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Comment on letter "Local {$\mathcal{PT}$} symmetry violates the no-signaling principle" by Yi-Chan Lee et al, Phys. Rev. Lett. 112, 130404 (2014)

Miloslav Znojil

Published Apr 6, 2014• 0 citations
OpenAlex

Abstract

It is shown that the toy-model-based considerations of {\em loc. cit.} (see also arXiv:1312.3395) are based on an incorrect, manifestly unphysical choice of the Hilbert space of admissible quantum states. A two-parametric family of all of the eligible correct and potentially physical Hilbert spaces of the model is then constructed. The implications of this construction are discussed. In particular, it is emphasized that contrary to the conclusions of {\em loc. cit.} there is no reason to believe that the current form of the $\mathcal{PT}$-symmetric quantum theory should be false as a fundamental theory.

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Comment on letter "Local {PT\mathcal{PT}PT} symmetry violates the no-signaling principle" by Yi-Chan Lee et al, Phys. Rev. Lett. 112, 130404 (2014)
It is shown that the toy-model-based considerations of {\em loc. cit.} (see also arXiv:1312.3395) are based on an incorrect, manifestly unphysical choice of the Hilbert space of admissible quantum states. A two-parametric family of all of the eligible correct and potentially physical Hilbert spaces of the model is then constructed. The implications of this construction are discussed. In particular, it is emphasized that contrary to the conclusions of {\em loc. cit.} there is no reason to believe that the current form of the PT\mathcal{PT}PT-symmetric quantum theory should be false as a fundamental theory.
Yi-Chan Lee et al. claim (cf. Phys. Rev. Lett. 112, 130404 (2014)) that the “recent extension of quantum theory to non-Hermitian Hamiltonians” (which is widely known under the nickname of “PT-symmetric quantum theory”) is “likely false as a fundamental theory”. By their opinion their results “essentially kill any hope of PT-symmetric quantum theory as a fundamental theory of nature”. In our present text we explain that their toy-model-based considerations are misleading and that they do not imply any similar conclusions.
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.
One of the most characteristic mathematical features of the PT\mathcal{PT}PT-symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states H=H(H)\mathcal{H}=\mathcal{H}(H)H=H(H). Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but PT\mathcal{PT}PT-symmetric user-friendly family of JJJ-parametric ``discrete anharmonic'' quantum Hamiltonians $H=H(\stackrel{P\vec}{\ensuremath{\lambda}})$ is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\in}{\mathcal{D}}^{(\mathrm{physical})}$. In contradiction to conventional wisdom, the systems are then shown to be stable with respect to admissible perturbations, inside the domain D(physical){\mathcal{D}}^{(\mathrm{physical})}D(physical), even in the immediate vicinity of the phase-transition boundaries $\ensuremath{\partial}{\mathcal{D}}^{(\mathrm{physical})}$.
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 .
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.