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arXiv:2007.13461 · Journal of Physics A Mathematical and Theoretical

Local preservation of no-signaling in multiparty PT -symmetric evolutions

Asmita Kumari, Ujjwal Sen

Published Mar 14, 2022• 2 citations
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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.

DOI
10.1088/1751-8121/ac5dae

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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.
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 P T -symmetric quantum mechanics, the extension of conventional quantum mechanics to the non-Hermitian Hamiltonian invariant under the combined parity ( P ) and time reversal ( T ) symmetry, has been successfully applied to a variety of fields—such as solid state physics, mathematical physics, optics, quantum field theory. Recently, the extension of P T -symmetrical theory to entangled quantum systems has been challenged, in that P T formulation within the conventional Hilbert space violates the no-signaling principle. Here, we revisit the derivation of non-signaling principle in the framework of C P T inner product prescription. Our results preserve the no-signaling principle for a two-qubit system, reaffirm the invariance of the entanglement, and reproduce the Clauser–Horne–Shimony–Holt (CHSH) inequality. We conclude that P T -symmetric quantum mechanics satisfies the requirements for a fundamental theory and provides a consistent description of quantum systems.
Bender and Boettcher explored a quantum theory based on a non-Hermitian PT­ symmetric Hamiltonian , where PT is the operator of the space-time reflection and demonstrated that the PT-symmetric Hamiltonian can possess entirely real spectra. In this thesis, we point out that in the framework of PT-symmetric quantum mechanics; the calculation of matrix elements in the Hilbert space is ill-defined. We point out the importance of using CPT inner product in PT-symmetric systems. We manifested our assessment using the CPT inner product prescription for the entangled wave function of the composite system. We show that for a composite system with a local PT-symmetry, it preserves the no-signaling condition and the orthogonality of the states.The reduced density matrix is diagonal and independent of the non-Hermitian parameter . We reaffirm the consistency of PT-symmetric quantum mechanics as a candidate for a fundamental theory .
Bell’s inequality is investigated in parity-time (PT) symmetric quantum mechanics, using a recently developed form of the inequality by Maccone, with two PT-qubits in the unbroken phase with real energy spectrum. It is shown that the inequality produces a bound that is consistent with the standard quantum mechanics even after using Hilbert space equipped with CPT inner product and therefore, the entanglement has identical structure with standard quantum mechanics. Consequently, the no-signaling principle for a two-qubit system in PT-symmetric quantum theory is preserved.