Approximating quantum channels by completely positive maps with small Kraus rank

Cécilia Lancien1 and Andreas Winter2,3

1Institut Fourier & CNRS, Université Grenoble Alpes, 38610 Gières, France
2Departament de Física: Grup d’Informació Quàntica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
3Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain

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We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space $\mathrm{A}$ to states on some output Hilbert space $\mathrm{B}$ can be compressed into one with order $d\log d$ Kraus operators, where $d=\max(|\mathrm{A}|,|\mathrm{B}|)$, hence much less than $|\mathrm{A}||\mathrm{B}|$. In the case where the channel's outputs are all very mixed, this can be improved to order $d$. We discuss the optimality of this result as well as some consequences.

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Cited by

[1] Bin Yan and Nikolai A. Sinitsyn, "Randomized channel-state duality", arXiv:2210.03723, (2022).

[2] Cécilia Lancien and Christian Majenz, "Weak approximate unitary designs and applications to quantum encryption", Quantum 4, 313 (2020).

[3] Hachem Kadri, Stéphane Ayache, Riikka Huusari, Alain Rakotomamonjy, and Liva Ralaivola, "Partial Trace Regression and Low-Rank Kraus Decomposition", arXiv:2007.00935, (2020).

[4] Anne Broadbent, Carlos E. González-Guillén, and Christine Schuknecht, "Quantum private broadcasting", Physical Review A 105 2, 022606 (2022).

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