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Einstein, incompleteness, and the epistemic view of quantum states

Nicholas Harrigan, Robert W. Spekkens

arXiv:0706.2661v1quant-ph

TL;DR

The paper asks whether the quantum state is part of reality or instead represents knowledge, and formalizes this distinction through a classification of hidden-variable models. It then applies the classification to locality and Einstein’s incompleteness arguments, showing that ψ-ontic theories are nonlocal by simpler arguments than Bell’s theorem and interpreting Einstein’s later argument as favoring an epistemic quantum state.

  • Problem

    The paper addresses the insufficiently emphasized distinction between quantum states as ontic reality and as epistemic representations of knowledge, including its significance for hidden-variable theory and Einstein’s incompleteness arguments.

  • Method

    The authors define ontological models and classify them according to whether quantum states are ψ-complete, ψ-ontic, or ψ-epistemic, then analyze locality arguments and Einstein’s historical reasoning.

  • Results

    The paper shows that locality fails for ψ-ontic theories through arguments simpler than Bell’s theorem and argues that Einstein’s 1935 incompleteness argument supports an epistemic status for quantum states.

  • Takeaways & Limitations

    The ψ-ontic/ψ-epistemic distinction reframes nonlocality and suggests that Einstein sought a completion in which quantum states represented knowledge rather than reality.

Abstract

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Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. Indeed, representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the status of something real, the failure of locality can be established through an argument considerably more straightforward than Bell's theorem. The historical significance of this result becomes evident when one recognizes that the same reasoning is present in Einstein's preferred argument for incompleteness, which dates back to 1935. This fact suggests that Einstein was seeking not just any completion of quantum theory, but one wherein quantum states are solely representative of our knowledge. Our hypothesis is supported by an analysis of Einstein's attempts to clarify his views on quantum theory and the circumstance of his otherwise puzzling abandonment of an even simpler argument for incompleteness from 1927.

I. INTRODUCTION

The paper distinguishes whether quantum states are ontic or epistemic and uses that distinction to reinterpret hidden-variable models, locality arguments, and Einstein’s changing incompleteness argument. It proposes that Einstein preferred the 1935 argument because it supported an epistemic view of quantum states.

  • Quantum-state classifications: ψ-ontic models associate each complete physical state with one pure quantum state, whereas ψ-epistemic models allow ontic states compatible with multiple pure states.Only ψ-epistemic models treat the quantum state as representing an observer’s knowledge rather than reality itself.
  • Research context: Among explicit hidden-variable models, ψ-ontic constructions have predominated, while ψ-epistemic models remain comparatively underexplored despite their potential usefulness for explaining quantum phenomena.The passage notes the Kochen–Specker proposal as an exception that works only for two-dimensional Hilbert spaces.
  • Quantum-state classifications: The classification distinguishes ψ-complete models, where the quantum state alone describes reality, from hidden-variable models that supplement or reinterpret that description.The paper calls the quantum-state-only view ψ-complete and contrasts it with ψ-ontic and ψ-epistemic hidden-variable models.
  • Einstein’s incompleteness arguments: Einstein’s 1927 argument used one measurement, whereas his later argument used a choice between two measurements, introducing counterfactual reasoning that drew substantial criticism.The paper notes that the alleged motivation of simultaneously defeating the uncertainty principle lacks textual support.
  • Einstein’s incompleteness arguments: The authors propose that Einstein adopted the more complicated 1935 argument because it supported the stronger conclusion that quantum states are epistemic, not merely incomplete.They connect this interpretation to Einstein’s later correspondence and summaries, which appealed to the Schrödinger argument rather than the EPR argument.
  • Locality and epistemic states: The paper argues that locality can be ruled out for ψ-ontic theories without Bell’s theorem, while Bell’s argument is needed for ψ-epistemic hidden-variable theories.It identifies Einstein’s 1935 reasoning as an argument that locality conflicts with ψ being ontic, thereby supporting an epistemic characterization.
  • Ontological-model framework: Ontological models represent preparations by probability distributions over ontic states and measurements by outcome probabilities conditioned on those states.The framework uses Λ for the ontic state space, p(λ|P) for preparation distributions, and p(k|λ,M) for measurement response functions.
  • Ontological-model framework: The ontological-model framework is broad but does not encompass every realist interpretation of quantum theory, although most previously analyzed models appear compatible with it.The stated scope boundary concerns realist interpretations not suited to the framework or its proposed extension.

