Rupert H. Levene

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.

Compactness properties of operator multipliers

Abstract We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C ∗ -algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C ∗ -algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollstandig beschrankte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universitat des Saarlandes, Saarbrucken, 1982].

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 CT⊗B(H) are analyzed in detail.

Commutants of weighted shift directed graph operator algebras

We consider non-selfadjoint operator algebras

Manifolds of Hilbert space projections

\mathfrak{L}(G,\lambda)

Complexity and Capacity Bounds for Quantum Channels

generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs

Schur multipliers of Cartan pairs

G

Positive extensions of Schur multipliers

. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for

Schur idempotents and hyperreflexivity

\mathfrak{L}(G,\lambda)

On the topological stable rank of non-selfadjoint operator algebras

in the case of the single vertex graph with two edges and a suitable choice of left weight function