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Algorithmic complexity of quantum capacity

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Abstract

We analyze the notion of quantum capacity from the perspective of algorithmic (descriptive) complexity. To this end, we resort to the concept of semi-computability in order to describe quantum states and quantum channel maps. We introduce algorithmic entropies (like algorithmic quantum coherent information) and derive relevant properties for them. Then we show that quantum capacity based on semi-computable concept equals the entropy rate of algorithmic coherent information, which in turn equals the standard quantum capacity. Thanks to this, we finally prove that the quantum capacity, for a given semi-computable channel, is limit computable.

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Notes

  1. Semi-density matrix is a broader notion with respect to the density matrix, in that it conceives positive matrices with trace not necessarily equal to 1, although finite.

  2. Throughout, the paper the \(\log \) is intended on base 2.

  3. Example: Let us consider the subset A of all limit computable numbers between 0 and 1 which are represented as binary numbers. We can enumerate them by a Turing machine as \(\psi _1, \psi _2, \ldots , \psi _k, \ldots \). We denote by \(\epsilon _k=0. \psi _k (1) \psi _k (2) \ldots \psi _k (n) \ldots \) the kth element of A being \(\psi _k(n)\) the nth digit of \(\epsilon _k\). Now, consider a number \(\lambda \in [0,1]\) whose nth digit is equal to \(\psi _n (n)+1\) modulo 2. It is obvious that \(\lambda \) is different of any \(\epsilon _k\), and so \(\lambda \) is non-limit computable. The set of such \(\lambda \)’s is clearly uncountable.

  4. Following [4], the embedding is obtained by turning the last bit of each canonical basis element to 0.

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Authors and Affiliations

  1. Department of Mathematics, Islamic Azad University, Varamin-Pishva Branch, Pishva, 33817-7489, Iran

    Samad Khabbazi Oskouei

  2. School of Science and Technology, University of Camerino, Via M. delle Carceri 9, 62032, Camerino, Italy

    Stefano Mancini

  3. INFN–Sezione Perugia, Via A. Pascoli, 06123, Perugia, Italy

    Stefano Mancini

Authors
  1. Samad Khabbazi Oskouei
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  2. Stefano Mancini
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Correspondence to Stefano Mancini.

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Oskouei, S.K., Mancini, S. Algorithmic complexity of quantum capacity. Quantum Inf Process 17, 94 (2018). https://doi.org/10.1007/s11128-018-1859-0

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