Local-hidden-state models for T-states using finite shared randomness
aa r X i v : . [ qu a n t - ph ] S e p epl draft Local-hidden-state models for T-states using finite shared ran-domness (a)
Yuan-Yuan Zhang and
Fu-Lin Zhang (b)
Department of Physics, School of Science, Tianjin University - Tianjin 300072, China
PACS – Entanglement and quantum nonlocality (e.g. EPR paradox, Bell’s inequalities,GHZ states, etc.)
PACS – Foundations of quantum mechanics; measurement theory
PACS – Entanglement measures, witnesses, and other characterizations
Abstract – The study of local models using finite shared randomness originates from the consid-eration about the cost of classically simulating entanglement in composite quantum systems. Weconstruct explicitly two families of local-hidden-state (LHS) models for T-states, by mapping theproblem to the Werner state. The continuous decreasing of shared randomness along with entan-glement, as the anisotropy increases, can be observed in the one from the most economical modelfor the Werner state. The construction of the one for separable states shows that the separableboundary of T-states can be generated from the one of the Werner state, and the cost is 2 classicalbits.
Introduction. –
Nonclassical correlations in compos-ite quantum systems and their hierarchy are fundamentalissues in quantum information [1–5]. Many concepts ofthese correlations can be traced back to the early days ofquantum mechanics, and play key roles in several quantuminformation processes. On the other hand, the tasks inquantum information also provide points of view to studythe correlations. An important example is the work ofWiseman et al. [6], in which they define Bell nonlocalityand Einstein-Podolsky-Rosen (EPR) steering according totwo tasks, and prove that the former is a sufficient condi-tion for the latter and entanglement is necessary for bothof them.In the tasks of Wiseman et al. [6], two observers, Aliceand Bob, share a bipartite entangled state. Alice can affectthe postmeasured states left to Bob by choosing differentmeasurements on her half. Such ability is termed steering by Schr¨odinger [7]. EPR steering from Alice to Bob ex-ists when Alice can convince Bob that she has such ability,which is equivalent to the fact that unnormalized postmea-sured states can not be described by a local-hidden-state(LHS) model. Further, their state is Bell nonlocal, whenthe two observers can convince Charlie, a third person,that the state is entangled. This is demonstrated by theinexistence of local-hidden-variable (LHV) model explain- (a)
EPL, 127 (2019) 20007 (b)
Corresponding author: fl[email protected] ing correlations of outcomes of their joint local measure-ment. A LHS model is a particular case of a LHV model,of which the hidden variable is a single-particle state andone of the response functions is the probability of mea-surement on the state.Construction of local models, especially the optimalones, provides a division between the quantum and classi-cal worlds, in the sense of whether the nonclassical correla-tions exist. However, it is an extremely difficult problemto explicitly derive optimal models. Only a few modelsbeyond Werner’s results [8] have been reported, such asthe ones in [9–12], most of which are for states with highsymmetries. Our recent work [13] shows the possibilityof generating local models for states with a lower symme-try, from the ones with a high symmetry. Namely, we ob-tain the optimal models for T-states (Bell diagonal states),given by Jevtic et al. [11] based on the steering ellipsoid[14], by mapping the problem to the one of the Wernerstate.On the other hand, Bowles et al. [10] raise the issueof constructing local models using finite shared random-ness. This comes from their consideration about the costof classically, measured by classical bits encoding the localvariable, simulating the correlations in an entangled state.They give a series of LHV models for Werner states usingfinite shared randomness, and prove the existence of theones for entangled states admitting a LHV model. Thesep-1uan-Yuan Zhang and Fu-Lin Zhangresults inspire a method for constructing LHV models forentangled states, in which the problem of finding a localmodel for an infinite set of measurements is mapped tothe one of a finite set of measurements [15–17]. In addi-tion, the concepts of superlocality [18, 19] and superun-steerability [20] stemm from the study of shared classicalrandomness required to simulate local correlations.In the present work, we study local models for T-statesby extending our strategy in [13] to the case with finiteshared randomness. They are LHS models, as the sharedlocal variables are sets of discrete states on the Blochsphere and Bob’s response function is his measurementprobability on these states. Expressing the discrete dis-tributions for Werner states in
Protocols 1 and of [10]in terms of Dirac delta functions, we derive a family ofLHS models for T-states by using the mapping in [13].The one generated from the most economical LHV modelfor Werner state is discussed in detail, which provides anexample to observe the continuously changing shared ran-domness with entanglement. Besides, we construct a LHSmodel, not belonging to the two protocols in [10], for thecritical separable Werner state, by decomposing it intoproduct states. It can be transformed into the LHS mod-els for the critical separable T-states by a generalizationof the original mapping in [13]. This shows the possibilityof generating the separable boundary for a class of stateswith a low symmetry, and decomposing them into productstates, from a higher symmetric case. Preliminaries. –
LHS model.
