The von Neumann entropy and the subentropy of a mixed quantum state are upper and lower bounds, respectively, on the accessible information of any ensemble consistent with the given mixed state. Here we define and investigate a set of quantities intermediate between entropy and subentropy.

arXiv:quant-ph/0207010Quantum Information and ComputationJan 1, 20039 citations

Quantum Information and Computation

Upper bounds on mixing rates

Élliott H. Lieb, Anna Vershynina

Published Nov 1, 2013• 3 citations

Publisher

Abstract

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.

DOI: 10.26421/qic13.11-12-5

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S.R. Nichols, William K. Wootters

arXiv:quant-ph/0207010Quantum Information and ComputationJan 1, 20039 citations

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.

\{(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.

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.