Upper bounds on entangling rates of bipartite Hamiltonians

Bravyi, Sergey

doi:10.1103/physreva.76.052319

Cited by 76 publications

(124 citation statements)

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“…We then use the powerful formalism of quasiadiabatic continuation [44] to relate such a state to the ground state of a spectrally gapped long-range interacting Hamiltonian. This strategy is made possible by the recent proof of Kitaev's small incremental entangling (SIE) conjecture [43,45], and by significant recent improvements in Lieb-Robinson bounds [4] for long-range interacting systems [46,47].…”

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confidence: 99%

Entanglement Area Laws for Long-Range Interacting Systems

et al. 2017

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We prove that the entanglement entropy of any state evolved under an arbitrary 1=r α long-rangeinteracting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α > D þ 1. We also prove that for any α > 2D þ 2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions. DOI: 10.1103/PhysRevLett.119.050501 Quantum many-body systems often have approximately local interactions, and this locality has profound effects on the entanglement properties of both ground states and the states created dynamically after a quantum quench. For example, the entanglement entropy, defined as the entropy of the reduced state of a subregion, often scales as the boundary area of the subregion for ground states of shortrange interacting Hamiltonians [1]. This "area law" of entanglement entropy is in sharp contrast to the behavior of thermodynamic entropy, which typically scales as the volume of the system. While the study of area laws originates from black hole physics [2,3], area laws have received considerable attention recently in the fields of quantum information and condensed matter physics. In particular, area laws are known to be closely related to the velocity of information propagation in quantum lattices [4], quantum critical phenomena [5], bulk-boundary correspondence [6], efficient classical simulation of quantum systems [7], topological order [8], and many-body localization [9].However, the description of many-body systems in terms of local interactions is often only an approximation, and not always a good one; in numerous systems of current interest, ranging from frustrated magnets and spin glasses [10,11] to atomic, molecular, and optical systems [12][13][14][15][16][17], longrangeinteractions are ubiquitous and lead to qualitatively new physics, e.g., giving rise to novel quantum phases and dynamical behaviors [18][19][20][21][22][23][24][25], and enabling speedups in quantum information processing [26][27][28][29][30]. Particles in these systems generally experience interactions that decay algebraically (∼1=r α ) in the distance (r) between them. As might be expected, α controls the extent to which the system respects notions of locality developed for short-range interacting systems: For α sufficiently small, it is well established [19] that locality may be completely lost, and for α sufficiently large there is ample numerical and analytical evidence [31][32][33][34] that area laws may persist. However, there is currently no general and rigorous understanding of when area laws do or do not survive the presence of long-range interactions.The modern understanding of area laws draws heavily from several rigor...

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confidence: 99%

Entanglement Area Laws for Long-Range Interacting Systems

et al. 2017

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“…For von Neumann entropy Audenaert [4] provided a completely different proof of SIM with constant 2. Numerical evidence [5] suggests that the optimal constant in SIM for von Neumann entropy is 1, although no proof is known for it at the time of writing this paper. Unfortunately, Audenaert's proof can not be easily generalized to functions other than logarithmic, so in this paper we have sacrificed the constant but generalized the problem to a large class of functions.…”

Section: Resultsmentioning

confidence: 93%

“…Bravyi [5] proved that Small Incremental Mixing, Theorem 2.2, with a constant c in front of the Shannon entropy implies Small Incremental Entangling, Theorem 2.1, with a constant 2c. In Lemma 1 [5] Bravyi proved that there exists a state µ aAB , such that a state appearing in (2.1) can be written as…”

Section: Sie Follows From Simmentioning

confidence: 99%

Entanglement rates for Rényi, Tsallis, and other entropies

Vershynina

2019

Journal of Mathematical Physics

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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 entropy taken as measures of entanglement in the system.

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“…The same question for closed bipartite system evolving under a unitary dynamics was answered by Bravyi in [1] and by Acoleyen et al in [2] and Audenaert [3] for the general case in the presence of ancillas. Let us say that two parties, Alice and Bob, have access to systems A and B respectively together with ancilla systems a and b respectively.…”

Section: Introductionmentioning

confidence: 76%

Entanglement rates for bipartite open systems

Vershynina

2015

Phys. Rev. A

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We provide 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

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Upper bounds on entangling rates of bipartite Hamiltonians

Cited by 76 publications

References 12 publications

Entanglement Area Laws for Long-Range Interacting Systems

Entanglement Area Laws for Long-Range Interacting Systems

Entanglement rates for Rényi, Tsallis, and other entropies

Entanglement rates for bipartite open systems

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