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N-box and quantum pigeonhole paradoxes occur exactly when one weak value is anomalous, so both need coherence of the pre- and post-selected states.

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2026-07-31 19:14 UTC pith:KFTJ5NTT

load-bearing objection Clean iff link between non-logical N-box/pigeonhole PPS paradoxes and one anomalous weak value, so both need coherence; algebra checks out. the 2 major comments →

arxiv 2607.24282 v1 pith:KFTJ5NTT submitted 2026-07-27 quant-ph

Coherence as a resource for N-box and quantum pigeonhole paradoxes

Som Kanjilal , Rafael Wagner , Ernesto F. Galv\~ao This is my paper
classification quant-ph
keywords quantum coherencepre- and post-selectionN-box paradoxquantum pigeonhole principleweak valuesABL ruleBargmann invariantsLeggett–Garg
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Pre- and post-selection paradoxes look like classical counting or location rules breaking under quantum conditioning. This paper treats the non-logical versions of the N-box and quantum pigeonhole scenarios, allowing mixed pre- and post-selected states, and asks what nonclassical resource they actually need. Under the scenario’s operational non-disturbance rule, each paradox is equivalent to a single weak-value anomaly: the excluded box for N-box, and a summed same-box operator for the pigeonhole. Because an anomalous weak value already forces coherence in the intermediate measurement basis, no coherence-free fragment of quantum theory can produce either paradox. The size of the anomaly is exactly the size of the classicality-bound violation, so the departure from an incoherent explanation is quantitative, not just qualitative.

Core claim

For the N-box scenario the sum of ABL probabilities equals one minus the real part of the weak value of the never-measured box; a paradox occurs if and only if that real part is negative. For the pigeonhole scenario the corresponding sum equals the real part of the weak value of the summed same-box operator; a paradox occurs if and only if that real part falls below the operator’s smallest eigenvalue. In both cases the anomaly requires coherence of both the pre- and post-selected states in the intermediate basis, so a coherence-free subtheory cannot yield the paradox.

What carries the argument

Under operational non-disturbance, the ABL joint probabilities reduce via an anticommutator identity to the real part of a weak value (or a third-order Bargmann invariant). That single identity converts the classicality inequality into a statement about weak-value anomaly and thereby into a coherence witness.

Load-bearing premise

The post-selection success rate must be the same whether or not an intermediate box or pair check was performed; without that non-disturbance rule the paradox need not force coherence and classical toy models can fake it.

What would settle it

Prepare pre- and post-selected states that are diagonal in the intermediate basis (or compute their weak value of the excluded box / summed same-box operator) and check whether the ABL sum can still violate the classical bound; any such violation would refute the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any experimental claim of an N-box or pigeonhole paradox must exhibit an anomalous weak value of the paradox-specific operator and therefore coherence in both boundary states.
  • The numerical size of the classicality-bound violation is fixed by how negative (or how sub-minimal) that weak value is, giving a direct figure of merit for mixed-state realizations.
  • Logical PPS paradoxes sit at the extreme of the same identities (weak value −1 or 0), so the non-logical and logical cases share one coherence mechanism.
  • Incoherent subtheories of quantum theory are ruled out as explanations of these PPS statistics once non-disturbance is imposed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same weak-value-to-ABL reduction may classify other non-logical PPS effects (Cheshire cat, Hardy) by a single operator anomaly rather than a full contextuality proof.
  • If generalized noncontextuality inequalities can be written directly from these ABL bounds, the non-logical paradoxes would become proofs of contextuality beyond the logical case the paper leaves open.
  • Robustness curves for mixed states already supplied in the examples give concrete noise thresholds experimental groups can target without needing pure pre- and post-selection.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The manuscript analyses two non-logical pre- and post-selection (PPS) paradoxes — the N-box paradox and the quantum pigeonhole paradox — with arbitrary mixed pre- and post-selected states. Building on Maroney's operational analysis of the three-box paradox, the authors define each scenario (Defs. II.1, V.1) with an operational non-disturbance condition (Eqs. 14, 51) and a no-cheating/three-particle constraint. The central technical move is to use non-disturbance plus Hermiticity to rewrite the joint probability p(b, pass) as Re Tr[Π_b ρ_F ρ_I] (Eq. 25), so that summing over the intermediate measurements converts the ABL statistics into the real part of a single weak value: Σ_b p(b|...) = 1 − Re⟨Π_{N+1}⟩_w (Eq. 33) for N-box, and Σ p(same_ij|...) = Re⟨Θ⟩_w (Eq. 63) for pigeonhole, where Θ = Σ Π^same_ij has spectrum {1,3}. Invoking the result of Ref. [38] that weak-value anomaly requires coherence of both states in the eigenbasis, they conclude (Theorems IV.1, VI.1) that no coherence-free subtheory of quantum theory can reproduce either paradox, and (Corollaries IV.2, VI.2) that each paradox occurs iff the relevant single weak value is anomalous. Mixed-state examples (IV.4, VI.4) exhibit non-logical paradox families with explicit parameter regions.

