Entanglement without distillable secret key. Constructs an entangled state on $\mathbb C^{10}\otimes\mathbb C^{10}$ with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on $M_{21}(\mathbb C)$ whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.
released 2026-09-27 | 5 theorems · 18 lemmas · 29 proofs · 23,171 words |
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We construct an entangled state on $\mathbb C^{10}\otimes\mathbb C^{10}$ with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on $M_{21}(\mathbb C)$ has a square that is not entanglement breaking.