Quantum Compression Relative to a Set of Measurements

Andreas Bluhm; Lukas Rauber; Michael M. Wolf

doi:10.1007/s00023-018-0660-z

Quantum Compression Relative to a Set of Measurements

Andreas Bluhm, Lukas Rauber, Michael M. Wolf

4 Citations (Scopus)

Abstract

In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven, and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed-point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and Bézout’s theorem. The latter approach allows lifting the results from the single-copy level to the case of multiple copies and from completely positive to merely positive maps.

Original language	English
Journal	Annales Henri Poincare
Volume	19
Issue number	6
Pages (from-to)	1891-1937
Number of pages	47
ISSN	1424-0637
DOIs	https://doi.org/10.1007/s00023-018-0660-z
Publication status	Published - 1 Jun 2018

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title = "Quantum Compression Relative to a Set of Measurements",

abstract = "In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven, and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed-point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and B{\'e}zout{\textquoteright}s theorem. The latter approach allows lifting the results from the single-copy level to the case of multiple copies and from completely positive to merely positive maps.",

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T1 - Quantum Compression Relative to a Set of Measurements

AU - Bluhm, Andreas

AU - Rauber, Lukas

AU - Wolf, Michael M.

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Y1 - 2018/6/1

N2 - In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven, and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed-point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and Bézout’s theorem. The latter approach allows lifting the results from the single-copy level to the case of multiple copies and from completely positive to merely positive maps.

AB - In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven, and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed-point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and Bézout’s theorem. The latter approach allows lifting the results from the single-copy level to the case of multiple copies and from completely positive to merely positive maps.

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DO - 10.1007/s00023-018-0660-z

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JO - Annales Henri Poincare

JF - Annales Henri Poincare

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Quantum Compression Relative to a Set of Measurements

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