define a sequence {ρ̂ [N ] , N = 0, 1, . . . } for
a general state ρ̂ according to Eq. (4.27), then the P -function of each ρ̂ [N ] from this sequence is zero
outside the origin, which does not apply to the P -function of the limit of this sequence.
It is also interesting to note that the convergence of the series (4.25) is not directly related to
the behavior of the initial terms. It can happen (e.g. for a weak thermal state or a weakly squeezed
vacuum state) that the initial subsequent terms decrease quickly but after some time, they start
to grow and the series diverges. At the same time, for weak signal states (compared to the LO)
j
these first terms provide an increasingly good approximation to the photon counting probability P m
.
The situation is thus similar to the one in perturbation theory: even though a perturbation series
diverges, its several (or many) initial terms may give a good approximation.
2
as we have mentioned, the P -function of the state |ni is equal to the nth derivative of the Dirac δ-function, which
is very singular in the origin; however, outside the origin it is equal to zero in the whole complex plane
31
Three problems from quantum optics
To verify the result (4.23), we have performed a number of numerical simulations in which for
j
a given state ρ̂ we have calculated the probabilities Pm
for fixed j and all possible m in two ways – one
used the exact expression in terms of the Wigner functions djmm0 and the other used our result (4.25).
We have truncated the series after one, three and five terms, respectively, and have observed whether
j
the increasing number of terms approximates the exact probability Pm
with an increasing accuracy.
Indeed, it was really so even for the squeezed state for which the series (4.25) does not converge. The
results of the simulation can be seen in Fig. 4.2.
4.5
Strong local oscillator
It remains to say what the conditions are under which homodyne detection measures the field quadrature “well”. As we have seen in Eqs. (4.4) and (4.5) already, this happens for large local oscillator
amplitudes A. To see what this means in a given situation can be seen from analyzing the series (4.25)
where the first term hx|ρ̂|xi should be dominant as it corresponds to the ideal quadrature measurement. It has turned out that the condition is as follows: the mean number of photons n in the signal
state should be much less than the amplitude of the local oscillator A. It is thus not enough if n is
much smaller than the mean photon number in the local oscillator (which is A2 ). In fact the condition
is stronger – is can also be expressed in that the mean photon number in the signal state is much less
than the fluctuation of the mean photon number in the local oscillator. If this condition were not
satisfied, by measuring the total photon number 2j one could approximately determine how many
photons originate from the signal state. However, this would necessarily disturb the measurement
of the quadrature x̂ϕ as the photon number operator does not commute with the quadrature. This
way, the strong local oscillator condition follows from the complementarity principle.
4.6
Conclusion
Together with Barry Sanders I have analyzed balanced homodyne detection via calculating the probability of detecting given numbers of photons at homodyne detector beam splitter outputs. We have
derived the POVM by a direct calculation for the first time by two different methods. We have shown
that for a strong local oscillator the homodyne detection provides a projective measurement of the
field quadrature and the corresponding POVM is proportional to the projector |xihx|. In addition,
the calculation that employs the Glauber-Sudarshan P -representation yields correction terms useful
if the local oscillator is not too strong. We have performed numerical simulations that confirmed the
theoretical results including the correction terms.
32
Chapter 4. Homodyne detection
(a) 0.00045
Photon counting probability
0.0004
0.00035
0.0003
0.00025
0.0002
0.00015
0.0001
5e-05
0
10
20
30
40
50
60
70
m
(b) 0.0005
Photon counting probability
0.00045
0.0004
0.00035
0.0003
0.00025
0.0002
0.00015
0.0001
5e-05
0
-5e-05
-20
-15
-10
-5
0
5
10
15
20
60
80
m
(c)
7e-05
Photon counting probability
6e-05
5e-05
4e-05
3e-05
2e-05
1e-05
0
-1e-05
-80
-60
-40
-20
0
20
40
m
j
Figure 4.2: Simulations of the homodyne detection probability distribution P m
for (a) coherent state |γi with
γ = 2 for j = 190, (b) squeezed state with squeezing parameter s = 4.5 for j = 219.5, and (c) number state
|6i for j = 183.5. The exact probabilities are shown in black, and the truncated ones are shown in green, blue
and red, respectively, according to the increasing number of terms in Eq. (4.25) taken into account. The red
curves are so close to the black ones in (b) and (c) that they almost cover them in the plots.
