Quantum Channel Capacities

Markus Grassl, Zhengfeng Ji, Zhaohui Wei +1 more

The quantum-channel capacity gives the ultimate limit for the rate at which quantum data can be reliably transmitted through a noisy quantum channel. Degradable quantum channels are among the few channels whose quantum capacities are known. Given the quantum capacity of a degradable channel, it remains challenging to find a practical coding scheme which approaches capacity. Here we discuss code designs for the detected-jump channel, a degradable channel with practical relevance describing the physics of spontaneous decay of atoms with detected photon emission. We show that this channel can be used to simulate a binary classical channel with both erasures and bit flips. The capacity of the simulated classical channel gives a lower bound on the quantum capacity of the detected-jump channel. When the jump probability is small, it almost equals the quantum capacity. Hence using a classical capacity-approaching code for the simulated classical channel yields a quantum code which approaches the quantum capacity of the detected-jump channel.

Masanori Ohya, Dėnes Petz, Noboru Watanabe

Archishna Bhattacharyya, Arthur Mehta, Yuming Zhao

An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the entanglement-assisted zero-error capacity under some restrictions is uncomputable; indicative of the fact that quantum channel capacities may generally be undecidable.

Fatih Özaydin, Şahin Kaya Özdemir, Masato Koashi +1 more

Superadditivity of quantum capacity of communication channels is one of the most interesting findings of the field. Yard and Smith, finding a relation between the private capacity and the assisted quantum capacity, showed a striking example of superadditivity, i.e. two channels of zero quantum capacity could achieve a positive quantum capacity when used together [1]. The four dimensional channels they used are a 50% erasure channel (therefore zero quantum capacity, due to no-cloning theorem) and a Horodecki channel (again zero quantum capacity due to incapability of sharing free entanglement). In this work we present the more general cases of superadditivity. Directly calculating the lower bounds of joint quantum capacities without using the relation between private capacity and assisted quantum capacity, we examine scenarios considering erasure channels of arbitrary probabilities and different Horodecki channels, and discuss the roles of degradability and anti-degradability as well as the role of the private capacity in superadditivity. We also derive an upper bound for the joint quantum capacity for the superactivation case.

Peter P. Rohde