Significance. If the results hold — and my checks indicate they do — this is a useful, cleanly executed extension of the Pusey–Leifer logical-case analysis to non-logical PPS paradoxes with mixed states, where the previous non-orthogonality criterion no longer applies. The strengths are concrete: the key identities (Eqs. 25, 33, 63) are parameter-free and derived in a few transparent lines from the ABL rule and the imposed operational constraints; the pigeonhole result is a genuinely new quantitative statement (paradox iff Re⟨Θ⟩_w < 1, with Θ's spectrum computed exactly); and the worked families in Examples IV.4 and VI.4 give explicit, checkable robustness regions, including the nontrivial threshold p ≈ 0.866. I independently verified the anticommutator step (Eq. 20), the incoherence no-go (incoherent ρ_I makes Tr[ρ_F Π_b ρ_I] = p_b Tr[ρ_F Π_b] ≥ 0, so the projector weak values are a genuine distribution), the spectrum of Θ (eigenvalue = number of coinciding pairs, hence {1,3}), and the formulas of Examples IV.4 and VI.4; all reproduce. The reinterpretation via Schmid's strict classicality (Sec. IV A) is a nice conceptual fit. The work is incremental relative to [2, 26, 38] but the increment is实

major comments (2)
  1. [Abstract; Corollaries IV.2 and VI.2; Eqs. (14), (51)] The iff statements of Corollaries IV.2 and VI.2, and hence the headline claim that the paradoxes 'require coherence', hold only within the constrained state space defined by the operational non-disturbance conditions (14) and (51). The paper is aware of this and builds the conditions into Defs. II.1/V.1, which is the right formal choice, but the abstract and conclusions state the conclusion unconditionally, and a casual reader could take away 'any N-box/pigeonhole statistics violating (12)/(49) certify coherence' without the caveat that violation of (14) voids the ABL-to-weak-value identity (Eq. 25) and admits classical disturbance models (as Maroney showed). I ask for two things: (i) an explicit sentence in the abstract and in Sec. VII stating that the equivalence paradox ⟺ anomalous weak value is conditional on operational non-disturbance; (ii) some characterization of the class of mix
  2. [Sec. V A, Eq. (50)] Definition V.1 includes Eq. (50), the requirement that sequential pair-checks never find all three pairs different. Unlike the N-box no-cheating condition (13), the sequential joint probability in (50) is never given an explicit formula (the analogue of Eqs. (1)–(4) for Lüders-sequential measurement), and the claim that it vanishes for any pre-selection is asserted without proof. The claim is correct and elementary — the Π^diff_ij are all diagonal in C and their product projects onto the empty set of strings with no coinciding pair — but one line showing Π^diff_23 Π^diff_13 Π^diff_12 = 0 (or the support argument after the first two outcomes) would close the gap and make the operational meaning of (50) as clean as the rest of the manuscript.
minor comments (7)
  1. [Examples IV.4, VI.4; Fig. 3 caption] Numbering inconsistency: the logical-case examples are labeled Example IV.3 and Example VI.3, but the text of Examples IV.4 and VI.4 refers to 'the same as in Theorem IV.3' and 'Theorem VI.3' (also Fig. 3 caption, 'the logical pigeonhole paradox of Theorem VI.3', and 'Theorem VI.3' in Example VI.4). These should be Example IV.3/VI.3.
  2. [Sec. I outline; Fig. 2 caption] The outline (end of Sec. I) and Fig. 2 caption refer to 'Theorem IV.2' and 'Theorem VI.2', but these results are labeled Corollary IV.2 and Corollary VI.2 in the body. Please make the referencing uniform.
  3. [Corollary IV.2, Eq. (34)] Eq. (34) writes 'Re((Π_{N+1})_w) < 0' with the weak-value angle brackets missing; compare Eq. (33). Also 'ie.,' should be 'i.e.,' in the proof of Corollary IV.2.
  4. [Example VI.4] Example VI.4 states 0 ≤ p_I, p_F < 1 as the domain but then evaluates the limit 'at p_I = p_F = 1'; either extend the domain or phrase as a limit.
  5. [Sec. II A; Example IV.3] In Eq. (9) and Example IV.3 the overlap is written |⟨ψ_I|ψ_F⟩|² in one place and |⟨ψ_F|ψ_I⟩|² in another; harmless, but uniform convention would help. Similarly the weak value in Eq. (36) is computed via ⟨ψ_I|ψ_F⟩⟨ψ_F|3⟩⟨3|ψ_I⟩ — worth one phrase noting the numerator convention matches Eq. (15), Tr[ρ_F Π_3 ρ_I].
  6. [Figs. 2 and 3] Figures 2 and 3: the axis labels ('Σ:', 'Re:') are cryptic and the 'Optimal' annotation is undefined in the captions; please spell out that 'optimal' means saturation of the algebraic bound (Σ = 2 resp. Σ = 0).
  7. [Sec. III] Sec. III: the statement that for pure states non-orthogonality plus non-parallelism is 'equivalent to set coherence of {ρ_I, ρ_F}' cites [34, 52, 68]; adding the original Designolle et al. [33] here would be appropriate since set coherence is their notion.

