Searcharxiv⌕ Search

arXiv subjects

Douglas Farenick

Publications and source records attributed to Douglas Farenick.

At least 19 recordsLinked to original sources
arXiv2026

Geometry of essential matrix ranges and the Smith-Ward problem for operator systems

A $d$-tuple of bounded linear selfadjoint operators acting on an infinite-dimensional separable Hilbert space is said to have the Smith-Ward property if the identity map of the image of the operator system in the Calkin algebra has a completely positive lift. In this paper, we focus on noncommutative geometric properties of a finite dimensional operator system with the goal of understanding how geometric information encoded by the essential matrix range of a spanning set of linear basis for the operator system implies the Smith-Ward property. Some geometric objects of special interest in this paper include one form of noncommutative complex Euclidean ball and maximal noncommutative cubes and polydiscs, as well as some extremal compact matrix convex sets, $K^{\rm min}$ or $K^{\rm max}$, determined by a given compact convex subset $K$ of $\mathbb R^d$.

math.OA↗
arXiv2026

Representations of noncommutative cubes and prisms

Representations of the operator system determined by the canonical generators of the free product of two cyclic groups of order $2$ and $k$, or $d$ cyclic groups of order $2$, are studied for the purpose of shedding light on the noncommutative geometry of noncommutative $d$-cubes and $k$-prisms. By way of the duality of the categories NCConv and OpSys of noncommutative convex sets and operator systems, respectively, an analysis of noncommutative extreme points, exactness, the lifting property, automatic complete positivity, controlled completely positive extensions, tensor products, and operator system duality is undertaken. Of note is the pairing of two classical dilation theorems of Halmos and Mirman to give a complete description of the noncommutative triangular prism in terms of joint unitary dilations.

math.OA↗
arXiv2026

Generating $\psl{2}{q}$ by elements of prime orders $2$ and $p$

For primes $p\geq 5$, we determine those $q$ for which the projective special linear group $\psl{2}{q}$ over the finite field of order $q$ is $(2,p)$-generated -- that is, there exist two elements of $\psl{2}{q}$ of orders $2$ and $p$, respectively, that generate $\psl{2}{q}$.

math.GR↗
arXiv2025

Categorical relations and bipartite entanglement in tensor cones for Toeplitz and Fejér-Riesz operator systems

The present paper aims to understand separability and entanglement in tensor cones, in the sense of Namioka and Phelps, that arise from the base cones of operator system tensor products. Of particular interest here are the Toeplitz and Fejér-Riesz operator systems, which are, respectively, operator systems of Toeplitz matrices and Laurent polynomials of bounded degree (that is, trigonometric polynomials), and which are related in the operator system category through duality. Some additional categorical relationships established in this paper for Toeplitz and Fejér-Riesz operator systems. Of independent interest is a single matrix criterion, similar to the criterion involving the Choi matrix, for a linear map of the Fejér-Riesz operator system to be completely positive.

math.OA↗
arXiv2024

Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples

Using works of T.~Ando and L.~Gurvits, the well-known theorem of P.R.~Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces recast as a result for $d$-tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition. Such operator $d$-tuples satisfying this matrix-positivity condition are called, herein, Toeplitz-contractive, and a characterisation of the Toeplitz-contractivity condition is presented. The matrix-positivity condition leads to definitions of new distance-measures in several variable operator theory, generalising the notions of norm, numerical radius, and spectral radius to $d$-tuples of operators (commuting, for the spectral radius) in what appears to be a novel, asymmetric way. Toeplitz contractive operators form a noncommutative convex set, and a scaling constant $c_d$ for inclusions of the minimal and maximal matrix convex sets determined by a stretching of the unit circle $S^1$ across $d$ complex dimensions is shown to exist.

math.FA↗
arXiv2023

Toeplitz separability, entanglement, and complete positivity using operator system duality

