On the Mixed-Unitary Rank of Quantum Channels

• Published in: Communications in mathematical physics 2022-09, Vol.394 (2), p.919-951
• Main Authors: Girard, Mark, Leung, Debbie, Levick, Jeremy, Li, Chi-Kwong, Paulsen, Vern, Poon, Yiu Tung, Watrous, John
• Format: Article
• Language: English
• Subjects: Channels; Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum phenomena; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-022-04412-y

In the theory of quantum information, the mixed-unitary quantum channels , for any positive integer dimension n , are those linear maps that can be expressed as a convex combination of conjugations by n × n complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank  N and the Choi rank  r of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality N ≤ r 2 - r + 1 is satisfied for every mixed-unitary channel (as is the equality N = 2 when r = 2 ), and we exhibit the first known examples of mixed-unitary channels for which N > r . Specifically, we prove that there exist mixed-unitary channels having Choi rank d + 1 and mixed-unitary rank 2 d for infinitely many positive integers d , including every prime power d . We also examine the mixed-unitary ranks of the mixed-unitary Werner–Holevo channels.