33
Chapter 5
Fermion coherent states
5.1
Introduction
As we have mentioned in Chapter 2, coherent states of light occupy an important position in quantum
optics for their useful physical and mathematical properties, and they provide important representations such as the Glauber-Sudarshan P -representation. These properties of coherent states are
connected with the boson nature of the electromagnetic field – the fact that the quanta of the field
are subject to Bose-Einstein statistics.
There are many similarities and analogies between the fields of bosons and those of fermions,
that is, particles subject to the Fermi-Dirac statistics and the Pauli exclusion principle. One can
perform similar interference experiments with the same results with both types of particles, define
coherence for both of them etc. Therefore a natural question arises if it is possible to extend the
definition of coherent states to fermion fields. Indeed, there is a method of introducing fermion
coherent states that employs so-called Grassmann variables [22, 23, 24]. The resulting states have
formally all the desired properties of coherent states; however, they lack any physical interpretation
and do not satisfy the basic axioms for vectors in the Hilbert space of physical states. This way, the
Grassmann coherent states are an interesting mathematical structure rather than a physical object
with a direct relation to reality.
For this reasons my Australian colleagues and I were thinking of defining coherent states without
using the Grassmann numbers. We have analyzed this question in the work [52] and we have shown
that generalizing coherent states to fermion fields is very problematic and that it is not possible
to define fermion coherent states analogous to their boson counterpart in the Hilbert space. In
addition, we have proved several theorems that hold for fermion correlation functions and have no
direct analogy for bosons. These theorems are valid due to the Pauli exclusion principle and show
the distinctive properties of fermion fields regarding multi-particle correlations. Our results from [52]
are presented and explained in the following sections.
5.2
The options for introducing coherent states of light
In this section, we have a closer look at some properties of boson coherent states and will show why
their generalization to fermions is problematic. For concreteness, we will talk about electromagnetic
field, but the definitions that follow can be applied to other boson fields as well.
Coherent states of boson fields can be defined in several equivalent ways and we will consider here
four of them, the equivalence of which was shown in our work [52]. As we have mentioned, coherent
states of light are close to states of the classical field with a well-defined amplitude and phase. In
the classical theory of the electromagnetic field one can define a so-called phasor, which is a complex
amplitude of the field that has a well-defined value for classical states of light. As the phasor is
replaced by the annihilation operator â in the quantum theory, it is natural to define coherent states
34
Chapter 5. Fermion coherent states
as those states for which the value of â is well-defined, that is, as the eigenstates of the annihilation
operator. This leads to the following definition:
Definition 1 Coherent state of a given mode of the electromagnetic field is an eigenstate of the
annihilation operator â of the mode. A multimode coherent state is the eigenstate of all annihilation
operators (that are linear combinations of the single-mode annihilation operators â k ).
The second way of defining coherent states is related to their more general conception as results
of some group action on a fixed state. This way, coherent states of light emerge by the action
†
∗
of elements of the Heisenberg-Weyl group HW(1) = {eαâ −α â+iϕ | α ∈ C, ϕ ∈ R} on the vacuum
state |0i. As the physical operation corresponding to the elements of the Heisenberg-Weyl group
is displacement in the phase space, we can understand coherent states as displaced vacuum states
according to the following definition:
Definition 2 Coherent state is the vacuum state displaced in the phase space, that is, the result
†
∗
of the action of the displacement operator D(α) = eαâ −α â on the vacuum |0i.
