1,008 research outputs found
Martin Boundary Theory of some Quantum Random Walks
In this paper we define a general setting for Martin boundary theory
associated to quantum random walks, and prove a general representation theorem.
We show that in the dual of a simply connected Lie subgroup of U(n), the
extremal Martin boundary is homeomorphic to a sphere. Then, we investigate
restriction of quantum random walks to Abelian subalgebras of group algebras,
and establish a Ney-Spitzer theorem for an elementary random walk on the fusion
algebra of SU(n), generalizing a previous result of Biane. We also consider the
restriction of a quantum random walk on introduced by Izumi to two
natural Abelian subalgebras, and relate the underlying Markov chains by
classical probabilistic processes. This result generalizes a result of Biane.Comment: 29 page
New scaling of Itzykson-Zuber integrals
We study asymptotics of the Itzykson-Zuber integrals in the scaling when one
of the matrices has a small rank compared to the full rank. We show that the
result is basically the same as in the case when one of the matrices has a
fixed rank. In this way we extend the recent results of Guionnet and Maida who
showed that for a latter scaling the Itzykson-Zuber integral is given in terms
of the Voiculescu's R-transform of the full rank matrix
The strong asymptotic freeness of Haar and deterministic matrices
In this paper, we are interested in sequences of q-tuple of N-by-N random
matrices having a strong limiting distribution (i.e. given any non-commutative
polynomial in the matrices and their conjugate transpose, its normalized trace
and its norm converge). We start with such a sequence having this property, and
we show that this property pertains if the q-tuple is enlarged with independent
unitary Haar distributed random matrices. Besides, the limit of norms and
traces in non-commutative polynomials in the enlarged family can be computed
with reduced free product construction. This extends results of one author (C.
M.) and of Haagerup and Thorbjornsen. We also show that a p-tuple of
independent orthogonal and symplectic Haar matrices have a strong limiting
distribution, extending a recent result of Schultz.Comment: 12 pages. Accepted for publication to Annales Scientifique de l'EN
Low entropy output states for products of random unitary channels
In this paper, we study the behaviour of the output of pure entangled states
after being transformed by a product of conjugate random unitary channels. This
study is motivated by the counterexamples by Hastings and Hayden-Winter to the
additivity problems. In particular, we study in depth the difference of
behaviour between random unitary channels and generic random channels. In the
case where the number of unitary operators is fixed, we compute the limiting
eigenvalues of the output states. In the case where the number of unitary
operators grows linearly with the dimension of the input space, we show that
the eigenvalue distribution converges to a limiting shape that we characterize
with free probability tools. In order to perform the required computations, we
need a systematic way of dealing with moment problems for random matrices whose
blocks are i.i.d. Haar distributed unitary operators. This is achieved by
extending the graphical Weingarten calculus introduced in Collins and Nechita
(2010)
Area law for random graph states
Random pure states of multi-partite quantum systems, associated with
arbitrary graphs, are investigated. Each vertex of the graph represents a
generic interaction between subsystems, described by a random unitary matrix
distributed according to the Haar measure, while each edge of the graph
represents a bi-partite, maximally entangled state. For any splitting of the
graph into two parts we consider the corresponding partition of the quantum
system and compute the average entropy of entanglement. First, in the special
case where the partition does not "cross" any vertex of the graph, we show that
the area law is satisfied exactly. In the general case, we show that the
entropy of entanglement obeys an area law on average, this time with a
correction term that depends on the topologies of the graph and of the
partition. The results obtained are applied to the problem of distribution of
quantum entanglement in a quantum network with prescribed topology.Comment: v2: minor typos correcte
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