Conditions for compatibility of quantum state assignments

• The paper studies conditions under which N parties’ possibly different quantum state assignments are compatible, treating the assignments as beliefs about one system.Source: §I · p. 1
• Two procedures assess compatibility: one compares the parties’ prior beliefs, while the other compares the measurement probabilities predicted by their states.Source: §I · p. 2
• The prior-belief procedure yields the Brun, Finkelstein, and Mermin (BFM) compatibility criterion, whereas the measurement procedure yields a hierarchy of criteria including post-Peierls compatibility.Source: §I · p. 2
• The BFM criterion is not equivalent to any measurement-based criterion, so measurements generally cannot confirm BFM compatibility or reveal BFM incompatibility.Source: §I · p. 2
• For commuting density operators with basis-diagonal measurements, the criteria specialize to classical probabilities, where BFM and post-Peierls compatibility become equivalent unlike in the quantum case.Source: §I · p. 2
• The paper derives BFM compatibility in Sec. III, develops measurement-based criteria and POVM generalizations in Sec. IV, studies three-state post-Peierls examples in Sec. V, and concludes with discussion in Sec. VI.Source: §I · p. 2

• Dutch-book consistency is an internal condition on one party’s fair bets and enforces the usual probability axioms without fixing particular probability assignments.Source: §II · p. 3
• Strong consistency strengthens ordinary consistency by making zero probability equivalent to a firm belief that an outcome is impossible.Source: §II · p. 3
• N parties’ beliefs are compatible when at least one alternative is possible for everyone; they are contradictory when every alternative is ruled out by at least one party.Source: §II · p. 3
• Under strong consistency, classical compatibility can be expressed through probability assignments: some alternative must receive nonzero probability from every party.Source: §II · p. 4
• Ordinary consistency is insufficient for determining compatibility from probabilities because parties may assign zero probability to outcomes they nevertheless consider possible.Source: §II · p. 4
• Compatible beliefs need not assign identical probabilities, but parties who combine compatible beliefs can acquire concordant assignments with the same support.Source: §II · p. 4

• For an ODOP measurement, party α assigns outcome k probability p_k^(α)=tr(ρ_αΠ_k)=⟨k|ρ_α|k⟩, with commuting states covering the classical case.Source: §III · p. 4
• Strong consistency identifies each state’s null subspace with outcomes its party regards as impossible and its support with the remaining possible outcomes.Source: §III · pp. 4–5
• BFM compatibility holds exactly when the intersection of all parties’ supports contains a nonzero vector, equivalently when some density operator contradicts none of their firm beliefs.Source: §III · p. 5
• In the classical case, the BFM condition reduces to at least one common eigenvector having nonzero eigenvalue for every party.Source: §III · p. 5
• Strong consistency is essential: with merely consistent assignments, zero probabilities need not represent impossibility, so the assignments do not determine whether a noncontradictory posterior exists.Source: §III · p. 5
• BFM-compatible parties who share their firm beliefs can reach concordant state assignments with the same support, though their density operators generally need not be identical.Source: §III · p. 5
• Unlike the original BFM derivation, this derivation uses only the parties’ beliefs and does not appeal to a real state of affairs, fitting a Bayesian interpretation of quantum mechanics.Source: §III · p. 5

• The hierarchy distinguishes compatible versus concordant measurement probabilities and universal versus existential measurement requirements.Source: A. Compatibility conditions · p. 6

A. Compatibility conditions

• ES requires every measurement outcome to receive either nonzero probability from all parties or zero probability from all parties, equivalently requiring identical supports.Source: A. Compatibility conditions · p. 6
• ES implies BFM compatibility but is strong and often violated in practice, while naturally arising when parties combine BFM-compatible beliefs.Source: A. Compatibility conditions · p. 6
• PP requires every measurement to contain at least one outcome assigned nonzero probability by all parties; its two-party form is Peierls’s criterion.Source: A. Compatibility conditions · p. 6
• The remaining conditions require one measurement: W demands concordance for every outcome, whereas W′ demands only one jointly nonzero outcome.Source: A. Compatibility conditions · pp. 6–7
• The logical relations among the conditions hold in both classical and quantum cases, but their substantive equivalences differ between those settings.Source: A. Compatibility conditions · p. 7

