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

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.

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