Decomposition of a system in pseudo-Hermitian quantum mechanics

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">T</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> -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.