Ciara Morgan

“Pretty Strong” Converse for the Quantum Capacity of Degradable Channels

We exhibit a possible road toward 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 above-mentioned asymptotic jump of the fidelity at the quantum capacity then decreases from 1 to 0.

Operational meaning of quantum measures of recovery

Several information measures have recently been defined that capture the notion of recoverability. In particular, the fidelity of recovery quantifies how well one can recover a system

Polar Codes in Network Quantum Information Theory

A

Invalidity of a strong capacity for a quantum channel with memory

of a tripartite quantum state, defined on systems

An improved rate region for the classical-quantum broadcast channel

ABC

Efficient achievability for quantum protocols using decoupling theorems

, by acting on system

Decoupling with random diagonal unitaries

C

Implementing Unitary 2-Designs Using Random Diagonal-unitary Matrices

alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the type I error is constrained to be no larger than a constant.

Towards a strong converse for the quantum capacity (of degradable channels)

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 this paper, 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 so-called quantum simultaneous decoder. Thus, this paper paves the way for developing network classical-quantum information theory further without requiring a quantum simultaneous decoder.

Unitary 2-designs from random X- and Z-diagonal unitaries

The strong capacity of a particular channel can be interpreted as a sharp limit on the amount of information which can be transmitted reliably over that channel. To evaluate the strong capacity of a particular channel one must prove both the direct part of the channel coding theorem and the strong converse for the channel. Here we consider the strong converse theorem for the periodic quantum channel and show some rather surprising results. We first show that the strong converse does not hold in general for this channel and therefore the channel does not have a strong capacity. Instead, we find that there is a scale of capacities corresponding to error probabilities between integer multiples of the inverse of the periodicity of the channel. A similar scale also exists for the random channel.