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arXiv:1703.03529 · Journal of Physics A Mathematical and Theoretical

No-signaling principle and Bell inequality in P T -symmetric quantum mechanics

George Japaridze, Dipendra Pokhrel, Xiao-Qian Wang

Published Mar 6, 2017• 11 citations
arXivHTMLPDF

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.

DOI
10.1088/1751-8121/aa649c

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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.
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.
Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We have developed a modified formulation of NHQM based on the geometry of Hilbert spaces which is consistent with the conventional quantum mechanics for Hermitian systems. Using this formulation the validity of these principles can be shown in a simple and uniform approach.
In quantum mechanics, nonlocality (a violation of a Bell inequality) is intimately linked to complementarity, by which we mean that consistently assigning values to different observables at the same time is not possible. Nonlocality can only occur when some of the relevant observables do not commute, and this noncommutativity makes the observables complementary. Beyond quantum mechanics, the concept of complementarity can be formalized in several distinct ways. Here we describe some of these possible formalizations and ask how they relate to nonlocality. We partially answer this question by describing two toy theories which display nonlocality and obey the no-signaling principle, although each of them does not display a certain kind of complementarity. The first toy theory has the property that it maximally violates the Clauser, Horn, Shimony, and Holt (CHSH) inequality, although the corresponding local observables are pairwise jointly measurable. The second toy theory also maximally violates the CHSH inequality, although its state space is classical and all measurements are mutually nondisturbing: if a measurement sequence contains some measurement twice with any number of other measurements in between, then these two measurements give the same outcome with certainty.
The Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in the vacuum state of a relativistic scalar quantum field is revisited by making use of the Hilbert space $\mathcal{H}\ensuremath{\bigotimes}{\mathcal{H}}_{AB}$, where H\mathcal{H}H and HAB{\mathcal{H}}_{AB}HAB​ stand, respectively, for the Hilbert space of the scalar field and of a generic bipartite quantum mechanical system. The construction of Hermitian, field-dependent, dichotomic operators is devised as well as the Bell-CHSH inequality. Working out the ABABAB part of the inequality, the resulting Bell-CHSH correlation function for the quantum field naturally emerges from unitary Weyl operators. Furthermore, introducing a Jaynes-Cummings type Hamiltonian accounting for the interaction between the scalar field and a pair of qubits, the quantum corrections to the Bell-CHSH inequality in the vacuum state of the scalar field are evaluated till the second order in perturbation theory.