Download - Quantum Discord and Classical Correlations of Two Bosonic Modes in the Two-Reservoir Model
Journal of Russian Laser Research, Volume 35, Number 1, January, 2014
QUANTUM DISCORD AND CLASSICAL CORRELATIONS
OF TWO BOSONIC MODES
IN THE TWO-RESERVOIR MODEL
Aurelian Isar
Horia Hulubei National Institute for Research and Development
in Physics and Nuclear Engineering
P.O. Box MG-6, Bucharest-Magurele, Romania∗E-mail: [email protected]
Abstract
Within the framework of the theory of open systems based on completely positive quantum dynamicalsemigroups, we give a description of the continuous variable quantum discord for a system consistingof two bosonic modes embedded in two independent thermal environments. We describe the evolutionof discord in terms of the covariance matrix for Gaussian input states. In the case of an entangledinitial squeezed thermal state, we analyze the evolution of the Gaussian quantum discord, which isa measure of all quantum correlations in the bipartite state, including entanglement, and show thatquantum discord decays asymptotically in time under the effect of the thermal reservoirs. We describealso the evolution of classical correlations and show that they also decay asymptotically in time.
Keywords: Gaussian states, open systems, Gaussian quantum discord, classical correlations.
1. Introduction
The realization of quantum information processing tasks depends on the generation and manipulation
of nonclassical states. In quantum information theory of continuous variable systems, Gaussian states,
in particular, two-mode Gaussian states, play a key role since they can be easily created and controlled
experimentally [1].
Quantum entanglement represents the indispensable physical resource for the description and per-
formance of quantum information processing tasks [2, 3]. However, entanglement does not describe all
the nonclassical properties of quantum correlations. Zurek [4, 5] defined the quantum discord as a mea-
sure of quantum correlations, which includes the entanglement of bipartite systems and can also exist
in separable states. Recently, an operational interpretation was given to quantum discord in terms of
consumption of the entanglement in an extended quantum state merging protocol [6, 7].
Quantum coherence and quantum correlations – entanglement and discord – of quantum systems are
inevitably influenced during their interaction with the external environment, and, in order to describe
realistically quantum information processes, it is necessary to take decoherence and dissipation into
consideration. In [8–11], we studied the dynamics of the Gaussian quantum discord of two bosonic modes
(two harmonic oscillators) coupled to a common thermal environment, employing the theory of open
systems based on completely positive quantum dynamical semigroups. Recently [12], we analyzed the
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Manuscript submitted by the author in English on November 19, 2013.
1071-2836/14/3501-0062 c©2014 Springer Science+Business Media New York
Volume 35, Number 1, January, 2014 Journal of Russian Laser Research
dynamics of the Gaussian quantum discord of two modes coupled to two independent thermal reservoirs
for initial two-mode squeezed vacuum states. In the present work, we extend this study to the general
case of initial two-mode squeezed thermal states. The initial state of the subsystem is assumed of a
Gaussian form, and the evolution under the quantum dynamical semigroup assures the preservation in
time of the Gaussian form of the state.
This paper is organized as follows.
In Sec. 2, we write the Markovian master equation for an open system interacting with a general
environment and the evolution equation for the covariance matrix. For this equation, we give its general
solution, i.e., we derive the variances and covariances of coordinates and momenta corresponding to
a one-mode Gaussian state. Using this solution, we obtain in Sec. 3 the evolution of the covariance
matrix for a Gaussian state of the two-mode bosonic system, where each mode interacting with its own
thermal reservoir. We analyze the time evolution of the Gaussian quantum discord, which is a measure
of all quantum correlations in the bipartite state, including entanglement. Then, using the asymptotic
covariance matrix, we show that quantum discord decays asymptotically in the limit of long times under
the effect of the thermal baths. We describe also the evolution of classical correlations and show that
they also decay asymptotically in time. We give our summary in Sec. 4.
