ОПТИКА И СПЕКТРОСКОПИЯ, 2011, том 111, № 4, с. 690-699
ТОМОГРАФИЯ КВАНТОВЫХ СОСТОЯНИЙ
УДК 535.14
GENERAL BELL-CHSH TYPE AND ENTROPIC INEQUALITIES BASED
ON QUANTUM TOMOGRAMS
© 2011 L. V. Akopyan* and V. I. Manko**
*Moscow Institute of Physics and Technology (State University), Moscow 141700, Russia **P.N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia e-mail: loran.akopyan@phystech.edu Received April 18, 2011
Absract—Review of Bell-CHSH type and entropic inequalities in composite quantum correlated systems in the probability representation of states is presented. The upper bounds for some new Bell-CHSH type inequalities within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-CHSH type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states. Joint tomographic entropies of multiqubit systems are studied. Limitations on inequalities for tomographic entropies are obtained. A negative result of possible connection between the violation of entropic and Bell-CHSH type inequalities in multi-partite states is reported.
1. INTRODUCTION
Composite quantum systems can have strong quantum correlations which are manifested by the violation of Bell-CHSH inequalities [1, 2]. In [3], some connection of Bell-CHSH inequalities with tomo-graphic probability representation was studied. The development of this study was presented in [4—6]. To understand the origin of Bell-CHSH inequalities in tomographic probability representation of bipartite qudit states, we focus on the origin of the Bell-CHSH type inequalities in corresponding factorized joint probability distributions and their convex sums in standard classical probability theory.
The Bell-CHSH inequalities and their violation in quantum mechanics are considered as specific properties of non-local quantum correlations. In [7, 8], it be-cameclear that the Bell inequalities can be associated with properties of some probability distributions in classical probability theory.
The aim of this work is to establish general framework for Bell-CHSH type inequalities of composite random systems with specific joint-probability distributions. We show that there exist families of joint-probability distributions depending on extra parameters which violate Bell-CHSH type inequalities. We apply tomographic probability representation [9, 10] of quantum qudit states and show that the joint-probability distributions (tomograms) describing the mul-tiqudit states can violate Bell-CHSH type inequalities.
In addition, we study Shannon and von Neumann entropies [11, 12] for multiqudit states in order to understand whether there exists a certain connection between the upper bounds of Bell-CHSH type inequalities [13] and entropies, i.e., whether or not the knowledge of systems entropy and Bell-number is sufficient for classification of its separable and entangled states.
Some of the basic inequalities of information theory [14] are violated by tomographic entropies. Entropic Bell inequalities were studied in [15—17].
Since [18], we know that Bell-CHSH type numbers do not provide us with enough information on separability and entanglement of the composite system. We know this because there is no one-to-one correspondence between the separability of states and values of Bell-CHSH form [19].
In this work, we have also considered the properties of entropies within the framework of tomographic approach to the quantum states following [18—22] as well as to discuss the connection of these properties with Bell-CHSH type inequalities (see, for example [13, 19].
The paper is organized as follows.
In Section 2, we present the tomographic probability representation approach to studying multipartite systems. We discuss the origin of Bell-CHSH type inequalities and their relation to quantum non-local correlations. The violation of Bell-CHSH type inequalities is linked to the general problem of separability and entanglement of quantum states. We illustrate the theory on the example of two-qubit system and point out several cases of separable and entangled states based on the analysis of Bell-CHSH type inequality for the system and its upper bounds. In Section 3, we study the qubit-qutrit system in both classical and quantum tomographic probability representations of multipartite states. We establish both classical and quantum upper bounds for Bell-CHSH type inequalities in qubit-qutrit system. Several examples of parameters violating classical Bell-CHSH type inequalities are presented and their domains are calculated. Finally, in Sections 4 and 5, two-qutrit and three-qubit states are studied in terms of establishing the upper
bounds of possible Bell-CHSH type inequalities analogous to Tsirelson bound in two-qubit case.
In Section 6, tomographic entropies and their general properties in terms of entropie inequalities are reviewed. Tomographic entropy for joint-probability distributions is introduced and studied for two-qubit system. Violation of classical entropic inequalities are explained. In Section 7, the possibility for a special connection between tomographic entropy and Bell-CHSH type inequalities is discussed. Finally, we summarize the results of our work in Conclusion.
