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Decomposition of a system in pseudo-Hermitian quantum mechanics

Himanshu Badhani, Sibasish Ghosh

TL;DR

This work tackles defining subsystems in pseudo-Hermitian quantum mechanics without relying on a fixed tensor-product Hilbert space. It develops an algebraic approach using commuting C*-subalgebras and GNS representations to define subsystems that are independent of the inner-product structure, showing that different metric operators $G$ yield inequivalent but physically meaningful subsystem decompositions. The authors establish a classification of representations via intertwining isometries $T_U$ and show that subsystem structure depends on the chosen metric up to equivalence classes $[G]$, while full statistics remain invariant. They prove that subsystem tomography is possible in any metric space and that the no-signalling principle holds regardless of whether the metric is tensor-product, thereby putting all metric choices on equal footing for composite systems. The results extend operational quantum information concepts to PT-symmetric and pseudo-Hermitian settings and offer a robust framework for analyzing subsystems without assuming a priori Hilbert-space factorization.

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.

Decomposition of a system in pseudo-Hermitian quantum mechanics

TL;DR

This work tackles defining subsystems in pseudo-Hermitian quantum mechanics without relying on a fixed tensor-product Hilbert space. It develops an algebraic approach using commuting C*-subalgebras and GNS representations to define subsystems that are independent of the inner-product structure, showing that different metric operators yield inequivalent but physically meaningful subsystem decompositions. The authors establish a classification of representations via intertwining isometries and show that subsystem structure depends on the chosen metric up to equivalence classes , while full statistics remain invariant. They prove that subsystem tomography is possible in any metric space and that the no-signalling principle holds regardless of whether the metric is tensor-product, thereby putting all metric choices on equal footing for composite systems. The results extend operational quantum information concepts to PT-symmetric and pseudo-Hermitian settings and offer a robust framework for analyzing subsystems without assuming a priori Hilbert-space factorization.

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 -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.
Paper Structure (15 sections, 3 theorems, 29 equations, 2 figures)

This paper contains 15 sections, 3 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Underlying vector space and the metric
  3. Overview of pseudo-Hermitian quantum mechanics
  4. Open-system perspective on pseudo-Hermitian Hamiltonians
  5. The C*-algebra
  6. Cyclic representations in different metric spaces
  7. Partitions of a system
  8. Tomography and the no-signalling principle
  9. An example
  10. The no-signalling principle
  11. Conclusion
  12. Lindbladian dynamics in $\mathcal{H}_G$
  13. GNS constrcution
  14. Combining the two representations
  15. Spin tomography

Key Result

Theorem 1

For a system $\{ \mathcal{A},\omega\}$, any two cyclic representations $\pi$ and $\pi'$, with representation spaces $\mathcal{H}_G$ and $\mathcal{H}_{G'}$ respectively, correspond to the same bi-partitioning into system 1 and 2 if and only if they belong to the equivalence class given by: Here $U_1$ and $U_2$ are unitary operators corresponding to the two subsystems.

Figures (2)

  • Figure 1: Unitary maps between different metric Hilbert spaces. The state $\psi$ in $\mathcal{H}_{G'}$ can be mapped to a state in $\mathcal{H}$ either through Hermitisization ($\eta'$) or through the Hilbert space $\mathcal{H}_G$ ($\eta T_U$). The resulting states are related by a unitary $U$ in $\mathcal{H}$, which implies $\eta'=U\eta T_U$
  • Figure 2: Spin tomography of a spin 1/2 particle. The filter $F$ lets particles of a certain velocity pass, which are then unitarily rotated by the magnetic field in the x-y plane. As they pass through the second magnetic field in z-direction, the two arms are detected, corresponding to $m=\pm1/2$ values.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Corollary 1.1
  • proof
  • Theorem 2