gaussian quantum steering of two bosonic modes in a ... · gaussian quantum steering of two bosonic...

GAUSSIAN QUANTUM STEERING OF TWO BOSONIC MODESIN A THERMAL ENVIRONMENT

TATIANA MIHAESCU1,2, AURELIAN ISAR1,3

1Department of Theoretical Physics,Horia Hulubei National Institute for Physics and Nuclear Engineering,

Reactorului 30, RO-077125, POB-MG6, Bucharest-Magurele, RomaniaE-mail: [email protected], [email protected]

2Faculty of Physics, University of Bucharest, P.O.Box MG-11, Bucharest-Magurele, Romania3Academy of Romanian Scientists, 54 Splaiul Independentei, RO-050094, Bucharest, Romania

Received November 27, 2016

Abstract. Einstein-Podolsky-Rosen steerability of quantum states is a propertythat is different from entanglement and Bell nonlocality. We describe the time evolu-tion of a measure that quantifies steerability for arbitrary bipartite Gaussian states in asystem consisting of two bosonic modes embedded in a common thermal environment.We work in the framework of the theory of open quantum systems. If the initial state ofthe subsystem is taken of Gaussian form, then the evolution under completely positivequantum dynamical semigroups assures the preservation in time of the Gaussian formof the state. We study the Gaussian quantum steering in terms of the covariance ma-trix under the influence of the thermal environment and find out that the thermal noiseand dissipation destroy the steerability between the two parts. We make a comparisonwith other quantum correlations for the same system, and show that, unlike Gaussianquantum discord, which is decreasing asymptotically in time, the Gaussian quantumsteerability suffers a ”sudden death” behaviour - it vanishes in a finite time, like quan-tum entanglement.

Key words: Quantum steering, squeezed thermal states, open systems.

PACS: 03.65.Yz, 03.67.Bg, 03.67.Mn.

1. INTRODUCTION

In the past decades the field of quantum information theory has developedmainly due to the great advances in implementation of information processing tasksusing states of quantum nature allowing for impressing groundbreaking applicationsin quantum technologies, like sensing, imaging and metrology. The genuine proper-ties of quantum states in comparison with classical ones consist in quantum correla-tions, such as entanglement, Bell nonlocality, quantum discord and quantum steering[1–3].

Quantum entanglement has attracted a major interest since the discovery ofquantum teleportation, based on the property of nonseparability of quantum systems,in the sense that the subsystems cannot be completely described by only local states

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Article no. 107 Tatiana Mihaescu, Aurelian Isar 2

[4]. Another much studied manifestation of quantum correlations is the Bell nonlo-cality, which cannot be explained by the local hidden variable model, and this leadto disproving the local causality as a true feature of the world. While these two typesof quantum correlations coincide for pure states, the rigourous study in the case ofmixed states shows that the violation of Bell inequalities takes place in a restrictedset of states, which form the subset of the entangled states.

Steering is also a type of quantum nonlocality first identified in the Einstein -Podolsky - Rosen paper [5], which is distinct from both nonseparability and Bell non-locality, allowing for new practical applications such as one-sided device-independentquantum key distribution [6]. To infer the steerability between two parties is equiva-lent with verifying the shared entanglement distribution by an untrusted party. That isto say, Alice has to convince Bob (who does not trust Alice) that the state they share isentangled, by performing local measurements and classical communications. Thus,the operational definition of steering is obtained if we weaken the condition in thedefinition of Bell nonlocality, namely, in bipartite setting the protocol of attestingentanglement allows just one of the parties to be untrusted [7].

In the present paper we work with continuous variable states [2, 8], in parti-cular with Gaussian states, which are easily produced and controlled in laboratories.During their evolution, the systems interact with the environment, and the phenom-ena of decoherence and dissipation take place. In some previous papers [9–24], itwas studied decoherence and dynamics of quantum entanglement and quantum dis-cord in continuous variable open systems. In order to study the evolution of quantumsystems in weak interaction with the environment we use the Kossakowski - Lind-blad master equation, which is valid in the Markovian approximation of the theoryof open systems based on completely positive dynamical semigroups. Within thismodel we consider the system of two uncoupled bosonic modes interacting with acommon thermal reservoir and describe the behaviour of the Gaussian quantum steer-ing, when the initial state of the system is a squeezed thermal state. We show that thesuppression of the Gaussian steering takes place in a finite time, for all temperaturesof the thermal reservoir and all values of the squeezing parameter, this behaviour be-ing similar to the well-known phenomenon of entanglement sudden death. This kindof evolution of Gaussian steering and entanglement dynamics is in contrast with thedynamics of quantum discord, which decreases to zero asymptotically in time.