2. Master Equation for Open Quantum Systems
We study the dynamics of a subsystem composed of two identical bosonic modes, each one in weak
interaction with its local thermal environment. In the axiomatic formalism based on completely positive
quantum dynamical semigroups, the irreversible time evolution of an open system is described by the
following Kossakowski–Lindblad quantum Markovian master equation for the density operator ρ(t) in
the Schrodinger representation († denotes the Hermitian conjugation) [13–15]
dρ(t)
dt= − i
�[H, ρ(t)] +
1
2�
∑j
(2Vjρ(t)V
†j − {ρ(t), V †
j Vj}+), (1)
where H denotes the Hamiltonian of the open system, and the operators Vj , V†j , defined on the Hilbert
space of H represent the interaction of the open system with the environment.
We are interested in the set of Gaussian states; therefore, we introduce such quantum dynamical
semigroups that preserve this set during the time evolution of the system. Consequently, H is taken
to be a polynomial of the second degree in the coordinates x, y and momenta px, py of the two bosonic
modes, and Vj , V†j are taken as polynomials of the first degree in these canonical observables.
We introduce the following 4×4 bimodal covariance matrix:
σ(t) =
⎛⎜⎜⎜⎜⎝
σxx(t) σxpx(t) σxy(t) σxpy(t)
σxpx(t) σpxpx(t) σypx(t) σpxpy(t)
σxy(t) σypx(t) σyy(t) σypy(t)
σxpy(t) σpxpy(t) σypy(t) σpypy(t)
⎞⎟⎟⎟⎟⎠ , (2)
with the correlations of operators A1 and A2 defined by using the density operator ρ of the initial state
of the quantum system as follows:
σA1A2(t) =1
2Tr[ρ(A1A2 +A2A1)(t)]− Tr[ρA1(t)]Tr[ρA2(t)]. (3)
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A Gaussian channel is a map that converts the Gaussian states to the Gaussian states, and the
evolution given by Eq. (1) is such a channel. The evolution of the initial covariance matrix σ(0) of the
system, under the action of a general Gaussian channel, can be characterized by two matrices X(t) and
Y (t)
σ(t) = X(t)σ(0)XT(t) + Y (t), (4)
where Y (t) is a positive operator [16], with T denoting the transposed matrix. These two matrices com-
pletely characterize the action of environment, and Eq. (4) guarantees that σ(t) is a physical covariance
matrix for all finite times t.
In the case of one bosonic mode (harmonic oscillator) with the general Hamiltonian
H =1
2mp2x +
mω2
2x2 +
μ
2(xpx + pxx), (5)
where the parameter μ determines the damping ratio, we obtain from Eq. (1) the following system of
equations for the elements of the covariance matrix:
s(t) =
(σxx(t) σxpx(t)
σxpx(t) σpxpx(t)
), (6)
representing the quantum correlations of the canonical observables of a single-mode system [15,17]
ds(t)
dt= Z1s(t) + s(t)ZT
1 + 2D1, (7)
where from now we assume, for simplicity, m = ω = � = 1,
Z1 =
(−(λ1 − μ) 1
−1 −(λ1 + μ)
), D1 =
(Dxx Dxpx
Dxpx Dpxpx
), (8)
with Dxx, Dxpx , Dpxpx the diffusion coefficients, and λ1 is the dissipation constant. The solution of Eq. (7)
is given by [17]
s(t) = X1(t)[s(0)− s(∞)]XT1 (t) + s(∞), (9)
where the matrix X1(t) = exp(Z1t) has to fulfill the condition limt→∞X1(t) = 0.
