Completely Bounded Maps and Operator Algebras

Paulsen, Vern I.

doi:10.1017/cbo9780511546631

Cited by 1,223 publications

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“…||·|| ♦ is the dual norm to ||·|| cb , the norm of complete boundedness. We refer to [23] and [24] for additional information on these norms. It is an important fact that we need not maximize over all non-negative integers in (40), the norm stabilizes at the value n = d = dim H, i.e.…”

Section: A Existence Of Good Netsmentioning

confidence: 99%

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Quantum capacity of a class of compound channels

2008

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In this paper we address the issue of universal or robust communication over quantum channels. Specifically, we consider memoryless communication scenario with channel uncertainty which is an analog of compound channel in classical information theory. We determine the quantum capacity of finite compound channels and arbitrary compound channels with informed decoder. Our approach in the finite case is based on the observation that perfect channel knowledge at the decoder does not increase the capacity of finite quantum compound channels. As a consequence we obtain coding theorem for finite quantum averaged channels, the simplest class of channels with long-term memory. The extension of these results to quantum compound channels with uninformed encoder and decoder, and infinitely many constituents remains an open problem.

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Section: A Existence Of Good Netsmentioning

confidence: 99%

“…Two different proofs of this fact can be found in [23] and [24]. It is well known, and not difficult to show, that for N ∈ C(H, K) the relation…”

Section: A Existence Of Good Netsmentioning

confidence: 99%

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Quantum capacity of a class of compound channels

2008

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“…Thus, the map σ → V M σV * M is tracepreserving, completely positive ( [31]). Its composition with the partial trace, given by M(σ) := Tr C,H,I,A (V M σV * M ), is a quantum operation.…”

Section: Mathematical Description Of Qtmsmentioning

confidence: 99%

Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

Benatti

Krüger

Müller

et al. 2006

Commun. Math. Phys.

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In classical information theory, entropy rate and algorithmic complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qubit descriptions of qubit strings that, run by a universal quantum Turing machine, reproduce them as outputs.

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“…[22], [54] or [67]]), we have strived to make our presentation accessible to a reader unfamiliar with those techniques. This explains in particular why we present the scalar valued (i.e.…”

Section: Q2mentioning

confidence: 99%

Are Unitarizable Groups Amenable?

Pisier

2005

Progress in Mathematics

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We give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group G are similar to unitary ones, is the group amenable? In §5, we give a new proof of Haagerup's theorem that, on non-commutative free groups, there are Herz-Schur multipliers that are not coefficients of uniformly bounded representations. We actually prove a refinement of this result involving a generalization of the class of Herz-Schur multipliers, namely the class M d (G) which is formed of all the functions f : G → C such that there are bounded functions ξ i : G → B(H i , H i−1 ) (H i Hilbert) withWe prove that if G is a non-commutative free group, for any d ≥ 1, we haveand hence there are elements of M d (G) which are not coefficients of uniformly bounded representations. In the case d = 2, Haagerup's theorem implies that M 2 (G) = M 4 (G).

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Completely Bounded Maps and Operator Algebras

Cited by 1,223 publications

References 0 publications

Quantum capacity of a class of compound channels

Quantum capacity of a class of compound channels

Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

Are Unitarizable Groups Amenable?

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