The Quantum Formalization Drive

A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood.…

We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on…

A host algebra of a topological group G is a C^-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value αon such a semigroup…

We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type B_d(l) consists of symmetric polynomials in its commuting generators. The classification of the blocks of the degenerate cyclotomic Hecke algebras is an easy consequence. We then deduce that the center of an integral block of parabolic…

We give the first composable security proof for continuous-variable quantum key distribution with coherent states against collective attacks. Crucially, in the limit of large blocks the secret key rate converges to the usual value computed from the Holevo bound. Combining our proof with either the de Finetti theorem or the Postselection technique then shows…

We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bijection between continuous and smooth…

We investigate Gaussian quantum states in view of their exceptional role within the space of all continuous variables states. A general method for deriving extremality results is provided and applied to entanglement measures, secret key distillation and the classical capacity of Bosonic quantum channels. We prove that for every given covariance matrix the…

Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is in a certain class of unital qubit channels, with the other completely arbitrary. This qubit class includes the depolarizing channel. As a byproduct this proves that the Holevo bound is the ultimate information capacity of such qubit channels…

We study the symmetric-side-channel-assisted private capacity of a quantum channel, for which we provide a single-letter formula. This capacity is additive, convex, and, for degradable channels, equal to the unassisted private capacity. While a channel's (unassisted) capacity for for private classical communication may be strictly larger than its quantum…

We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the "data-processing bound," is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier…

We derive a simple relation between a quantum channel's capacity to convey coherent (quantum) information and its usefulness for quantum cryptography.

We establish the classical capacity of optical quantum channels as a sharp transition between two regimes---one which is an error-free regime for communication rates below the capacity, and the other in which the probability of correctly decoding a classical message converges exponentially fast to zero if the communication rate exceeds the classical…

The long-standing conjectures of the optimality of Gaussian inputs for Gaussian channel and Gaussian additivity are solved for a broad class of covariant or contravariant Bosonic Gaussian channels (which includes in particular thermal, additive classical noise, and amplifier channels) restricting to the class of states with finite second moments. We show…

Optical channels, such as fibers or free-space links, are ubiquitous in today's telecommunication networks. They rely on the electromagnetic field associated with photons to carry information from one point to another in space. As a result, a complete physical model of these channels must necessarily take quantum effects into account in order to determine…

Schur–Weyl duality says that on the tensor power $V^{\otimes N}$ of the natural representation of $\mathfrak{gl}_m(\mathbb C)$, the symmetric group $\Sigma_N$ acting by place permutations generates the full commutant of $\mathfrak{gl}_m$. It is the bridge between representations of general linear groups and of symmetric groups. Two extensions matter in…

A quantum circuit normally applies a sequence of gates to a prepared state. Verstraete, Wolf and Cirac proposed a different physical route: couple a quantum system to a memoryless environment so that dissipation drives it toward a state recording the circuit's computation. Their dissipative quantum computation construction places a clock beside the…

Many constructions in differential topology and infinite-dimensional Lie theory start from a continuous object and need a smooth one: a continuous section of a smooth bundle, a continuous map into a manifold, a continuous homotopy between smooth maps. Steenrod's approximation theorem (Steenrod 1951, §6.7) says that, for finite-dimensional bundles, a…

The Lie superalgebra $\widehat{\mathfrak{gl}}_{\infty|\infty}$, a central extension of the Lie superalgebra of infinite matrices of finite support, appears in the representation theory of the $W_{1+\infty}$ vertex algebra and in Howe-type dualities with $\mathfrak{gl}_n$ (Kac–Radul; Cheng–Lam). Its irreducible quasi-finite highest weight modules of positive…

We give an overview of relative tensor products (RTPs) for von Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and index. Then we consider the analytic construction, with particular emphasis on explaining why the RTP is not generally…

The walled Brauer algebra organizes the linear operators that commute with a general linear group acting on a tensor product of vector spaces and their duals. Its diagrams mix crossings and contractions, with a wall distinguishing the two tensor factors. This makes it useful in mixed tensor representation theory, where the ordinary symmetric group algebra…

The Birman–Murakami–Wenzl (BMW) algebras were introduced by Birman and Wenzl and by Murakami to study the Kauffman link polynomial; they are deformations of the Brauer algebras and act as centralizer algebras of quantum groups of orthogonal and symplectic type on tensor powers of the natural module. Their cyclotomic generalization was introduced by…

Sending classical messages through a quantum device requires choosing input states that remain distinguishable after noise. The one-shot classical capacity measures the best information rate when each channel use is signaled with a separate input state. For many channels the optimization is difficult because it ranges over all ensembles of quantum states.…

Many infinite-dimensional Lie groups arising in geometry and physics — loop groups, gauge groups, groups of diffeomorphisms and of volume-preserving diffeomorphisms — have interesting central and abelian extensions, and these extensions are usually found first at the level of Lie algebras, as continuous 2-cocycles. Whether such a Lie algebra extension comes…

A transformation achieving the optimal symmetric N-to-M cloning of coherent states is presented. Its implementation only requires a phase-insensitive linear amplifier and a network of beam splitters. An experimental demonstration of this continuous-variable cloner should therefore be in the scope of current technology. The link between optimal quantum…

We consider two examples of dynamical semigroups obtained by singular perturbations of a standard generator which are special case of unbounded completely positive perturbations studied in detail in [10]. In the section 2 we propose a generalization of an example from [1] aimed to give a positive answer to a conjecture of Arveson. In the section 3 we…