For such a limit to exist, Z1 must have only eigenvalues with negative real parts. In the underdamped
case ω > μ and Ω2 ≡ ω2 − μ2, Eq. (9) takes the form
s(t) = X1(t)s(0)XT1 (t) + Y1(t), (10)
where
X1(t) = e−λ1t
(cosΩt+ (μ/Ω) sinΩt (1/Ω) sinΩt
−(1/Ω) sinΩt cosΩt− (μ/Ω) sinΩt
)(11)
and
Y1(t) = −X1(t)s(∞)XT1 (t) + s(∞). (12)
In the case where the asymptotic state is the Gibbs state corresponding to a harmonic oscillator in
the thermal equilibrium at temperature T1, the values at infinity are obtained from the equation
Z1s(∞) + s(∞)ZT1 = −2D1, (13)
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where the matrix of diffusion coefficients becomes
D1 =
([(λ1 − μ)/2] coth(2kT1)
−1 0
0 [(λ1 + μ)/2] coth(2kT1)−1
). (14)
3. Dynamics of the Gaussian Quantum Discord
A two-mode Gaussian state is entirely specified by its covariance matrix (2), which is a real, symmetric,
and positive matrix with the following block structure:
σ(t) =
(A C
CT B
), (15)
where A, B, and C are 2×2 Hermitian matrices. The matrices A and B denote the symmetric covariance
matrices for the individual reduced one-mode states, while the matrix C contains the cross correlations
between the modes. The covariance matrix (15) (where all first moments have been set to zero by
means of local unitary operations that do not affect the Gaussian discord) contains four local symplectic
invariants in the form of the determinants of the block matrices A,B,C and covariance matrix σ.
Consider a system of two identical bosonic modes coupled to two local thermal baths. If the initial
two-mode 4×4 covariance matrix is σ(0), its subsequent evolution is given by
σ(t) =(X1(t)⊕X2(t)
)σ(0)
(X1(t)⊕X2(t)
)T+
(Y1(t)⊕ Y2(t)
), (16)
where X1,2(t) and Y1,2(t) are given by Eqs. (11) and (12) and the similar ones, corresponding to each
bosonic mode immersed in its own thermal reservoir characterized by temperatures T1, respectively T2,
and dissipation constants λ1 and, respectively, λ2.
Quantum discord was introduced [4, 5] as a measure of all quantum correlations in a bipartite state,
including – but not restricted to – entanglement. Quantum discord was defined as the difference between
two quantum analogs of classically equivalent expressions of the mutual information, which is a measure
of total correlations in a quantum state. For pure entangled states, quantum discord coincides with the
entropy of entanglement. Quantum discord can be different from zero also for some mixed separable
states, and therefore the correlations in such separable states with positive discord are an indicator of the
quantumness. States with zero discord represent essentially a classical probability distribution embedded
in a quantum system.
For an arbitrary bipartite state ρ12, the total correlations are expressed by quantum mutual informa-
tion [18]
I(ρ12) =∑i=1,2
S(ρi)− S(ρ12), (17)
where ρi represents the reduced density matrix of subsystem i, and S(ρ) = −Tr(ρ ln ρ) is the von Neumann
entropy. Henderson and Vedral [19] proposed a measure of bipartite classical correlations C(ρ12) based
on a complete set of local projectors {Πk2} on the subsystem 2 — the classical correlation in the bipartite
quantum state ρ12 can be given by
C(ρ12) = S(ρ1)− inf{Πk2}{S(ρ1|2)}, (18)
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where S(ρ1|2) =∑
k pkS(ρk1) is the conditional entropy of subsystem 1, and inf{S(ρ1|2)} represents the
minimum value of the entropy with respect to a complete set of local measurements {Πk2}. Here, pk is
the measurement probability for the kth local projector, and ρk1 denotes the reduced state of subsystem
1 after the local measurements. Then the quantum discord is defined by
D(ρ12) = I(ρ12)− C(ρ12). (19)
Originally, the quantum discord was defined and evaluated mainly for finite dimensional systems.
Recently [20,21], the notion of discord has been extended to the domain of continuous variable systems,
in particular, to the analysis of bipartite systems described by two-mode Gaussian states. Closed formulas
were derived for bipartite thermal squeezed states [20] and all two-mode Gaussian states [21].