2. BELL-CHSH TYPE INEQUALITIES IN QUANTUM TOMOGRAPHY
For simplicity, the following assumes all relevant state spaces are finite dimensional. The tomogram of qubit described by a 2 x 2 density matrix p was given [23, 24]
w
(рДФ ) = Diag (iTp Û ) =
Pllcos 2 - sineRe(p12.-) + p22sin 2 Pli cos2 e + sineRe(pi2<?+ P22 cos2 e
(1)
M (ni, n 2) =
w +
1
Г'" i
w +
1
Г'" 2
1 - w (+ i, ni ) 1 - w (+ 2 ' П2
(2)
The elements of the composite bipartite spin tomogram with separable or entangled density matrix p can be calculated via the formula
w (mi, Û1, m2Û2 )
= (mi, m lu/ ® Ûlp Ûi
>гт,
mi, m2
(3)
where U is the unitary rotation operator of the SU(2) irreducible representation of spin 1/2, 9 and 8 are the
Euler angles. The elements of the tomogram w (±1 ,U
are the probabilities of detecting thequbit in spin up or down states. The term "tomogram" originates from
the matrix of the operator U(n) those elements are parameterized in Euler angles. Hence, the quantum-to-mographic distribution (1) depends on the direction n(sin 0 cos 9, sin 0 sin 9, cos 0) along which the states of qubit are measured.
The tomogram (1) describes the qubit states just as well as a density matrix does [10]. However, it describes a qubit state with just one direction of measurement. To describe a qubit state with two directions of measurements, we fixate two unit vectors n1 and n2, and build a 2 x 2 stochastic matrix out of corresponding probability vectors (1)
The matrix (2) describes the quantum state of qubit just as completely as its quantum tomogram (1) and just as well as a density matrix or a wave function of the state (see for example, [6]). It allows to rewrite averages for quantum correlations in multipartite systems in a way that leads directly to discovery of new type of Bell-CHSH inequalities (see for example, [13]).
where Ub U2 are unitary rotation operators pertaining to the Hilbert spaces of first and second qudits respectively.
For qubits, the probability distribution (3) will result in tomographic vector w(nb n 2) with four components. To build the stochastic matrix of the states of two-qubits M(ab a2, a3, a4), one needs to fixate four directions a1, a 2, a3, a 4 two for the first qubit and two for the second. It will result in a 4 x 4 stochastic matrix with columns taken as w(aj, ak) with j = 1,2 and k = 3,4.
Based on quantum tomograms for bipartite systems (3), it is possible to derive a useful way of seeing entanglement in quantum multipartite states [6, 19]: if the stochastic matrix of the composite system M (a1, a 2, a 3, a 4) violates the tomographic Bell-CHSH type inequalities then the composite state is entangled and the composite state is separable if M(a1, a 2, a 3, a 4) satisfies the inequality.
The definitions (1)—(3) are easily generalized over arbitrary qudit states by considering af-dimensional Hilbert spaces and directly generalizing the tomographic distributions over the f-dimensions. For example, the stochastic matrix for a qutrit is a 3 x 3 matrix with three arbitrary measurement directions, the stochastic matrix of a two-qubit system will be obviously given either by formula (3) or by simple tensor product of two matrices of (2) and is a 4 x 4 matrix and alike.
In the probabilistic approach to quantum mechanics [9, 10], the mixed quantum state is separable if its tomogram can be presented as convex sum of simply separable tomograms
w (12)(mb m2, ni, n 2) = ^ PtwkVi, n^wfm n 2) (4)
k
with some probability weights pk .
In order to detect entanglement, one can use the similarity of classical states and those of quantum separable states. Since the probability representation of states is a universal method of describing both classical and non-classical states, one could begin with constructing separable classical states and observe how their statistical properties are changed in quantum tomography.
We regard Bell-CHSH type inequalities exactly as such statistical properties of tomographic probability distributions. To this end, we begin with constructing
Bell-CHSH type inequalities first in classical statistical mechanics. Then entering quantum mechanics, it can be shown that they can be violated by quantum tomograms [3].
Given a tomographic joint-probability distribution of ú-partite system
w(mbm2, ...md,nbn2,... nd) = = wi(mi, ni) ® w2(m2, n2) ® ... ® w¿(md, nd),
one can define the correlation function between arbitrary local observables m1, m2, ..., md (for example, spin projections) as follows
(mm...md) =
\ 1 2 d/ n1,n2,.nd
^ w(mi,m2,.md,ni,n2,...ndM
m2 ...md
(6)
o(2 0 2) =
(+ + + +
+ + + +
Iifài —
+ 1 +1 +1 -1
-1 -1 -1 +1
-1 -1 -1 +1
+ 1 +1 +1 -1
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