A contextual advantage for conclusive exclusion: repurposing the Pusey-Barrett-Rudolph construction

The task of conclusive exclusion for a set of quantum states is to find a measurement such that for each state in the set, there is an outcome that allows one to conclude with certainty that the state in question was not prepared. Defining classicality of statistics as realizability by a generalized-noncontextual ontological model, we show that there is a quantum-over-classical advantage for how well one can achieve conclusive exclusion. This is achieved in an experimental scenario motivated by the construction appearing in the Pusey-Barrett-Rudolph theorem. We derive noise-robust noncontextuality inequalities bounding the conclusiveness of exclusion, and describe a quantum violation of these. Finally, we show that this bound also constitutes a classical causal compatibility inequality within the bilocality scenario, and that its violation in quantum theory yields a novel possibilistic proof of a quantum-classical gap in that scenario.

💡 Research Summary

The paper investigates the task of conclusive exclusion—given a set of quantum states ({\rho_k}_{k=1}^n), one seeks a measurement with (n) outcomes such that outcome (E_k) certifies that (\rho_k) was not prepared. The performance is quantified by (1-\frac{1}{n}\sum_k P(E_k|\rho_k)); perfect exclusion corresponds to a value of 1.

The authors build on the construction used in the Pusey‑Barrett‑Rudolph (PBR) theorem and define an extended experimental scenario, dubbed the “PBR+ scenario”. Two independent multi‑sources, (S_A) and (S_B), each prepare a qubit in one of two possible bases: ({|0\rangle,|+\rangle}), ({|0\rangle,|-\rangle}), ({|1\rangle,|+\rangle}) or ({|1\rangle,|-\rangle}). By combining the settings of the two sources, sixteen product states are generated, four of which belong to each of four non‑orthogonal product‑state sets (P_{0+},P_{0-},P_{1+},P_{1-}). For each set a specific entangled measurement (M_T) (with (T\in{0+,0-,1+,1-})) is defined; its four projectors ({E_k^T}) are orthogonal to the corresponding states (\rho_k^T). Consequently, quantum theory predicts perfect conclusive exclusion for each of the four tasks, giving an average success rate (CE = 1).

The central conceptual framework is generalized non‑contextuality. In an ontological model each quantum state (|\psi\rangle) is represented by a probability distribution (\mu_\psi(\lambda)) over an ontic space (\Lambda); each measurement effect (E) is represented by a response function (\xi(E|\lambda)). Non‑contextuality requires that operationally equivalent preparations or measurements be represented by the same ontic objects, irrespective of the experimental context.

A key operational identity, (\frac12|0\rangle\langle0|+\frac12|1\rangle\langle1| = \frac12|+\rangle\langle+|+\frac12|-\rangle\langle-|), shows that the Z‑source and X‑source produce the same average quantum state. Under non‑contextuality this forces the ontic distributions for (|0\rangle) and (|+\rangle) to have non‑trivial overlap; the same reasoning applies to the other three pairs ((|0\rangle,|-\rangle), (|1\rangle,|+\rangle), (|1\rangle,|-\rangle)). Because the product‑state sets are Cartesian products of the single‑qubit ontic supports, any overlap at the single‑qubit level yields a region where all four product states overlap. In such a region an ontic state is compatible with every preparation, so no measurement outcome can conclusively rule out any particular (\rho_k^T). Hence, any non‑contextual ontological model must satisfy a bound on the average conclusive‑exclusion success rate:

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