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

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 (input dimension 4, Choi rank 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 the superactivation of 1-shot quantum zero-error capacity. Finally, we consider multi-dimensional construction which increases the estimate for quantum zero-error capacity of channels having vanishing n-shot capacity.