Frozen external records

The external paper asserts an explicit entangled density operator

whose range contains no nonzero product vector. For its admitted completed local-instrument protocols on every ${\rho }^{\otimes n}$, Eve retains a purification and the entire public transcript. The asserted distance from every ideal secret bit is at least $1/5$ in unhalved trace norm, equivalently at least $1/10$ in trace distance. Its stated distillable key is zero.

The protocols allow joint processing of all copies and authenticated two-way public communication without a uniform communication bound. The input is the only shared private resource. This is not a statement about every entangled state or every operational enlargement of that protocol model OpenAI272.

The source specifies PPT maps

without requiring trace preservation, and asserts that

Its state is $\rho =Z/\mathrm{Tr}Z$, not generally $Z/10$. The pair therefore supplies a counterexample to the unrestricted two-map PPT-composition assertion, provided its advertised positivity and range properties hold OpenAI272.

A different construction in the source gives a trace-preserving PPT channel

whose square is asserted not to be entanglement breaking. The source does not claim that dimensions $10$ or $21$ are minimal. Failure at the second iterate does not assert failure of eventual entanglement breaking OpenAI272.

These three records are not premises of the local theorems below. In particular, the notation $\rho ,{\Phi }_1,{\Phi }_2,\Theta$ in a record does not assert that their external construction has been checked here.