B. Classifying ontological models of quantum theory: heuristics

The paper distinguishes ontological models by whether ψ is complete and whether it is ontic or epistemic, yielding three—not four—classes.

  • ψ-complete models identify the ontic state space with quantum states, so each pure ψ gives a complete description of reality.
  • ψ-supplemented models treat ψ as part of the ontic state while requiring additional hidden variables to specify the complete reality.
  • ψ-epistemic models treat ψ as encoding a probability distribution over underlying ontic states rather than as an ontic variable itself.
  • A model is ψ-ontic when distinct quantum states correspond to non-overlapping epistemic states, and ψ-epistemic when some such distributions overlap.
  • Because every ψ-complete model is ψ-ontic, ψ-complete and ψ-epistemic cannot coexist; the resulting classes are ψ-complete, ψ-supplemented, and ψ-epistemic.

D. Examples

The paper next presents examples from the literature belonging to each class of ontological model.

  • Examples are introduced to illustrate the three classes of ontological models defined earlier.

1. The Beltrametti-Bugajski model

The Beltrametti-Bugajski model takes quantum states themselves as the ontic states and reproduces quantum statistics through its measurement rule.

  • The model sets the ontic state space equal to projective Hilbert space, with each prepared ψ represented by a sharp distribution centered on the corresponding ontic state.
  • It is therefore ψ-complete, because the possible states of reality are identified directly with the possible quantum states.
  • Measurement outcome probabilities depend indeterministically on the ontic state and the POVM associated with the measurement.
  • The model’s predicted statistics reproduce the quantum probabilities trivially.
  • For two-dimensional Hilbert space, the ontic states can be parameterized by unit Bloch vectors.

2. The Bell-Mermin model

The Bell-Mermin model uses two Bloch-sphere variables, one aligned with ψ and one uniformly distributed, while reproducing the Born rule through an indicator function.

  • The model’s ontic state space is a Cartesian product of two unit-sphere spaces, requiring two Bloch-vector variables to specify reality.
  • The ψ-associated distribution is sharp in the first variable and uniform, ψ-independent, in the second.
  • For a projective measurement, the φ outcome occurs exactly when the sum of the two ontic vectors has positive inner product with the φ Bloch vector.
  • The model reproduces the quantum-mechanical Born rule by integrating its epistemic distributions against the measurement indicator function.
  • Because its ontic space differs from projective Hilbert space, the Bell-Mermin model is ψ-incomplete and ψ-ontic, hence ψ-supplemented.

3. The Kochen-Specker model

The Kochen-Specker model represents quantum states by probabilistic distributions over a spherical ontic state space and uses hemisphere-based measurement indicators. Its overlapping distributions for nonorthogonal states make it ψ-epistemic and ψ-incomplete.

  • The model takes the ontic state space to be the unit sphere, with each quantum state represented by a distribution determined by its Bloch vector.
  • The epistemic distribution assigns cos θ within the hemisphere centered on the state vector and zero outside it.
  • A projector measurement yields a positive outcome when the ontic state lies in the hemisphere centered on the projector’s Bloch vector.
  • The model reproduces quantum statistics through overlaps between epistemic states and indicator functions.
  • Because its quantum-state distributions are non-sharp and overlap for nonorthogonal states, the model is ψ-incomplete and ψ-epistemic.

