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arXiv:0802.2132 · Physical Review A

Bell-inequality violation versus entanglement in the presence of local decoherence

Abraham G. Kofman, Alexander N. Korotkov

Published May 22, 2008• 31 citations
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Abstract

We analyze the effect of local decoherence of two qubits on their entanglement and the Bell-inequality violation. Decoherence is described by Kraus operators, which take into account dephasing and energy relaxation at an arbitrary temperature. We show that in the experiments with superconducting phase qubits the survival time for entanglement should be much longer than for the Bell-inequality violation.

DOI
10.1103/physreva.77.052329

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We analyze the effect of local decoherence of two qubits on their entanglement and the Bell-inequality violation. Decoherence is described by Kraus operators, which take into account dephasing and energy relaxation at an arbitrary temperature. We show that in the experiments with superconducting phase qubits the survival time for entanglement should be much longer than for the Bell-inequality violation.
We analyze conditions for violation of the Bell inequality in the Clauser-Horne-Shimony-Holt form, focusing on Josephson phase qubits. We start the analysis with maximum violation in the ideal case, and then take into account the effects of local measurement errors and decoherence. Special attention is paid to configurations of the qubit measurement directions in the pseudospin space lying within either horizontal or vertical planes; these configurations are optimal in certain cases. Besides local measurement errors and decoherence, we also discuss the effect of measurement crosstalk, which affects both the classical inequality and the quantum result. In particular, we propose a version of the BI which is insensitive to the crosstalk.
A well-known manifestation of quantum entanglement is that it may lead to correlations that are inexplicable within the framework of a locally causal theory---a fact that is demonstrated by the quantum violation of Bell inequalities. The precise relationship between quantum entanglement and the violation of Bell inequalities is, however, not well understood. While it is known that entanglement is necessary for such a violation, it is not clear whether all entangled states violate a Bell inequality, even in the scenario where one allows joint operations on multiple copies of the state and local filtering operations before the Bell experiment. In this paper we show that all entangled states, or more precisely, all not-fully-separable states of arbitrary Hilbert space dimension and arbitrary number of parties, violate a Bell inequality when combined with another state which on its own cannot violate the same Bell inequality. This result shows that quantum entanglement and quantum nonlocality are in some sense equivalent, thus giving an affirmative answer to the aforementioned open question. It follows from our result that two entangled states that are apparently useless in demonstrating quantum nonlocality via a specific Bell inequality can be combined to give a Bell violation of the same inequality. Explicit examples of such activation phenomenon are provided.
We discuss the relations between the violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality for systems of two qubits on the one side and entanglement of formation, local filtering operations, and the entropy and purity on the other. We calculate the extremal Bell violations for a given amount of entanglement of formation and characterize the respective states, which turn out to have extremal properties also with respect to the entropy, purity, and several entanglement monotones. The optimal local filtering operations leading to the maximal Bell violation for a given state are provided, and the special role of the resulting Bell diagonal states in the context of Bell inequalities is discussed.
Violations of a Bell inequality are reported for an experiment where one of two entangled qubits is stored in a collective atomic memory for a user-defined time delay. The atomic qubit is found to preserve the violation of a Bell inequality for storage times up to 21 micros, 700 times longer than the duration of the excitation pulse that creates the entanglement. To address the question of the security of entanglement-based cryptography implemented with this system, an investigation of the Bell violation as a function of the cross correlation between the generated nonclassical fields is reported, with saturation of the violation close to the maximum value allowed by quantum mechanics.
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.
By introducing a quantitative ``degree of commutativity'' in terms of the angle between spin observables we present two tight quantitative trade-off relations in the case of two qubits. First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to $\ensuremath{\pi}∕2$ (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach $\ensuremath{\pi}∕2$ the maximal value obtainable for the correlations in the Bell inequality decreases and thus the non-violation increases. As expected, the extremes of these relations are found in the case of anticommuting local observables where, respectively, the bounds of 222\sqrt{2}22​ and 2\sqrt{2}2​ hold for the expectation value of the Bell operator. The trade-off relations show that noncommmutativity gives ``a more than classical result'' for entangled states, whereas ``a less than classical result'' is obtained for separable states. The experimental relevance of the trade-off relation for separable states is that it provides an experimental test for two qubit entanglement. Its advantages are twofold: in comparison to violations of Bell inequalities it is a stronger criterion and in comparison to entanglement witnesses it needs to make less strong assumptions about the observables implemented in the experiment.