2. EVOLUTION OF A TWO-MODE BOSONIC SYSTEM INTERACTING WITH ATHERMAL RESERVOIR

We consider a system of two bosonic modes with the vector R= x,px,y,pyof canonical variables of coordinates and momenta, in a Gaussian state defined byGaussian characteristic function and quasi-probability distributions. The canonical

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3 Gaussian quantum steering of two bosonic modes in a thermal environment Article no. 107

commutation relations can be expressed in the form [Ri,Rj ] = i~Ωij , i, j = 1, ..,4,with the symplectic matrix

Ω =2⊕

k=1

(0 1−1 0

). (1)

A Gaussian state is completely characterized by the first and second moments ofcanonical variables, however for the study of quantum entanglement and steeringonly the 4×4 covariance matrix σ of second moments σij = Tr[(RiRj +RjRi)ρ]/2is sufficient to know, where ρ is the density operator. The bipartite covariance matrixcan be written in the following block form:

σ =

(A CCT B

), (2)

where A and B are the 2× 2 uni-modal covariance matrices and C is the correla-tion matrix. The Robertson-Schrodinger uncertainty principle imposes the followingrestriction on the covariance matrix:

σ+i

2Ω≥ 0. (3)

In the Markovian approximation and for an weak interaction with the environment,the most general equation which describes an irreversibile evolution of an open quan-tum system is the following Kossakowski-Lindblad master equation for the densityoperator ρ [25–27]:

dρ(t)

dt=− i

~[H,ρ(t)] +

1

2~∑k

(2Bkρ(t)B†k−ρ(t),B†kBk+

), (4)

where H is the Hamiltonian of the system and Bk are operators which describe theinteraction with the environment. The Hamiltonian of the two uncoupled bosonicmodes is

H =1

2m(p2x+p2y) +

m

2(ω2

1x2 +ω2

2y2), (5)

where m is the mass, ω1, ω2 are the frequencies of the two modes, and the operatorsBk are taken polynomials of the first degree in coordinates and momenta, in order toassure the preservation in time of the Gaussianity of the considered states [28].

By using the master equation (4), we obtain the following time evolution of thecovariance matrix [28]:

σ(t) =M(t)(σ(0)−σ(∞))MT(t) +σ(∞), (6)

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with M(t) = exp(Y t), where [28]

Y =

−λ 1

m 0 0−mω1 −λ 0 0

0 0 −λ 1m

0 0 −mω2 −λ

(7)

and λ is the dissipation coefficient. σ(∞) is the covariance matrix correspondingto the asymptotic Gibbs state of the two bosonic modes in thermal equilibrium attemperature T. Its diagonal elements are given by (we take ~ = 1) [27]:

mω1σxx(∞) =σpxpx(∞)

mω1=

1

2coth

ω1

2kT,

mω2σyy(∞) =σpypy(∞)

mω2=

1

2coth

ω2

2kT, (8)

where k is the Boltzmann constant, while all the non-diagonal elements are 0.

3. DYNAMICS OF GAUSSIAN QUANTUM STEERING

In the case of bipartite Gaussian states one restricts the local measurementsperformed on Alice’s side to the set of Gaussian measurements. Such measurementscan be described by a positive operator with covariance matrix ΓRA , satisfying

ΓRA +i

2ΩA ≥ 0, ΩA =

(0 1−1 0

), (9)

and are generally implemented via symplectic transformations and homodyne detec-tion [29]. When measuring the part of Alice’s state we obtain the outcome a andthe conditional state of Bob ρaB is described by the covariance matrix B−C(A+ΓRA)−1CT.

It has been shown in Ref. [7, 30] that the steerability A→ B is present if andonly if the following relation does not hold:

σ+i

20A⊕ΩB ≥ 0, (10)

or equivalently

A> 0, and ∆Bσ +

i

2ΩB ≥ 0, (11)

where ∆Bσ = B−CA−1CT is the Schur complement of A in covariance matrix σ.

Since the covariance matrix of a reduced state is always positive, the only conditionfor certifying the absence of steering is to check whether ∆B

σ is a genuine covariancematrix. For this, one has to compute the Williamson form of ∆B

σ using appropriatesymplectic transformation such that ST∆B

σ S = Diag(νB,νB), and verify the uncer-tainty relation for its symplectic eigenvalue νB ≥ 1/2. Accordingly, a measure can

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be proposed of how much a state with covariance matrix σ is A→ B steerable withGaussian measurements, by quantifying the amount by which the condition (10) isviolated, as follows [31]:

GA→B(σ) := max0,− ln2νB. (12)

Clearly, also the B→A steerability can be quantified in an equable manner by com-puting the symplectic eigenvalues of the Schur complement of B in the covariancematrix σ. The quantity in Eq. (12) vanishes if and only if the state σ is not steerableby Gaussian measurements and is invariant under local symplectic transformations.

Note the similarity of this quantity with the negativity which measures the de-gree of entanglement of a state. In that situation the symplectic eigenvalues of thepartially transposed covariance matrix σ are relevant, which quantify the violation ofthe positive partial transpose (PPT) criterion [32, 33]. However, steering is funda-mentally different from entanglement, being in general asymmetric with respect tothe interchange between steerable parties.