We first give a brief review of the conceptsof EPR steering and LHS model, under the context of two-qubit system and projective measurements. An arbitrarytwo-qubit state shared by Alice and Bob can be writtenas ρ AB = 14 ( I ⊗ I + ~a · ~σ ⊗ I + I ⊗ ~b · ~σ + X ij T ij ~σ i ⊗ ~σ j ) , (1)where I is the unit operator, ~a and ~b are the Bloch vectorsfor Alice and Bob’s qubit, ~σ = ( σ x , σ y , σ z ) is the vectorof the Pauli operators, and T ij is correlation matrix. Wefocus on the case in which Alice makes a projection mea-surement on her part. The measurement operator of Aliceuniquely corresponds to a unit vector ~x and a outcome a = ± ~xa = 12 ( I + a~x · ~σ ) . (2)After the measurement, Bob’s state becomes ρ ~xa = Tr(Π ~xa ⊗ I ρ AB )= 14 [(1 + a~a · ~x ) I + ( ~b + aT T ~x ) · ~σ ] , (3)where T T is transposed T . The set of ρ ~xa is called anassemblage.A LHS model is defined as ρ LHS = Z ω ( ~λ ) p ( a | ~x, ~λ ) ρ ~λ d~λ. (4) Here, ρ ~λ is a hidden state depending on the hidden vari-able ~λ with the distribution function ω ( ~λ ). And, p ( a | ~x, ~λ )is a response function simulating the probability of Alice’soutcome, with p ( a | ~x, ~λ ) > p (1 | ~x, ~λ )+ p ( − | ~x, ~λ ) = 1.If there exists a LHS model satisfying ρ ~xa = ρ LHS (5)for all the measurements, the outcomes of Alice’s mea-surements and Bob’s collapsed state can be simulated bya LHS strategy without any entangled state [6]. On thecontrary, if a LHS model satisfying (5) does not exist, ρ AB is termed steerable from Alice to Bob.Without loss of generality, we may take a hidden vari-able to the unit Bloch vectors and the local hidden statesto be corresponding pure qubit states [12] as ρ ~λ = | ~λ ih ~λ | = 12 ( I + ~λ · ~σ ) . (6)Then, d~λ is the surface element on the Bloch sphere. Wecan take p ( a | ~x, ~λ ) = 12 (cid:2) af ( ~x, ~λ ) (cid:3) , (7)with f ( ~x, ~λ ) ∈ [ − , ρ LHS = Z ω ( ~λ )14 (cid:20) I + ~λ · ~σ + af ( ~x,~λ )+ af ( ~x,~λ ) ~λ · ~σ (cid:21) d~λ. (8)Consequently, the equation (5) requires Z ω ( ~λ ) d~λ = 1 , (9a) Z ω ( ~λ ) f ( ~x, ~λ ) d~λ = ~a · ~x, (9b) Z ω ( ~λ ) ~λd~λ = ~b, (9c) Z ω ( ~λ ) f ( ~x, ~λ ) ~λd~λ = T T ~x. (9d)The spin correlation matrix can always be diagonalized bylocal unitary operations, which preserve steerability or un-steerability. Hence, we consider the diagonalized T , that T = Diag { T x , T y , T z } , and omit its superscript T in thefollowing parts of this article. Constructing a LHS modelfor a state ρ AB is equivalent to finding a pair of ω ( ~λ ) and f ( ~x, ~λ ) fulfilling these requirements. T-states.
