Perfect Discrimination of Non-Orthogonal Quantum States via Adaptive Post-Measurement Queries

Abstract

A set of pairwise non-orthogonal quantum states cannot be perfectly discriminated, and this remains true even when one bit of classical partial information is available prior to the measurement. Contrary to the usual intuition that earlier information is at least as valuable as later information, we show that the same bit can be more useful when it arrives after the measurement. We present a framework for state discrimination in which the sender provides classical information in response to a request from the receiver, which we refer to as a query. In some cases, pairwise non-orthogonal states can be perfectly discriminated when the query is allowed to depend on the measurement outcome. We give a general method for finding the optimal strategy in this setting, which turns out to be the standard minimum-error discrimination problem for an auxiliary ensemble.

The paper studies state discrimination assisted by one truthful binary answer from the sender, comparing queries made before measurement, fixed queries answered afterward, and post-measurement queries chosen adaptively from the measurement outcome. The distinction is that adaptivity changes which classical information is requested, rather than merely delaying the same information. For an ensemble $\{q_x,\rho_x\}_{x=1}^n$, the optimal adaptive strategy is reduced to minimum-error discrimination of an auxiliary ensemble containing every pairwise mixture $\rho_{(x,y)}=(q_x\rho_x+q_y\rho_y)/(q_x+q_y)$, with prior $(q_x+q_y)/(n-1)$. The optimal success probabilities satisfy $P_{\mathrm{adaptive}}=(n-1)P_{\mathrm{aux}}$. The proof groups measurement outcomes by their two most likely labels; a subsequent binary query separates those labels. Uniform trine qubit states illustrate perfect adaptive discrimination: an antidistinguishing measurement excludes one state, and the answer identifies the remaining candidate. Any pre-measurement query instead achieves only $(4+\sqrt{3})/6\approx0.955$. Lifted trines retain a strict adaptive advantage without perfect discrimination. Further constructions give perfect adaptive discrimination of $d+1$ pairwise nonorthogonal states in dimension $d$, and of $\lfloor3d/2\rfloor$ states using orthogonal blocks. The latter exceeds the stated outcome-independent bound $n\le d+1$ for $d>3$. An operator-norm bound and explicit optimal measurements establish these constructions; a separate argument proves $n\le d+1$ for $d\le3$. The maximal achievable number in general dimension remains open, as does a characterization of ensembles benefiting from adaptivity. Tetrahedral and lifted-BB84 qubit examples favor pre-measurement queries; whether any ensemble of at least four qubit states exhibits an adaptive advantage is unresolved.