Coherent information of a quantum channel or its complement is generically positive

Satvik Singh and Nilanjana Datta

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

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The task of determining whether a given quantum channel has positive capacity to transmit quantum information is a fundamental open problem in quantum information theory. In general, the coherent information needs to be computed for an unbounded number of copies of a channel in order to detect a positive value of its quantum capacity. However, in this paper, we show that the coherent information of a $\textit{single copy}$ of a $\textit{randomly selected channel}$ is positive almost surely if the channel's output space is larger than its environment. Hence, in this case, a single copy of the channel typically suffices to determine positivity of its quantum capacity. Put differently, channels with zero coherent information have measure zero in the subset of channels for which the output space is larger than the environment. On the other hand, if the environment is larger than the channel's output space, identical results hold for the channel's complement.

If a quantum channel's output space is larger than its environment, then the information leakage by the channel to its environment is expected to be smaller in comparison to the amount of information that is sent to the output. Hence, such a channel should be able to transmit quantum information at a net positive rate. Surprisingly, this intuition fails to hold in general, and examples of quantum channels with large output spaces are known to exist that nevertheless have no capacity to transmit quantum information. However, we show that even though this intuition is not always correct, it is 'almost always' correct. In other words, whenever the output space of a channel is larger than its environment, one can be 'almost sure' that the channel has the ability to transmit quantum information at a strictly positive rate.

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Cited by

[1] Satvik Singh and Nilanjana Datta, "Fully undistillable quantum states are separable", arXiv:2207.05193.

[2] Satvik Singh and Nilanjana Datta, "Detecting positive quantum capacities of quantum channels", npj Quantum Information 8, 50 (2022).

[3] D. -S. Wang, "On quantum channel capacities: an additive refinement", arXiv:2205.07205.

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