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arXiv:0704.0964 · Physical Review A

Upper bounds on entangling rates of bipartite Hamiltonians

Sergey B. Bravyi

Published Nov 21, 2007• 64 citations
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Abstract

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.

DOI
10.1103/physreva.76.052319

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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 HHH coupling two finite dimensional particles AAA and BBB, the entangling rate is shown to be upper bounded by c log(d)\ensuremath∥H\ensuremath∥c\text{ }\text{log}(d)\ensuremath{\Vert}H\ensuremath{\Vert}c log(d)\ensuremath∥H\ensuremath∥, where ddd is the smallest dimension of the interacting particles, \ensuremath∥H\ensuremath∥\ensuremath{\Vert}H\ensuremath{\Vert}\ensuremath∥H\ensuremath∥ is the operator norm of HHH, and ccc 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 ccc 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 provide an upper bound on the maximal rate at which irreversible quantum dynamics can generate entanglement in a bipartite system. The generator of irreversible dynamics consists of a Hamiltonian and dissipative terms in Lindblad form. The relative entropy of entanglement is chosen as a measure of entanglement in an ancilla-free system. We provide an upper bound on the entangling rate which has a logarithmic dependence on a dimension of a smaller system in a bipartite cut. We also investigate the rate of change of quantum mutual information in an ancilla-assisted system and provide an upper bound independent of dimension of ancillas.
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 XXX. 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.
We investigate the fundamental limits of entanglement generation under bipartite Hamiltonian dynamics when only finite physical resources-specifically, bounded energy variance-are available. Using the relative entropy of entanglement, we derive a closed analytical expression for the instantaneous entanglement generation rate for arbitrary pure states and Hamiltonians expressed in the Schmidt basis. We find that constraints based solely on the mean energy of the Hamiltonian are insufficient to bound the entanglement generation rate, whereas imposing a variance constraint ensures a finite and well-defined maximum. We fully characterize the Hamiltonians that achieve this optimal rate, establishing a direct relation between their imaginary components in the Schmidt basis and the structure of the optimal initial states. For systems without ancillas, we obtain a closed-form expression for the maximal rate in terms of the surprisal variance of the Schmidt coefficients and identify the family of optimal states and Hamiltonians. We further extend our analysis to scenarios where Alice and Bob may employ local ancillary systems: using a matrix-analytic framework and a refined description of the Hamiltonians allowed by the physical constraints, we derive an explicit optimization formula and characterize the attainable enhancement in entanglement generation.
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.
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,ρ1),(p,ρ2)}\{(1\!-\!p,ρ_1),(p,ρ_2)\}{(1−p,ρ1​),(p,ρ2​)} by c ∥H∥h2(p)c\,\|{H}\|h_{2}(p)c∥H∥h2​(p) (with the binary entropy h2h_{2}h2​). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality ∥[A,log⁡B]∥1 ⁣≤ ⁣c h2(p)\|[A,\log B]\|_{1}\!\leq\! c\,h_{2}(p)∥[A,logB]∥1​≤ch2​(p) for positive trace-class operators A ⁣≤ ⁣BA\!\leq\! BA≤B with TrA ⁣= ⁣p\text{Tr} A\!=\!pTrA=p and TrB ⁣= ⁣1\text{Tr} B\!=\!1TrB=1 on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be c ⁣= ⁣1c\!=\!1c=1. Here, I prove the latter conjecture using an exact integral representation of the commutator [A,log⁡B][A,\log B][A,logB] 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 (2log⁡d ⁣+ ⁣1)∥H∥(2\log d\!+\!1)\|H\|(2logd+1)∥H∥. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.