The third definition is related to the behavior of coherent states on a beam splitter. If a general
state of light is mixed with the vacuum on a beam splitter, the output states will in general be
entangled or at least correlated. For example, for a single-photon input state |1i, the output twomode state is t|1i|0i + r|0i|1i with t and r being the transmissivity and reflectivity of the beam
splitter, respectively. This state is entangled because it cannot be expressed as a product of states
of the two output modes. However, if the input state is a coherent state, the beam splitter outputs
will be unentangled coherent states as we have mentioned in Chapter 4, Eq. (4.17). This property
makes coherent states useful for complicated optical experiments in which many beams are derived
from the same coherent source of light. This way we arrive at the following definition:
Definition 3 Coherent state is a pure state that produces unentangled outputs when mixed with the
vacuum on a beam splitter.
In 1963 R. J. Glauber developed the theory of coherent states of light based on the properties
of normally-ordered correlation functions [53]. These correlation functions (or shortly correlators)
are defined by
G(n) (x1 , . . . , xn , yn , . . . , y1 ) ≡ hψ̂ † (x1 ) · · · ψ̂ † (xn )ψ̂(yn ) · · · ψ̂(y1 )i ,
(5.1)
where ψ̂ † (x) and ψ̂(x) is the creation and annihilation operator at the space-time point x, respectively.
The normal ordering means that all the creation operators are to the left of the annihilation operators
in Eq. (5.1).
The normally-ordered correlators describe coherence properties of the field related to photodetection in which photons are absorbed in the detector. Of a particular importance are correlators
with repeated arguments for which yi = xi . The correlator
G(n) (x1 , . . . , xn ) ≡ G(n) (x1 , . . . , xn , xn , . . . , x1 )
expresses the probability density of finding a particle at the point x1 , another particle at x2 , etc., up
to the nth particle at xn . It is possible to measure these correlators directly using detectors placed
in the field. One can show that for states of the classical electromagnetic field with a well-defined
phase and amplitude the correlators factorize – it is possible to express them as products of functions
of their arguments. For example, the correlator (5.1) factorizes if there exists a function f (x) such
that
G(n) (x1 , . . . , xn , yn , . . . , y1 ) = f ∗ (x1 ) · · · f ∗ (xn )f (yn ) · · · f (y1 )
(5.2)
holds. Then it is natural to define coherent states of a quantized field as those factorizing the
correlators, which leads to this definition:
Definition 4 Coherent state is a state for which the normally-ordered correlators factorize.
35
Three problems from quantum optics
5.3
Fermion analogy of the boson coherent state
Before attempting to define fermion coherent states, we briefly mention fermion creation and annihilation operators ĉ†i , ĉi . Similarly as in the case of bosons, these operators raise and lower the particle
number in the ith mode by one. However, in contrast to the boson commutation relations (2.2), the
fermion field operators satisfy the following anticommutation relations:
{ĉi , ĉ†j } = δij 1̂,
{ĉi , ĉj } = {ĉ†i , ĉ†j } = 0 .
(5.3)
Here the anticommutator is defined as {Â, B̂} ≡ ÂB̂ + B̂ Â. One consequence of these relations is
that (ĉ† )2 = 0, that is, it is not possible to create more than one fermion in a given mode, which
is a possible way of expressing the Pauli exclusion principle. Hence, for each mode there are only
two states with a definite number of particles: the vacuum |0i and the occupied state |1i. The field
operator action on these states is as follows,
ĉ† |0i = |1i,
ĉ|0i = 0,
ĉ† |1i = 0
ĉ|1i = |0i.
(5.4)
(5.5)
When attempting to generalize coherent states to fermions using the definitions from the previous
section, one meets serious difficulties. Consider Definition 1 first. As can be verified easily using the
relations (5.5), the only eigenstate of the fermion annihilation operator in the Hilbert space is the
vacuum |0i. Indeed, when acting by the annihilation operator on a general pure state |ψi = γ|0i+δ|1i,
one obtains δ|0i, which is a multiple of |ψi only if δ = 0, which means that |ψi = |0i. According to
such definition, the only fermion coherent state would be the vacuum, which is not very interesting.