The Gaussian quantum discord of a general two-mode Gaussian state ρ12 can be defined as the
quantum discord where the conditional entropy is restricted to generalized Gaussian positive operator-
valued measurements (POVM) on mode 2, and in terms of symplectic invariants it is given by (the
symmetry between modes 1 and 2 is broken) [21]
D = f(√
β)− f(ν−)− f(ν+) + f(√ε), (20)
where
f(x) =x+ 1
2log
x+ 1
2− x− 1
2log
x− 1
2, (21)
ε =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
2γ2 + (β − 1)(δ − α) + 2|γ|√γ2 + (β − 1)(δ − α)
(β − 1)2, if (δ − αβ)2 ≤ (β + 1)γ2(α+ δ),
αβ − γ2 + δ −√γ4 + (δ − αβ)2 − 2γ2(δ + αβ)
2β, otherwise,
(22)
α = 4 detA, β = 4 detB, γ = 4 detC, δ = 16 detσ, (23)
and ν∓ are the symplectic eigenvalues of the state given by
2ν2∓ = Δ∓√Δ2 − 4δ, (24)
where Δ = α+β+2γ. Notice that the Gaussian quantum discord depends only on | detC|, i.e., entangled(detC < 0) and separable states are treated on equal footing.
The evolution of the Gaussian quantum discord D (20) is illustrated in Fig. 1, where we represent
the dependence of the Gaussian discord D on time t, damping parameter μ, temperatures T1 and T2, and
dissipation constants λ1 and λ2 of the two thermal reservoirs, when the diffusion coefficients are given
by Eq. (14) and a similar one for the second reservoir.
We assume that the initial Gaussian state is a two-mode squeezed thermal state, with the covariance
matrix of the form [22]
σst(0) =
⎛⎜⎜⎜⎜⎝a 0 c 0
0 a 0 −c
c 0 b 0
0 −c 0 b
⎞⎟⎟⎟⎟⎠ , (25)
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Fig. 1. Gaussian quantum discord D versus time t and temperature T2 (via coth 1/2kT2) for an initial squeezedthermal state with r = 0.8, λ1 = 0.3, λ2 = 0.2, μ = 0.1, n1 = 1, n2 = 3, coth 1/2kT1 = 1.1 (left) and r = 1, λ1 =0.03, λ2 = 0.05, μ = 0.02, n1 = 2, n2 = 4, coth 1/2kT1 = 2 (right). We assume m = ω = � = 1.
with the matrix elements given by
a = n1 cosh2 r + n2 sinh
2 r +1
2cosh 2r, b = n1 sinh
2 r + n2 cosh2 r +
1
2cosh 2r,
(26)
c =1
2(n1 + n2 + 1) sinh 2r,
where n1, n2 are the average number of thermal photons associated with the two modes, and r denotes
the squeezing parameter. In the particular case n1 = 0 and n2 = 0, (25) becomes the covariance matrix
of the two-mode squeezed vacuum state. A two-mode squeezed thermal state is entangled when the
squeezing parameter r satisfies the inequality r > rs [22], where
cosh2 rs =(n1 + 1)(n2 + 1)
n1 + n2 + 1. (27)
The Gaussian discord has nonzero values for all finite times, and this fact certifies the existence of
nonclassical correlations in two-mode Gaussian states. The dynamics of discord of the two modes strongly
depends on the parameters characterizing the system and the coefficients describing the interaction of
the system with both reservoirs: quantum discord increases with increase in the squeezing parameter
and decreases with increase in the damping parameter, temperatures, and dissipation constants. The
evolution of D manifests a monotonically decreasing behavior. The Gaussian discord asymptotically
decreases to zero for large times. This behavior has to be compared with that of the entanglement,
which was recently shown [23] to manifest, in the same system and for the same initial Gaussian states,
a suppression at some finite moment of time.
A similar behavior of the discord was also obtained in the case of two bosonic modes interacting with
a common thermal bath, and for this system the logarithmic negativity has again the evolution leading
to a sudden death of the entanglement [24–27].
