Abstract:
This work presents a consistent method for defining subsystems on a Hilbert space whose underlying vector space admits a tensor-product decomposition, while the Hilbert space itself lacks such a decomposition due to its inner-product structure. The inner product on a Hilbert space determines the measure of distance on it, governed by a "metric operator". Such Hilbert spaces frequently arise in pseudo-Hermitian quantum mechanics, where evolution is governed by a non-Hermitian Hamiltonian with real eigenvalues. In recent years, such Hamiltonian evolutions have been realized in various quantum systems and found useful in several quantum technologies, including quantum metrology and quantum batteries.
Defining subsystems in these Hilbert spaces is non-trivial, since the partial trace operation—the conventional method for constructing a reduced state—applies only to Hilbert spaces with tensor-product decomposition. This study employs a quantum walk as a toy model to demonstrate a method for constructing subsystems. The analysis first focuses on a specific model of quantum walks and provides a non-trivial extension of the partial trace operation to compute the coin state and its dynamics. The discussion then turns to the freedom in choosing the metric operator for a given Hamiltonian and argues, on operational grounds, that the choice of the metric operator determines how a system may be decomposed into subsystems, making this decomposition observer-dependent. Finally, a unified framework for defining subsystems in such Hilbert spaces is presented, and the earlier arguments are justified through the algebraic formulation of quantum mechanics. The findings show that time-independent pseudo-Hermitian quantum mechanics represents a straightforward extension of standard quantum mechanics, even for composite quantum systems.
