We prove upper bounds on the rate, called "mixing 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\sqrt{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.We prove upper bounds on the rate, called "mixing 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\sqrt{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.

Quantum Information and ComputationNov 1, 20133 citations

Upper bounds on mixing rates

Élliott H. Lieb, Anna Vershynina

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.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.

Jan 1, 20135 citations

Sharp Quantum Entropy Mixing Rates and the Operator Layer Cake Theorem

Alexander Stottmeister

At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble $\{(1\!-\!p,ρ_1),(p,ρ_2)\}$ by $c\,\|{H}\|h_{2}(p)$ (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.At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble $\{(1\!-\!p,ρ_1),(p,ρ_2)\}$ by $c\,\|{H}\|h_{2}(p)$ (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.

quant-pharXiv:2609.39814Oct 1, 20260 citations

Upper bounds on entangling rates of bipartite Hamiltonians

Sergey B. Bravyi

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.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.

arXiv:0704.0964Physical Review ANov 21, 200764 citations

Entanglement rates for Rényi, Tsallis, and other entropies

Anna Vershynina

We provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution. A large class of entropy measures in considered, which includes Rényi and Tsallis entropies. The result is derived from a general bound on the trace-norm of a commutator, which can be expected to find other implementations. We apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with Rényi and Tsallis entanglement entropies taken as measures of entanglement in the system.We provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution. A large class of entropy measures in considered, which includes Rényi and Tsallis entropies. The result is derived from a general bound on the trace-norm of a commutator, which can be expected to find other implementations. We apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with Rényi and Tsallis entanglement entropies taken as measures of entanglement in the system.

arXiv:1803.07117Journal of Mathematical PhysicsFeb 1, 201914 citations

Asymptotic entropic uncertainty relations

Radosław Adamczak, Rafał Latała, Zbigniew Puchała +1 more

We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.

arXiv:1412.7065Journal of Mathematical PhysicsMar 1, 201618 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.