The general quantity proposed in Ref. [31, 34], while easily computable for anarbitrary number of modes, has a particularly simple form when the steered party hasone mode:

GA→B(σ) = max0, 12

lndetA

4detσ= max0,S(A)−S(σ), (13)

where S is the Renyi-2 entropy, which for Gaussian states reads S = 12 ln(16detσ).

Now we study the dynamics of the Gaussian quantum steering for the consi-dered system, by taking an initial squeezed thermal state, with the covariance matrixof the form:

σ(0) =

a 0 c 00 a 0 −cc 0 b 00 −c 0 b

, (14)

witha = n1 cosh2 r+n2 sinh2 r+ 1

2 cosh2r,b = n1 sinh2 r+n2 cosh2 r+ 1

2 cosh2r,c = 1

2(n1 +n2 + 1)sinh2r,(15)

where n1, n2 are the mean photon numbers of the two modes and r is the squeezingparameter. The state is entangled if r > rs, where cosh2 rs = (n1+1)(n2+1)/(n1+n2 + 1) [35]. If ω1 = ω2 and n1 = n2, then we have a symmetry between the twomodes and therefore A→ B and B → A steerability are equivalent in the sense ofabove mentioned measure.

In Figs.1 it is illustrated the evolution of the GaussianA→B quantum steeringG for an initial squeezed vacuum state (n1 = n2 = 0) and a squeezed thermal state,respectively, for a non-zero equilibrium temperature of the thermal bath, as a function

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Fig. 1 – (Color online). Gaussian quantum steeringGA→B versus time t and squeezing parameter r foran initial squeezed vacuum state with n1 = n2 = 0 (left), and for an initial squeezed thermal state withn1 = n2 = 1 (right), in a thermal environment with temperature C ≡ coth(ω/2kT ) = 2, dissipationcoefficient λ= 0.1 and ω = 1 (ω1 = ω2).

of time t and squeezing parameter r. While the initial squeezed vacuum state isalways steerable, the initial squeezed thermal state is not steerable for all r > 0, likein the case of the entanglement. We notice that the Gaussian steering is increasingwith the squeezing parameter r and it is suppressed in a finite time.

Fig. 2 – (Color online). Gaussian quantum steering GA→B versus time t and temperature C ≡coth ω

2kT for an initial squeezed vacuum state with r = 2, λ= 0.1 and ω = 1.

In Fig. 2 it is shown the dynamics of the Gaussian A→ B steering for aninitial squeezed vacuum state as a function of time t and temperature C of the ther-mal reservoir. We observe that the Gaussian steering decreases with increasing thetemperature. Thermal noise and dissipation destroy in a finite time the steerabilitybetween the two parts.

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Fig. 3 – (Color online). Gaussian quantum steering GA→B (left) and Gaussian quantum steeringGB→A (right), versus time t and squeezing parameter r for an initial squeezed thermal state (n1 = 2,n2 = 0), temperature C = 2, dissipation coefficient λ= 0.1 and ω = 1.

The asymmetry in Gaussian steering is captured in Figs. 3 and this feature isthe result of the asymmetry between the modes in the initial squeezed thermal state.

Fig. 4 – (Color online). Gaussian quantum steeringG≡GA→B =GA→B (green plot) and logarithmicnegativity N (red plot) versus time t and squeezing parameter r for an initial squeezed vacuum state,for C = 2, λ= 0.1 and ω = 1.

Compared to the Gaussian quantum discord, which is decreasing asymptoti-cally in time, the Gaussian quantum steering suffers a sudden death behaviour likequantum entanglement. This can be seen in Fig. 4 where it is represented the depen-dence on time and squeezing parameter of the logarithmic negativity (red plot) andGaussian A→B quantum steering (green plot) for an initial squeezed vacuum state.

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4. SUMMARY AND CONCLUSIONS

We investigated the dynamics of the Gaussian quantum steering of two uncou-pled bosonic modes in interaction with a common thermal bath in the Markovianapproximation. For initial squeezed vacuum states and squeezed thermal states wehave shown that a sudden suppression of quantum steering takes place, for all tem-peratures of the thermal reservoir and all values of the squeezing parameter, a phe-nomenon similar to the entanglement sudden death. This behaviour is different fromthe time evolution of the Gaussian quantum discord, which tends to zero asympto-tically in time [36]. An initial squeezed thermal state is not steerable for all valuesof the squeezing parameter r, while a squeezed vacuum state is steerable for all val-ues of r. The Gaussian steering is increasing with the squeezing parameter and isdecreasing with increasing the temperature of the thermal bath. We mention that thesuppression of steering in a finite time takes place for all temperatures, including zerotemperature. The asymmetric Gaussian steering between the two modes has been il-lustrated in the case of a squeezed thermal state with different photon numbers of themodes, in particular it was shown that for definite values of the squeezing parameter,the system exhibits only an one-way steering.

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