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Revista Colombiana de Fı́sica, Vol.44, No.1, 2012.
An Accidental Relationship Between a Relative Quantum Phase
Distribution and Concurrence
Una relación accidental entre la distribución de fase cuántica y la concurrencia
R. D. Guerrero M. a R. R. Rey-González a , K. M. Fonseca Romero a . ⋆
a Departamento
de Fı́sica, Universidad Nacional de Colombia, Sede Bogotá.
Recibido diciembre 21 de 2011; aceptado febrero 20 de 2012.
Resumen
Consideramos algunos estados particulares de dos sistemas continuos acoplados, para los cuales tiene
sentido el uso de la concurrencia como un cuantificador del enredamiento, y mostramos que existe una
conexión accidental entre la distribución de probabilidad de la fase relativa de su función de Husimi y
su concurrencia. Mostramos además un sistema cuántico abierto donde se pueden producir tales estados.
Por lo tanto, la medición de la función de Husimi permite la cuantificación de la concurrencia para esos
estados particulares.
Palabras Claves: enredamiento, medidas de enredamiento, decoherencia.
Abstract
We consider some particular states of two coupled continuous systems, for which it is meaningful to use the
concurrence as an entanglement quantifier. We show there is an accidental connection between probability
distribution of the relative phase of their Husimi function, and their concurrence. Moreover, weshow a
quantum open system where these states can be produced. Hence, measurement of the Husimi function
allows for a quantification of entanglement for these particular states.
Keywords: entanglement, entanglement measures, decoherence.
PACS: 03.65.Ud,03.67.Mn,03.65.Yz.
c
2012.
Revista Colombiana de Fı́sica. Todos los derechos reservados.
1. Introduction
Due to its role in fundamental physics and information processing, quantum entanglement is one of the
most studied features of quantum mechanics nowadays.
Although, there are many aspects of quantum entanglement which are not completely understood, in 2×2 systems the entanglement of formation, and concurrence
are considered bona fide quantifiers of entanglement.
On the experimental side, quantum state tomography
[1,2,3], entanglement witnesses [4,5,6], and the use of
⋆ [email protected]
two copies of the state [7,8,9,10] have been used to detect and/or estimate entanglement.
Quantum state estimation is a current important research subject since the accurate reconstruction of a
state provides all the information available of the system
(prepared in that state). Although, most of the experiments on quantum information employ quantum tomography, a quantum state estimation technique, to reconstruct the state, there are many ongoing efforts devoted
to the establishment of optimal measurement schemes.
For example, it is well-known that, in the case of a fi-
Rev. Col. Fı́s., 44, No.1, 2012.
nite number of states, that projection-valued measurements of mutually unbiased basis [11], constitute an optimal strategy. Once a state is reconstructed, its entanglement can be calculated, at least in principle. In fact,
many experimental demonstrations of entanglement use
quantum tomographic methods. In cavity quantum electrodynamics experiments it is possible to measure the
Husimi function [12], or the Wigner function, whose
knowledge is equivalent to the knowledge of the density
matrix.
Concurrence and other measures of entanglement, being nonlinear functions of the density operator, can not
be directly measured. Therefore, the search of observables related to entanglement, including entanglement
witnesses, is an important goal. In this paper we show
a result in that direction. For some states, it is possible to establish a relation between the concurrence and
the probability distribution of the relative phase of the
Husimi function. In sections 2 and 3 we briefly review
the concurrence and the Husimi function, while in section 4 we show how they are related, for some particular states, which arise, for example, in the dissipative
system described in section 5. Some final conclusions
are given in section 6.
3. Husimi Quasi-probability Function
For two modes the Husimi function, a semipositive
defined quasi-probability function, is written as
1
(4)
Q(α, α∗ , β, β ∗ ) = 2 hα, β|ρ|α, βi ,
π
where |αi (|βi) is a coherent state of the mode A (B),
which is given by
∞
i
X
|α|2 α
e− 2 √ |ii
|αi =
i!
i=0
in the Fock basis of mode A. If we write the labels of
the coherent states as
we expect some joint information of the state of the
two-mode system to be encoded in the probability distribution of the relative phase θ
θ = φa − φb ,
which is obtained after integration over |α|, |β| y φa . If
we write the density operator in the Fock basis |iji and
calculate the Husimi function we get
∞
1 X ij
Q(α, α∗ , β, β ∗ ) = 2
ρkl hαβ|iji hkl|αβi
π
2. Concurrence
1
= 2
π
It is well known that concurrence is a good entanglement quantifier for states of two-qubit systems [13]. In
particular, if the state of a system can be written, in an
orthonormal basis, as
X
ρ=
ρij,kl |iji hkl|
i,j,k,l=0
∞
X
ij
i(k+l−i−j)φa i(j−l)θ
e
,
ρij
kl qkl (α, β)e
i,j,k,l=0
ij
where qkl
(α, β) = e−|α|
2
i+k
|β|j+l
−|β|2 |α|
√
√ .
i!k!
j!l!
After inte-
gration over |α|, |β| y φa , we obtain the phase difference probability distribution P (θ)
∞
1 X
P (θ) =
pjl ei(j−l)θ ,
2π
ijkl∈{0,1}
j,l=0
then its concurrence is given by
C(ρ) = máx {0, λ1 − λ2 − λ3 − λ4 },
β = |β|eiφa −iθ ,
α = |α|eiφa ,
with pjl =
(1)
P∞
i,k=0
ρij
kl
) Γ(1+ j+l
Γ(1+ i+k
√ 2
√ 2 ) δ(k+l−i−j).
i!k!
j!l!
Here Γ(x) denotes, as usual, Euler’s gamma function.
In order to make a connection with Wootters concurrence, we restrict ourselves to zero and one excitations,
where the probability distribution P (θ) reduces to
1 01 iθ
1
−iθ
+
ρ10 e + ρ10
P (θ) =
01 e
2π 8
1
1
01
+ |ρ01
=
10 | cos θ + arg(ρ10 ) .
2π 4
where the numbers λi are the eigenvalues, in descending order, of the nonnegative-defined operator
√
√
ρ (σy ⊗ σy ) ρ∗ (σy ⊗ σy ) ρ.
where ρ∗ is the complex conjugate of the density operator in the usual basis, and σy stands for the second
Pauli matrix. If the state is pure and given by
4. Relation Between Phase Difference Distribution
and Concurrence
|ψi = c00 |00i + c01 |01i + c10 |10i + c11 |11i , (2)
its concurrence reduces to
C(|ψi) = 2|c00 c11 − c01 c10 |.
We begin by considering pure states. By comparison
of the concurrence for a pure state (3) and the phase-
(3)
28
R. D. Guerrero et al.: An Accidental Relationship between...
0, the probability distribution of the phase-difference,
obtained from the Husimi function of ρ, is given by
1
C
P (θ) =
+ cos θ + arg(ρ01
(8)
10 ) ,
2π
4
where C is the concurrence of ρ. From a practical point
of view, in order to “measure” entanglement, it would
suffice to measure the element ρ01
10 , and the populations
11
ρ00
00 and ρ11 , provided we know that there are at the
most two total excitations, and that one of the populations vanishes.
difference distribution, in particular the absence of terms
involving the states |00i and |11i, we see that a possible
relation exists only if c00 = 0, or if c11 = 0, because in
∗
that case ρ10
01 = c10 c01 , and C = 2|c10 c01 |. Hence, we
have
C
1
+ cos (θ + arg(c01 ) − arg(c10 )) ,
P (θ) =
2π
8
if c00 = 0 or c11 = 0.
Our next goal is to extend this result to more general
density matrices. We can consider states with ρ00
00 (or
ρ11
11 ). In order to obtain physically meaningful density
matrices (in particular semidefinite positive and Hermiij
tian), it is necessary that ρ00
ij = 0 = ρ00 . Indeed, if
00
ρ00 = 0 the matrix representation of the state, in the
basis |01i , |00i , |10i , |11i, given by


