Automatic hermiticity for mixed states

TL;DR

This paper extends a mechanism for rendering non-Hermitian Hamiltonians Hermitian via a modified inner product to mixed states, using density matrices and exploring implications in future-included quantum theories.

Contribution

It demonstrates that the Hermitization mechanism for non-normal Hamiltonians applies to mixed states through density matrices, including in future-included quantum frameworks.

Findings

The mechanism works for mixed states using density matrices.

Introduction of skew density matrices combining past and future states.

Approximate equivalence of skew density matrices to ensembles of past states.

Abstract

We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product $I_Q$ that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator $Q$. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time $T_A$ but also a future state at the final time $T_B$ is given, we study a couple of candidates for it, and introduce a ``skew density matrix'' composed of both ensembles of the future and past states such that the trace of the product of it and an operator ${\cal O}$ matches a normalized matrix element of ${\cal O}$. We argue that the skew density matrix defined with $I_Q$ at the present time $t$ for large $T_B-t$ and large $t-T_A$ approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product $I_{Q_J}$ for large $t-T_A$.