One obtains the same result also from Definition 3 related to the behavior of coherent states on
a beam splitter. As can be shown easily, when any state other than vacuum is split on a beam
splitter, the output states will always be entangled. Again, the vacuum would be the only fermion
coherent state according to such definition.
One could also use the generalization of the boson displacement operator to fermion fields. It
turns out that the action of this operator,
D(α) = eαĉ
† −α∗ ĉ
,
(5.6)
on the vacuum produces an arbitrary pure single-mode state. This would not yield a reasonable
definition of fermion coherent states as all single-mode states would be coherent. However, if the
magnitude of the displacement |α| is small, one can define an approximate fermion coherent state
as |αi = D(α)|0i that also approximately satisfies Definitions 1 and 3. Multimode approximate
coherent states can then be obtained by a consequent action of single-mode displacement operators
on the vacuum. However, at this point one meets other difficulties. The reason is that displacement
operators of different modes do not commute and hence it matters in what order the displacements in
the individual modes are performed. By changing the order of the displacement operators one obtains
a different state in general, which is illustrated in Fig. 5.1. One could try to solve this ambiguity
by averaging the multimode state over all possible orderings (permutations) of the displacement
operators, which would yield, however, a state with no more than one fermion and so one could not
obtain in this way multi-particle states at all.
When we tried to define fermion coherent states by generalizing Definition 4, i.e., as states that
factorize the correlation functions, we found out that the result is similar as in the case of Definitions
1 – 3, that is, that the fermion coherent states cannot be reasonably defined in an analogous way as
for bosons. We have also found a number of interesting properties of fermion correlation functions
that are connected with the Pauli exclusion principle. Some of them can be expected intuitively
while others may be surprising. In the next section we explain the statements that we proved in [52]
and we will show that fermion correlation functions cannot be factorized up to some exceptions.
36
Chapter 5. Fermion coherent states
(a)
(b)
line 1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
line 1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
2
1.5
1.5
1
x
1
0.5
0
-0.5
-1
-1.5
-2 -2
-1.5
-1
-0.5
0
0.5
1
1.5
x
2
0.5
0
-0.5
-1
-1.5
y
-2 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
y
Figure 5.1: The square of modulus of the complex degree of coherence, |γ(x, y)| 2 defined in Eq. (5.7), for two
approximate complex fermion coherent states that were obtained from the vacuum by the consequent action
of displacement operators of forty modes. The modes and the corresponding amplitudes α k are the same
for both figures (a) and (b) and the only difference is the ordering of the displacement operators. As these
operators do not commute, one obtains physically different states with different observable properties.
5.4
Properties of fermion correlators
According to the first proposition we proved, the probability of finding n fermions at n points, two
of which approach each other, goes to zero. This is a direct consequence of the fermion anticommutation relations and the Pauli principle. The proposition can be formulated as follows:
Proposition 1 For any fermion field state, the fermion correlator G(n) (x1 , . . . , xn ) tends to zero
whenever two points xi , xj approach each other.
This proposition is demonstrated in Fig. 5.2 for the approximate fermion coherent state.
One consequence of Proposition 1 is that a correlation function of order higher than one (i.e.,
for n > 1) cannot factorize except the case in which it is identically equal to zero. Indeed, if there
existed a function f (x) such that Eq. (5.2) holds, then f (x) would have to be zero because it holds
G(n) (x1 , . . . , xn ) → 0 for xi → xj . The following proposition is hence valid:
Proposition 2 If the fermion correlator G(n) (x1 , . . . , xn , yn , . . . , y1 ) factorizes for some for n > 1,
then it is identically equal to zero.
As we can see, the definition of coherent states based on correlator factorization does not have much
sense for fermions as the correlators of the second or higher order cannot be factored non-trivially.
One of the important characteristics of the coherence properties of boson and fermion fields is
the normalized first-order correlator with mixed arguments. It is called complex degree of coherence
and is defined as
G(1) (x, y)
γ(x, y) = p
.
