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 ($d_A$=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.

20 pages, in v.2 the examples of channels are made explicit (without uncertain parameter) and their quantum zero-error capacity is estimated, in v.3 the multi-dimensional generalization is added and open questions are formulated, in v.5 some of these questions are solved

References in corpus (5)

• Quantum Communication With Zero-Capacity Channels
• Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function
• The structure of degradable quantum channels
• Unbounded number of channel uses are required to see quantum capacity
• On superactivation of zero-error capacities and reversibility of a quantum channel

Cited by in corpus (5)

• Memory effects can make the transmission capability of a communication channel uncomputable
• On capacity of quantum channels generated by irreducible projective unitary representations of finite groups
• On non-commutative operator graphs generated by reducible unitary representation of the Heisenberg-Weyl group
• Maximum privacy without coherence, zero-error
• Unextendible entanglement of quantum channels