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

Victor V. Albert

doi:10.22331/q-2019-06-06-151

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.

Highlights

Introduction & outlineA quantum channel A is the most general map between two quantum systems
Channels enjoy a range of applications, primarily in the quantum information community [1], and in studies of matrix product states [2, 3], entanglement renormalization [4, 5], computability theory [6], and even biological inference processes [7]
There have been two different decompositions associated with general channels: [23, Thm. 8] being a fine-grained blockdecomposition of the asymptotic subspace—the subspace that survives under repeated application of the channel—and [25, Thm. 2] being a coarser algebraic decomposition of the Kraus operators into blocks of noiseless subsystems

Summary

Introduction & outline

A quantum channel A ( called a quantum markov chain, Kraus map, or completely-positive tracepreserving map) is the most general map between two quantum systems. Channels enjoy a range of applications, primarily in the quantum information community [1], and in studies of matrix product states [2, 3], entanglement renormalization [4, 5], computability theory [6], and even biological inference processes [7]. Whenever one studies such maps, a key question to ask is: What information from an initial state ρ survives under repeated application of A?.

Asymptotics of channels

Faithful channels

Spectral restrictions

Limited Noether-type theorem

Extending to general channels

Adding fixed-point structure

Decoherence-free subspaces

Irreducible channels

Noiseless subsystems

How to find PA

Application

From channels to Lindbladians