(5.7)
G(1) (x, x)G(1) (y, y)
The complex degree of coherence has a direct physical meaning: in a double-slit experiment in which
equal slits are placed at the points x and y the visibility of the interference fringes is given by |γ(x, y)|
(see Fig. 5.3). If the field has a large coherence of the first order (or second order coherence according
to the terminology in [21]), then the fringe visibility is high and |γ(x, y)| is close to unity. We have
found an interesting consequence of such coherence for multi-particle correlators. If |γ(x, y)| = 1 for
some x, y, it is not possible to find a fermion at the point x and another one at y. The following
proposition formulates this even more generally:
37
Three problems from quantum optics
line 1
600
500
400
300
200
100
700
600
500
400
300
200
100
0
2
1.5
1
x
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
1
0.5
1.5
2
y
Figure 5.2: The second-order correlator |G(2) (x, y)|2 for a multimode approximate complex fermion coherent
state, i.e., the vacuum displaced in the individual modes with a small displacement |α|. The dip along the
line x = y is a common feature of this correlator regardless of the state and is a consequence of the fact that
G(2) (x, y) → 0 when x → y. For this reason, it is not possible to factorize this correlator into a product
of a function of a and a function of y.
x
detector
y
Figure 5.3: Fringe visibility in a double-slit experiment with equal slits located at the points x, y is given
by the magnitude of the complex degree of coherence γ(x, y) when a quasimonochromatic beam of bosons or
fermions incides on the slits.
Proposition 3 Let |γ(x, y)| = 1 holds for the fermion state |ψi and a pair of points x, y. Then
G(n) (x, y, x3 , . . . , xn , yn , yn−1 , . . . , y1 ) = 0 holds for any n > 1 and arbitrary points x3 , . . . , xn ,
y1 , . . . , y n .
There is an interesting corollary of Proposition 3 for fields that have a full first-order coherence,
e.g., that satisfy |γ(x, y)| = 1 for all x, y. According to Proposition 3 it is then not possible to find two
fermions at two different points and hence the field contains one fermion at most. This is expressed
by the following proposition:
Proposition 4 Let |γ(x, y)| = 1 for some state |ψi for all x, y. Then the state has support over only
the vacuum and single-particle states. That is, the probability of finding two fermions in the field is
identically zero.
Proposition 4 imposes a strong condition on the coherence of fermion fields: a field that contains
more than one fermion cannot exhibit a full first-order coherence. Moreover, for such fields even
the first-order correlator cannot factorize as from G(1) (x, y) = f (x)g(y) it follows that |γ(x, y)| = 1,
which is not possible according to Proposition 4.
38
Chapter 5. Fermion coherent states
None of Propositions 1 – 4 is valid for bosons. Hence it is clear that there is a fundamental
difference between both types of particles that does not vanish even when considering fields with
very low occupation numbers. If the field contains more than one fermion and correlators of order
larger than one are in question, the difference will always be present.
We have demonstrated some of the propositions mentioned above on the example of approximate
fermion coherent states introduced in Sec. 5.3. It has turned out that even for very small values
|α| the second-order correlators do not factorize, which is in agreement with Proposition 1 and is
illustrated in Fig. 5.2.
5.4.1
Correlators of chaotic states
We illustrate the general properties of fermion correlators on the example of chaotic states [54]. These
states have the maximum entropy of all states satisfying certain conditions, e.g. having a given energy
or mean particle numbers in the individual modes. The best-known example of a chaotic state is the
thermal state that has the maximum entropy for a given energy.
For noninteracting bosons and fermions, the single-mode density matrix of a chaotic state can be
expressed as
¶n
∞ µ
1 X
M
ρ̂B =
|nihn|,
1+M
1+M
¶n
1 µ
X
M
ρ̂F = (1 − M )
|nihn| ,
1−M
n=0
(5.8)
n=0
respectively.
Now consider a multi-mode chaotic state of bosons or fermions. Let N denote the number
of occupied modes that will be labeled by 1, . . . , N , and let Mi be the mean number of particles in
the ith mode. We have shown that the first-order correlator is the same for both bosons and fermions:
(1)
GB,F (x, y) =
N
X
Mi ϕ∗i (x)ϕi (y) .
