In defense of the epistemic view of quantum states: a toy theory

I. INTRODUCTION

• A toy theory based on incomplete knowledge reproduces many quantum phenomena, supporting the thesis that both pure and mixed quantum states are epistemic rather than ontic.The authors emphasize conceptual understanding, not merely mathematical derivation, and present the theory as evidence for the epistemic view.Source: §I · pp. 1–2
• The toy theory cannot derive quantum theory because it concerns incomplete knowledge of local, noncontextual hidden variables, but it remains useful for identifying interpretation-independent features of incomplete knowledge.The authors leave open what quantum states represent knowledge about and describe this as the central obstacle to completing the epistemic research program.Source: §I · p. 3

II. THE KNOWLEDGE BALANCE PRINCIPLE

• The theory’s central principle requires knowledge and ignorance to be equal in every state of maximal knowledge.Knowledge is measured by the maximum number of answered questions in any canonical set, while ignorance is the remainder.Source: §II · pp. 3–4
• The simplest system has four ontic states because its minimal canonical set contains two yes/no questions, with composite systems built from elementary systems under additional locality assumptions.For N elementary systems, the canonical set has 2N questions and 2^(2N) ontic states; the balance applies to whole systems and subsystems.Source: §II · p. 4

A. Epistemic states

• An elementary system has six maximal-knowledge epistemic states, mapped by convention to the six pure states of a qubit, while its nonmaximal state corresponds to the completely mixed state.The six mappings are 1∨2↔|0⟩, 3∨4↔|1⟩, 1∨3↔|+⟩, 2∨4↔|−⟩, 2∨3↔|+i⟩, and 1∨4↔|−i⟩.Source: A. Epistemic states · pp. 4–5
• Compatibility requires intersecting ontic bases to yield a valid epistemic state, ensuring agreement among maximally informed observers and paralleling quantum support compatibility.Disjointness analogizes orthogonality, while nondisjoint states provide a toy-theory analogue of nonorthogonality.Source: A. Epistemic states · p. 5
• Classical fidelity between toy-theory epistemic states exactly matches quantum fidelity for the analogous qubit states.The corresponding values include 0 for disjoint states, 1/2 for selected nonorthogonal pure states, 1/√2 for a pure and mixed state, and 1 for identical states.Source: A. Epistemic states · p. 5
• The mixed epistemic state has multiple convex decompositions into pure states, mirroring the completely mixed qubit state because pure toy states represent incomplete knowledge.Unlike quantum convex sums, toy-theory convex combinations require disjoint states with a valid union and do not support unequal weights.Source: A. Epistemic states · pp. 5–6
• Four coherent binary operations combine disjoint epistemic states by selecting low-low, high-high, high-low, or low-high ontic-state indices.They produce the states a∨c, b∨d, b∨c, and a∨d, respectively.Source: A. Epistemic states · p. 6
• The high-low and low-high operations are noncommutative, paralleling quantum coherent superpositions whose relative phases are π/2 and 3π/2.For example, (1∨2)+3(3∨4)=2∨3, whereas reversing the arguments gives 1∨4.Source: A. Epistemic states · p. 7
• The toy operations parallel corresponding quantum-state relations under the proposed state associations, but two relations cannot be matched simultaneously, showing that a toy elementary system is not simply a constrained qubit.Changing the associations fixes one pair of mismatches while causing another pair to fail.Source: A. Epistemic states · p. 7
• Unlike qubit superpositions, toy-theoretic operations are undefined for nondisjoint epistemic states and offer only four operation types rather than a continuum of relative weights and phases.Quantum qubit states can be coherently superposed for arbitrary pairs and with continuously varying weights and phases.Source: A. Epistemic states · p. 7

• Reversible transformations are exactly permutations of the four ontic states, forming the 24-element group S4.Source: B. Transformations · p. 8
• The permutation (123)(4) appears as a 120° Bloch-sphere rotation, while (13)(24) appears as a 180° rotation.Source: B. Transformations · p. 8
• Some permutations correspond to reflections or combined rotations and reflections, analogous to anti-unitary maps in Hilbert space.These transformations arise because the toy theory permits discrete transformations, whereas quantum evolution is assumed continuous in time.Source: B. Transformations · p. 8
• The allowed transformations form the symmetry group of a tetrahedron whose vertices represent the four ontic states.Source: B. Transformations · p. 8

