Limited Holism and Real-Vector-Space Quantum Theory

1 Introduction

• An n-component measurement may involve an interaction among the chosen components, unlike concurrent separate measurements that cannot interact.The components are selected within a conceptual partition of the system.Source: §1 · p. 2
• A theory is n-locally tomographic when measurements on one through n components determine the composite state.Source: §1 · p. 2
• For more than n components, n-local tomography constrains the theory; the paper focuses on n = 2, called bilocal tomography.Complex-Hilbert-space quantum theory is locally tomographic, whereas real-Hilbert-space quantum theory is bilocally tomographic.Source: §1 · p. 2
• The tomography condition must hold regardless of how a system is partitioned, although the arguments assume a basic subsystem partition.Source: §1 · p. 2

2 Basic concepts

• The state is represented by a minimal probability list sufficient to calculate the probability of a general outcome linearly.Using unnormalized states makes this representation convenient for parameter counting.Source: §2 · p. 3
• K_A is the minimum number of probabilities needed to determine system A's state, assuming arbitrary mixtures of preparations.Each k_A corresponds to a filter-type measurement, with p_kA the probability of passing that filter.Source: §2 · p. 3
• For the example given, joint probabilities for filter outcomes on two subsystems are enough to determine the composite state.The passage identifies this as local tomography.Source: §2 · p. 4
• N_A is the maximum number of preparations distinguishable in a single-shot measurement, and equals Hilbert-space dimension in quantum theory.Source: §2 · p. 4

3 Local tomography

• Local tomography requires one-component measurement statistics to determine a composite state; with local independence, K_AB = K_AK_B.The proof extends the counting relation iteratively to systems with more components.Source: §3 · pp. 4–5
• Measurements operationally equivalent on A alone are assumed to remain equivalent when A is combined with B.This noncontextuality assumption is unnecessary in the foliable joint-probabilities framework.Source: §3 · p. 4
• Local independence assumes all K_AK_B contributions in the local counting are independent, with none redundant.The text describes this assumption as obtainable from more basic ideas.Source: §3 · p. 5
• The assumptions that N is specified, distinguishability is multiplicative, and K increases with N yield a form for K with positive-integer r.The exact functional form is not stated in the supplied passages.Source: §3 · p. 5
• The inequality from local counting alone does not establish that every locally accessible contribution is independent.Global degrees of freedom can make K_AB large even when some local contributions are redundant.Source: §3 · p. 5
• For complex Hilbert-space quantum theory, N^2 linearly independent projectors span the Hermitian operators, so K = N^2.Their probabilities suffice to determine the state.Source: §3 · p. 6
• Tensor-product projectors extend the local-tomography result from bipartite systems to systems with arbitrarily many components.Source: §3 · p. 6

4.1 Counting contributions

• For ABC, bilocal measurements access local K_AK_BK_C parameters plus pairwise joint-measurement excesses, bounding K_ABC by their total.Pairwise excesses count only parameters beyond those already accessible through local measurements.Source: §4.1 · p. 7

4.2 Latent parameters and iteration

• Bilocal independence and auxiliary assumptions yield a latent parameter L_A depending only on N_A, alongside K_A locally measurable parameters.When L_A is an integer, it represents additional hidden or latent parameters.Source: §4.2 · p. 8
• Applying the K/L relations iteratively expands K_ABC into products of local and latent parameters across the three systems.The expansion includes K_AK_BK_C and terms mixing K and L across systems.Source: §4.2 · p. 8

4.3 Functional form

• The resulting family is K=(N^r+N^s)/2 and L=(N^r−N^s)/2, with r≥s>0; complex theory has r=s=2 and real theory r=2,s=1.The added assumptions rule out pathological solutions involving different prime factors raised to different powers.Source: §4.3 · p. 9

4.4 Proof that real-vector-space quantum theory is bilocally tomographic

• Real Hilbert-space states have K=N(N+1)/2 independent real parameters and L=N(N−1)/2 latent parameters, but counting alone does not prove bilocal tomography.Source: §4.4 · p. 10
• A complete linearly independent set of real symmetric observables reconstructs the state through expectation values, with required measurements executable on one or two components.The construction also yields a complete set of linearly independent projection operators representing bilocal measurements.Source: §4.4 · pp. 10–11

• The analysis does not pursue general results when N_AB differs from N_A N_B.Source: §5 · p. 12
• For the restricted case, N_AB is taken to be αN_AN_B, with α a positive integer.Source: §5 · p. 12
• In this special case, r=s gives a locally tomographic solution.Source: §5 · p. 12

