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

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 channeldetermine 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.

Thermodynamically-efficient local computation and the inefficiency of quantum memory compression

“…Specifically, using a generalized notion of quantum sufficient statistic [24][25][26][27], we show that a local operation on part of a system is efficient if and only if it unitarily preserves the minimal sufficient statistic of the part for arXiv:2001.02258v3 [quant-ph] 1 Feb 2020 the whole. Our geometric interpretation of this also draws on recent progress on fixed points of quantum channels [28][29][30][31].…”

Symmetries and monotones in Markovian quantum dynamics

“…The resulting inequality, due to the commutation relations [M, E t ] = [N , E t ] = 0, expresses the 6 In fact, monotone metrics are also closely related to generalized relative entropies [24], as every monotone Riemannian metric arises from a generalized relative entropy [25]. Hence the previous discussion can be generalized to D corresponding to a relative entropy, as in our example (11).…”

“…[23] for an accessible introduction to the subject). While the latter is uniquely determined from the monotonicity property under classical stochastic maps (up to a normalization constant), the same does not hold for its quantum counterparts, which show a rich variety 6 .…”

“…An accessible discussion regarding the connection between symmetries and conserved quantities for Markovian dynamics can be found in the papers by Baumgartner and Narnhofer [5], and by Albert and Jiang [4]. Symmetries have also been discussed for the closely related case of iterated quantum channels [6]. From the viewpoint of quantum resource theories [7], consequences of symmetries in quantum dynamics have been systematically considered in the theory of reference frames and asymmetry [8,9].…”

Absorption in Invariant Domains for Semigroups of Quantum Channels

“…Various fields of applications to quantum physics motivate the study of these channels and, in particular, of absorption operators and fixed points (see [1,13,32,33,37]), but the approach we shall follow here comes from classical and quantum probability. In a first step, a central theme which stimulated this work is the study of the fixed points of a quantum channel: This theme has been widely investigated, with satisfactory results for the case of positive recurrent channels, i.e., channels with an invariant faithful density (see [5,7,9,10,19,27]), but information is lacking and yet useful in the general case [1,2,24]. Once the construction of the mathematical object clearly appeared, we realized that it could be a useful tool to tackle the different aforementioned subjects.…”