• A universal state inverter would map every pure epistemic state to its disjoint state, paralleling universal inversion of pure qubit states.Source: C. No universal state inverter · pp. 8–9
• The required pairings are inconsistent with any permutation because the first two conditions imply 1 ↔4 and 2 ↔3, contradicting the third.Source: C. No universal state inverter · p. 9
• In the Bloch-ball representation, universal inversion is inversion through the origin, which cannot be achieved by the toy theory’s rotations and reflections.Source: C. No universal state inverter · p. 9

• The knowledge-balance principle permits only reproducible measurements that partition the four ontic states into two sets of two.Source: D. Measurements · p. 9
• Measurements distinguishing singleton ontic states are forbidden because a positive singleton outcome would produce knowledge violating the balance principle.Source: D. Measurements · p. 9
• The three valid partitions correspond respectively to the qubit bases {|0⟩, |1⟩}, {|+⟩, |−⟩}, and {|+i⟩, |−i⟩}.Source: D. Measurements · p. 9
• An outcome is certain when the initial epistemic state lies within that outcome’s ontic base; otherwise, repeated trials yield equal frequencies for the two outcomes.Source: D. Measurements · p. 9

E. Measurement update rule

• The update requires unknown disturbance: after observing an outcome, the ontic state may undergo one of two outcome-dependent permutations, whose identity is unknown.Source: E. Measurement update rule · p. 10
• The possible measurement-induced permutations depend on the measurement and outcome, not on the initial epistemic state or correlations with other systems.Source: E. Measurement update rule · p. 10
• For an uncorrelated system, a reproducible maximally informative measurement updates the epistemic state to the ontic states associated with the observed outcome.This parallels updating a quantum state to the eigenvector associated with the measurement outcome.Source: E. Measurement update rule · p. 10

F. Noncommutativity of measurements

• Measurement order matters: reversing the two partition measurements changes a certain 1 ∨2 outcome into equal frequencies of 1 ∨2 and 3 ∨4.Source: F. Noncommutativity of measurements · p. 11

G. Interference

• The toy theory represents coherent superposition through binary operations distinct from convex combination, making interference compatible with an epistemic interpretation.The coherent combination of disjoint pure epistemic states is neither their ‘or’ nor their ‘and’.Source: G. Interference · p. 11
• (1, 0) replaces the expected (1/2, 1/2) distribution in the third interference experiment, reproducing destructive interference.The first two experiments each yield (1/2, 1/2), whereas the coherent combination yields zero probability for |−⟩.Source: G. Interference · p. 11

A. Epistemic states

• Applying the knowledge-balance principle separately to subsystems rules out composite states that would give either subsystem more than one answered question.A state can satisfy the composite constraint while violating the corresponding constraint for an individual subsystem.Source: A. Epistemic states · p. 12
• Maximal knowledge for two systems has two forms analogous to quantum product states and maximally entangled states.The distinction arises because the knowledge-balance principle permits knowledge of individual states or knowledge of relations, but not both maximally.Source: A. Epistemic states · pp. 12–13
• Uncorrelated composite states are products of valid subsystem epistemic states and update only the marginal of the measured subsystem.These states correspond to quantum product states such as |0⟩|0⟩ and |+i⟩|−i⟩.Source: A. Epistemic states · pp. 12–13
• Correlated maximal-knowledge states encode complete relational information while leaving each subsystem’s ontic state individually unknown.There are 24 such states, one for each permutation of the four ontic states, and measurements on one system refine the other’s description.Source: A. Epistemic states · pp. 13–14

B. Remote steering

• Quantum steering appears to threaten local causality when |0⟩, |1⟩, |+⟩, and |−⟩ are treated as mutually exclusive realities.The concern arises because Alice can apparently determine different final states for Bob by choosing different measurements.Source: B. Remote steering · p. 14
• The toy theory reproduces the steering updates for measurements distinguishing 1∨2 from 3∨4 or 1∨3 from 2∨4.Each outcome produces a corresponding correlated epistemic state for the pair, paralleling the quantum examples.Source: B. Remote steering · pp. 14–15
• The toy theory does not reproduce correlations that violate Bell inequalities.Its steering analogue is explicitly local and therefore lacks this quantum feature.Source: B. Remote steering · p. 14
• Alice’s measurement choice changes her description of Bob without changing Bob’s ontic state, providing a local analogue of remote steering.Different measurements narrow the possible states of B in different ways, while B remains in ontic state 1.Source: B. Remote steering · p. 15