• The authors conjecture that these axioms suffice for quantum theory on a real Hilbert space.Source: §6 · p. 12
• The proposed axioms require information-capacity equivalence, additive information capacity, bilocal tomography, continuity between pure states, and simplicity.Source: §6 · p. 12
• The proposal parallels the axioms for standard complex-Hilbert-space quantum theory, substituting bilocal for local tomography.Source: §6 · p. 12

7 Discussion

• Real-vector-space quantum theory is not locally tomographic but is bilocally tomographic, relaxing reductionism without moving to unrestricted holism.The theory makes maximal use of the freedom allowed by bilocal tomography through bilocal independence.Source: §7 · pp. 12–13
• For four binary systems, bilocal measurements suggest 138 independent parameters, but the theory has K = 136, so two counted parameters are redundant.In real-vector-space quantum theory, measurements on different pairings can yield the same parameter.Source: §7 · p. 13
• In bilocally but not locally tomographic theories satisfying Eqs. (26), (27), and (28), counting each two-pair measurement as independent overcounts four-system parameters.The term L_A L_B L_C L_D is added three times rather than once.Source: §7 · pp. 13–14
• For 3-local tomography, it remains unclear whether a natural independence equation exists, and the appendix’s proposed approach does not count parameters.The authors identify finding a theory that is 3-locally but not bilocally tomographic as a useful next example.Source: §7 · p. 14

Appendix 1: The function L(N)

• The appendix defines h(N_A, N_B) as the number of parameters accessible through a joint measurement on AB beyond separate measurements, and seeks a function L satisfying K_AB = K_A K_B + L_A L_B.The derivation assumes K_A = K(N_A), K_AB ≥ K_A K_B, and bilocal independence.Source: Appendix 1: The function L(N) · pp. 14–15
• The number of parameters accessible only by a joint measurement on AD takes the product form L_A L_D.The appendix defines L_A using h(N_A, x)/h(x, x) when some h(x, x) is positive.Source: Appendix 1: The function L(N) · p. 15
• If h(x, x) is zero for every integer x, then h(N_A, N_D) is always zero and L can be set to zero, leaving no hidden parameters.The text identifies ordinary quantum mechanics as an instance of this case.Source: Appendix 1: The function L(N) · p. 15
• At this stage, L(N) is known to be the square root of an integer, but the appendix has not established that it is an integer.The text states L(N) = h(N, N).Source: Appendix 1: The function L(N) · p. 15
• For three systems, the parameter count has the form K_ABC = K_A K_B K_C + K_A L_B L_C + K_B L_A L_C + K_C L_A L_B.This follows by substituting the two-system expression into the three-system relation.Source: Appendix 1: The function L(N) · p. 16

Appendix 2: Functional forms

• For local tomography, multiplicativity and strict monotonicity imply K(N) = N^r for a positive integer r.The proof uses prime factorization and shows that monotonicity requires the same exponent for every prime.Source: Appendix 2: Functional forms · pp. 16–17
• For bilocal tomography, K(N) + L(N) = N^r and K(N) − L(N) = N^s, with integers r ≥ s > 0.The argument applies multiplicativity and monotonicity separately to the sum and difference, first establishing that K(N) − L(N) is nonnegative.Source: Appendix 2: Functional forms · p. 17

independence

• The n-local ideality approach expresses a condition as a sum over system partitions, with a real coefficient associated with each partition.The discussion focuses on the equation's mathematical form rather than its measurement interpretation.Source: independence · p. 18
• The simple local case recovers Eq. (12) with α=1 without permutation invariance or novelty; the 2-local case uses novelty to obtain α=1, β=−2, recovering Eq. (29).Source: independence · pp. 18–19
• The general symmetric 3-local ansatz leaves its coefficient equations underdetermined, so it does not uniquely specify 3-local ideality.The derivation concludes that its equations do not uniquely determine the coefficients.Source: independence · p. 19
• Strict inclusion requires n-local ideality to imply (n + 1)-local ideality without a reverse implication; the authors apply it at n=2 to constrain Eq. (66).They note that local ideality already implies bilocal ideality and then impose inclusion from bilocal to 3-local ideality.Source: independence · p. 19
• ε=1/2, α=1, β=1/3, γ=−4/3, and δ=4 satisfy the inclusion requirement and yield a candidate 3-local ideality equation.The equation is proposed as a way to express 3-local independence in a 3-locally tomographic theory.Source: independence · p. 20
• The approach does not establish consistency of inclusion for n>1, even at n=2, and parameter counting does not yield fractional coefficients such as 1/3.The authors present the approach as one possible starting point for analyzing n-local tomographic theories.Source: independence · p. 20