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arXiv:1404.6166

Violation of Invariance of Entanglement Under Local PT Symmetric Unitary

Arun Kumar Pati

Published Apr 24, 2014• 6 citations
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Abstract

Entanglement is one of the key feature of quantum world and any entanglement measure must satisfy some basic laws. Most important of them is the invariance of entanglement under local unitary operations. We show that this is no longer true with local ${\cal {PT}}$ symmetric unitary operations. If two parties share a maximally entangled state, then under local ${\cal {PT}}$ symmetric unitary evolution the entropy of entanglement for pure bipartite states does not remain invariant. Furthermore, we show that if one of the party has access to ${\cal {PT}}$-symmetric quantum world, then a maximally entangled state in usual quantum theory appears as a non-maximally entangled states for the other party. This we call as the "entanglement mismatch" effect which can lead to the violation of the no-signaling condition.

DOI
10.48550/arxiv.1404.6166

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Entanglement is one of the key feature of quantum world and any entanglement measure must satisfy some basic laws. Most important of them is the invariance of entanglement under local unitary operations. We show that this is no longer true with local PT{\cal {PT}}PT symmetric unitary operations. If two parties share a maximally entangled state, then under local PT{\cal {PT}}PT symmetric unitary evolution the entropy of entanglement for pure bipartite states does not remain invariant. Furthermore, we show that if one of the party has access to PT{\cal {PT}}PT-symmetric quantum world, then a maximally entangled state in usual quantum theory appears as a non-maximally entangled states for the other party. This we call as the "entanglement mismatch" effect which can lead to the violation of the no-signaling condition.
Abstract Recently, it was shown that a violation of the no-signaling principle can be obtained by using local PTPTPT PT -symmetric evolutions on shared quantum states that are entangled and pure. The violation can be removed by using a CPTCPTCPT CPT inner product instead of the traditional one. We show that within the traditional inner product quantum mechanics, local PTPTPT PT -symmetric evolutions can lead to violation of the no-signaling principle for separable and even for classically correlated bipartite shared quantum states. For classically correlated states, specially chosen PTPTPT PT -symmetric operations from a set of zero volume can also preserve the principle. The classically correlated states used for the violation have the same correlation, when only a single basis is considered for measurements, as the maximally entangled states that were erstwhile used for the same purpose. This, along with the fact that states with no classical correlation cannot be used for violation of no-signaling within PTPTPT PT -symmetric evolutions, can potentially lead to the identification of minimal resources necessary for obtaining the violation. The violation obtained here can again be removed by using a CPTCPTCPT CPT inner product instead of the traditional one. Graphic abstract
Abstract Violation of the no-signaling principle may occur in PT -symmetric evolutions, that is, evolutions that stem from Hamiltonians that are symmetric with respect to parity and time-reversal, of bipartite entangled states. The violation can be avoided by using a non-conventional inner product. We show that even within the formalism that utilizes the conventional inner product between physical state vectors in a separable Hilbert space, it is possible to obtain instances of local preservation of the no-signaling principle for evolutions corresponding to PT -symmetric non-Hermitian Hamiltonians with real eigenvalues, of multiparty entangled states, whose bipartite versions still violate the principle. The phenomenon can be witnessed already by using the Greenberger–Horne–Zeilinger state. Interestingly, the generalized W states do not support such a local preservation of no-signaling.
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Given a bipartite quantum state ρ\rhoρ with subsystems AAA and BBB of arbitrary dimensions, we study the entanglement detecting capabilities of locally noneffective, or cyclic, unitary operations [Fu, Europhys. Lett., vol. 75]. Local cyclic unitaries have the special property that they leave their target subsystem invariant. We investigate the distance between ρ\rhoρ and the global state after local application of such unitaries as a possible indicator of entanglement. To this end, we derive and discuss closed formulae for the maximal such distance achievable for three cases of interest: (pseudo)pure quantum states, Werner states, and two-qubit states. What makes this criterion interesting, as we show here, is that it surprisingly displays behavior similar to recent anomalies observed for non-locality measures in higher dimensions, as well as demonstrates an equivalence to the CHSH inequality for certain classes of two-qubit states. Yet, despite these similarities, the criterion is not itself a non-locality measure. We also consider entanglement detection in bound entangled states.