Nathaniel Johnston

We study the distinguishability of quantum states under local operations with classical communication (LOCC), separable, and positive-partial-transpose (PPT) measurements, focusing on \emph{locally diagonal orthogonally invariant} (LDOI) states---those invariant under local diagonal orthogonal twirling. This class includes many important families such as Werner states, isotropic states, X-states, and Dicke states. We show that optimal PPT and separable measurements for distinguishing LDOI states can always be taken to be LDOI, and the LOCC supremum can be approached by LDOI LOCC POVMs, enabling a dimensional reduction from $n^4$ to $O(n^2)$ in the associated optimization problems. We establish efficiently computable bounds on the distinguishability of orthonormal LDOI bases and prove that for a broad class of such bases---including all two-qubit cases---the LOCC supremum equals the PPT and separable optima. More generally, we show the gap between PPT and LOCC distinguishability is at most $(n-2)/(2n^2)$ for local dimension $n$.

The problem of quantum state classification asks how accurately one can identify an unknown quantum state that is promised to be drawn from a known set of pure states. In this work, we introduce the notion of $k$-\emph{learnability}, which captures the ability to identify the correct state using at most $k$ guesses, with zero error. We show that deciding whether a given family of states is $k$-learnable can be solved via semidefinite programming. When there are $n$ states, we present polynomial-time (in $n$) algorithms for determining $k$-learnability for two cases: when $k$ is a fixed constant or the dimension of the states is a fixed constant. When both $k$ and the dimension of the states are part of the input, we prove that there exist succinct certificates placing the problem in NP, and we establish NP-hardness by a reduction from the classical $k$-clique problem. Together, our findings delineate the boundary between efficiently solvable and intractable instances of quantum state classification in the perfect (zero-error) regime.