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On the quantum capacity of the qubit depolarizing channel

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The qubit depolarizing channel, defined in terms of a single depolarizing parameter p, is a simple and widely studied quantum channel. However, an exact determination of its quantum capacity has remained elusive so far, and we can only give lower and upper bounds on its capability to transmit quantum data faithfully. In this seminar talk, I will discuss two upper bounds on the quantum capacity of the qubit depolarizing channel (in the sequel denoted D_p), which constitute the best known upper bounds for low and high values of the depolarizing parameter p, respectively.

The first bound is the approximate degradability bound derived by Sutter et al. (arXiv:1412.0980). This powerful bound uses the fact that D_p is “almost degradable” for low values of p, and hence its quantum capacity is close to its (computable) channel coherent information. In joint work with Debbie Leung and Graeme Smith (soon to be posted on the arXiv), we analyze this bound in greater detail. We prove that the complementary channel of the depolarizing channel (with a tweaked depolarizing parameter) serves as an “approximate degrading map”, and renders the depolarizing channel O(p^2)-approximate degradable. As a result, the corresponding upper bound derived by Sutter et al. on the quantum capacity is tangent to the channel coherent information, which explains the power of the approximate degradability approach. We can lift this result to all generalized Pauli channels, and more generally to “low-noise” channels, i.e., channels that are close to the identity channel with respect to the diamond norm.

For large values of the depolarizing parameter p, the best known upper bound has recently been derived in arXiv:1701.03081, which is joint work with Nilanjana Datta and Graeme Smith. One of our main results in arXiv:1701.03081 is a general upper bound on the one-way distillable entanglement of a mixed bipartite state, based on convex decompositions of this state into degradable and antidegradable states. Since the depolarizing channel belongs to the class of so-called teleportation-simulable channels, quantum data transmission is equivalent to entanglement distillation using its Choi state. Hence, an upper bound on the quantum capacity of D_p is obtained from evaluating our bound for the Choi state of D_p, which is a so-called isotropic state. Exploiting this symmetry, we were able to phrase our general upper bound as a non-convex optimization problem that can be solved numerically, yielding the advertised bound.

This talk is part of the CQIF Seminar series.

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