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arXiv:1505.00416 · Physical Review A

Entanglement rates for bipartite open systems

Anna Vershynina

Published Aug 6, 2015• 3 citations
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Abstract

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.

DOI
10.1103/physreva.92.022311

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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.
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We prove an upper bound on the maximal rate at which a Hamiltonian interaction can generate entanglement in a bipartite system. The scaling of this bound as a function of the subsystem dimension on which the Hamiltonian acts nontrivially is optimal and is exponentially improved over previously known bounds. As an application, we show that a gapped quantum many-body spin system on an arbitrary lattice satisfies an area law for the entanglement entropy if and only if any other state with which it is adiabatically connected (i.e., any state in the same phase) also satisfies an area law.
We prove continuity of distillable entanglement, the optimal asymptotic rate at which copies of a bipartite state can be converted into maximally entangled pairs by local operations and classical communication (LOCC) with vanishing error. We establish an explicit asymptotic continuity bound that holds for arbitrary bipartite states and depends logarithmically on the system dimension. This bound has the same structure as known continuity bounds for conditional entropy, entanglement cost, and relative entropy of entanglement. Our proof constructs an explicit LOCC protocol that converts either of two arbitrary bipartite states into the other at an ebit cost that vanishes as their trace distance tends to zero.
We determine the relative entropy of entanglement of a bipartite mixed state ρABρ_{AB}ρAB​ obtained by tracing out one subsystem of a tripartite Haar-random pure state ∣ψ⟩ABC|ψ\rangle_{ABC}∣ψ⟩ABC​, finding ER(ρAB)=log⁡dAdBmax⁡(dA,dB,dC)+O(1)E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)ER​(ρAB​)=logmax(dA​,dB​,dC​)dA​dB​​+O(1). Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation EF(ρAB)E_F(ρ_{AB})EF​(ρAB​) and the mutual information I(A:B)I(A:B)I(A:B). The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.
The speed limit on entanglement is defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary and arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For the unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.