4. Connections between the models

The section connects Bell-Mermin, Beltrametti-Bugajski, and Kochen-Specker models, then formalizes locality through separability and local causality. Kochen-Specker achieves epistemic indeterminism without Bell-Mermin’s extra ontological variable.

  • Connections between the models: Bell-Mermin supplements the Beltrametti-Bugajski model with a hidden variable that uniquely determines projective-measurement outcomes.
  • Connections between the models: Reparameterizing Bell-Mermin’s variables shows that its operational predictions depend only on u, while v does not affect the indicator functions.
  • Connections between the models: Kochen-Specker is Bell-Mermin with v eliminated, preserving empirical predictions while reducing the ontic state space.
  • Connections between the models: Kochen-Specker renders indeterminism epistemic without increasing ontic-state-space size, unlike Bell-Mermin’s added ontological overhead.
  • Locality framework: Locality is defined as the conjunction of separability and local causality, with local causality requiring screening-off by a complete specification in region C.
  • Locality framework: ψ-complete models are nonseparable because composite Hilbert spaces use tensor products rather than Cartesian products, so they are not local.
  • Locality framework: Even setting nonseparability aside, ψ-complete state updates after local measurements imply failure of local causality.

A. ψ-ontic models of quantum theory are nonlocal

The paper uses steering of an entangled two-level system to show that any ψ-ontic model reproducing quantum statistics violates locality. This argument is simpler than Bell’s theorem and connects directly to Einstein’s incompleteness reasoning.

  • The argument is stronger than Einstein’s 1927 argument but weaker than Bell’s theorem, and is identified with Einstein’s 1935 incompleteness argument.
  • Alice’s two measurement choices remotely prepare Bob in one of two alternative pairs of pure states, while leaving the choice of pair under her control.
  • The theorem states that every ψ-ontic ontological model reproducing quantum statistics violates locality.
  • The proof assumes separability so that Bob’s ontic state is well-defined, while explicitly requiring no independent realism assumption.
  • Assuming locality makes Bob’s ontic-state probabilities independent of Alice’s measurement, but the resulting constraints force overlap between epistemic states of distinct pure states.
  • That overlap implies the model is ψ-epistemic, contradicting ψ-onticity and establishing nonlocality.

A. The EPR incompleteness argument

Einstein’s 1927 argument uses a single-particle diffraction experiment to challenge ψ-completeness under locality. The argument yields a contradiction for ψ-complete models but does not extend to general ψ-ontic models with additional hidden variables.

  • Quantum prediction: Quantum mechanics assigns zero probability to detections at both distinct screen regions A and B.The argument contrasts this prediction with the probabilities inferred under ψ-completeness and locality.
  • The contradiction: Under locality and ψ-completeness, conditional detection probabilities factorize, yielding a nonzero probability for simultaneous detections at A and B.With λ identified with ψ, the model infers p(1A|ψ)=p(1B|ψ)=1/4, contradicting quantum mechanics.
  • The contradiction: The logical structure is that locality, quantum statistics, and ψ-completeness jointly imply a contradiction.Thus the argument establishes incompatibility between these three assumptions.
  • Scope of the argument: The 1927 argument cannot rule out locality for general ψ-ontic models supplemented by hidden variables.When λ=(ψ,ω), conditioning on the hidden variable changes the relevant detection probabilities and blocks the contradiction.

C. Einstein’s 1935 incompleteness argument

Einstein’s 1935 argument adopts a notion of completeness equivalent to ψ-completeness and uses entanglement, steering, separability, and local causality. Its stronger conclusion rules out locality for both ψ-complete and ψ-supplemented models, leaving ψ-epistemic models as the remaining possibility for locality.