The state (1) is called a T-state, when theBloch vectors, ~a and ~b , vanish. In our recent work [13],we present an approach to derive the optimal LHS modelfor T-states. We first assume the correlation matrix onthe EPR-steerable boundary being T and T = tT with t ≥
0. That is, the T-state with t > ≤ t ≤ T − and defining the unit vector ~λ ′ = T − ~λ | T − ~λ | − ,p-2ocal-hidden-state models for T-states using shared randomnesswhere |·| is the Euclidean vector norm. Then the condition(9d) is rewritten as Z ω ′ ( ~λ ′ ) 1 | T ~λ ′ | f ( ~x, ~λ ) ~λ ′ d~λ ′ = t~x, (10)where ω ′ ( ~λ ′ ) is the distribution function of the new definedhidden variable ~λ ′ , and d~λ ′ is a surface element on itsunit sphere. These variables are connected by a Jacobiandeterminant as d~λ = | det T || T − ~λ | d~λ ′ , ω ( ~λ ) d~λ = ω ′ ( ~λ ′ ) d~λ ′ . (11)In the optimal LHS model of the critical Werner state[6,8], with T = − Diag[1 / , / , / ω ′ ( ~λ ′ ) = 12 π | T ~λ ′ | , f ( ~x, ~λ ) = sgn( ~x · ~λ ′ ) . (12)We find that, these relations give exactly the optimal LHSmodel for an arbitrary T-state and leads to the criticalcondition [11, 13, 21] Z π | T ~λ ′ | d~λ ′ = 1 . (13)An explicit expression for this integral can be found in thework of Jevtic et al. [11]. LHS models for T-States with finite shared ran-domness. –
We now generate the LHS models for T-states with finite shared randomness, using our approachreviewed above. The formulas in the above section are de-rived based on continuous local variables. To utilize theseresults, we represent the distribution of the finite hiddenvariables as delta functions. We mainly concentrate onthe details of the two models corresponding to the mosteconomical one of the Werner state and the one for theseparable Werner state.
LHS model on the icosahedron.
In the most econom-ical model simulating an entangled Werner state, Aliceand Bob share i = { , ..., } uniformly distributed, cor-responding to 12 vertices of the icosahedron representedby the normalized vectors ~v i . That is, the distribution isgiven by ω ( ~λ ) = X i δ ( ~λ − ~v i ) . (14)The radius of a sphere inscribed inside the icosahedron is l = q (5 + 2 √ /
15. The icosahedron is symmetric under ~v i → − ~v i , and its vertices satisfy the properties P j sgn( ~v j · ~v i ) ~v j = 2 γ~v i with γ = 1 + √
5. Then, the vector l~x canalways be represented as a convex decomposition l~x = P i ω i ~v i with ω i ≥ P i ω i = 1. Defining the function f ( ~x, ~λ ) = − X i ω i sgn( ~v i · ~λ ) , (15) one can obtain Z (cid:20)X j δ ( ~λ − ~v j ) (cid:21)(cid:20) − X i ω i sgn( ~v i · ~λ ) (cid:21) ~λd~λ = − γl ~x, (16)which is the relation (9d) for the Werner state.To fulfill the condition (9d), equivalently the equation(10), for T-states, an intuitive construction is given by ω ′ ( ~λ ′ ) = X i S δ ( ~λ ′ − ~v i ) | T ~λ ′ | , (17) f ( ~x, ~λ ) = X i ω i sgn( ~v i · ~λ ′ ) , (18)where S is a constant determined by the normalizationcondition (9a). They lead to the visibility parameter in(10) being t = S γl γl P i | T ~v i | . (19)Straightforward calculation gives the distribution of ~λ as ω ( ~λ ) = X i S δ ( ~λ − T ~v i | T ~v i | − ) | T ~v i | . (20)Then, both the integrals in (9b) and (9c) can be easilychecked to be zero, by using the symmetries ω ( − ~λ ) = ω ( ~λ )and f ( ~x, − ~λ ) = − f ( ~x, ~λ ). Therefore, the functions (18)and (20) represent a LHS model for the T-state with thevisibility parameter in (19). And the extension to smalleramounts of t is straightforward.Obviously, in our LHS models for T-states, the hiddenvariable i = { , ..., } distributes nonuniformly, whoseprobability is proportional to | T ~v i | . The correspondingunit vectors ~λ ′ locate on ~v i , and the Bloch vector of hid-den states ~λ on T ~v i | T ~v i | − . Both the distribution andvisibility parameter, given in (19), covered by the model,depend on the orientation of the icosahedron. A naturalquestion is which orientation is optimal, in the sense ofmaximizing the parameter t , or equivalently S . Optimal icosahedron.
