The Functional Analysis of Quantum Information Theory

Gupta, Ved Prakash; Mandayam, Prabha; Sunder, V. S.

doi:10.1007/978-3-319-16718-3

Cited by 24 publications

(15 citation statements)

References 49 publications

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“…Condition ( 15) is equivalent to (17) however, we will maintain (17) to achieve in our exposition a format more according to the new situation, that is, we would like to use only the involution ♮ in the equations involved.…”

Section: Admissible Kraus J-positive Maps and Quantum J-channelsmentioning

confidence: 99%

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$J$-states and quantum channels between indefinite metric spaces

Felipe-Sosa¹,

Felipe²

2020

Preprint

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In the present work, we introduce and study states and quantum channels or quantum operators on spaces equipped with an indefinite metric. We will limit the analysis exclusively to the matricial framework. As it will be observed from our considerations, the idea of to build states and quantum channels of indefinite character leads basically to replace, in the customary analysis of these concepts, the usual adjoint of a matrix by its J-adjoint which is defined through a J-metric where the corresponding matrix J is a fundamental symmetry of Mn(C). In particular, for quantum operators, we include in our paper, the general case in the which, they map J1-states into J2-states where J1 = J2 are two arbitrary fundamental symmetries. In the middle of this program, we carry out a founding study in this context, of the completely positive maps between two different spaces each one provided of an indefinite metric.

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Section: Admissible Kraus J-positive Maps and Quantum J-channelsmentioning

confidence: 99%

“…Suppose that Ψ is an admissible Kraus J-positive map then there is ν and a matrix vector ( (17) holds and Ψ has the representation (16). Define Φ(•) = JΨ(J •), so from ( 16) and (17) it is easy to recover ( 14) and ( 15) which imply that Φ is quantum channel. Hence, the theorem 18 shows that Ψ is a completely J-positive map.…”

Section: Definition 20mentioning

confidence: 99%

$J$-states and quantum channels between indefinite metric spaces

Felipe-Sosa¹,

Felipe²

2020

Preprint

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“…Z. Luo and B. Zheng [10] proved that ( ⊕X k ) E has the BCP if and only if each summand X k has the BCP, where E is either a Lorentz sequence space, a separable Orlicz sequence space, or ℓ ∞ . It was also shown in [6] that if (Ω, Σ, µ) is a separable measure space, then the space of Bochner integrable functions L p (µ, X) has the BCP if and only if X has the BCP.…”

Section: Introductionmentioning

confidence: 99%

“…is completely isometrically isomorphic to a completely 1-complemented subspace of B(H) in the operator space sense ([6]). Since L ∞ [0, 1] fails the BCP[10], Theorem 3.1 tells us that B(H) and L ∞ [0, 1] form a counterexample for Question 2 in the introduction.Next, we present some necessary conditions for B(X, Y ) with the BCP.…”

mentioning

confidence: 99%

Ball covering property from commutative function spaces to non-commutative spaces of operators

Minzeng¹,

Li²,

Jimeng³

et al. 2021

Preprint

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A Banach space is said to have the ball-covering property (abbreviated BCP) if its unit sphere can be covered by countably many closed, or equivalently, open balls off the origin. Let K be a locally compact Hausdorff space and X be a Banach space. In this paper, we give a topological characterization of BCP, that is, the continuous function space C 0 (K) has the (uniform) BCP if and only if K has a countable π-basis. Moreover, we give the stability theorem: the vector-valued continuous function space C 0 (K, X) has the (strong or uniform) BCP if and only if K has a countable π-basis and X has the (strong or uniform) BCP. We also explore more examples for BCP on non-commutative spaces of operators B(X, Y ). In particular, these results imply that B(c 0 ), B(ℓ 1 ) and every subspaces containing finite rank operators in B(ℓ p ) for 1 < p < ∞ all have the BCP, and B(L 1 [0, 1]) fails the BCP. Using those characterizations and results, we show that BCP is not hereditary for 1-complemented subspaces (even for completely 1-complemented subspaces in operator space sense) by constructing two different counterexamples.

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“…An operator system is a self-adjoint unital subspace of B(H) for some complex Hilbert space H. Choi and Effros [5] obtained an abstract characterization of an operator system and quite recently this abstraction proved very useful in the development of the theory of tensor products in the category of operator systems in a series of papers [7,9,[13][14][15][16]. A short survey of this development is available in [10,Ch. 4].…”

Section: Introductionmentioning

confidence: 99%

Operator System Nuclearity via -Envelopes

Gupta

Luthra

2016

J. Aust. Math. Soc.

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We prove that an operator system is (min, ess)-nuclear if its $C^{\ast }$-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).

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The Functional Analysis of Quantum Information Theory

Cited by 24 publications

References 49 publications

$J$-states and quantum channels between indefinite metric spaces

$J$-states and quantum channels between indefinite metric spaces

Ball covering property from commutative function spaces to non-commutative spaces of operators

Operator System Nuclearity via -Envelopes

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