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arXiv:1210.2738 · Journal of Physics A Mathematical and Theoretical

Quantum channels arising from abstract harmonic analysis

Jason Crann, Matthias Neufang

Published Jan 10, 2013• 11 citations
arXivHTMLPDF

Abstract

We present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups G . Beginning with a representation of the measure algebra M ( G ), we unify and elaborate on recent counter-examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions, as well as present an application to the noiseless subsystems method of quantum error correction. Using a representation of the completely bounded Fourier multiplier algebra M cb A ( G ), we provide a new class of counter-examples to the recently solved asymptotic quantum Birkhoff conjecture, along with a systematic method of producing the examples using a geometric representation of Schur maps. Further properties of our channels including duality, quantum capacity, and entanglement preservation are discussed along with potential applications to additivity conjectures.

DOI
10.1088/1751-8113/46/4/045308

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We present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups G . Beginning with a representation of the measure algebra M ( G ), we unify and elaborate on recent counter-examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions, as well as present an application to the noiseless subsystems method of quantum error correction. Using a representation of the completely bounded Fourier multiplier algebra M cb A ( G ), we provide a new class of counter-examples to the recently solved asymptotic quantum Birkhoff conjecture, along with a systematic method of producing the examples using a geometric representation of Schur maps. Further properties of our channels including duality, quantum capacity, and entanglement preservation are discussed along with potential applications to additivity conjectures.
In this paper, we compute the exact values of the minimum output entropy and the completely bounded minimal entropy of very large classes of quantum channels acting on matrix algebras Mn. Our new and simple approach relies on the theory of locally compact quantum groups and our results use a new and precise description of bounded Fourier multipliers from L1 (G) into Lp(G) for 1 < p< \infty where G is a co-amenable locally compact quantum group and on the automatic completely boundedness of these multipliers that this description entails. Indeed, our approach even allows to use convolution operators on quantum hypergroups. This enable us to connect equally the topic of computation of entropies and capacities to subfactor planar algebras. We also give a upper bound of the classical capacity of each considered quantum channel which is already sharp in the commutative case. Quite surprisingly, we observe by direct computations that some Fourier multipliers identifies to direct sums of classical examples of quantum channels (as dephasing channel or depolarizing channels). Indeed, we show that the study of unital qubit channels can be seen as a part of the theory of Fourier multipliers on the von Neumann algebra of the quaternion group Q8. Unexpectedly, we also connect ergodic actions of (quantum) groups to this topic of computation, allowing some transference to other channels. We also connect the Quantum Harmonic analysis of Werner. Finally, we investigate entangling breaking and PPT Fourier multipliers and we characterize conditional expectations which are entangling breaking.
In this paper, we compute the exact values of the minimum output entropy and the completely bounded minimal entropy of very large classes of quantum channels acting on matrix algebras Mn\mathrm{M}_nMn​. Our new and simple approach relies on the theory of locally compact quantum groups and our results use a new and precise description of bounded Fourier multipliers from L1(G)\mathrm{L}^1(\mathbb{G})L1(G) into Lp(G)\mathrm{L}^p(\mathbb{G})Lp(G) for 1<p≤∞1 < p \leq \infty1<p≤∞ where G\mathbb{G}G is a co-amenable locally compact quantum group and on the automatic completely boundedness of these multipliers that this description entails. Indeed, our approach even allows to use convolution operators on quantum hypergroups. This enable us to connect equally the topic of computation of entropies and capacities to subfactor planar algebras. We also give a upper bound of the classical capacity of each considered quantum channel which is already sharp in the commutative case. Quite surprisingly, we observe by direct computations that some Fourier multipliers identifies to direct sums of classical examples of quantum channels (as dephasing channel or depolarizing channels). Indeed, we show that the study of unital qubit channels can be seen as a part of the theory of Fourier multipliers on the von Neumann algebra of the quaternion group Q8\mathbb{Q}_8Q8​. Unexpectedly, we also connect ergodic actions of (quantum) groups to this topic of computation, allowing some transference to other channels. We also connect the Quantum Harmonic analysis of Werner. Finally, we investigate entangling breaking and PPT\mathrm{PPT}PPT Fourier multipliers and we characterize conditional expectations which are entangling breaking.
This article analyzes the algebraic structure of the set of all quantum channels and its subset consisting of quantum channels that have Holevo representation. The regularity of these semigroups under composition of mappings is analyzed. It is also known that these sets are compact convex sets and, therefore, rich in geometry. An attempt is made to identify generalized invertible channels and also the idempotent channels. When channels are of the Holevo type, these two problems are fully studied in this article. The motivation behind this study is its applicability to the reversibility of channel transformations and recent developments in resource-destroying channels, which are idempotents. This is related to the coding-encoding problem in quantum information theory. Several examples are provided, with the main examples coming from pre-conditioner maps which assign preconditioners to matrices in numerical linear algebra. Thus, the known pre-conditioner maps are viewed as quantum channels in finite dimensions. In addition, the infinite-dimensional analogue of preconditioners is introduced and certain limit theorems are discussed; this is with an aim to analyze asymptotic methods in quantum channels analogous to problems in asymptotic linear algebra.
Abstract In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem (theorem 2.5), which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive, etc) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.
Abstract. We present and investigate a new class of quantum channels, what we call ‘universal collective rotation channels’, that includes the class of collective rotation channels as a special case. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this C ∗-algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We prove that there is an abundance of noiseless subsystems for every channel in this class and that the Young tableaux combinatorial machine may be used to explicitly compute these subsystems. 1.
We present and investigate a new class of quantum channels, what we call “universal collective rotation channels,” that includes the class of collective rotation channels as a special case. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this C*-algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We prove that there is an abundance of noiseless subsystems for every channel in this class and that the Young tableaux combinatorial machine may be used to explicitly compute these subsystems.