We develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors. We prove that the problem of computation of the commuting quantum chromatic ...number of a graph is solvable by an SDP algorithm and describe an hierarchy of variants of the commuting quantum chromatic number which converge to it. We introduce the tracial rank of a graph, a parameter that gives a lower bound for the commuting quantum chromatic number and parallels the projective rank, and prove that it is multiplicative. We describe the tracial rank, the projective rank and the fractional chromatic numbers in a unified manner that clarifies their connection with the commuting quantum chromatic number, the quantum chromatic number and the classical chromatic number, respectively. Finally, we present a new SDP algorithm that yields a parameter larger than the Lovász number and is yet a lower bound for the tracial rank of the graph. We determine the precise value of the tracial rank of an odd cycle.
For every convex body K⊆Rd, there is a minimal matrix convex set Wmin(K), and a maximal matrix convex set Wmax(K), which have K as their ground level. We aim to find the optimal constant θ(K) such ...that Wmax(K)⊆θ(K)⋅Wmin(K). For example, if B‾p,d is the unit ball in Rd with the ℓp norm, then we find thatθ(B‾p,d)=d1−|1/p−1/2|. This constant is sharp, and it is new for all p≠2. Moreover, for some sets K we find a minimal set L for which Wmax(K)⊆Wmin(L). In particular, we obtain that a convex body K satisfies Wmax(K)=Wmin(K) only if K is a simplex.
These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every d-tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most d. We also introduce new explicit constructions of these (and other) dilations.
Quantum no-signalling bicorrelations Brannan, Michael; Harris, Samuel J.; Todorov, Ivan G. ...
Advances in mathematics (New York. 1965),
07/2024, Letnik:
449
Journal Article
Recenzirano
Odprti dostop
We introduce classical and quantum no-signalling bicorrelations and characterise the different types thereof in terms of states on operator system tensor products, exhibiting connections with ...bistochastic operator matrices and with dilations of quantum magic squares. We define concurrent bicorrelations as a quantum input-output generalisation of bisynchronous correlations. We show that concurrent bicorrelations of quantum commuting type correspond to tracial states on the universal C*-algebra of the projective free unitary quantum group, showing that in the quantum input-output setup, quantum permutations of finite sets must be replaced by quantum automorphisms of matrix algebras. We apply our results to study the quantum graph isomorphism game, describing the game C*-algebra in this case, and make precise connections with the algebraic notions of quantum graph isomorphism, existing presently in the literature.
Tensor products of operator systems Kavruk, Ali; Paulsen, Vern I.; Todorov, Ivan G. ...
Journal of functional analysis,
07/2011, Letnik:
261, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems. After giving an axiomatic definition of tensor products in this category, we ...examine in detail several particular examples of tensor products, including a minimal, maximal, maximal commuting, maximal injective and some asymmetric tensor products. We characterize these tensor products in terms of their universal properties and give descriptions of their positive cones. We also characterize the corresponding tensor products of operator spaces induced by a certain canonical inclusion of an operator space into an operator system. We examine notions of nuclearity for our tensor products which, on the category of
C
⁎
-algebras, reduce to the classical notion. We exhibit an operator system
S
which is not completely order isomorphic to a
C
⁎
-algebra yet has the property that for every
C
⁎
-algebra
A, the minimal and maximal tensor product of
S
and
A are equal.
In this article, a quality of service (QoS) dependent variable sampling dynamic event-triggered control method is designed for a cyber–physical system (CPS) with delays and packets dropout to cope ...with non-ideal network environments, maintain the desired control performance and improve the communication efficiency. To achieve the variable period sampling, a sampler is designed based on the QoS of the wireless network by using the delta operator discretization method. Then, a variable period sampling scheme for the delta operator system converted from the CPS is designed. Furthermore, a dynamic event-triggered mechanism (DETM) is proposed using the variable period sampling signal, which can reduce event triggered data calculations and increase event triggered intervals. By utilizing the average dwell time (ADT) approach, sufficient conditions contains the explicit variable sampling period are derived for the derived switched CPS. Finally, the effectiveness of the designed method is verified by numerical examples.
•A novel QoS-based variable period sampling strategy is proposed to realize real-time adjustment to ensure the desired control performance of the CPS.•A DETM is derived using the variable period sampling signal for the CPS to improve the communication efficiency.•A switched delta operator system converted from the CPS is designed, and the sufficient conditions contain the explicit sampling period for system stability are obtained.
We provide new results regarding the localization of the solutions of nonlinear operator systems. We make use of a combination of Krasnosel’skiĭ cone compression–expansion type methodologies and ...Schauder-type ones. In particular we establish a localization of the solution of the system within the product of a conical shell and of a closed convex set. By iterating this procedure we prove the existence of multiple solutions. We illustrate our theoretical results by applying them to the solvability of systems of Hammerstein integral equations. In the case of two specific boundary value problems and with given nonlinearities, we are also able to obtain a numerical solution, consistent with our theoretical results.
Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An ...analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space.
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