Since it is a complex problemto perform general maximisations, we consider the specialcase with an axial symmetry that | T ,x | = | T ,y | with theaid of numerical calculation. Then, the relation between | T ,x | and | T ,z | can be written as a simple formula [11].And the orientation of the icosahedron can be representedby the intersection of Z-axis with the surface of the icosa-hedron. There are three types of special points on thesurface, which are vertices, midpoints of edges and centreof faces. We suspect that the maximum of S occurs atthese special points.Choosing a trajectory of the intersection, as shown inFig. 1, consisting of an edge and two medians of faces,one can plot the values of S versus the location of in-tersection (we omit the figures here). These curves in-dicate that the maximum of S on the trajectory occursat vertices when | T ,z | ≤ / Fig. 1: (Color online) The icosahedron in the construction ofLHS models in
Protocol 1 . Dashed blue lines show intersectionsof Z-axis with the surface of icosahedron during our rotation.Vertices, midpoints of edges and centre of faces on the dashedblue lines are marked as A i , B j and C k respectively.Fig. 2: (Color online) The solid curve shows the maximumof t in the LHS models based on the icosahedron, in companywith the values for ten thousand random orientation, and thedashed line is for the value of the Werner state.Fig. 3: (Color online) The solid curve shows the shared ran-domness in the LHS model based on the icosahedron in optimalorientation, and the dashed line is for the value of the Wernerstate. T z E n t ang l e m en t Fig. 4: (Color online) The solid curve shows entanglementof T-states admitting the optimal LHS model based on theicosahedron, the dashed line is for the value of the Wernerstate, and the dot-dashed curve is for T-states on the EPR-steerable boundary. when 1 / < | T ,z | . .
89, or at midpoints of edges when | T ,z | & .
89. These maximums, in the same sequence,can be analytically expressed as S A = 6 √ Z + √ X + 5 Z , (21) S C = √ p Xα + + Zβ − + p Xα − + Zβ + , (22) S B = 3 √ √ X + p Xα + + Zα − + p Xα − + Zα + , (23)where X = T ,x , Z = T ,z , α ± = 5 ± √ β ± =5 / ± √
5. We find that they are optimal among arbitraryorientations of the icosahedron, by comparing them withone hundred thousand randomly generated intersections.One-tenth of the random points are shown in Fig. 2, incompany with the corresponding maximums of t .Our construction provides a family of LHS models witha fixed dimensionality of the local variable. It is inter-esting to observe the continuously changing shared ran-domness, and its relation with the region of T-states ad-mitting our models. We plot the maximums of t in Fig.2, which measure how close our models get to the EPR-steerable boundary. The corresponding shared random-ness, measured by the entropy of the distribution (20) [10], H = − P i q i log q i with q i = | T ~v i | S/
12, is shown in Fig.3. Obviously, the visibility parameter and entropy showtwo opposite trends. The anisotropy of the correlation ma-trix enhances the maximums of t , while it decreases theentropy. Among the family of T-states, the LHS model forthe Werner state, with the maximum distance to the EPR-steerable boundary, requires the most shared randomness.This anomalous phenomenon prompts us to go back tothe original point: the cost of classically simulating thecorrelations of entangled states [10]. It is direct to derivethe entanglement, measured by concurrence [22], for axialsymmetric T-states as max { , (2 t | T ,x | + t | T ,z |− / } . Asp-4ocal-hidden-state models for T-states using shared randomnessshown by the solid line in Fig. 4, the entanglement revealsa similar tend as the number of classical bits to simulate it.The degree of entanglement reaches its maximum at thepoint of Werner states, T ,z = 1 /
2, and decreases with theanisotropy. Comparing with the maximums of t , one canfind that the points on the EPR-steerable boundary withsmall entanglement are easy to approach, in the sense ofthe cost of classically simulating the correlations of entan-gled states.In Fig. 4, a noteworthy point is the small interval withzero entanglement. This indicates that our LHS modelsare not the most economical ones, at least for the separablestate in the small interval. This is because a 4-dimensionallocal variable is sufficient to simulate a separable two-qubitstate, while the least shared bits in our construction forT-states is 2 .
96. We shall present more discussion aboutthe LHS models for separable T-states below.
Separable boundary.
