The Quantum Formalization Drive

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…

An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the classical and quantum capacities of the channel. Although our formula involves regularization, i.e. taking a limit over many copies of the channel, it reduces to a…

The concept of translation of an operator allows to consider the analogous of shift-invariant subspaces in the class of Hilbert-Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the corresponding sampling formulas. The key point here is the use of the Weyl transform, a unitary mapping between the space of…

We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum…

Let {\rho} the density matrix of a mixed Gaussian state. Assuming that one of the Robertson--Schr"odinger uncertainty inequalities is saturated by {\rho}, e.g. ({\Delta}^{{\rho}}X_1)^2({\Delta}^{{\rho}}P_1)^2={\Delta}^{{\rho}}(X_1,P_1)^2+(1/4)\hbar^2, we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of…

We show that Araki and Masuda's weighted non-commutative vector valued $L_p$-spaces [Araki & Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched R'enyi divergences with parameter $\alpha = \frac{p}{2}$. Using complex interpolation theory, we prove various fundamental properties of these divergences in…

The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method…

This article provides an elementary introduction to Gaussian channels and their capacities. We review results on the classical, quantum, and entanglement assisted capacities and discuss related entropic quantities as well as additivity issues. Some of the known results are extended. In particular, it is shown that the quantum conditional entropy is…

We prove that it is impossible to distill more entanglement from a single copy of a two-mode bipartite entangled Gaussian state via LOCC Gaussian operations. More generally, we show that any hypothetical distillation protocol for Gaussian states involving only Gaussian operations would be a deterministic protocol. Finally, we argue that the protocol…

We develop an approximation approach to infinite dimensional quantum channels based on detailed investigation of the continuity properties of entropic characteristics of quantum channels and operations (trace-nonincreasing completely positive maps) as functions of a pair ``channel, input state''. The obtained results are then applied to the following…

We address the problem of learning an unknown unitary transformation from a finite number of examples. The problem consists in finding the learning machine that optimally emulates the examples, thus reproducing the unknown unitary maximum fidelity. Learning a unitary is equivalent to storing it in the state of a quantum memory (the memory of the learning…

The coding theorem for the entanglement-assisted communication via infinite-dimensional quantum channel with linear constraint is extended to a natural degree of generality. Relations between the entanglement-assisted classical capacity and the $\chi$-capacity of constrained channels are obtained and conditions for their coincidence are given. Sufficient…

We formalize the task of unitary Schur sampling -- an extension of weak Schur sampling -- which is the process of measuring the Young label and the unitary group register of an input $m$ qudit state. Intuitively, this task is equivalent to applying the Schur transform, projecting onto the isotypic subspaces of the unitary and symmetric groups indexed by the…