On channels with positive quantum zero-error capacity having vanishing n-shot capacity

TL;DR

This paper constructs low-dimensional quantum channels demonstrating that perfect quantum information transmission may require many channel uses, revealing complex behaviors of zero-error capacities and superactivation phenomena.

Contribution

It explicitly constructs low-dimensional channels with positive zero-error capacity but zero n-shot capacity, and analyzes their properties and implications.

Findings

Channels with positive zero-error capacity can have zero n-shot capacity for all n

Superactivation of 1-shot quantum zero-error capacity is demonstrated in low dimensions

Channels close to classical-quantum channels can exhibit these properties

Abstract

We show that unbounded number of channel uses may be necessary for perfect transmission of quantum information. For any n we explicitly construct low-dimensional quantum channels ($d_A$=4, $d_E$=2 or 4) whose quantum zero-error capacity is positive but the corresponding n-shot capacity is zero. We give estimates for quantum zero-error capacity of such channels (as a function of n) and show that these channels can be chosen in any small vicinity (in the cb-norm) of a classical-quantum channel. Mathematically, this property means appearance of an ideal (noiseless) subchannel only in sufficiently large tensor power of a channel. Our approach (using special continuous deformation of a maximal commutative *-subalgebra of $M_4$) also gives low-dimensional examples of superactivation of 1-shot quantum zero-error capacity. Finally, we consider multi-dimensional construction which gives channels with greater values of quantum zero-error capacity and vanishing n-shot capacity.