arXiv subjects

Rupert H. Levene

Publications and source records attributed to Rupert H. Levene.

23 records · Page 2Linked to original sources

arXiv2015

Private algebras in quantum information and infinite-dimensional complementarity

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.

quant-ph↗

arXiv2014

Schur multipliers of Cartan pairs

We define the Schur multipliers of a separable von Neumann algebra M with Cartan masa A, generalising the classical Schur multipliers of $B(\ell^2)$. We characterise these as the normal A-bimodule maps on M. If M contains a direct summand isomorphic to the hyperfinite II_1 factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product $A \otimes_{eh} A$ are strictly contained in the algebra of all Schur multipliers.

math.OA↗

arXiv2014

Norms of idempotent Schur multipliers

Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers η_0 < η_1 < η_2 < ... < η_6 so that for every bounded, normal D-bimodule map Φ on B(H) either ||Φ|| > η_6, or ||Φ|| = η_k for some k <= 6. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm η_k for 0 <= k <= 6. We also show that the Schur idempotents which keep only the diagonal and superdiagonal of an n x n matrix, or of an n x (n+1) matrix, both have norm 2/(n+1) cot(pi/(n+1)), and we consider the average norm of a random idempotent Schur multiplier as a function of dimension. Many of our arguments are framed in the combinatorial language of bipartite graphs.

math.FA↗

arXiv2007

Manifolds of Hilbert space projections

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(λ) = \exp(i(λ_1ψ_1 + ... + λ_nψ_n)) for λin R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(λ)H^2(R) of L^2(R). For classes of admissible functions ψ_i the strong operator topology closures of M and M \cup M^\perp are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary equivalence. In particular the Fourier-Plancherel 2-sphere and the hyperbolic 3-sphere of Katavolos and Power appear as distinguished special cases admitting nontrivial unitary automorphisms groups which are explicitly described.

math.FA↗

arXiv2005

1-Hyperreflexivity and Complete Hyperreflexivity

The subspaces and subalgebras of B(H) which are hyperreflexive with constant 1 are completely classified. It is shown that there are 1-hyperreflexive subspaces for which the complete hyperreflexivity constant is strictly greater than 1. The constants for $\bC T \otimes B(H)$ are analyzed in detail.

math.FA↗