01 00 10 11
ρ ρ ρ ρ
 01 01 01 01 
 01
11 
 ρ00 0 ρ10
00 ρ00 


(5)
 01 00 10 11  ,
 ρ10 ρ10 ρ10 ρ10 


00 10 11
ρ
ρ
ρ
ρ01
11 11 11 11
5. A Physical System
Not only is (8) an interesting result, it also appears
in the evolution of physically motivated models. Let us
consider two quantum harmonic oscillators of natural
frequencies ω1 and ω2 , with an RWA-coupling g. We
assume that decoherence processes of the first oscillator are very slow as compared to those of the second,
that’s why we consider only the latter. The second oscillator loses excitations at the rate γ, corresponding
to a coupling with a zero-temperature bath. Physical
systems like coupled microcavities [14], optical-fibre
Fabry-Pérot cavities [15] coupled by a connecting fibre,
and coupled QED cavities [16] are possible physical realizations of the model considered here.
If we assume both oscillators to have the same natural frequency, the Liouville-von Neumann equation for
the density matrix ρ̂ of the total system is given by
dρ̂
= −i[ω b̂† b̂ + ω↠â + g(↠b̂ + âb̂† ), ρ̂]
(9)
dt
†
†
†
+ γ(2âρ̂â − â âρ̂ − ρ̂â â).
must have nonnegative determinants


