The Infimum Norm of Completely Positive Maps

Let A  be a unital C* -algebra, let L: A→B(H)  be a linear map, and let ∅: A→B(H)  be a completely positive linear map. We prove the property in the following:  is completely positive}=inf {||T*T+TT*||1/2:  L= V*TπV  which is a minimal commutant representation with isometry} . Moreover, if L=L* , then  is completely positive  . In the paper we also extend the result  is completely positive}=inf{||T||: L=V*TπV}  [3 , Corollary 3.12].

Physical Implementability of Linear Maps and Its Application in Error Mitigation

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

The Images of China and Britain in the Syrian Media

China and Britain have contrasting images in the official and unofficial Syrian media. By analysing relevant news stories, this study reveals that China’s involvement in the Syrian crisis as covered by the Syrian media is usually limited to governmental affairs, while Britain’s involvement covered by the Syrian media, especially the “revolutionary” outlet, figures in a relatively wider range of diverse nongovernmental happenings including activities of nongovernmental organizations (NGOs) and the media. Most importantly, the study finds that the “revolutionary” outlet Enab Baladi is apt to present Chinese involvement as negative, but presents British involvement as positive, while the government-backed news agency SANA portrays a completely positive image of China and a fundamentally negative image of Britain.

Canonical structure of A and B maps

In their seminal 1961 paper, Sudarshan, Mathews and Rau investigated properties of the dynamical A and B maps acting on n-dimensional quantum systems. The nature of dynamical maps in open quantum system evolutions has attracted great deal of attention in the later years. However, the novel paper on the A and B dynamical maps has not received its due attention. In this tutorial article, we review the properties of A and B forms associated with the dynamics of finite dimensional quantum systems. In particular, we investigate a canonical structure associated with the A form and establish its equivalence with the associated B form. We show that the canonical structure of the A form captures the completely positive (not completely positive) nature of the dynamics in a succinct manner. This feature is illustrated through physical examples of qubit channels.Quanta 2021; 10: 34–41.

An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

We propose an analytic center cutting plane method to determine whether a matrix is completely positive and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman et al. [Berman A, Dur M, Shaked-Monderer N (2015) Open problems in the theory of completely positive and copositive matrices. Electronic J. Linear Algebra 29(1):46–58]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia et al. [Xia W, Vera JC, Zuluaga LF (2020) Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. 32(1):40–56]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like [Formula: see text] for d × d matrices. The method is implemented in Julia and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl . Summary of Contribution: Completely positive matrices play an important role in operations research. They allow many NP-hard problems to be formulated as optimization problems over a proper cone, which enables them to benefit from the duality theory of convex programming. We propose an analytic center cutting plane method to determine whether a matrix is completely positive by solving an optimization problem over the copositive cone. In fact, we can use our method to solve any copositive optimization problem, provided we know the radius of a ball containing an optimal solution. We emphasize numerical performance and stability in developing this method. A software implementation in Julia is provided.

A note on completely positive relaxations of quadratic problems in a multiobjective framework

In a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.

Completely positive maps for imprimitive complex reflection groups

In 1994, M. Bożejko and R. Speicher proved the existence of completely positive quasimultiplicative maps from the group algebra of Coxeter groups to the set of bounded operators. They used some of them to define an inner product associated to creation and annihilation operators on a direct sum of Hilbert space tensor powers called full Fock space. Afterwards, A. Mathas and R. Orellana defined in 2008 a length function on imprimitive complex reflection groups that allowed them to introduce an analogue to the descent algebra of Coxeter groups. In this article, we use the length function defined by A. Mathas and R. Orellana to extend the result of M. Bożejko and R. Speicher to imprimitive complex reflection groups, in other words to prove the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive complex reflection groups to the set of bounded operators. Some of those maps are then used to define a more general inner product associated to creation and annihilation operators on the full Fock space. Recall that in quantum mechanics, the state of a physical system is represented by a vector in a Hilbert space, and the creation and annihilation operators act on a Fock state by respectively adding and removing a particle in the ascribed quantum state.