Variations on the Choi-Jamiolkowski isomorphism

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Frembs, Markus
Cavalcanti, Eric G
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2024
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Abstract

The Choi-Jamiolkowski isomorphism is an essential component in every quantum information theorist's toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C∗-algebras they arise from, which are distinguished by their order of composition. A number of important and novel insights stem from our analysis. We find that Choi's theorem, which asserts that Choi's version of the isomorphism [M.-D. Choi, Lin. Alg. Appl., 10, 285 (1975)] further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, crucially depends on the dynamical (compositional) structure in C∗-algebras. This in turn lies at the heart of the mismatch between the basis-dependence of Choi's version of the isomorphism, and the basis-independent version by Jamiolkowski [A. Jamiolkowski, Rep. Math. Phys., 3, 275 (1972)]. Here, we overcome this subtle but pervasive issue in a number of ways: first, we prove a version of Choi's theorem for Jamiolkowski's isomorphism, second, we define a basis-independent variant of Choi's isomorphism and, third, by making explicit the dynamical distinction between Jordan and C∗-algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.

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Journal of Physics A: Mathematical and Theoretical

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© 2024 The Author(s). Published by IOP Publishing Ltd. As the Version of Record of this article is going to be / has been published on a gold open access basis under a CC BY 4.0 licence, this Accepted Manuscript is available for reuse under a CC BY 4.0 licence immediately. Everyone is permitted to use all or part of the original content in this article, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by/4.0 Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions may be required. All third party content is fully copyright protected and is not published on a gold open access basis under a CC BY licence, unless that is specifically stated in the figure caption in the Version of Record.

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This publication has been entered in Griffith Research Online as an advance online version.

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Quantum information, computation and communication

Mathematical sciences

Physical sciences

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Frembs, M; Cavalcanti, EG, Variations on the Choi-Jamiolkowski isomorphism, Journal of Physics A: Mathematical and Theoretical, 2024

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