Secret key from every Bell-nonlocal behavior

Unsolved ID op_299c3becfd760123 Last edited 4 September 2026

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Fields: Quantum Cryptography
Quantum Resource Theory
Topics: Secret-key distillation
Device-independent cryptography
Bell nonlocality

Problem

Does every finite-alphabet Bell-nonlocal behavior have a strictly positive asymptotic secret-key rate against arbitrary individual nonsignalling attacks? Let \(P(a,b\mid x,y)\) be bipartite and nonsignalling, and suppose that it has no local decomposition of the form

\begin{equation} P(a,b\mid x,y) =\int_\Lambda \mu(d\lambda)\, P_A(a\mid x,\lambda)P_B(b\mid y,\lambda). \tag{1} \end{equation}

Thus Eq. (1) fails for every probability measure \(\mu\) and local response functions \(P_A,P_B\). Alice and Bob receive independent copies of \(P\); on each copy, an adversary may hold an arbitrary nonsignalling extension. The honest parties may choose their inputs, process all outputs locally, and communicate publicly. The question asks whether some such protocol always extracts secret key at a nonzero asymptotic rate.

Source

Gisin explicitly asks whether every Bell-nonlocal behavior can yield secret key against nonsignalling individual attacks [Gis09].

Progress

Gisin posed the implication from Bell nonlocality to positive secret key specifically for nonsignalling adversaries restricted to individual attacks [Gis09].
Bell nonlocality is necessary for secrecy in this adversarial model, and explicit nonlocal families admit positive-key protocols. These results do not prove sufficiency for every behavior that violates Eq. (1) [AGM06].
Some nonlocal correlations obtained from noisy two-qubit Werner states have zero key rate for the broad class of standard device-independent protocols that announce measurements before classical postprocessing. This is not an impossibility theorem for all protocols or for the full individual nonsignalling-adversary model [FBJLA21].
Bipartite bound information exists for finite classical sources: there are distributions with zero key rate and positive formation cost. Those sources have no Bell inputs and hence do not settle the implication for Bell-nonlocal behaviors [PGR26].

Comment

The unresolved implication concerns unrestricted protocols at the behavior level under arbitrary individual nonsignalling extensions. Impossibility for a standard protocol class and zero-key classical sources are strictly narrower statements.

References

[Gis09]: N. Gisin, “Bell Inequalities: Many Questions, a Few Answers,” in W. C. Myrvold and J. Christian (eds.), Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, The Western Ontario Series in Philosophy of Science 73, 125–138 (Springer, 2009).DOI arXiv
[AGM06]: A. Acín, N. Gisin, and L. Masanes, “From Bell’s Theorem to Secure Quantum Key Distribution,” Physical Review Letters 97, 120405 (2006).DOI arXiv
[FBJLA21]: M. Farkas, M. Balanzó-Juandó, K. Łukanowski, J. Kołodyński, and A. Acín, “Bell Nonlocality Is Not Sufficient for the Security of Standard Device-Independent Quantum Key Distribution Protocols,” Physical Review Letters 127, 050503 (2021).DOI arXiv
[PGR26]: J. Pauwels, N. Gisin, and R. Renner, “Bipartite Bound Information Exists,” arXiv:2607.25838 (2026).DOI arXiv

Secret key from every Bell-nonlocal behavior

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