01 00
ρ
ρ
01 01 
2
= −|ρ00
ρ01
det 
01 | ≥ 0,
01 ≥ 0,
01
ρ00 0
in order to be positive semidefinite. We have shown that
00
ρ01
01 ≥ 0 and that ρ01 = 0. Different orderings show
that all of the matrix elements ρ00
ij = 0. If we assume
=
0,
and
following
a
completely
analogous proρ11
11
cedure, we obtain ρ11
=
0.
We,
thereby,
consider the
ij
two-qubit density matrix


ρ01
ρ10
0
ρ00
00
00
00


 00 01 10 
 ρ01 ρ01 ρ01 0 


(6)
 00 01 10  ,
 ρ10 ρ10 ρ10 0 


0 0 0 0
The right hand side of this equation of motion contains
two terms: while the first is of unitary character, the
second is a dissipative one. The bosonic operators of
creation b̂ (â) and annihilation b̂† (↠) of one excitation
of the first (second) oscillator, satisfy the usual commutation relations. In absence of the coupling with the first
oscillator, the mean lifetime of the second oscillator’s
energy is the inverse of twice the dissipation rate γ.
At zero time, we assume the state of the total system |Ψ(0)i is separable with the second oscillator in its
ground state, and the first oscillator prepared in a pure
state which we choose as a linear combination of its
ground and first excited states
where we have assumed ρ11
11 = 0. For density matrices
of type (6) the concurrence reduces to
q
q
q
q
10 +
01 ρ10 − ρ01 ρ10 −
01 ρ10 ρ
C = ρ01
ρ
ρ
10 01
01 10 10 01
01 10 q
10 10
(7)
= 2 ρ01
10 ρ01 = 2 ρ01 ,
(cos(θ) |0i + sin(θ) |1i) ⊗ |0i .
where the positivity and hermiticity of the density matrix were used. Similar considerations lead to the same
ij
result in the case ρ11
ij = 0 = ρ11 .
Our main result is the following. For two-qubit physi11
cal density matrices ρ satisfying either ρ00
00 = 0 or ρ11 =
(10)
If we solve the Liouville-von Neumann equation (9)
with the initial condition (10), we find the total density matrix to be of the form (6), where the coefficients
ρij
kl depend on time as follows (the omitted coefficients
29
Rev. Col. Fı́s., 44, No.1, 2012.
are complex conjugates of the coefficients listed here,
or zero)
ΓSω
4 − Γ2 C ω
2
−Γt′
−
ρ00
=
1
−
sin
θe
00
4 − Γ2
ω̃
′ 1 − Cω
2
−Γt
ρ01
01 = 2 sin (θ)e
4 − Γ2
!
2 − Γ2 Cω + 2 ΓSω
2
10
−Γt′
−
ρ10 = sin (θ)e
4 − Γ2
ω̃
′
iΩt −
ρ00
01 = i sin(2θ)e
ρ00
10
ρ01
10
Γt′
2
sin(2θ) iΩt′ − Γt′
2
=
e
2
2
= 2i sin (θ)e
−Γt′
S̃ω
ω̃
distribution for the relative phase, is maximum for maximally entangled states, and zero for separable states.
7. Acknowledgements
We acknowledge partial funding from DIB-UNAL
(División de Investigaciones Bogotá - Universidad Nacional de Colombia)under project 10971.
S̃ω
ω̃
ΓS̃ω
C̃ω −
ω̃
!
ΓS̃ω
− C̃ω
ω̃
!
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.
We have defined rescaled quantities t′ = gt, Γ = γg
√
and Ω = ωg . Moreover, we have defined ω̃ = 4 − Γ2 ,
′
Cω = cos(ω̃t′ ), Sω = sin(ω̃t′ ), S̃ω = sin ω̃ t2 , and
′
C̃ω = cos ω̃ t2 . The nonvanishing elements of the
density matrix, were written in the underdamped case
(|Γ| < 2). As time goes on the entanglement between
both systems goes to zero while the phase difference
probability distribution becomes flat, that is, the less entangled are the oscillators the less the knowledge of the
relative phase.
6. Conclusions
We have shown that the probability distribution for
the relative phase of the Husimi function of a pair of
oscillators is related to the entanglement between them,
if the state of the system contains at most two total excitations, and the population of the state |00i (or of the
state |11i) is zero. These states can be experimentally
produced even taking into account dissipative effects.
Interestingly enough, the visibility of the probability
30