On State Distinguishing Without Inverse Queries

In the quantum state distinguishing problem, an algorithm is given copies of a state ψ or ϕ separated by trace distance epsilon, and tasked with deciding which is the case. It is a standard result that Θ(epsilon-2) copies are required, while having access to a unitary state preparation oracle and its inverse improves this to Θ(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 unknown close state requires Ω(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.