The above results can bestraightforwardly extended to any LHS model for theWerner state in
Protocols 1 and 2 of [10] using a 3-dimensional polyhedron with D vertices. We omit theseformulas for brevity.In this part, we focus on the case with a shared vari-able of dimension D = 4. In the results for Werner state[10], the tetrahedron, with 4 vertices, is without inversionsymmetry and hence be excluded from Protocols 1 and 2 .On the other hand, the maximum visibility parameter onecan simulate with D = 4 is the boundary of the separableWerner state [10]. Here our question is whether one cangenerate the boundary of the separable T-states from theone of the Werner state, as we do in the study of EPR-steering [13].To answer the above question, we restrict the responsefunction to the form f ( ~x, ~λ ) = ~x · ~η, (24)where ~η is Alice’s Bloch vector depending on ~λ . We termthe corresponding LHS model as a LHS model for sepa-rable state . The entanglement of the two-qubit state ρ AB is demonstrated by the inexistence of a LHS model with f ( ~x, ~λ ) in the above form.On can derive the solution for Werner states to the con-ditions (9), by decomposing the critical separable Wernerstate into four product states. Let the Bell states | Ψ ± i =( | i± | i ) / √ | Φ ± i = ( | i± | i ) / √
2. The criticalseparable Werner state is ρ wAB = (3 | Φ − ih Φ − | + | Φ + ih Φ + | + | Ψ + ih Ψ + | + | Ψ − ih Ψ − | ) /
6. We assume the normalized state | φ i i ∝ √ | Φ − i + e iθ i | Φ + i + e iθ i | Ψ + i + e iθ i | Ψ − i , (25)to be separable, and to satisfy ρ wAB = P i | φ i ih φ i | / | φ i = (sin α | i− cos α e iβ | i ) ⊗ (cos α | i +sin α e iβ | i ) , (26) | φ i = σ x ⊗ σ x | φ i , | φ i = σ y ⊗ σ y | φ i , and | φ i = σ z ⊗ σ z | φ i , where α = arccos(1 / √
3) and β = − π/
4. Alice’smeasurements on the decomposition ρ wAB = P i | φ i ih φ i | / ω ( ~λ ) = X i δ ( ~λ − ~v i ) , ~η = − ~λ, (27)with the 4 vertices of the tetrahedron ~v = (1 , − , / √ ~v = (1 , , − / √ ~v = ( − , , / √ ~v =( − , − , − / √
3. They satisfy
Z (cid:20) X i δ ( ~λ − ~v i ) (cid:21)(cid:20) ~x · ( − ~λ ) (cid:21) ~λd~λ = − ~x. (28)The other solution leads to a model on the mirror imageof the tetrahedron.We now turn to the T-states on the separable boundary.It is universal to consider a positive definite T , as anyminus sign can be merged into ~η ( ~λ ). Here we perform T − on the condition (9d), and define the unit vector ~λ ′′ = T − ~λ | T − ~λ | − and its distribution ω ′′ ( ~λ ′′ ). Thenthe condition (9d) for a separable state is rewritten as Z ω ′′ ( ~λ ′′ ) 1 | T ~λ ′′ | (cid:2) ( T − ~x ) · ~η (cid:3) ~λ ′′ d~λ ′′ = ~x. (29)Defining the unit vector ~η ′′ = T − ~η | T − ~η | − , one canfind that ( T − ~x ) · ~η = ~x · ~η ′′ | T ~η ′′ | − . From the integral(28), it was easy to find a pair of ω ′′ ( ~λ ′′ ) and ~η ′′ satisfying(29) as ω ′′ ( ~λ ′′ ) = X i δ ( ~λ ′′ − ~v i ) | T ~v i | , ~η ′′ = ~λ ′′ . (30)The normalization condition (9a) and the coordinates of ~v i lead to | T x | + | T y | + | T z | = 1 , (31)which is nothing but the separable boundary of T-states[23,24]. Then, in the space of λ , the distribution and Blochvector of Alice are ω ( ~λ ) = X i δ (cid:0) ~λ − √ T ~v i (cid:1) , ~η = ~λ. (32)Substituting them into the equations (9b) and (9c), onecan confirm both the integrals to be zero.In the LHS models for separable T-states, defined by(32), the shared variables are encoded on √ T ~v i , and areuniformly distributed. The amount of shared randomnessis 2 bits, which is not affected by the anisotropy of thecorrelation matrix. The models are optimal in the senseof reaching the separable boundary. However, the questionas to whether they are the most economical remains open.p-5uan-Yuan Zhang and Fu-Lin Zhang Summary. –
We study LHS models for T-states usingfinite shared randomness. The models are generated fromthe ones for Werner states, two of which are mainly dis-cussed. The first is derived by using our recent approach[13] on the most economical model for the Werner state.It provides an example to observe the continuously chang-ing shared randomness with an entangled state. With theincrease of anisotropy, the amount of shared classical bitsdrops along with entanglement, although the model getscloser to the EPR-steerable boundary. The second one isrestricted to simulate a separable state by a condition onAlice’s response function. It is derived from the one forthe Werner state by a generalized generating approach,and reaches exactly the separable boundary of T-states.The cost of classical randomness in this model is 2 bits,which is not affected by the anisotropy of the correlationmatrix.It would be interesting to consider the open questions orextensions below. First, our approach to derive the LHSmodels for T-states on the separable boundary is actuallyto decompose them into product states. Geneneralizingthis method may be a starting point to define T-statesin higher-dimensional systems, which has been raised inour recent work [13]. Second, in what region our modelusing the icosahedron is the most economical one? Third,what is the minimal cost to classically simulate a separablestate? This is a nontrivial question, as in LHS models onthe separable boundary, the amount of bits is differentfrom the entropy of states. This difference originates fromthe superposition of states in composite quantum systems,and may be interpreted as a kind of quantum correlation. ∗ ∗ ∗
This work is supported by the NSF of China (Grant No.11675119, No. 11575125 and No. 11105097).
