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Psi-Epistemic Theories: The Role of Symmetry
Scott Aaronson, Adam Bouland, Lynn Chua, George Lowther
TL;DR
The paper asks whether ψ-epistemic theories can remain viable when distributions overlap for every pair of non-orthogonal quantum states, without relying on Bell, Kochen-Specker, or PBR assumptions. It constructs such a theory in every finite dimension and proves that symmetry rules it out for d ≥3 in the studied ontic spaces. The conclusions leave broader ontic spaces and merely symmetric theories as open directions.
Problem
The paper studies whether ψ-epistemic theories can have overlapping distributions for all non-orthogonal states after dropping the Bell, Kochen-Specker, and PBR assumptions.
Source: §1 · p. 2Method
Results
A maximally nontrivial ψ-epistemic theory exists in every finite dimension without symmetry, whereas symmetric such theories do not exist for d ≥3 in the studied settings.
Source: §2 · p. 5, §4 · p. 19Takeaways & Limitations
Maximum nontriviality and symmetry cannot jointly be maintained in dimensions d ≥3 for ontic spaces CPd−1 or the related U(d) setting considered.
Source: §1 · p. 3, §4 · p. 19The paper leaves open whether merely symmetric theories exist for U(d) or larger ontic spaces acted on by the unitary group, and notes an unresolved Kakeya/Besicovitch-type issue.
Source: §4 · p. 19
Abstract
from arXiv · showhide
Formalizing an old desire of Einstein, "psi-epistemic theories" try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called "ontic states." Regardless of one's philosophical views about such theories, the question arises of whether one can cleanly rule them out, by proving no-go theorems analogous to the Bell Inequality. In the 1960s, Kochen and Specker (who first studied these theories) constructed an elegant psi-epistemic theory for Hilbert space dimension d=2, but also showed that any deterministic psi-epistemic theory must be "measurement contextual" in dimensions 3 and higher. Last year, the topic attracted renewed attention, when Pusey, Barrett, and Rudolph (PBR) showed that any psi-epistemic theory must "behave badly under tensor product." In this paper, we prove that even without the Kochen-Specker or PBR assumptions, there are no psi-epistemic theories in dimensions d>=3 that satisfy two reasonable conditions: (1) symmetry under unitary transformations, and (2) "maximum nontriviality" (meaning that the probability distributions corresponding to any two non-orthogonal states overlap). This no-go theorem holds if the ontic space is either the set of quantum states or the set of unitaries. The proof of this result, in the general case, uses some measure theory and differential geometry. On the other hand, we also show the surprising result that without the symmetry restriction, one can construct maximally-nontrivial psi-epistemic theories in every finite dimension d.
1 Introduction
The paper asks whether ψ-epistemic theories can reproduce quantum predictions while assigning overlapping distributions to every pair of non-orthogonal states. It defines symmetry and maximum nontriviality, then states results contrasting symmetric impossibility with unrestricted constructions.
Motivation
- ψ-epistemic theories represent quantum states as probability distributions over underlying ontic states, but existing results constrain their ability to reproduce quantum predictions.Bell, Kochen-Specker [2], and PBR [3] impose different restrictions on such theories.Source: §1 · pp. 1–2
Definitions
- Maximum nontriviality requires distributions for every pair of non-orthogonal states to have nontrivial overlap, while orthogonal states must have trivial overlap.This is weaker than requiring overlap to explain all measurement uncertainty.Source: §1 · p. 3
- Symmetry means each distribution depends only on the overlap magnitude |⟨ψ|λ⟩|, while strong symmetry additionally uses the same function for every state.Strong symmetry supports applying unitary transformations directly to ontic states while preserving the Schrödinger evolution.Source: §1 · p. 3
Main results
- The paper constructs a maximally nontrivial ψ-epistemic theory in arbitrary finite dimensions, but the construction is unnatural and nonsymmetric.It therefore answers positively whether maximum nontriviality is possible without symmetry.Source: §1 · p. 3
- For dimensions d ≥3, no symmetric maximally nontrivial theory exists, and a related impossibility result holds for strongly symmetric theories with ontic space U(d).Thus achieving maximum nontriviality requires a different ontic space or distributions that single out preferred Hilbert-space directions.Source: §1 · p. 3
2 Nonsymmetric, Maximally Nontrivial Theory
The section constructs a maximally nontrivial ψ-epistemic theory by creating overlap for each non-orthogonal state pair and combining these pairwise theories. The resulting theory works in every finite dimension but uses an unnatural, nonsymmetric ontic representation.
