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 $\Theta_{\rm NS}(K)$, where $\log_2\Theta_{\rm NS}(K)=C_{0,\rm NS}(K)$ and $C_{0,\rm NS}(K)$ denotes the asymptotic quantum no-signalling-assisted zero-error capacity. Among known upper bounds on $\Theta_{\rm NS}$ 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 $\Theta_{\rm NS}(K)\leq\Gamma(K)\leq\Sigma^{-}(K)\leq\Gamma(K)^2$. Using this multiplicativity, we show that neither $S_{0,\rm NS}(K)=\log_2\Sigma^{-}(K)$ nor its regularized extension $S_{0,\rm NS}(K)=\lim\nolimits_{n\to\infty}n^{-1}\log_2\Sigma^{-}(K^{\otimes n})$ holds in general. This answers an open question about the asymptotic simulation cost $S_{0,\rm NS}(K)$. For classical-quantum channels, we obtain sharp bounds. If $r$ is the effective output dimension, then $\Theta_{\rm NS}\leq\Gamma\leq(1+r^{1/2})\Theta_{\rm NS}/2$ and $\Gamma\leq\Sigma^{-}\leq(1+r^{1/2})\Gamma/2$. If $k$ is the minimum rank of the output support projections, then $\Sigma^{-}/\Theta_{\rm NS}\leq[(1+r^{1/2})/(1+k^{1/2})]^2$, and this bound is sharp for every $1\leq k\leq r$. Thus, we obtain the sharp bound $S_{0,\rm NS}-C_{0,\rm NS}\leq2\log_2[(1+r^{1/2})/(1+k^{1/2})]$.

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2610.21071 [quant-ph]

(or arXiv:2610.21071v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.21071 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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