REFERENCES[1]
Nielsen M. A. and
Chuang I. L. , Quantum Compu-tation and Quantum Information (Cambridge UniversityPress, Cambridge) 2000.[2]
Horodecki R., Horodecki P., Horodecki M. and
Horodecki K. , Rev. Mod. Phys. , (2009) 865.[3] Modi K., Brodutch A., Cable H., Paterek T. and
Vedral V. , Rev. Mod. Phys. , (2012) 1655.[4] Brunner N., Cavalcanti D., Pironio S., Scarani V. and
Wehner S. , Rev. Mod. Phys. , (2014) 419.[5] Augusiak R., Demianowicz M. and
Ac´ın A. , J. Phys.A: Math. Theor. , (2014) 424002.[6] Wiseman H. M., Jones S. J. and
Doherty A. C. , Phys.Rev. Lett. , (2007) 140402.[7] Schr¨odinger E. , Proc. Camb. Phil. Soc. , (1935) 555.[8] Werner R. F. , Phys. Rev. A , (1989) 4277.[9] Bowles J., V´ertesi T., Quintino M. T. and
BrunnerN. , Phys. Rev. Lett. , (2014) 200402.[10] Bowles J., Hirsch F., Quintino M. T. and
BrunnerN. , Phys. Rev. Lett. , (2015) 120401. [11] Jevtic S., Hall M. J., Anderson M. R., Zwierz M. and
Wiseman H. M. , J. Opt. Soc. Am. B , (2015) A40.[12] Bowles J., Hirsch F., Quintino M. T. and
BrunnerN. , Phys. Rev. A , (2016) 022121.[13] Zhang F.-L. and
Zhang Y.-Y. , Phys. Rev. A , (2019)062314.[14] Jevtic S., Pusey M., Jennings D. and
Rudolph T. , Phys. Rev. Lett. , (2014) 020402.[15] Cavalcanti D., Guerini L., Rabelo R. and
Skrzypczyk P. , Phys. Rev. Lett. , (2016) 190401.[16] Hirsch F., Quintino M. T., V´ertesi T., Pusey M. F. and
Brunner N. , Phys. Rev. Lett. , (2016) 190402.[17] Cavalcanti D. and
Skrzypczyk P. , Rep. Prog. Phys. , (2017) 024001.[18] Donohue J. M. and
Wolfe E. , Phys. Rev. A , (2015)062120.[19] Jebaratnam C., Aravinda S. and
Srikanth R. , Phys.Rev. A , (2017) 032120.[20] Das D., Bhattacharya B., Datta C., Roy A., Je-baratnam C., Majumdar A. S. and
Srikanth R. , Phys. Rev. A , (2018) 062335.[21] Nguyen H. C. and
Vu T. , EPL (Europhysics Letters) , (2016) 10003.[22] Wootters W. K. , Phys. Rev. Lett. , (1998) 2245.[23] Horodecki R. et al. , Phys. Rev. A , (1996) 1838.[24] Daki´c B., Vedral V. and
Brukner ˇC. , Phys. Rev.Lett. , (2010) 190502.(2010) 190502.