The Quantum Formalization Drive

An action of an affine algebraic group $G$ on an affine variety $X$ is observable when invariant regular functions see every nonzero $G$-stable ideal of the coordinate ring: if $I\subseteq k[X]$ is a nonzero $G$-stable ideal, then $I^G\neq(0)$. The notion goes back to Grosshans' theory of observable subgroups (F. D. Grosshans, Observable groups and Hilbert's fourteenth…

Quantum Max Cut asks for the highest energy of a two-qubit interaction Hamiltonian placed on the edges of a weighted graph. It is a quantum analogue of Max Cut: a classical cut assigns one of two labels to each vertex, while a quantum state can use correlations and entanglement across the qubits. Lee studies an approximation algorithm that first solves a…

The generalized Stieltjes transform of order $\alpha>0$ of a non-negative measure $\mu$ on $[0,\infty)$ is the function $z\mapsto\int_{[0,\infty)}\mu(du)/(u+z)^\alpha$. For $\alpha=1$ this is the classical Stieltjes transform, whose functions (Stieltjes functions) appear in moment problems, in the theory of completely monotone and Bernstein functions, in the spectral…

In quantum field theory the vacuum is entangled between any two regions of space, and the amount of entanglement between nearby regions is infinite: the algebras of observables of regions that touch admit no normal separable state at all. For regions at a positive distance the question becomes quantitative. How much entanglement does the vacuum carry…

The Brauer algebra packages pairings of points into an algebra whose multiplication records how pairings join when diagrams are stacked. It was introduced by Brauer as the centralizer algebra of orthogonal and symplectic groups acting on tensor powers of their natural representation (Brauer 1937), and is semisimple for generic $\delta$ (Wenzl 1988). It…

A quantum dynamical semigroup describes the irreversible evolution of an open quantum system: a family $T_t$, $t \ge 0$, of normal completely positive maps on the algebra $\mathfrak B(\mathcal H)$ of bounded operators, with $T_0 = \mathrm{Id}$ and $t \mapsto T_t[X]$ weakly continuous. When the generator is bounded its structure is given by the…

Let $V$ be the natural representation of $GL_n$ and $V^*$ its dual. The walled Brauer algebra $B_{r,s}(\delta)$ describes the endomorphisms of mixed tensor space $V^{\otimes r}\otimes V^{*\otimes s}$ that commute with $GL_n$, through mixed Schur–Weyl duality. It was introduced in this role independently by Turaev, Koike and…

A strongly continuous unitary representation turns each group element into a unitary operator on a Hilbert space. Averaging those operators against a smooth, compactly supported function gives a bounded operator. When every such average is trace class, its trace defines a distribution character. The trace condition is therefore a bridge from an operator…

Classical Max Cut asks for a partition of the vertices of a weighted graph that maximises the total weight of edges crossing it. Its best-known approximation algorithm, due to Goemans and Williamson, rounds a semidefinite relaxation and guarantees ratio $0.878$. Quantum Max Cut is the analogous problem for a family of local Hamiltonians: each edge contributes…

A quantum computer that runs long computations must protect its data with a quantum error-correcting code and must compute on the encoded data without letting errors spread. The simplest gates with that property are transversal gates: a code is laid out in several code blocks, and a transversal gate couples each physical subsystem of one block only to the…

We relate the structure of cyclotomic and degenerate cyclotomic BMW algebras, for arbitrary parameter values, to that for admissible parameter values. In particular, we show that these algebras are cellular. We characterize those parameter sets for affine BMW algebras over an algebraically closed field that permit the algebras to have non--trivial…

We study admissibility conditions for the parameters of degenerate cyclotomic BMW algebras. We show that the u-admissibility condition of Ariki, Mathas and Rui is equivalent to a simple module theoretic condition.

Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X \to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such that f(Y) =0. We characterize this condition geometrically as follows. The action $G\times X \to X$ is observable if and only if (1) there…

In this paper all of the classical constructions of A. Young are generalized to affine Hecke algebras of type A. It is proved that the calibrated irreducible representations of the affine Hecke algebra are indexed by placed skew shapes and that these representations can be constructed explicitly with a generalization of Young's seminormal construction of…

A representation of the exceptional Lie algebras is presented. It reflects a simple unifying view and it is realized in terms of Zorn-type matrices. The role of the underlying Jordan pair and Jordan algebra content is crucial in the development of the structure. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the…

We derive the class of covariant measurements which are optimal according to the maximum likelihood criterion. The optimization problem is fully resolved in the case of pure input states, under the physically meaningful hypotheses of unimodularity of the covariance group and measurability of the stability subgroup. The general result is applied to the case…

We start with Fannes' type and Winter's type tight continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quantity is continuous on…