A new proof is presented of a theorem of L.~Gurvits, which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system $C(S^1)^{(n)}$ of $n\times n$ Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems $C(S^1)^{(n)}\omin\B(\H)$ and $C(S^1)_{(n)}\omin\B(\H)$, where $\H$ is an arbitrary Hilbert space and $C(S^1)_{(n)}$ is the operator system dual of $C(S^1)^{(n)}$. Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from $\B(\H)$ when $\H$ has infinite dimension. In particular, we prove that normal positive linear maps $ψ$ on $\B(\H)$ are partially completely positive in the sense that $ψ^{(n)}(x)$ is positive whenever $x$ is a positive $n\times n$ Toeplitz matrix with entries from $\B(\H)$. We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T.~Ando to universality.

math.OA↗
arXiv2021

The operator system of Toeplitz matrices

A recent paper of A.~Connes and W.D.~van Suijlekom identifies the operator system of $n\times n$ Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than $n$. The present paper examines this identification in somewhat more detail by showing explicitly that the Connes--van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Consequences of this complete order isomorphism are also examined, yielding two special results of note: (i) that every positive linear map of the $n\times n$ complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the $n\times n$ Toeplitz matrices into the algebra of all $n\times n$ complex matrices is a unitary similarity transformation. This latter result gives a new proof of a theorem established earlier. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of $n\times n$ complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix generates an extremal ray in the cone of all continuous $n\times n$ Toeplitz-matrix valued functions $f$ on the unit circle $S^1$ whose Fourier coffecients $\hat f(k)$ vanish for $|k|\geq n$ .

math.FA↗
arXiv2021

Universality of Weyl Unitaries

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper on group theory and quantum mechanics, are $p\times p$ unitary matrices given by the diagonal matrix whose entries are the $p$-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by $\mathfrak u$ and $\mathfrak v$, satisfy $\mathfrak u^p=\mathfrak v^p=1_p$ (the $p\times p$ identity matrix) and the commutation relation $\mathfrak u\mathfrak v=ζ\mathfrak v\mathfrak u$, where $ζ$ is a primitive $p$-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if $u$ and $v$ are any $d\times d$ unitary matrices such that $u^p= v^p=1_d$ and $ u v=ζvu$, then there exists a unital completely positive linear map $ϕ:\mathcal M_p(\mathbb C)\rightarrow\mathcal M_d(\mathbb C)$ such that $ϕ(\mathfrak u)= u$ and $ϕ(\mathfrak v)=v$. We also show, moreover, that any two pairs of $p$-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent. When $p=2$, the Weyl matrices are two of the three Pauli matrices from quantum mechanics. It was recently shown that $g$-tuples of Pauli-Weyl-Brauer unitaries are universal for all $g$-tuples of anticommuting selfadjoint unitary matrices; however, we show here that the analogous result fails for positive integers $p>2$. Finally, we show that the Weyl matrices are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

math.OA↗
arXiv2020

Complete Order Equivalence of Spin Unitaries

This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from \emph{spin systems}, or \emph{spin unitaries}, which are finite sets $\mathcal S$ of selfadjoint unitary matrices such that any two unitaries in $\mathcal S$ anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all $m$-tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C$^*$-envelope of any operator system generated by a spin system of cardinality $2k$ or $2k+1$ is the simple matrix algebra $\mathcal M_{2^k}(\mathbb C)$. As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [Helton-Klep-McCullough-Schweighofer, "Dilations, linear matrix inequalities, the matrix cube problem and beta distributions", Mem. Amer. Math. Soc. 257(2019)] that the spin ball $\mathfrak B_m^{\rm spin}$ and max ball $\mathfrak B_m^{\rm max}$ coincide as matrix convex sets in the cases $m=1,2$. We also derive analogous results for countable spin systems and their C$^*$-envelopes.

math.OA↗
arXiv2020

Purity of the Embeddings of Operator Systems into their C$^*$- and Injective Envelopes

