arXiv subjects

Travis B. Russell

Publications and source records attributed to Travis B. Russell.

10 recordsLinked to original sources

arXiv2025

Products and factorization in operator systems

We study unital operator spaces endowed with a partially defined product. We give a matrix-norm characterization of such products that allows for a representation theorem where the partial product is realized as composition of operators on Hilbert space. We study product-respecting C*-covers, including a universal product C*-cover, and product quotients. We show that for the Haagerup tensor product of unital operator spaces remains injective, while projectivity holds relative to product quotients. Moreover, we identify the commuting tensor product as a complete product quotient of the Haagerup tensor product. Our framework yields new factorization norm formulas for a variety of product structures, as well as an intrinsic trace-extension criterion that resolves a question posed by Sinclair. Our work unifies and extends tensor products for operator systems, with applications in quantum information theory.

math.OA↗

arXiv2025

Operator system characterizations of SIC-POVMs and mutually unbiased bases

We show that a symmetric informationally-complete positive operator-valued measure exists in a given dimension $d$ if and only if there exists a $d^2$-dimensional operator system satisfying certain order-theoretic conditions. We also describe a method of constructing such an operator system and demonstrate that the first step of this construction can be carried out successfully. We obtain analogous results for the existence of $d+1$ mutually unbiased bases in a given dimension.

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arXiv2025

Real operator systems

Operator systems are the unital self-adjoint subspaces of the bounded operators on a Hilbert space. Complex operator systems are an important category containing the C*-algebras and von Neumann algebras, which is increasingly of interest in modern analysis and also in modern quantum physics (such as quantum information theory). They have an extensive theory, and have very important applications in all of these subjects. We present here the real case of the theory of (complex) operator systems, and also the real case of their remarkable tensor product theory, due in the complex case to Paulsen and his coauthors and students (such as Kavruk), building on pioneering earlier work of Kirchberg and others. We uncover several notable differences between the real and complex theory, including the absence of minimal and maximal functors in the category of real operator systems. We also develop very many foundational structural results for real operator systems, and elucidate how the complexification interacts with the basic constructions in the subject. In the final two sections of our paper we study real analogues of the Kirchberg conjectures (and of several important related problems that have attracted much interest recently), and study the deep relationships between them.

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arXiv2023

Synchronous Values of Games

We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix. We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy.

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arXiv2023

A synchronous NPA hierarchy with applications

We present an adaptation of the NPA hierarchy to the setting of synchronous correlation matrices. Our adaptation improves upon the original NPA hierarchy by using smaller certificates and fewer constraints, although it can only be applied to certify synchronous correlations. We recover characterizations for the sets of synchronous quantum commuting and synchronous quantum correlations. For applications, we show that the existence of symmetric informationally complete positive operator-valued measures and maximal sets of mutually unbiased bases can be verified or invalidated with only two certificates of our adapted NPA hierarchy.

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arXiv2022

Characterizations of Ordered Self-adjoint Operator Spaces

We describe how self-adjoint ordered operator spaces, also called non-unital operator systems in the literature, can be understood as $*$-vector spaces equipped with a matrix gauge structure. We explain how this perspective has several advantages over other notions of non-unital operator systems in the literature. In particular, the category of matrix gauge $*$-vector spaces includes injective objects and a Webster-Winkler-type duality theorem, both of which we show generally fail with other notions of non-unital operator systems. As applications, we characterize those subspaces of operator systems which are kernels of completely positive maps and define a new operator space structure on the matrix ordered dual of an operator system generalizing the classical notion of a base norm space.

math.OA↗

arXiv2020

No quantum Ramsey theorem for stabilizer codes

In this paper we study the quantum graphs of mixed-unitary channels generated by tensor products of Pauli operators, which we call Pauli channels. We show that most quantum graphs arising from Pauli channels have non-trivial quantum cliques or quantum anticliques which are stabilizer codes. However, a reformulation of Nik Weaver's quantum Ramsey theorem in terms of stabilizer codes and Pauli channels fails. Specifically, for every positive integer $n$, there exists an $n$-qubit Pauli channel for which any non-trivial quantum clique or quantum anticlique fails to be a stabilizer code.

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arXiv2020

Geometry of the set of synchronous quantum correlations

We provide a complete geometric description of the set of synchronous quantum correlations for the three experiment two outcome scenario. We show that these correlations form a closed set. Moreover, every correlation in this set can be realized using projection valued measures on a Hilbert space of dimension no more than 16.

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arXiv2019

Two-outcome synchronous correlation sets and Connes' embedding problem

We show that Connes' embedding problem is equivalent to the weak Tsirelson problem in the setting of two-outcome synchronous correlation sets. We further show that the extreme points of two-outcome synchronous correlation sets can be realized using a certain class of universal C*-algebras. We examine these algebras in the three-experiment case and verify that the strong and weak Tsirelson problems have affirmative answers in that setting.

math.OA↗

arXiv2017

Characterizations of ordered operator spaces

We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). Defining the gauge of an operator $T \in B(H)$ to be $\|Re(T)_+\|$, we demonstrate an abstract characterization of operator spaces up to complete gauge-isometry. Both of these characterizations preserve the structure of the self-adjoint, positive, and accretive operators, as well as the operator norm. We show that an operator space with a given matrix ordering of positive or accretive cones can be represented completely isometrically and completely order isomorphically if and only if each positive cone is normal, in the sense that $x \leq y \leq z$ implies that $\|y\| \leq \max(\|x\|,\|z\|)$ at each matrix level. This is achieved by showing that normal matrix ordered operator spaces are induced by gauges. We show that inducing gauges are not unique in general. Finally, we show that completely positive completely contractive linear maps on non-unital operator spaces extend to any containing operator system if and only if the operator space is induced by a unique gauge.

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