Upper bounds on mixing rates

Élliott H. Lieb, Anna Vershynina

Published Jan 1, 2013• 5 citations

Abstract

We prove upper bounds on the called rate, at which the von Neumann entropy of the expected density operator of a given ensemble of states changes under non-local unitary evolution. For an ensemble consisting of two states, with probabilities of p and 1 - p, we prove that the mixing rate is bounded above by 4√p(1-p) for any Hamiltonian of norm 1. For a general ensemble of states with probabilities distributed according to a random variable X and individually evolving according to any set of bounded Hamiltonians, we conjecture that the mixing rate is bounded above by a Shannon entropy of a random variable X. For this general case we prove an upper bound that is independent of the dimension of the Hilbert space on which states in the ensemble act.

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OpenAlex

Abstract

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For a quantum state undergoing unitary Schrödinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator [R. S. Ingarden, Quantum information theory, Rep. Math. Phys. 10, 43 (1976)RMHPBE0034-487710.1016/0034-4877(76)90005-7]. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system composed of a set of highly interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.For a quantum state undergoing unitary Schrödinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator [R. S. Ingarden, Quantum information theory, Rep. Math. Phys. 10, 43 (1976)RMHPBE0034-487710.1016/0034-4877(76)90005-7]. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system composed of a set of highly interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.

arXiv:1901.08956Physical review. EJul 2, 201918 citations

\{(1\!-\!p,ρ_1),(p,ρ_2)\}

c\,\|{H}\|h_{2}(p)

(with the binary entropy

h_{2}

). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality

\|[A,\log B]\|_{1}\!\leq\! c\,h_{2}(p)

for positive trace-class operators

A\!\leq\! B

with

\text{Tr} A\!=\!p

and

\text{Tr} B\!=\!1

on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be

c\!=\!1

. Here, I prove the latter conjecture using an exact integral representation of the commutator

[A,\log B]

obtained from the operator layer cake theorem, due to Cheng and Liu. The result gives a sharp dimension-independent limit on how rapidly unitary evolution of one component can change the entropy of a binary quantum ensemble. It follows that mixing rates satisfy small incremental mixing with the optimal constant, and entangling rates of bipartite Hamiltonians are bounded by

(2\log d\!+\!1)\|H\|

. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.

X

. For this general case we prove an upper bound that is independent of the dimension of the Hilbert space on which states in the ensemble act.

We discuss upper bounds on the rate at which unitary evolution governed by a nonlocal Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian

H

coupling two finite dimensional particles

A

and

B

, the entangling rate is shown to be upper bounded by $c\text{ }\text{log}(d)\ensuremath{\Vert}H\ensuremath{\Vert}$, where

d

is the smallest dimension of the interacting particles, $\ensuremath{\Vert}H\ensuremath{\Vert}$ is the operator norm of

H

, and

c

is a constant close to 1. Under certain restrictions on the initial state we prove an analogous upper bound for the ancilla-assisted entangling rate with a constant

c

that does not depend upon dimensions of local ancillas. The restriction is that the initial state has at most two distinct Schmidt coefficients (each coefficient may have arbitrarily large multiplicity). Our proof is based on analysis of a mixing rate---a functional measuring how fast entropy can be produced if one mixes a time-independent state with a state evolving unitarily.