Interpolation by completely positive maps

Li, Chi-Kwong; Poon, Yiu‐Tung

doi:10.1080/03081087.2011.585987

Cited by 29 publications

(21 citation statements)

References 12 publications

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“…Note that Theorem 3.16 provides one more (different) argument for Corollary 3.2 in [16], and different from the argument given in Remark 3.6. (3) as well.…”

Section: Resultsmentioning

confidence: 96%

“…Other conditions like trace preserving may be required as well, with direct applications to quantum operations. In this general formulation, these problems have been considered by C.-K. Li and Y.-T. Poon in [16], where solutions have been obtained in case when the given states (more generally, Hermitian matrices) commute. More general criteria for existence of solutions have been considered by Z. Huang, C.-K. Li, E. Poon, and N.-S. Sze in [10], while T. Heinosaari, M.A.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization

Ambrozie

Gheondea

2014

Linear and Multilinear Algebra

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We present certain existence criteria and parameterizations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to linear functionals on (Formula presented.) -subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson’s Hahn-Banach Type Theorem. A necessary and sufficient condition for the existence of solutions and a parametrization of the set of all solutions of the interpolation problem in terms of a closed and convex set of an affine space are obtained. Other linear affine restrictions, like trace preserving, can be included as well, hence covering applications to quantum channels that yield certain quantum states at prescribed quantum states. We also perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional. © 2014, © 2014 Taylor & Francis

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“…Note that Theorem 3.16 provides one more (different) argument for Corollary 3.2 in [16], and different from the argument given in Remark 3.6. (3) as well.…”

Section: Resultsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization

Ambrozie

Gheondea

2014

Linear and Multilinear Algebra

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“…Note that condition (c) is of independent interest, for it relates the fidelity between the initial states with the fidelity √ B 1 √ B 2 1 between the final states B 1 , B 2 , and can be generalized to give a necessary (but not sufficient) condition for the existence of a TPCP map sending k initial states to k final states (see equation (6) later, also [2]).…”

Section: Qubit Statesmentioning

confidence: 99%

Physical transformations between quantum states

Huang

Poon

et al. 2012

Journal of Mathematical Physics

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Given two sets of quantum states {A 1 , . . . , A k } and {B 1 , . . . , B k }, represented as sets as density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T , represented as a trace-preserving completely positive map, such that T (A i ) = B i for i = 1, . . . , k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered.

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“…While this papers original motivation arose from considerations of free optimization as it appears in linear systems theory, determining the matrix convex hull of a free set has an analog in quantum information theory, see [LP11]. In free optimization, the relevant maps are completely positive and unital (ucp).…”

Section: Tracial Setsmentioning

confidence: 99%

The tracial Hahn–Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra

Helton¹,

Klep²,

McCullough³

2017

J. Eur. Math. Soc.

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This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory.Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A 0 +A 1 x 1 +· · ·+A g x g 0 and their matrix convex solution sets {X : L(X) 0}, called free spectrahedra. The Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros-Winkler Theorem.The projection of a free spectrahedron in g + h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.

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Interpolation by completely positive maps

Cited by 29 publications

References 12 publications

An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization

An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization

Physical transformations between quantum states

The tracial Hahn–Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra

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