Superactivation of bipartite classical secret-key rates
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- Topics
Problem
Can two classical sources, each with zero secret-key rate between two honest parties, yield a positive secret-key rate when used jointly? Let \(p_{ABE}\) and \(q_{A'B'E'}\) be probability distributions on finite classical alphabets. For such a source \(p\), let \(K_{\mathrm{cl}}(p)\) denote its asymptotic secret-key rate, in bits per independent sample, under arbitrary local random processing and unlimited authenticated interactive public discussion. Eve holds exactly the specified classical variables and the entire public transcript; she is not supplied a quantum purification. Keys must become uniform, shared correctly, and independent of Eve’s information in total variation distance. The combined source is the independent product \(p\otimes q:=p_{ABE}\,q_{A'B'E'}\), with Alice holding \(AA'\), Bob holding \(BB'\), and Eve holding \(EE'\). The question is whether there is a pair satisfying
Source
Prettico and Acín explicitly ask whether bipartite classical information resources can be activated, and give evidence that the classical secret-key rate may be non-additive [PA13]. Pauwels, Gisin, and Renner restate this activation question in the outlook of their bipartite bound-information preprint [PGR26]. The strict condition in Eq. (1) isolates the superactivation form of that question.
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Acín, Cirac, and Masanes prove that bound information exists and can be activated with three honest parties: they give tripartite distributions from which no pair of honest parties can distill secret key, even with help from the third, but whose equal mixture yields a common secret key [ACM04]. Such multipartite non-distillability arguments group honest parties across bipartitions, which is impossible with only two honest parties [PGR26], so they do not settle Eq. (1).
Prettico and Acín construct two classical distributions, modeled on a quantum activation example, from which the advantage-distillation protocols they analyze extract no key individually, while a combined protocol yields positive key in part of the parameter range [PA13]. The individual zero-key rates remain conjectural, and the authors note that one or both distributions might be key-distillable. Failure of the tested protocols is not a converse against all public-discussion protocols, so this construction does not establish Eq. (1).
Pauwels, Gisin, and Renner’s July 2026 preprint, revised on 8 September 2026, gives an explicit bipartite bound-information source \(p_\star\) with \(A,B\in\{0,1\}\) and \(E\in\{0,1,\perp\}\). Its probability matrices \(M_e:=[p_\star(a,b,e)]_{a,b=0}^{1}\) are
\begin{equation} M_0=\frac1{36}\begin{pmatrix}5&2\\2&0\end{pmatrix},\qquad M_1=\frac1{36}\begin{pmatrix}0&2\\2&5\end{pmatrix},\qquad M_\perp=\frac1{36}\begin{pmatrix}5&4\\4&5\end{pmatrix}. \tag{2} \end{equation}For the source in Eq. (2), Theorem 1 of the preprint proves
\begin{equation} K_{\mathrm{cl}}(p_\star)=0 <I(A:B\downarrow E)_{p_\star} \leq I_{\mathrm{form}}(p_\star). \tag{3} \end{equation}Here \(I(A:B\downarrow E):=\inf_{E\to\bar E}I(A:B\mid\bar E)\) is the intrinsic information, minimized over classical stochastic maps, and \(I_{\mathrm{form}}\) is the formation cost: the minimal rate of preshared secret bits needed to generate the source by public discussion whose transcript can be simulated from Eve’s variable [PGR26].
In its outlook, the revised preprint notes that bipartite activation had been asked but could not be settled without a proven example, and proposes its sources as explicit resources on which to investigate activation [PGR26]. It supplies a candidate factor, not a pair with
\begin{equation} K_{\mathrm{cl}}(p_\star)=K_{\mathrm{cl}}(q)=0<K_{\mathrm{cl}}(p_\star\otimes q). \tag{4} \end{equation}The strict condition in Eq. (4) is stronger than increasing the rate of a source that already has positive key.
Grouping independent samples cannot activate a zero rate:
\begin{equation} K_{\mathrm{cl}}(p^{\otimes k})=k\,K_{\mathrm{cl}}(p) \qquad(k\geq1), \tag{5} \end{equation}since protocols on blocks of \(k\) samples and protocols on individual samples convert into each other with rates rescaled by \(k\). By Eq. (5), a pair satisfying Eq. (1) must combine genuinely different sources, not finite blocks of one zero-key source.
Comment
Bipartite bound information has an affirmative preprint result, while strict bipartite superactivation of the classical secret-key rate remains unresolved. The unresolved target is a pair satisfying Eq. (1) or a theorem that the full set of sources with \(K_{\mathrm{cl}}=0\) is closed under independent products. A classical bound-information source is not a quantum bound-key example, because the adversary and the allowed resources differ. Literature checked through 15 September 2026.