Pairwise construction
- For any two non-orthogonal quantum states |a⟩ and |b⟩, Lemma 1 establishes a ψ-epistemic theory in which their distributions µa and µb have nontrivial overlap.The construction mixes probability in Ea,b, where all λ yield the same measurement outcome for every measurement, preserving the measurement statistics.Source: §2 · p. 4
Convex combination
- A convex combination preserves ψ-epistemic validity and combines overlap guarantees from separate theories, provided each component receives nonzero weight.Separate copies of the ontic spaces are required because mixing response functions on one space introduces cross terms that generally violate the Born rule.Source: §2 · p. 5
Global construction
- The full construction uses finite 1/n-nets of Hilbert space and combines pairwise theories with weights forming an infinite sequence summing to one.The net states are perturbed so every pair within each net is non-orthogonal, allowing the combination to cover all non-orthogonal state pairs.Source: §2 · p. 5
- The resulting theory is maximally nontrivial: distributions overlap for every pair of non-orthogonal states.Its ontic space is CPd−1 × [0,1] × N, which has the same cardinality as CPd−1.Source: §2 · p. 5
Scope and interpretation
- Although a bijection can map the construction onto CPd−1, that mapping destroys its natural association between quantum states and ontic states and makes the theory less symmetric.The paper therefore treats the construction as mathematically valid but unnatural and nonsymmetric.Source: §2 · p. 5
3 Nonexistence of Symmetric, Maximally Nontrivial Theories
This section proves the nonexistence of symmetric, maximally nontrivial ψ-epistemic theories in dimensions d ≥ 3 for ontic spaces CP^{d−1} and U(d). Using measure-theoretic and differential-geometric arguments, it derives contradictions from positive-measure distribution overlaps and the Born rule.
3 Nonexistence of Symmetric, Maximally Nontrivial Theories
- Maximum nontriviality and symmetry force deficiency: for any measurement basis in d ≥3, the deficiency region has positive measure, yet orthogonal-state supports have measure zero there.This produces orthogonal states and a measurement for which the forbidden outcome would occur with nonzero probability, contradicting quantum mechanics.Source: §3 · p. 7
3.1 Proof of Generalized No-Go Theorem
- For CPd−1, Lemmas 8 and 9 provide d states with linearly independent tangent directions and a positive-measure region where all corresponding distributions are equivalent to Lebesgue measure.Each state is orthogonal to a distinct vector in a measurement basis, creating the measure-theoretic contradiction used in Theorem 2.Source: §3.1 · p. 13
- There exists no symmetric, maximally nontrivial ψ-epistemic theory in dimension d ≥3.The proof constructs states orthogonal to measurement outcomes whose distributions are all equivalent to Lebesgue measure on a positive-measure set, contradicting the Born rule because every ontic state must produce some outcome.Source: §3.1 · p. 13
3.2 Extending the Proof to Λ = U(d)
- For Λ = U(d), strong symmetry with left multiplication makes every distribution μψ characterized by one probability measure ν on CPd−1, while maximum nontriviality forces its absolutely continuous component to have positive measure.The construction then uses Lebesgue density points to obtain three states with positive-measure joint overlap.Source: §3.2 · pp. 14–16
- There are no strongly symmetric, maximally nontrivial ψ-epistemic theories with Λ = U(d) in dimension d ≥3.Three suitably chosen states—two nearly orthogonal and a third non-coplanar—admit an orthonormal basis where every basis outcome is orthogonal to at least one state, contradicting positive joint overlap under the Born rule.Source: §3.2 · pp. 16–18
- Compactness of CPd−1 lets the proof pass to convergent subsequences and fix a non-coplanar limiting state despite initially varying choices.A state in T is chosen non-coplanar with the limiting pair, and unitary invariance preserves this non-coplanarity.Source: §3.2 · p. 18
- A positive-measure set T ensures pairwise overlapping measures can be aligned and that any three suitably aligned states have positive-measure joint overlap.T consists of points that are Lebesgue density points of the relevant supports, enabling properties (A) and the alignment construction.Source: §3.2 · p. 18
- The argument combines properties (A) and (B) for three states, and the resulting contradiction rules out the theory.The states are selected from sequences approaching orthogonal states, while remaining nonorthogonal at every finite stage.Source: §3.2 · p. 19
4 Conclusions and Open Problems
The paper constructs maximally nontrivial ψ-epistemic theories in every finite dimension but rules out symmetric versions for d ≥3, while identifying several unresolved extensions and quantitative questions.
Conclusions
- Symmetric, maximally nontrivial ψ-epistemic theories do not exist for d ≥3, whereas the constructed theory is neither symmetric nor natural.The impossibility proof relies heavily on symmetry and uses d ≥3 to obtain a nonempty deficiency region and establish rψ = 1.Source: §4 · p. 19
Extensions
- The no-go result might extend beyond the studied settings because deficiency holds for any ψ-epistemic theory in d ≥3 without symmetry.Open cases include symmetric theories with ontic space U(d), or larger ontic spaces acted on by the d-dimensional unitary group.Source: §4 · p. 19
Scope
- The proof uses only zero probability for projections onto orthogonal states, so additional Born-rule properties could impose further constraints.The paper does not establish those additional constraints.Source: §4 · p. 19
Open problems
- Generalizing the proof to merely symmetric theories leads to an open Kakeya/Besicovitch-type question about whether orthogonal-state measures can evade a deficiency region.The analogous planar problem asks whether a set containing positive-measure circle pieces around every center can have a complement of positive Lebesgue measure, and remains open.Source: §4 · p. 19
- It remains open whether symmetric and nontrivial theories exist in dimensions d ≥3 and how the deficiency region scales with dimension.These questions concern weaker symmetry requirements and the dimension dependence of the obstruction.Source: §4 · p. 19
- The constructed theory’s overlap for non-orthogonal states is vanishingly small, like (ε/d)O(d), motivating the search for substantially larger overlaps.For d ≥3, the overlap cannot reach the maximum value ε2 according to Leifer and Maroney [5]. Other open questions concern whether ontic states can avoid uniquely recovering ψ and what happens in infinite-dimensional Hilbert spaces.Source: §4 · pp. 19–20