We study the issue of issue of purity (as a completely positive linear map) for identity maps on operators systems and for their completely isometric embeddings into their C$^*$-envelopes and injective envelopes. Our most general result states that the canonical embedding of an operator system $\mathcal R$ into its injective envelope $\mathcal I(\mathcal R)$ is pure if and only if the C$^*$-envelope of $\mathcal R$ is a prime C$^*$-algebra. To prove this, we also show that the identity map on any AW$^*$-factor is a pure completely positive linear map. For embeddings of operator systems into their C$^*$-envelopes, the issue of purity is seemingly harder to describe in full generality, and so we focus here on operator systems arising from the generators of discrete groups. The question of purity of the identity is quite subtle for operator system that are not C$^*$-algebras, and we have results only for two universal operator systems arising from discrete groups. Lastly, a previously unrecorded feature of pure completely positive linear maps is presented: every pure completely positive linear map on an operator system $\mathcal R$ into an injective von Neumann algebra $\mathcal M$ has a pure completely positive extension to any operator system $\mathcal T$ that contains $\mathcal R$ as an operator subsystem, thereby generalising a result of Arveson for the injective type I factor $\mathcal B(\mathcal H)$.

math.OA↗
arXiv2018

Approximately Clean Quantum Probability Measures

A quantum probability measure--or quantum measurement--is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al (2005) for finite-dimensional Hilbert space, and was studied subsequently by Kahn (2007) and Pellonpää (2011). The present paper provides new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of `approximately clean quantum probability measures' and characterise this property for measures whose range determines a finite-dimensional operator system.

quant-ph↗
arXiv2017

Bures Contractive Channels on Operator Algebras

In a unital C*-algebra with a faithful trace functional $τ$, the set $D_τ(A)$ of positive $ρ\in A$ of trace τ(ρ)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space $D_τ(A)$ in terms of the Bures metric. Linear maps on A that map $D_τ(A)$ back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space $D_τ(A)$, (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space $D_τ(A)$, and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra $A^{**}$.

quant-ph↗
arXiv2016

Fidelity of density operator in an operator-algebraic framework

Josza's definition of fidelity for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C$^*$-algebras $A$ that possess a faithful trace functional $τ$. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements $ρ\in A$ for which $τ(ρ)=1$. The second of our two settings is more operator theoretic: by fixing a faithful normal semifinite trace $τ$ on a semifinite von Neumann algebra $M$, we define and consider the fidelity of pairs of positive operators in $M$ of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of $A$ or of the predual $M_*$. Our results also yield a new proof of a theorem of Molnár on the structure of quantum channels on the trace-class operators that preserve fidelity.

quant-ph↗
arXiv2015

Spectra and Variance of Quantum Random Variables

We study essentially bounded quantum random variables and show that the Gelfand spectrum of such a quantum random variable coincides with the hypoconvex hull of its essential range. Moreover, a notion of operator-valued variance is introduced, leading to a formulation of the moment problem in the context of quantum probability spaces in terms of operator-theoretic properties involving semi-invariant subspaces and spectral theory. As an application of quantum variance, new measures of random and inherent quantum noise are introduced for measurements of quantum systems, modifying some recent ideas of Polterovich.

quant-ph↗
arXiv2015

Isometries of the Toeplitz Matrix Algebra

We study the structure of isometries defined on the algebra $\mathcal{A}$ of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either $A\mapsto UAU^*$ or $A\mapsto U\overline AU^*$, where $\overline A$ is the complex conjugation and $U$ is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is of the form $A\mapsto UAV$ where $U$ and $V$ are two unitary matrices. This implies, in particular, that every such an isometry is a complete isometry and that a unital linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is necessarily an algebra homomorphism.

math.FA↗
arXiv2013

Characterisations of the weak expectation property

We use representations of operator systems as quotients to deduce various characterisations of the weak expectation property (WEP) for C?*-algebras. By Kirchberg's work on WEP, these results give new formulations of Connes' embedding problem.

math.OA↗