Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

TL;DR

This paper derives an analytical formula for the asymptotic state of quantum channels, linking conserved quantities to initial states, and applies it to adiabatic transport and matrix product states, revealing new insights into their behavior.

Contribution

It introduces a Noether-like theorem for quantum channels and provides a formula for the asymptotic state, with applications to adiabatic processes and matrix product states.

Findings

Conserved quantities determine the asymptotic state dependence on initial conditions.

The spectral gap can close during adiabatic evolution, affecting transport.

Expectation values of matrix product states can be computed with reduced bond dimension.

Abstract

This work derives an analytical formula for the asymptotic state---the quantum state resulting from an infinite number of applications of a general quantum channel on some initial state. For channels admitting multiple fixed or rotating points, conserved quantities---the left fixed/rotating points of the channel---determine the dependence of the asymptotic state on the initial state. The formula stems from a Noether-like theorem stating that, for any channel admitting a full-rank fixed point, conserved quantities commute with that channel's Kraus operators up to a phase. The formula is applied to adiabatic transport of the fixed-point space of channels, revealing cases where the dissipative/spectral gap can close during any segment of the adiabatic path. The formula is also applied to calculate expectation values of noninjective matrix product states (MPS) in the thermodynamic limit, revealing that those expectation values can also be calculated using an MPS with reduced bond dimension and a modified boundary.