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arXiv:quant-ph/0104007 · Physical Review A

Mixedness in the Bell violation versus entanglement of formation

Sibasish Ghosh A, Guruprasad Kar, Aditi Sen, Ujjwal Sen

Published Sep 13, 2001• 48 citations
arXivHTMLPDF

Abstract

Recently, Munro, Nemoto, and White (The Bell Inequality: A Measure of Entanglement?, quant-ph/0102119) tried to indicate that the reason behind a state $\ensuremath{\rho}$ having a higher amount of entanglement (as quantified by the entanglement of formation) than a state ${\ensuremath{\rho}}^{\ensuremath{'}},$ but producing the same amount of Bell violation, is due to the fact that the amount of mixedness (as quantified by the linearized entropy) in $\ensuremath{\rho}$ is higher than that in ${\ensuremath{\rho}}^{\ensuremath{'}}.$ We counter their argument with examples. We extend these considerations to the von Neumann entropy. Our results suggest that the reason as to why equal amount of Bell violation requires different amounts of entanglement cannot, at least, be explained by mixedness alone.

DOI
10.1103/physreva.64.044301

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Recently, Munro, Nemoto, and White (The Bell Inequality: A Measure of Entanglement?, quant-ph/0102119) tried to indicate that the reason behind a state \ensuremathρ\ensuremath{\rho}\ensuremathρ having a higher amount of entanglement (as quantified by the entanglement of formation) than a state \ensuremathρ\ensuremath′,{\ensuremath{\rho}}^{\ensuremath{'}},\ensuremathρ\ensuremath′, but producing the same amount of Bell violation, is due to the fact that the amount of mixedness (as quantified by the linearized entropy) in \ensuremathρ\ensuremath{\rho}\ensuremathρ is higher than that in \ensuremathρ\ensuremath′.{\ensuremath{\rho}}^{\ensuremath{'}}.\ensuremathρ\ensuremath′. We counter their argument with examples. We extend these considerations to the von Neumann entropy. Our results suggest that the reason as to why equal amount of Bell violation requires different amounts of entanglement cannot, at least, be explained by mixedness alone.
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 relation between the violation of the Bell-CHSH inequalities and entanglement properties of quantum states is not clear so one may consider the mixedness of the system to understand the entanglement properties better than the Bell-CHSH inequality. In this respect, we prove that if the mixedness of the state measured by the linear entropy is less than 2/3 but strictly greater than zero then the two qubit states are entangled. But if the linear entropy is greater or equal to 2/3 then the state may or may not be entangled. Further we show that our entanglement criterion detects larger set of entangled state than Bell-CHSH inequality and Santos's entropic criterion [Phys. Rev. A 69, 022305 (2004)]. Lastly we illustrate our result by citing few examples.
For two qubits belonging to Alice and Bob, we derive an approach to set up the bound of the Bell operator in the condition that Alice and Bob continue to perform local vertical measurements. For pure states we find that if the entanglement of the two qubits is less than 0.2644 (measured with von Neumann entropy) the violation of the Bell inequality will never be realized, and only when the entanglement is equal to 1 can the maximal violation (22)(2\sqrt{2})(22​) occur. For a specific form of mixed states, we prove that the bound of the Bell inequality depends on the concurrence. Only when the concurrence is greater than 0.6 can the violation of the Bell inequality occur and the maximal violation can never be achieved. We suggest that the bound of the Bell operator in the condition of local vertical measurements may be used as a measure of the entanglement.
Abstract. Entanglement is a critical resource used in many current quantum information schemes. As such entanglement has been extensively studied in two qubit systems and its entanglement nature has been exhibited by violations of the Bell inequality. Can the amount of violation of the Bell inequality be used to quantify the degree of entanglement? What do Bell inequalities indicate about the nature of entanglement? Entanglement was recognized early as one of the key features of quantum mechanics [1, 2]. Entanglement can be described as the correlation between distinct subsystems which cannot be created by local actions on each subsystem separately. The advantage o ered by quantum entanglement relies on the crucial premise that it cannot be reproduced by any classical theory [3-5]. Despite the fact that the possibility of quantum entanglement was acknowledged almost as soon as quantum theory was discovered, it is only in recent years that consideration has been given to nding methods to quantify it [6-17]. Historically the Bell inequal-ities were seen as a means of determining whether a two qubit system is entangled.
There is no doubt about the fact that entanglement and nonlocality are distinct resources. It is acknowledged that a clear illustration of this point is the difference between maximally entangled states and states that maximally violate a Bell inequality. We give strong evidence that this anomaly may be an artifact of the measures that have been used to quantify nonlocality. By reasoning that the numeric value of a Bell function is akin to a witness rather than a quantifier, we define a measure of nonlocality and show that, for pairs of qutrits and of four-level systems, maximal entanglement does correspond to maximal nonlocality in the same scenario that gave rise to the discrepancy.