B. Relations among quantum compatibility conditions

• W is satisfied by any set of quantum state assignments, making it the weakest condition, while BFM is stronger than PP and PP is stronger than W′.This ordering differs sharply from the classical hierarchy because quantum mechanics permits a much broader range of measurements.Source: B. Relations among quantum… · p. 7
• BFM implies PP because a common support vector guarantees a nonzero probability for at least one outcome of every projective measurement.The converse fails: two nonorthogonal pure states satisfy PP but have only the zero vector in their support intersection, so they violate BFM.Source: B. Relations among quantum… · p. 7
• W′ does not imply PP: two orthogonal pure states violate PP yet satisfy W′ for a measurement basis containing a suitable superposition.Thus existence of one jointly possible measurement outcome is weaker than compatibility across every measurement.Source: B. Relations among quantum… · p. 7
• W follows from W′ because one can construct an orthonormal basis whose every vector has nonzero overlap with every state support.The construction perturbs basis vectors away from the set of vectors orthogonal to at least one support, repeating rotations until all basis vectors have the required overlaps.Source: B. Relations among quantum… · p. 8

C. Generalized measurements

• Adding POVMs cannot change W or W′ compatibility, because the single-outcome uninformative measurement makes all states compatible under both criteria.Additional measurements can otherwise alter measurement-based compatibility, including making some ODOP-compatible states incompatible under PP.Source: C. Generalized measurements · p. 9
• PP-POVM compatibility is equivalent to PP-ODOP compatibility after embedding the states in an arbitrarily larger Hilbert space, via rank-one refinement and Neumark extension.For two parties, the condition is dimension-independent, so PP-ODOP and PP-POVM coincide; two nonorthogonal pure states satisfy both but violate BFM.Source: C. Generalized measurements · p. 9
• For qubit pure states, PP-POVM compatibility holds exactly when their Bloch-vector convex hull excludes zero, equivalently when the generated state convex set excludes the maximally mixed state.This establishes that PP-ODOP and PP-POVM are not equivalent for two-state systems.Source: C. Generalized measurements · pp. 9–10
• Quantum generalized measurements produce the hierarchy ES ⇒ BFM ⇒ PP-POVM ⇒ PP-ODOP ⇒ W′ ⇔ W.Only PP changes under the distinction between projective measurements and POVMs; ES, W′, and W retain their stated compatibility behavior.Source: C. Generalized measurements · pp. 9–10
• In the classical coarse-grained setting, W moves from the strongest to the weakest compatibility condition, paralleling the quantum hierarchy’s reversal of W’s role.The uninformative measurement makes all classical probabilities compatible under W and W′.Source: C. Generalized measurements · p. 10

V. THREE-PARTY POST-PEIERLS COMPATIBILITY IN THREE DIMENSIONS

• PP is the only compatibility criterion lacking a simple general mathematical condition for arbitrary sets of density operators.For three or more Hilbert-space dimensions, the general PP condition for N states is highly nontrivial.Source: §V · p. 10
• For two parties, PP compatibility holds exactly when the density operators are nonorthogonal, equivalently when tr(ρ_Aρ_B) ≠ 0 or ρ_Aρ_B ≠ 0.The same condition applies to both ODOP and POVM measurements and follows from the equivalence between orthogonality and reliable one-shot discrimination.Source: §V · p. 10
• The three-party, three-dimensional analysis restricts attention to rank-one or rank-two states because full-rank density operators do not affect PP compatibility.The cases are organized by the number of pure states, with mixed-state cases treated separately from the three-pure-state case.Source: §V · p. 10

A. Mixed and pure states

• Three rank-two density operators in three dimensions are PP-POVM incompatible exactly when their support-orthogonal vectors are mutually orthogonal, so PP-POVM and PP-ODOP coincide.The required POVM elements must be positive multiples of the support-orthogonal projectors and can sum to the identity only as an ODOP.Source: A. Mixed and pure states · p. 11
• Pairwise PP compatibility does not imply joint PP compatibility: three states can be pairwise compatible yet incompatible when considered together.This establishes that multipartite compatibility imposes constraints beyond checking every pair.Source: A. Mixed and pure states · p. 11
• For one pure state and two rank-two states with different supports, PP incompatibility reduces to the incompatibility of three pure states in the two-dimensional subspace R.The reduction uses the intersection state |χ⟩ and the support states |φ2⟩ and |φ3⟩ lying in R.Source: A. Mixed and pure states · pp. 11–12
• In that mixed-state case, PP-POVM incompatibility holds exactly when |ψ1⟩ is orthogonal to |χ⟩ and the convex hull of |ψ1⟩⟨ψ1|, |φ2⟩⟨φ2|, and |φ3⟩⟨φ3| contains R/2.For PP-ODOP compatibility, the corresponding condition is that |ψ1⟩, |φ2⟩, and |φ3⟩ include an orthogonal pair.Source: A. Mixed and pure states · p. 12
• For two pure states and one rank-two state, the paper determines only the PP-ODOP condition: incompatibility occurs if and only if ⟨ψ1|S3|ψ2⟩ = 0.The PP-POVM conditions were not determined for this case.Source: A. Mixed and pure states · p. 12

