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Unbounded number of channel uses may be required to detect quantum capacity

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Cubitt, Toby 
Elkouss, David 
Matthews, William 
Ozols, Maris 
Perez-Garcia, David 


Transmitting data reliably over noisy communication channels is one of the most important applications of information theory, and is well understood for channels modelled by classical physics. However, when quantum effects are involved, we do not know how to compute channel capacities. This is because the formula for the quantum capacity involves maximizing the coherent information over an unbounded number of channel uses. In fact, entanglement across channel uses can even increase the coherent information from zero to non-zero. Here we study the number of channel uses necessary to detect positive coherent information. In all previous known examples, two channel uses already sufficed. It might be that only a finite number of channel uses is always sufficient. We show that this is not the case: for any number of uses, there are channels for which the coherent information is zero, but which nonetheless have capacity.


This is the author accepted manuscript. The final version is available from NPG at


Physical sciences, Theoretical physics

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Nature Communications

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DE and DP acknowledge nancial support from the European CHIST-ERA project CQC (funded partially by MINECO grant PRI-PIMCHI-2011-1071) and from Comunidad de Madrid (grant QUITEMAD+-CM, ref. S2013/ICE-2801). TSC is supported by the Royal Society. MO acknowledges nancial support from European Union under project QALGO (Grant Agreement No. 600700). SS acknowledges the support of Sidney Sussex College. This work was made possible through the support of grant #48322 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily re ect the views of the John Templeton Foundation.