Quantum state exclusion with many copies

Quantum state exclusion is the task of identifying at least one state from a known set that was not used in the preparation of a quantum system. A set of quantum states is said to admit state exclusion if there exists a measurement whose outcomes can be put in one-to-one correspondence with the states in the set, such that each outcome rules out its corresponding state with certainty (while possibly also ruling out other states), and each outcome occurs with nonzero probability for at least one state in the set. State exclusion, however, is not always possible in the single-copy setting. In this paper, we investigate whether access to multiple identical copies of the system enables state exclusion. We prove that for any set of three or more pure states, state exclusion becomes possible with a finite number of copies. Moreover, we show that the number of copies required may be arbitrarily large: in particular, for every natural number $N$, we construct sets of states for which state exclusion remains impossible with $N$ or fewer copies.

💡 Research Summary

The paper investigates quantum state exclusion (also called antidistinguishability) in the regime where the experimenter has access to multiple identical copies of the unknown quantum system. In the single‑copy scenario a set of states S={ρ₁,…,ρ_k} is said to admit perfect exclusion if there exists a POVM {Π_j} such that Tr(ρ_j Π_j)=0 for every j (the outcome j certifies that state ρ_j was not prepared) and each outcome occurs with non‑zero probability for at least one state in S. While some sets of non‑orthogonal pure states are antidistinguishable, many are not (e.g. three qubit states {|0⟩,|+⟩,|1⟩}).

The central questions are: (i) does providing a finite number N of identical copies of the system enable exclusion for sets that are not antidistinguishable in the single‑copy case? and (ii) how large must N be in the worst case?

The authors first recall known characterizations: a necessary fidelity bound (∑_{i<j}F(ρ_i,ρ_j) ≤ k(k−2)/2), a sufficient condition for pure states (existence of positive coefficients t_i with ∑ t_i|ψ_i⟩⟨ψ_i|=I), and a complete Gram‑matrix criterion (the Gram matrix must be (k−1)-incoherent). For three pure states a simple pairwise‑overlap condition (inequalities (8)–(9)) is available.

Main Result 1 – Universal activation (Theorem 7).
Let S={|ψ₁⟩,…,|ψ_k⟩} with k≥3 and define c = max_{i≠j}|⟨ψ_i|ψ_j⟩|. Lemma 6 states that if all pairwise overlaps are ≤ t_k = 1/√2·(k−2)^{−1/(k−1)} then S is already antidistinguishable. If c > t_k, consider the N‑copy ensemble S^{⊗N}. Its overlaps shrink as |⟨ψ_i|ψ_j⟩|^N ≤ c^N. Choosing N ≥ ⌈ln t_k / ln c⌉ guarantees c^N ≤ t_k, so Lemma 6 applies and S^{⊗N} becomes antidistinguishable. Hence any set of three or more pure states becomes antidistinguishable after a finite number of copies; the required N depends only on the maximal pairwise overlap.

Main Result 2 – Unbounded copy requirement.
For every integer N≥1 the authors construct explicit families of pure‑state triples that are not antidistinguishable with N or fewer copies but become antidistinguishable with N+1 copies. The construction uses states with carefully chosen mutual angles so that the N‑copy overlaps remain above the Lemma 6 threshold, while the (N+1)‑copy overlaps fall below it. This demonstrates that the number of copies needed can be arbitrarily large; there is no universal finite bound that works for all ensembles.

An illustrative example is the set S={|0⟩,|+⟩,|1⟩}. It fails the antidistinguishability condition for a single copy, but the two‑copy set { |0⟩⊗2, |+⟩⊗2, |1⟩⊗2 } satisfies the three‑state overlap inequalities and admits a concrete 3‑outcome POVM (explicit matrices given in Eq. 12).

The paper also discusses the relationship to other “activation” phenomena: unambiguous discrimination becomes possible with finitely many copies, while some tasks (e.g., local entanglement transformation) never become possible regardless of copy number. The results place state exclusion alongside these examples, showing that for pure‑state ensembles of size ≥3 the activation always occurs, yet the activation depth can be arbitrarily deep.

Implications include: (a) in quantum cryptographic protocols where exclusion of certain states is a security primitive, providing multiple copies can restore security even when single‑copy exclusion fails; (b) the findings enrich the resource theory of antidistinguishability, suggesting copy number as a quantifiable resource; (c) the explicit constructions may guide experimental designs where collective measurements on several copies are feasible.

In summary, the authors prove that antidistinguishability is universally activatable for any set of three or more pure states, but the required number of copies may be unbounded, establishing both a positive activation theorem and a matching negative bound.