(5.9)
i=1
Here ϕi (x) are the spatialP
mode functions that connect the mode and point annihilation operators
via the relation ψ̂(xi ) =
k ϕk (xi )âk . Eq. (5.9) shows that the coherence properties of the first
order are not influenced by the boson of fermion nature of the particles. This can be expected as the
first-order coherence is not connected with multi-particle correlations and therefore it should not be
influenced by the exchange interaction of identical particles.
In contrast, higher-order correlators do depend on the type of particles. The n th -order correlators
of the boson and fermion chaotic state are
X par(P ) (1)
(n)
(1)
(1)
GB,F (x1 , . . . , xn , yn , . . . , y1 ) =
χB,F GB,F (x1 , yP (1) )GB,F (x2 , yP (2) ) · · · GB,F (xn , yP (n) ) . (5.10)
P
For fermions, this is an exact result while for bosons it is valid to a high precision if M i ¿ 1 for all
i. The sum runs over all permutations P of the indexes 1, 2, . . . , n, par(P ) denotes the parity of P
and χB = 1 and χF = −1 is the boson and fermion sign factor, respectively.
Eq. (5.10) shows that multiparticle correlators are different for fermions and bosons. If x i →
xj , i 6= j, the correlator GF goes to zero due to the factor (−1)par(P ) , which is in a full agreement
with Proposition 1.
5.5
Conclusion
As we have shown, introducing fermion coherent states is connected with large difficulties. It is not
possible to define states having analogous properties as boson coherent states even in a single mode.
39
Three problems from quantum optics
When attempting to generalize approximate fermion coherent states to multi-mode situation, inconsistencies arise due to the non-invariance of the states with respect to changing the mode ordering.
Further, we have shown that approximate fermion coherent states cannot factorize correlation functions. All these difficulties are connected with the Pauli exclusion principle and the anticommuting
properties of the field operators.
Thus it seems that the only reasonable option for defining fermion coherent states is to use the
Grassmann variables. However, then the physically significant quantities such as amplitude and
phase as well as the inner product of such states loose their meaning. Is may happen that when
investigating the Grassmann coherent states further, a closer relationship of these states with the
physical reality will be discovered. However, it is probably not reasonable to try to introduce fermion
coherent states without the Grassmann variables.
40
Bibliography
[1] C. Kittel and H. Kroemer, Thermal Physics (W. H. Freeman and Company, New York, 1980);
C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum mechanics (John Wiley, New York, 1977).
[2] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge
University Press, Cambridge, 2000).
[3] L. Vaidman, Phys. Rev. A 49, 1473 (1994).
[4] S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998).
[5] A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble and E. S. Polzik, Science
282, 706 (1998).
[6] D. Bouwmeester, A. K. Ekert and A. Zeilinger (eds.), The Physics of Quantum Information
(Springer, New York, 2000).
[7] C. H. Bennett and G. Brassard, Quantum cryptography, public key distribution and coin tossing,
in Proc. Int. Conf. Computers, Systems and Signal Processing (Bangalore, 1984).
[8] P. W. Shor and J. Preskill, Phys. Rev. Lett 85, 441 (2000).
[9] D. Deutsch and R. Jozsa, Proc. Royal Soc. London, 439A (1992).
[10] P. Shor, Proc. 35th Ann. Symp. on Foundations of Computer Science, 124 (1994).
[11] S. D. Bartlett and B. C. Sanders, J. Mod. Opt. 50, 2331 (2003).
[12] L. K. Grover, Proc. of the 28th Ann. ACM Symp. on the Theory of Computing, 212 (1996).
[13] W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982).
[14] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).
[15] H. P. Yuen and J. H. Shapiro, in Coherence and Quantum Optics IV, L. Mandel and E. Wolf
eds. (Plenum, New York, 1978).