Circularity Check

1 steps flagged

No significant circularity: the iff identities are derived algebraically from ABL + non-disturbance; coherence follows from a verifiable lemma (self-cited but independently checkable).

specific steps
  1. self citation load bearing [Theorem IV.1 proof; also Theorem VI.1; Sec. III]
    "Using the results from Ref. [38], if either ρ_F or ρ_I is incoherent with respect to the basis B={Π_b}^{N+1}_{b=1}, then the weak values ⟨Π_b⟩_w form a valid probability distribution... Consequently, 0≤⟨Π_{N+1}⟩_w≤1"

    The step from ‘no anomalous weak value’ to ‘coherence is necessary for the paradox’ rests on a lemma whose authors overlap with the present paper. This is load-bearing for the resource claim, but not circular in the strong sense: the lemma is a short, independently verifiable fact about mixed-state weak values (and is partially re-derived in the incoherent expansions Eqs. 17–19), not an unverified uniqueness theorem that forbids alternatives by fiat.

full rationale

The central equalities (Eq. 33 for N-box; Eq. 63 for pigeonhole) are obtained by expanding the operational non-disturbance conditions, using Hermiticity to extract real parts of Bargmann invariants, summing projectors, and rewriting via the ABL rule. These steps are ordinary quantum-probability algebra, not true by renaming or by fitting. Non-disturbance is imposed in Defs. II.1 and V.1 as a scenario constraint (explicitly motivated by Maroney’s classical-toy-model analysis), not derived from the paradox conclusion, so the setup is not self-definitional. The only overlapping-author load is Ref. [38] (Wagner–Galvão), used for “incoherence ⇒ projector weak values form a probability distribution.” That lemma is parameter-free, stated with assumptions that do not include the PPS paradoxes, and is re-derivable in a few lines (as the paper’s own incoherent-case calculation already sketches); it is ordinary prior-work citation, not a self-sealing uniqueness theorem. No fitted inputs are called predictions. Score 1 only to mark the minor self-citation; the derivation chain is otherwise self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claims sit inside finite-dimensional quantum theory with the ABL conditional rule and two operational constraints taken from Maroney-style analyses. No free parameters are fitted. The only external lemma used as a black box for the coherence direction is the known fact that anomalous weak values require coherence. Scenario definitions (what counts as a paradox) are stipulative but standard for the subfield.

axioms (6)
  • domain assumption Finite-dimensional quantum theory: states in D(H), Lüders update, Born rule joint probabilities (Eqs. 1–5).
    All ABL statistics and weak values are computed inside QT; no alternative GPT dynamics are derived.
  • domain assumption ABL rule as the correct conditional probability for intermediate outcomes given successful post-selection (Eq. 6 / 46).
    Standard in PPS literature; used to define the left-hand sides of the paradox inequalities.
  • domain assumption Operational non-disturbance: p(b,pass)+p(¬b,pass)=p(pass) for each intermediate measurement (Eq. 14; Eq. 51).
    Imposed in Defs. II.1 and V.1 following Maroney; converts ABL sums into Re(weak values) and blocks disturbance-based classical models.
  • domain assumption No-cheating / single-system constraints (Eq. 13; Eq. 50) and p(pass)≠0.
    Part of scenario definitions so that classical counting bounds (Σ≤1 or Σ≥1) are the intended classicality criteria.
  • standard math Anomalous weak values require coherence of pre/post states in the eigenbasis (cited from Wagner & Galvão, Ref. [38]).
    Used to pass from weak-value anomaly to the coherence no-go in Theorems IV.1 and VI.1.
  • ad hoc to paper Paradox ⇔ violation of classical counting inequalities (12) or (49).
    Defines the operational meaning of ‘paradox’ for non-logical mixed-state cases; logical 0-1 cases are recovered as extremes.

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read the original abstract

Pre- and post-selection (PPS) paradoxes are striking demonstrations of quantum nonclassicality. Logical PPS paradoxes, where inferences made with the Aharonov-Bergmann-Lebowitz (ABL) rule are exactly 0 or 1, are linked to contextuality. Non-logical paradoxes lack this strong signature. In this work, we analyse more general, non-logical PPS scenarios involving mixed pre- and post-selected states. We show that two such scenarios, the $N$-box and quantum pigeonhole paradoxes, require coherence of both pre- and post-selected states in the basis of the intermediate measurement. This is done by showing each paradox holds if and only if there is weak-value anomaly for a single, paradox-specific operator, together with the fact that weak-value anomaly requires coherence. This clarifies the role of different notions of nonclassicality in these scenarios, highlighting the required quantitative departures from (strictly) classical explanations provided by incoherent sub-theories of quantum theory.

Figures

Figures reproduced from arXiv: 2607.24282 by Ernesto F. Galv\~ao, Rafael Wagner, Som Kanjilal.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

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Reference graph

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