We exhibit a possible road towards a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a "pretty strong" converse: When the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 1 √ 2 , asymptotically. A similar result can be shown for the private (classical) capacity.Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse: The abovementioned asymptotic jump of the fidelity at the quantum capacity is then from 1 down to 0.Index Terms-quantum information, private classical information, channel coding, strong converse, smooth entropies, errorrate trade-off I. INTRODUCTIONCommunication via noisy channels is one of the information processing tasks by which, following the fundamental work of Shannon [42], we have learned to quantify information and noise. One of the most important models considered from these early days of information theory is that of a discrete memoryless channel, for which Shannon gave his famous single-letter formula for the capacity (i.e., the maximum communication rate achievable by asymptotically error-free block coding).The analogous model in quantum Shannon theory is the memoryless quantum channel N ⊗n (for asymptotically large integer n), given by a completely positive and trace preserving (cptp) map N : L(A ′ ) → L(B), with Hilbert spaces A ′ and B that we assume to be finite dimensional throughout this paper.The quantum capacity Q(N ) of N is informally defined as the maximum rate at which quantum information can be

Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols, and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and that in the Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after repetitions achieves a Θ(d − )approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times.

Abstract-Polar coding is a method for communication over noisy classical channels which is provably capacity-achieving and has an efficient encoding and decoding. Recently, this method has been generalized to the realm of quantum information processing, for tasks such as classical communication, private classical communication, and quantum communication. In the present work, we apply the polar coding method to network classical-quantum information theory, by making use of recent advances for related classical tasks. In particular, we consider problems such as the compound multiple access channel and the quantum interference channel. The main result of our work is that it is possible to achieve the best known inner bounds on the achievable rate regions for these tasks, without requiring a socalled quantum simultaneous decoder. Thus, this paper paves the way for developing network classical-quantum information theory further without requiring a quantum simultaneous decoder.