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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 + ↠√ , 2 p̂ = 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. 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