Entangled states with zero distillable secret key

Distillable secret key $K_D$ is the asymptotic rate of shared classical bits that are uniform and independent of an adversary. The question asks whether this quantity is faithful on entangled bipartite states: does every entangled state contain some distillable secrecy?

Distillable entanglement implies distillable key, but the converse fails: PPT bound-entangled states can have positive key. Consequently the absence of distillable entanglement does not establish zero key, and the NPT bound-entanglement conjecture E1 is a different problem.

Horodecki–Sikorski–Das–Wilde's published 2026 resource theory explicitly allows for the unresolved possibility of entangled zero-key states. Calling this version “mathematical bound key” avoids silently identifying it with other conjectures involving a positive key-formation cost, whose definition requires additional care.

A proof that $K_D$ is positive for every entangled state, or one explicit entangled state with a rigorous zero-key converse against arbitrary LOCC protocols, resolves this sharp question. No finite-copy failure or numerical lack of a protocol establishes the required asymptotic zero.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

published: The 2026 resource theory of quantum secret key explicitly treats the existence of entangled zero-key states as unresolved. Evidence