# Detecting mixed-unitary quantum channels is NP-hard

Colin Do-Yan Lee and John Watrous

Institute for Quantum Computing and School of Computer Science, University of Waterloo, Canada

### Abstract

A quantum channel is said to be a $\textit{mixed-unitary}$ channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel $\Phi$, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or not $\Phi$ is a mixed-unitary channel. This hardness result holds even under the assumption that $\Phi$ is not within an inverse-polynomial distance (in the dimension of the space upon which $\Phi$ acts) of the boundary of the mixed-unitary channels.

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### Cited by

[1] David W Kribs, Jeremy Levick, Katrina Olfert, Rajesh Pereira, and Mizanur Rahaman, "Nullspaces of entanglement breaking channels and applications", arXiv:2007.15893, Journal of Physics A: Mathematical and Theoretical 54 10, 105303 (2021).

[2] Otfried Gühne, Yuanyuan Mao, and Xiao-Dong Yu, "Geometry of Faithful Entanglement", Physical Review Letters 126 14, 140503 (2021).

[3] Mark Girard, Debbie Leung, Jeremy Levick, Chi-Kwong Li, Vern Paulsen, Yiu Tung Poon, and John Watrous, "On the mixed-unitary rank of quantum channels", arXiv:2003.14405.

[4] Uma Girish, Ran Raz, and Wei Zhan, "Quantum Logspace Algorithm for Powering Matrices with Bounded Norm", arXiv:2006.04880.

[5] Cristhiano Duarte, Barbara Amaral, Marcelo Terra Cunha, and Matthew Leifer, "Investigating Coarse-Grainings and Emergent Quantum Dynamics with Four Mathematical Perspectives", arXiv:2011.10349.

The above citations are from Crossref's cited-by service (last updated successfully 2021-08-01 13:20:54) and SAO/NASA ADS (last updated successfully 2021-08-01 13:20:55). The list may be incomplete as not all publishers provide suitable and complete citation data.