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Fig. 2. Classical correlations C versus time t and temperature T2 (via coth 1/2kT2) for an initial squeezed thermalstate with with r = 0.8, λ1 = 0.3, λ2 = 0.2, μ = 0.1, n1 = 1, n2 = 3, coth 1/2kT1 = 1.1 (left) and r = 1,λ1 = 0.03, λ2 = 0.05, μ = 0.02, n1 = 2, n2 = 4, coth 1/2kT1 = 2 (right). We assume m = ω = � = 1.
3.1. Classical Correlations
The measure of classical correlations for a general two-mode Gaussian state ρ12 can also be calculated;
it reads [21]
C = f(√α)− f(
√ε), (28)
while the expression of the quantum mutual information, which measures the total correlations, is
I = f(√α) + f(
√β)− f(ν−)− f(ν+). (29)
In Fig. 2, we illustrate the evolution of classical correlations C as a function of time t and temperature
T2 for an initial Gaussian state, taken in the form of a two-mode squeezed thermal state (25). This
quantity manifests a qualitative behavior similar to that of the Gaussian discord — it has nonzero values
for all finite times and, in the limit of infinite time it tends asymptotically to zero, corresponding to
the thermal product (separable) state, with no correlations at all. One can also see that the classical
correlations decrease with increase in the temperature of the thermal bath. One can also remark that,
in the considered cases, the values of classical correlations are larger than those of quantum correlations,
represented by the Gaussian quantum discord.
3.2. Asymptotic State
We assume that the asymptotic state of the considered open system is the Gibbs state corresponding
to two independent quantum harmonic oscillators in thermal equilibrium with their own reservoirs at
temperatures T1 and T2. On general grounds, one expects that the effect of decoherence is dominant in
the long-time regime, so that no quantum correlations are expected to be left at infinity. Indeed, using
the diffusion coefficients given by Eq. (14) and a similar one for the second thermal bath, we obtain from
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Volume 35, Number 1, January, 2014 Journal of Russian Laser Research
Eq. (13) the following elements of the asymptotic matrices A(∞) and B(∞) :
σxx(∞) = σpxpx(∞) = 2−1 coth(2kT1)−1, σxpx(∞) = 0,
(30)σyy(∞) = σpypy(∞) = 2−1 coth(2kT2)
−1, σypy(∞) = 0,
while all the elements of the entanglement matrix C(∞) are zero. Then the discord expression (20)
becomes zero in the limit of large times and, correspondingly, the equilibrium asymptotic state has
always zero discord in the case of two identical bosonic modes immersed in two independent thermal
reservoirs, corresponding to the thermal product (separable) state, with no correlation at all, as in the
case of the same system immersed in a common thermal bath.
4. Summary
Within the framework of the theory of open quantum systems based on completely positive quantum
dynamical semigroups, we investigated the Markovian dynamics of the Gaussian quantum discord for
a subsystem composed of two bosonic modes, each one embedded in its own thermal bath. We have
presented and discussed the influence of the reservoirs on the dynamics of quantum discord in terms
of the covariance matrix for squeezed thermal initial states. We assumed that the asymptotic state of
the considered open system is the Gibbs state corresponding to two independent quantum harmonic
oscillators, each one in thermal equilibrium with its thermal bath.
The Gaussian discord has nonzero values for all finite times, and its dynamics strongly depends on the
parameters characterizing the system (squeezing parameter and damping parameter) and the coefficients
describing the interaction of the system with both reservoirs (temperatures and dissipation constants).
The values of the Gaussian discord asymptotically decrease to zero for large times. We described also
the time evolution of classical correlations. Presently there is a large debate relative to the physical
interpretation of quantum correlations – quantum entanglement and quantum discord. The present
results, in particular, the existence of quantum discord and the possibility of maintaining it in thermal
environments for long times, might be useful in controlling quantum correlations in open systems and
also for applications in the field of quantum information processing and communication.
Acknowledgments
The author acknowledges financial support of the Romanian Ministry of Education and Research
under Projects Nos. CNCS-UEFISCDI PN-II-ID-PCE-2011-3-0083 and PN 09 37 01 02/2010.
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