Abstract Recently, it was shown that a violation of the no-signaling principle can be obtained by using local
PT PT -symmetric evolutions on shared quantum states that are entangled and pure. The violation can be removed by using a
CPT CPT inner product instead of the traditional one. We show that within the traditional inner product quantum mechanics, local
PT 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
PT 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
PT 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
CPT 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.
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 symmetric unitary operations. If two parties share a maximally entangled state, then under local
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-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.
We show that the no-signaling principle can be violated with classical inseparable beams in the presence of a parity-time (PT) symmetric subsystem. Thus, the problems associated to PT-symmetric quantum theories recently discovered by Lee et al. [Phys. Rev. Lett. 112, 130404 (2014)] are not exclusive to quantum mechanics, but already exist in the classical case. The possibility to implement local optical PT-symmetric subsystems via light-matter interactions enables the experimental exploration of local PT symmetry and subtle quantum concepts via classical analogues.
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 .
Bender et al. [Phys. Rev. Lett. 80, 5243 (1998)] have developed PT-symmetric quantum theory as an extension of quantum theory to non-Hermitian Hamiltonians. We show that when this model has a local PT symmetry acting on composite systems, it violates the nonsignaling principle of relativity. Since the case of global PT symmetry is known to reduce to standard quantum mechanics A. Mostafazadeh [J. Math. Phys. 43, 205 (2001)], this shows that the PT-symmetric theory is either a trivial extension or likely false as a fundamental theory.