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arXiv:1611.05586 · International Journal of Quantum Information

Bell-CHSH violation under global unitary operations: Necessary and sufficient conditions

Nirman Ganguly, Amit Mukherjee, Arup Roy, Some Sankar Bhattacharya, Biswajit Paul, Kaushiki Mukherjee

Published Jun 1, 2018• 13 citations
arXivHTMLPDF

Abstract

The relation between Bell-CHSH violation and factorization of Hilbert space is considered here. That is, a state which is local in the sense of the Bell-CHSH inequality under a certain factorization of the underlying Hilbert space can be Bell-CHSH nonlocal under a different factorization. While this question has been addressed with respect to separability, the relation of the factorization with Bell-CHSH violation has remained hitherto unexplored. We find here that there is a set containing density matrices, which do not exhibit Bell-CHSH violation under any factorization of the Hilbert space brought about by global unitary operations. Using the Cartan decomposition of [Formula: see text], we characterize the set in terms of a necessary and sufficient criterion based on the spectrum of density matrices. Sufficient conditions are obtained to characterize such density matrices based on their bloch representations. For some classes of density matrices, necessary and sufficient conditions are derived in terms of bloch parameters. Furthermore, an estimation of the volume of such density matrices is achieved in terms of purity. The criterion is applied to some well-known class of states in two qubits. Since both local filtering and global unitary operations influence the Bell-CHSH violation of a state, a comparative study is made between the two operations. The inequivalence of the two operations (in terms of increasing Bell-CHSH violation) is exemplified through their action on some classes of states.

DOI
10.1142/s0219749918500405

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The relation between Bell-CHSH violation and factorization of Hilbert space is considered here. That is, a state which is local in the sense of the Bell-CHSH inequality under a certain factorization of the underlying Hilbert space can be Bell-CHSH nonlocal under a different factorization. While this question has been addressed with respect to separability, the relation of the factorization with Bell-CHSH violation has remained hitherto unexplored. We find here that there is a set containing density matrices, which do not exhibit Bell-CHSH violation under any factorization of the Hilbert space brought about by global unitary operations. Using the Cartan decomposition of [Formula: see text], we characterize the set in terms of a necessary and sufficient criterion based on the spectrum of density matrices. Sufficient conditions are obtained to characterize such density matrices based on their bloch representations. For some classes of density matrices, necessary and sufficient conditions are derived in terms of bloch parameters. Furthermore, an estimation of the volume of such density matrices is achieved in terms of purity. The criterion is applied to some well-known class of states in two qubits. Since both local filtering and global unitary operations influence the Bell-CHSH violation of a state, a comparative study is made between the two operations. The inequivalence of the two operations (in terms of increasing Bell-CHSH violation) is exemplified through their action on some classes of states.
The action of global unitary operations can have intriguing effects on the non-local manifestations of a quantum state. States which do not violate Bell-CHSH inequality, can violate it when transformed by a global unitary. This phenomena assumes significance when one exercises to augment non-local resources(in terms of violation of Bell-CHSH inequality) from these seemingly useless states(i.e., in terms of non-local tasks). However, equally intriguing is the existence of states from which no non-local resource can be generated with a global unitary. The present work confirms the existence of such a set, pertaining to two qubit states. The set exhibits counter intuitive features by containing within it some entangled states which remain Bell-CHSH local on the action of any unitary. Furthermore through an analytic characterization of the set we lay down a generic prescription through which one can operationally identify states which can generate non-local resources under the action of a unitary on the composite system. It has also been shown that our prescription remains valid for any linear Bell inequality.
We investigate the detection problem of quantum nonlocal correlation by two qubit detectors. The detectors with an initial product state interact with a massless scalar field in the vacuum state, and then the out-state of the detectors are correlated after the interaction. Under the perturbative treatment in the second order of the coupling, the detectors' state can be entangled but satisfies the Bell-CHSH inequality. It is known that the violation of the Bell-CHSH inequality for such an entangled state is obtained after a local filtering operation. In this paper, we construct the optimal filtering operation for the qubit detectors and derive the success probability of the filtering operation, which characterizes the reliability of revealing the Bell-CHSH nonlocality by the filtering operations. By applying the optimal filtering, it is shown that the detected Bell-CHSH nonlocality depends on the coherence of the detectors' state and the spontaneous emission of scalar particles from each detector. We also comment on a trade-off relation between the success probability and the size of the parameter region showing quantum correlation.
We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.
Unitary operators are employed to investigate the violation of the Bell-CHSH inequality. The ensuing modifications affecting both classical and quantum bounds are elucidated. The relevance of a particular class of unitary operators whose expectation values are real is pointed out. For these operators, the classical and quantum bounds remain unaltered, being given, respectively, by 2 and 2 2 . As examples, we discuss the explicit realization of phase space Bell-CHSH inequality violation and the Weyl unitary operators for a real scalar field in relativistic Quantum Field Theory.
We investigate the behavior of the maximal violations of the CHSH inequality and Vèrtesi's inequality under the local filtering operations. An analytical method has been presented for general two-qubit systems to compute the maximal violation of the CHSH inequality and the lower bound of the maximal violation of Vértesi's inequality over the local filtering operations. We show by examples that there exist quantum states whose non-locality can be revealed after local filtering operation by the Vértesi's inequality instead of the CHSH inequality.