10,051 research outputs found
Entanglement, quantum randomness, and complexity beyond scrambling
Scrambling is a process by which the state of a quantum system is effectively
randomized due to the global entanglement that "hides" initially localized
quantum information. In this work, we lay the mathematical foundations of
studying randomness complexities beyond scrambling by entanglement properties.
We do so by analyzing the generalized (in particular R\'enyi) entanglement
entropies of designs, i.e. ensembles of unitary channels or pure states that
mimic the uniformly random distribution (given by the Haar measure) up to
certain moments. A main collective conclusion is that the R\'enyi entanglement
entropies averaged over designs of the same order are almost maximal. This
links the orders of entropy and design, and therefore suggests R\'enyi
entanglement entropies as diagnostics of the randomness complexity of
corresponding designs. Such complexities form a hierarchy between information
scrambling and Haar randomness. As a strong separation result, we prove the
existence of (state) 2-designs such that the R\'enyi entanglement entropies of
higher orders can be bounded away from the maximum. However, we also show that
the min entanglement entropy is maximized by designs of order only logarithmic
in the dimension of the system. In other words, logarithmic-designs already
achieve the complexity of Haar in terms of entanglement, which we also call
max-scrambling. This result leads to a generalization of the fast scrambling
conjecture, that max-scrambling can be achieved by physical dynamics in time
roughly linear in the number of degrees of freedom.Comment: 72 pages, 4 figures. Rewritten version with new title. v3: published
versio
Asymptotic entropy and green speed for random walks on countable groups
We study asymptotic properties of the Green metric associated with transient
random walks on countable groups. We prove that the rate of escape of the
random walk computed in the Green metric equals its asymptotic entropy. The
proof relies on integral representations of both quantities with the extended
Martin kernel. In the case of finitely generated groups, where this result is
known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91--112]),
we give an alternative proof relying on a version of the so-called fundamental
inequality (relating the rate of escape, the entropy and the logarithmic volume
growth) extended to random walks with unbounded support.Comment: Published in at http://dx.doi.org/10.1214/07-AOP356 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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