On State Distinguishing Without Inverse Queries

Abstract

In the quantum state distinguishing problem, an algorithm is given copies of a state $|\psi ⟩$ or $|\varphi ⟩$ separated by trace distance $\epsilon$, and tasked with deciding which is the case. It is a standard result that $\Theta ({\epsilon }^{-2})$ copies are required, while having access to a unitary state preparation oracle and its inverse improves this to $\Theta ({\epsilon }^{-1})$ queries. We study the intermediate case with forward-only access, capturing scenarios where the oracle is a physical process that cannot be reversed. Tang and Wright recently showed that distinguishing a known state from an \emph{unknown} close state requires $\Omega ({\epsilon }^{-2})$ forward queries in high dimension; we strengthen this result by proving the same bound even when both states are known, as in the state distinguishing setting. Moreover, our lower bound uses a direct hybrid argument, avoiding advanced compressed oracle techniques used in the prior work. In contrast, we give a dimension-independent algorithm solving a continuous-time variant of state distinguishing with only $O({\epsilon }^{-1})$ total forward evolution time. The algorithm uses quantum Zeno dynamics to confine evolution to a known two-dimensional subspace. Thus, a quadratic advantage survives without inverse access for a natural continuous-time analogue of state distinguishing.