[16] H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-25, 179 (1979); IT-26, 78 (1980);
H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[17] H. P. Yuen and V. W. Chan, Opt. Lett. 8, 177 and 8, 345 (1983).
[18] P. M. Radmore and S. M. Barnett, Methods in Theoretical Quantum Optics (Oxford University
Press, 1997).
[19] S. L. Braunstein, Phys. Rev. A 42, 474 (1990).
41
Three problems from quantum optics
[20] K. Banaszek and K. Wódkiewicz, Phys. Rev. A 55, 3117 (1997).
[21] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press,
Cambridge, 1995).
[22] J. L. Martin, Proc. Roy. Soc. Lond. A 251, 543 (1959); Y. Ohnuki and T. Kashiwa, Prog. Theor.
Phys. 60, 548 (1978); J. R. Klauder and B.-S. Skagerstam, Coherent States (World Scientific,
Singapore, 1985).
[23] F. A. Berezin, The Method of Second Quantization (Academic Press, New York, 1966).
[24] K. E. Cahill, R. J. Glauber, Phys. 59, 1538 (1999).
[25] I. M. Gelfand and N. Y. Vilenkin, Generalized functions, Vol. 4 – Applications of Harmonic
Analysis (Academic Press, New York, 1964).
[26] F. Gieres, Rep. Prog. Phys. 63, 1893 (2000).
[27] S. L. Braunstein, arXiv: quant-ph/9904002.
[28] R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[29] E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[30] J. R. Klauder, Phys. Rev. Lett. 16, 534 (1966).
[31] K. Mølmer, Phys. Rev. A 55, 3195 (1997).
[32] T. Rudolph and B. C. Sanders, Phys. Rev. Lett. 87, 077903 (2001).
[33] C. W. Helstorm, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
[34] A. D. Smith, arXiv: quant-ph/0001087.
[35] R. Cleve, D. Gottesman and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999).
[36] D. Gottesman, Phys. Rev. A 61, 042311 (2000).
[37] T. Tyc and B. C. Sanders, Phys. Rev. A 65, 042310 (2002).
[38] T. Tyc, D. Rowe and B. C. Sanders, J. Phys. A 36, 7625 (2003).
[39] A. M. Lance, T. Symul, W. P. Bowen, T. Tyc, B. C. Sanders and P. K. Lam, New J. Phys. 5,
4 (2003).
[40] A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, T. Tyc, T. C. Ralph and P. K. Lam, Phys.
Rev. A 71, 33814 (2005).
[41] A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders and P. K. Lam, Phys. Rev. Lett. 92, 177903
(2004).
[42] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).
[43] C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[44] C. Weedbrook et al., Phys. Rev. Lett. 93, 170504 (2004); N. Treps et al., Science 301, 940
(2003); N. Treps et al., Phys. Rev. Lett. 88, 203601 (2002).
[45] B. Schumacher, Phys. Rev. A 51, 2738 (1995).
42
Bibliography
[46] S. Reynaud, A. Heidmann, E. Giacobono and C. Fabre, Quantum fluctuations in optical systems,
in Progress in Optics XXX, E. Wolf ed. (Elsevier, 1992)
[47] U. Leonhardt, Measuring the quantum state of light (Cambridge University Press, Cambridge,
1997).
[48] T. Tyc and B. C. Sanders, J. Phys. A 37, 7341 (2004).
[49] J. J. Sakurai, Modern quantum mechanics (Addison-Wesley, 1985); L. D. Landau and E. M.
Lifshitz, Quantum mechanics (Pergamon Press, 1977).
[50] B. Yurke, S. L. McCall and J. R. Klauder, Phys. Rev. A 33, 4033 (1986).
[51] D. J. Rowe, B. C. Sanders and H. de Guise, J. Math. Phys. 42, 2315 (2001).
[52] T. Tyc and B. C. Sanders, in the review process in New Journal of Physics.
[53] R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[54] R. Glauber, in Quantum Optics, edited by S. Kay and A. Maitland. (Academic Press, New York,
1970).
43