Entanglement with zero distillable secret key in local dimension ten
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.
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