B. Three pure states

• The paper has numerical evidence that PP-POVM and PP-ODOP coincide for this three-state case, but it proves conditions only for ODOPs.The PP-POVM equivalence remains unproved.Source: B. Three pure states · p. 12
• The derivation assumes three distinct, pairwise PP-compatible states and seeks an ODOP whose every outcome definitely contradicts one of them.Pairwise compatibility means no pair is orthogonal, while joint PP-ODOP incompatibility is a stronger condition.Source: B. Three pure states · p. 12
• The states violate PP-ODOP exactly when angles θk with 0 < θk < π/2 realize the three squared overlaps as a = (sin θ1 cos θ2)^2, b = (sin θ2 cos θ3)^2, and c = (sin θ3 cos θ1)^2.These parameterizations arise from an orthonormal-basis representation of the three state vectors.Source: B. Three pure states · pp. 12–13
• Introducing xk = sin^2 θk converts the overlap relations into x1(1 − x2) = a, x2(1 − x3) = b, and x3(1 − x1) = c.Existence of solutions with 0 < xk < 1 yields the algebraic inequalities used in the final criterion.Source: B. Three pure states · p. 13
• For fixed c with 0 < c < 1, the ellipse in the a-b plane is centered at a = b = 1/2, and its principal axes form angles of 45° with the a and b axes.The ellipse intersects the a axis at a = 1 − c and the b axis at b = 1 − c.Source: B. Three pure states · p. 14
• Three pure states are PP-ODOP incompatible if and only if their squared pairwise inner products a, b, and c satisfy a + b + c < 1 and (a + b + c − 1)^2 > 4abc.The criterion is symmetric in all three squared inner products.Source: B. Three pure states · p. 14

VI. DISCUSSION

• The BFM criterion is one of five criteria: it concerns compatible prior beliefs, whereas the other four use measurements and none is equivalent to BFM.BFM-compatible parties can agree on a joint state without abandoning firm prior beliefs.Source: §VI · p. 14
• The PP criterion asks whether a measurement can guarantee disagreement for every outcome, with the answer depending on whether POVMs or only ODOPs are allowed.This creates distinct measurement-based compatibility conditions.Source: §VI · pp. 14–15
• The W criterion, equivalent to W′, imposes no necessary compatibility constraint on a finite set of density operators because some measurement can always enable agreement.Consequently, BFM compatibility is neither sufficient nor necessary in the W sense.Source: §VI · p. 15
• Quantum mechanically BFM is stronger than PP, whereas classically BFM and PP are equivalent; conversely, W is weakest quantum mechanically but strongest classically for fine-grained ODOP measurements.The classical specialization uses commuting density operators and measurements diagonal in their common eigenbasis.Source: §VI · p. 15
• The BFM derivation requires strong Dutch-book consistency: without it, parties may agree on a joint state even when their prior assignments are not BFM consistent.Thus strong consistency is an assumption, not merely a technical convenience.Source: §VI · p. 15

APPENDIX A: BAYESIAN PROBABILITIES AND THE DUTCH-BOOK ARGUMENT

• A Bayesian probability p is operationally defined by willingness to accept a fair bet at odds (1 − p)/p to 1, for any payoff.The bettor pays stake px and receives payoff x if the event occurs.Source: Appendix A · p. 16
• Dutch-book consistency requires nonnegative probabilities, probability one for believed certainties, additivity for exclusive events, and Bayes’s rule for conjunctions.Violating any rule permits bets producing a sure loss according to the bettor’s own possibilities.Source: Appendix A · p. 16
• Consistency requires pE∨F = pE + pF, while Bayes’s rule requires pE∧F = pE|FpF.Source: Appendix A · pp. 17–18
• Strong consistency forbids assignments where all possible outcomes have nonpositive gains and at least one has a loss, forcing probability one to represent an event regarded as certain.Ordinary consistency alone cannot establish this converse certainty condition.Source: Appendix A · p. 18
• Standard probability rules make expected gains zero, preventing both sure losses and the strong-consistency pattern of nonpositive gains with some losses.The Dutch-book argument concerns internal coherence of belief assignments rather than repeated-bet averages.Source: Appendix A · pp. 18–19
• In quantum mechanics, strong consistency is crucial because a pure-state assignment is interpreted as certainty about an ODOP outcome containing that state.Ordinary consistency only requires a pure state when the bettor is already certain about an ODOP outcome.Source: Appendix A · p. 19