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arXiv:1608.07730 · Quantum Information Processing

Algorithmic complexity of quantum capacity

Samad Khabbazi Oskouei, Stefano Mancini

Published Mar 2, 2018• 7 citations
arXivHTMLPDF
DOI
10.1007/s11128-018-1859-0

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Based on Arimoto’s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut–Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that, to reach ε accuracy, the iteration complexity of the algorithm is upper bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state { ρ x } x ∈ X is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O ( m 3 log n log ε ε ) , and O ( m 3 log ε log ( 1 − δ ) ε D ( p * | | p N 0 ) ) in the special case, where m is the output dimension, D ( p * | | p N 0 ) is the relative entropy of two distributions, and δ is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.
Quantum channel capacities are fundamental to quantum information theory. Their definition, however, does not limit the computational resources of sender and receiver. In this work, we initiate the study of computational quantum capacities. These quantify how much information can be reliably transmitted when imposing the natural requirement that en- and decoding have to be computationally efficient. We focus on the computational two-way quantum capacity and showcase that it is closely related to the computational distillable entanglement of the Choi state of the channel. This connection allows us to show a stark computational capacity separation. Under standard cryptographic assumptions, there exists a quantum channel of polynomial complexity whose computational two-way quantum capacity vanishes while its unbounded counterpart is nearly maximal. More so, we show that there exists a sharp transition in computational quantum capacity from nearly maximal to zero when the channel complexity leaves the polynomial realm. Our results demonstrate that the natural requirement of computational efficiency can radically alter the limits of quantum communication.