  • Completeness: Einstein defined completeness as a one-to-one correlation between ψ and the real state of the system.This corresponds to the paper’s definition of ψ-completeness.
  • Entanglement and steering: The 1935 argument begins with an entangled bipartite system and uses measurement choice on A to associate different quantum states with subsystem B.This is the steering phenomenon described in the paper.
  • Locality assumptions: Einstein combines separability with local causality to argue that B’s real state cannot depend on which measurement is performed on spacelike-separated A.He treats the real state of the joint system as precisely the real states of A and B.
  • Conclusion: Einstein’s reasoning shows that one ontic state can correspond to multiple quantum states, rather than merely showing that ψ is incomplete.That structure is characteristic of the ψ-epistemic conclusion emphasized by the paper.
  • Conclusion: The 1935 argument rules out locality for ψ-complete and ψ-supplemented models, making ψ-epistemic models the only models retaining any hope of preserving locality.The paper presents this as a stronger conclusion than the nominal claim of ψ-incompleteness.

V. HISTORICAL IMPLICATIONS

The paper explains Einstein’s shift from a simpler 1927 argument to the more complex 1935 version as a pursuit of the stronger ψ-epistemic conclusion. It also argues that Einstein’s later writings support an epistemic interpretation, while historical confirmation remains incomplete.

  • The puzzle: The central historical puzzle is why Einstein replaced a one-measurement 1927 argument with a two-measurement 1935 argument.The change was permanent in his later publications and writings.
  • Alternative explanations: The two-measurement argument cannot be explained solely by a desire to defeat the uncertainty principle.Einstein explicitly de-emphasized the uncertainty principle in his own writings.
  • Experimental significance: The 1935 setup supplies experimental distinctions that can rule out separable mixed-state explanations unavailable to the simpler 1927 setup.Such explanations would require additional evidence, including interference, to establish coherence in the 1927 case.
  • A possible explanation: The paper’s preferred explanation is that Einstein sought a ψ-epistemic model, since the 1935 argument rules out both ψ-complete and ψ-supplemented local models.The 1927 argument lacks this stronger reach, and deBroglie-Bohm theory provides a local explanation of that thought experiment.
  • Einstein’s later views: Einstein’s later statements treating ψ as an ensemble or as knowledge support the paper’s identification of his view with an epistemic interpretation.The paper argues that ensemble talk grounds probabilities in an observer’s knowledge.
  • Relation to Bell’s theorem: Bell’s theorem later showed that adequate theories must violate locality, but ψ-ontic models can already be shown nonlocal by Einstein’s simpler argument.The paper assigns Bell’s theorem the additional task of addressing ψ-epistemic models.
  • Historical limitation: The historical interpretation remains provisional and would require a careful reexamination of Einstein’s papers and correspondence.The paper presents further archival reassessment as a needed step.

VI. THE FUTURE OF ψ-EPISTEMIC MODELS

The paper presents ψ-epistemic models as motivated by information-theoretic phenomena, while emphasizing that their general existence remains unresolved. Contextuality results do not settle the ψ-ontic/ψ-epistemic distinction, and the operational framework has scope limitations.

  • Motivations: Information-theoretic phenomena such as teleportation, no-cloning, and entanglement can be derived in toy theories with hidden variables and epistemic analogues of ψ.This provides motivation for ψ-epistemic models independent of locality.
  • Open problem: Whether ψ-epistemic ontological models exist beyond pure states and projective measurements in two dimensions remains unclear.The paper identifies more general cases as an open challenge.
  • Contextuality: Bell–Kochen–Specker results establish the necessity of contextuality in dimensions three and higher but do not determine whether a model is ψ-ontic or ψ-epistemic.The two distinctions are independent.
  • Existing models: Existing examples include ψ-ontic contextual models such as deBroglie-Bohm theory and ψ-epistemic constructions restricted to two-dimensional Hilbert spaces.The paper distinguishes the classification of quantum-state status from contextuality.
  • Construction challenges: Some proposed ψ-epistemic models are approximate, restricted to limited bases, or may lack desired symmetry properties.The paper discusses these as unresolved construction challenges rather than settled impossibility results.
  • Scope boundary: The operational framework may not capture interpretations in which systems are inseparable from apparatus or fundamentally relational.The paper therefore limits any no-go result proved within that framework.
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