Secret key from all entangled states: does every entangled state have positive distillable secret key? (Kruger-Werner Problem 24)

Theorem 1.1: there is an explicit entangled density operator $\rho$ on $\mathbb C^{10}\otimes\mathbb C^{10}$ whose range contains no nonzero product vector, such that for every $n$ and every admitted completed bit-output protocol on $\rho^{\otimes n}$ the output is at trace-norm distance at least $1/5$ from an ideal secret bit; hence $K_D(\rho)=0$. The state is the normalized Choi matrix of a composition of two PPT maps; a coherence inequality for such states bounds secrecy by disagreement probability. So the answer to Problem 24 is no. Not shown: minimality of dimension ten, or a zero-key example for protocol models beyond the paper's admitted class.

Theorem 1.1: there is an explicit entangled density operator $\rho$ on $\mathbb C^{10}\otimes\mathbb C^{10}$ whose range contains no nonzero product vector, such that for every $n$ and every admitted completed bit-output protocol on $\rho^{\otimes n}$ the output is at trace-norm distance at least $1/5$ from an ideal secret bit; hence $K_D(\rho)=0$. The state is the normalized Choi matrix of a composition of two PPT maps; a coherence inequality for such states bounds secrecy by disagreement probability. So the answer to Problem 24 is no. Not shown: minimality of dimension ten, or a zero-key example for protocol models beyond the paper's admitted class.