OP-0056SolvedQuantum information

Additivity of the minimum output entropy of quantum channels

Everyone expected additivity. Hastings produced a random counterexample, and the classical capacity of a quantum channel became genuinely hard.

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Problem statement

For quantum channels $\Phi, \Psi$ , is the minimum output von Neumann entropy additive,

S_{\min}(\Phi \otimes \Psi) = S_{\min}(\Phi) + S_{\min}(\Psi)?

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Background

Additivity was the central conjecture of quantum Shannon theory in the 2000s, equivalent to additivity of the Holevo capacity and hence to the classical capacity of a quantum channel being given by a single-letter formula. Shor proved the equivalence of several additivity conjectures. Hastings disproved additivity in 2009 using random channels in high dimension.

Why it matters

The failure of additivity means entangled inputs across channel uses can strictly increase the transmissible classical information, and that the classical capacity has no known finite-dimensional characterisation. It is the reason capacity computation remains open.

Known partial results

Disproved: Hastings (2009), with the violation exponentially small in the dimension and established non-constructively.
No explicit counterexample of moderate dimension is known; producing one is an open sub-problem.
Whether the classical capacity is computable at all remains open, and related quantities have been shown uncomputable.

References

[1]M. B. Hastings, Superadditivity of communication capacity using entangled inputs, Nature Physics 5 (2009), 255–257. link

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