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A revisit to maximally and non-maximally entangled mixed states in the light of Tsirelson's bound

Sovik Roy, Biplab Ghosh

Published Jun 11, 2015• 0 citations
OpenAlex

Abstract

In this paper we have investigated different classes of maximally and non-maximally entangled mixed states from the view point of Bell -violation, entanglement and mixedness. We have shown that mixedness and entanglement of Werner state is parameter independent whereas that of Ishizaka class, Munro class of maximally entangled mixed states and our proposed class of non- maximally entangled mixed states are dependent on certain parameter ranges. We have also shown that our non maximally entangled class of mixed states have structural similarity with the rank -$2$ form of Ishizaka class of maximally entangled mixed states.

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In this paper we have investigated different classes of maximally and non-maximally entangled mixed states from the view point of Bell -violation, entanglement and mixedness. We have shown that mixedness and entanglement of Werner state is parameter independent whereas that of Ishizaka class, Munro class of maximally entangled mixed states and our proposed class of non- maximally entangled mixed states are dependent on certain parameter ranges. We have also shown that our non maximally entangled class of mixed states have structural similarity with the rank -222 form of Ishizaka class of maximally entangled mixed states.
Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy. Along the way, we determine the states that, for a given value of entropy, achieve maximal violation of Bell's inequality.
The maximally entangled mixed states of Munro et al. [Phys. Rev. A 64, 030302 (2001)] are shown to exhibit interesting features vis \'a vis conditional entropic measures. The same happens with the Ishizaka and Hiroshima states [Phys. Rev. A 62, 022310 (2000)], whose entanglement degree cannot be increased by acting on them with logic gates. Special types of entangled states that do not violate classical entropic inequalities are seen to exist in the space of two qubits. Special meaning can be assigned to the Munro et al. special participation ratio of 1.8.
We study the efficiency of two-qubit mixed entangled states as resources for quantum teleportation. We first consider two maximally entangled mixed states, viz., the Werner state\cite{werner}, and a class of states introduced by Munro {\it et al.} \cite{munro}. We show that the Werner state when used as teleportation channel, gives rise to better average teleportation fidelity compared to the latter class of states for any finite value of mixedness. We then introduce a non-maximally entangled mixed state obtained as a convex combination of a two-qubit entangled mixed state and a two-qubit separable mixed state. It is shown that such a teleportation channel can outperform another non-maximally entangled channel, viz., the Werner derivative for a certain range of mixedness. Further, there exists a range of parameter values where the former state satisfies a Bell-CHSH type inequality and still performs better as a teleportation channel compared to the Werner derivative even though the latter violates the inequality.
We compute the measure of entanglement for some classes of states belonging to the simplex of Bell-diagonal states of two qutrits introduced by Baumgartner et al. [Phys. Rev. A 74, 032327 (2006)]. We discuss also partial characterization of that simplex in terms of negativity and linear entropy. A numerical exploration of the entropy-entanglement plane indicates that, in the analogy to the two-qubit case [Munro et al., Phys. Rev. A 64, 030302(R) (2001)], two-qutrit Werner states do not maximize entanglement for a given entropy.
We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity, and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. Furthermore we characterize all nearly entangled states closest to the maximally mixed state and derive a lower bound on the volume of separable mixed states.