Decomposition of a system in pseudo-Hermitian quantum mechanics

TL;DR

This paper presents a method to define and analyze subsystems in pseudo-Hermitian quantum mechanics, showing that subsystem decomposition is possible independently of the metric's tensor product structure, with implications for quantum theory foundations.

Contribution

It introduces a metric-independent approach to subsystem decomposition in pseudo-Hermitian quantum mechanics using algebraic quantum mechanics.

Findings

Subsystems can be well-defined in all metric spaces, regardless of tensor product form.

Different metric choices can lead to inequivalent subsystem decompositions.

Subsystems are tomographically reconstructible and obey no-signalling.

Abstract

This work outlines a consistent method of identifying subsystems in finite-dimensional Hilbert spaces, independent of the underlying inner-product structure. Such Hilbert spaces arise in $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics, where a non-Hermitian Hamiltonian is made self-adjoint by changing the inner product using the so-called ``metric operator". This is the framework of pseudo-Hermitian quantum mechanics. For composite quantum systems in this framework, defining subsystems is generally considered feasible only when the metric operator is chosen to have a tensor product form so that a partial trace operation can be well defined. In this work, we use arguments from algebraic quantum mechanics to show that the subsystems can be well-defined in every metric space - irrespective of whether or not the metric is of tensor product form. This is done by identifying subsystems with a decomposition of the underlying C*-algebra into commuting subalgebras. Although the choice of the metric is known to have no effect on the system's statistics, we show that different choices of the metric can lead to inequivalent subsystem decompositions. Each of the subsystems can be tomographically constructed and these subsystems satisfy the no-signalling principle. With these results, we put all the choices of the metric operator on an equal footing for composite systems.