Multiplicative Semidefinite Programming Upper Bounds on Quantum No-Signalling-Assisted Zero-Error Capacity

Abstract

We study zero-error communication and exact simulation of quantum channels assisted by quantum no-signalling correlations. For a channel with Kraus operator space $K$, we introduce a semidefinite programming parameter $\Gamma (K)$ that is exactly multiplicative under tensor products and upper-bounds ${\Theta }_{\mathrm{NS}}(K)$, where ${\log }_2{\Theta }_{\mathrm{NS}}(K)=C_{0,\mathrm{NS}}(K)$ and $C_{0,\mathrm{NS}}(K)$ denotes the asymptotic quantum no-signalling-assisted zero-error capacity. Among known upper bounds on ${\Theta }_{\mathrm{NS}}$ given by semidefinite programs, $\Gamma$ is the tightest. We also prove that the previously introduced simulation lower bound ${\Sigma }^{-}(K)$ is exactly multiplicative and satisfies ${\Theta }_{\mathrm{NS}}(K)\le \Gamma (K)\le {\Sigma }^{-}(K)\le \Gamma (K{)}^2$. Using this multiplicativity, we show that neither $S_{0,\mathrm{NS}}(K)={\log }_2{\Sigma }^{-}(K)$ nor its regularized extension $S_{0,\mathrm{NS}}(K)={\lim }_{n\to \infty }{n}^{-1}{\log }_2{\Sigma }^{-}(K^{\otimes n})$ holds in general. This answers an open question about the asymptotic simulation cost $S_{0,\mathrm{NS}}(K)$. For classical-quantum channels, we obtain sharp bounds. If $r$ is the effective output dimension, then ${\Theta }_{\mathrm{NS}}\le \Gamma \le (1+r^{1/2}){\Theta }_{\mathrm{NS}}/2$ and $\Gamma \le {\Sigma }^{-}\le (1+r^{1/2})\Gamma /2$. If $k$ is the minimum rank of the output support projections, then ${\Sigma }^{-}/{\Theta }_{\mathrm{NS}}\le [(1+r^{1/2})/(1+k^{1/2}){]}^2$, and this bound is sharp for every $1\le k\le r$. Thus, we obtain the sharp bound $S_{0,\mathrm{NS}}-C_{0,\mathrm{NS}}\le 2{\log }_2[(1+r^{1/2})/(1+k^{1/2})]$.

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