Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p Close to 0
Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Rényi entropies of channels are not generally additive for p > 1, we demonstrate here by a careful random selection argument that also at p = 0, and consequently for sufficiently small p, there exist counterexamples.
An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Rényi entropy is non-additive for all p ≲ 0.11. We conjecture however that violations of additivity exist for all p < 1.
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- 2.King, C.: Maximization of capacity and p-norms for some product channels. http://arXiv.org/list/quant-ph/0103086 (2001)
- 5.King, C.: Announced at the 1st joint AMS-PTM meeting, Warsaw 31 July – 3 Aug (2007)Google Scholar
- 7.Winter, A.: The maximum output p-norm of quantum channels is not multiplicative for any p > 2. http://arXiv.org/abs/:0707.0402[quant-ph], (2007)
- 8.Hayden, P.: The maximal p-norm multiplicativity conjecture is false. http://arXiv.org/abs/:0707.3291[quant-ph], (2007)
- 12.Walgate, J., Scott, A.J.: Generic local distinguishability and completely entangled subspaces . http://arXiv.org/abs/:0709.4238[quant-ph], (2007)
- 15.Duan, R.Y., Shi, Y.: Entanglement between Two Uses of a Noisy Multipartite Quantum Channel Enables Perfect Transmission of Classical Information. http://arXiv.org/abs/:0712.3700[quant-ph], (2007)