C. Epistemic states of nonmaximal knowledge

• Nonmaximal knowledge for two systems is constrained into two types of valid epistemic state.Unlike the single-system case, two systems can contain eight ontic states while answering one of four canonical questions.Source: C. Epistemic states of nonmaximal… · p. 15
• Uncorrelated nonmaximal states combine a pure epistemic state for one subsystem with a mixed state for the other.They are analogous to density operators such as |1⟩⟨1|⊗ I/2 and I/2 ⊗|+⟩⟨+|.Source: C. Epistemic states of nonmaximal… · p. 15
• Correlated nonmaximal states require updating one subsystem’s epistemic description after some measurements on the other.For the state [(1∨2)·(1∨2)]∨[(3∨4)·(3∨4)], measuring 1∨2 versus 3∨4 on A selects the corresponding correlated state.Source: C. Epistemic states of nonmaximal… · p. 15
• The toy theory distinguishes perfect from imperfect correlations, analogous to quantum entanglement versus classical correlation.Some measurements update the nonmeasured marginal, while others do not, so the correlation is not universally perfect.Source: C. Epistemic states of nonmaximal… · p. 15
• States of nonmaximal knowledge are mixed states and can have multiple convex decompositions.The completely ignorant pair state is analogous to I/2 ⊗ I/2.Source: C. Epistemic states of nonmaximal… · p. 15
• The toy theory’s perfect correlations do not violate Bell inequalities.Thus the analogy with quantum correlated states has a specific limitation.Source: C. Epistemic states of nonmaximal… · p. 15

D. Transformations

• Local permutations preserve whether two systems are correlated, whereas other allowed permutations can change the degree of correlation like entangling operations.The latter are analogous to entangling quantum operations, including a transformation analogous to controlled NOT.Source: D. Transformations · p. 16

E. No cloning

• The toy theory forbids cloning because permutations preserve the number of overlapping ontic states, but cloning would reduce that overlap.The example’s initial and final epistemic-state fidelities are 1/2 and 1/4, respectively.Source: E. No cloning · p. 17
• A knowledge restriction is needed for nondisjoint pure epistemic states, making the toy theory more quantum-like than classical theories regarding pure-state cloning.No-cloning for nondisjoint epistemic states alone does not require restricted knowledge.Source: E. No cloning · p. 17

F. No broadcasting

• Quantum broadcasting of pure states reduces to cloning because a state with pure marginals must be uncorrelated.Consequently, broadcasting nonorthogonal pure states is impossible.Source: F. No broadcasting · p. 17
• In the toy theory, pure epistemic states cannot be marginals of correlated states because maximal knowledge of each subsystem already exhausts the knowledge allowed for the composite.The knowledge-balance principle therefore rules out the classical measure-and-prepare broadcasting protocol.Source: F. No broadcasting · pp. 17–18
• The paper does not develop the analogue of no-broadcasting for mixed epistemic states.Only the pure-state case is considered here.Source: F. No broadcasting · p. 18

G. Measurements

• Maximally informative measurements partition the sixteen ontic states into four disjoint pure epistemic states, yielding product, correlated, or mixed measurement structures.Product measurements decompose into measurements on individual systems, whereas joint measurements contain correlated epistemic states and cannot be implemented separately on spatially separated systems.Source: G. Measurements · pp. 18–19
• Product measurements are analogous to product bases, while fully correlated measurements are analogous to the Bell basis.The toy theory also permits measurements combining uncorrelated and correlated epistemic states.Source: G. Measurements · pp. 18–19

H. Mutually unbiased measurements

• A single elementary system has three mutually unbiased partitionings, matching the three mutually unbiased bases available for a single qubit.Mutual unbiasedness is defined through equal pairwise classical fidelities between elements of two partitionings.Source: H. Mutually unbiased measurements · p. 19
• Two elementary systems admit five mutually unbiased partitionings, and the paper conjectures that this correspondence with qubit bases extends to any number of systems.The conjecture identifies the number of toy-theory partitionings with the number of quantum mutually unbiased bases for the same number of qubits.Source: H. Mutually unbiased measurements · p. 19

I. Dense coding

• Shared correlations let Alice communicate two classical bits by sending one elementary system, whereas an uncorrelated single system communicates only one bit.Alice selects among four local permutations, and Bob distinguishes the resulting four epistemic states after receiving her system.Source: §I · p. 20
• The toy theory interprets dense coding as access to two bits about a relation between correlated systems, not knowledge of one system’s four ontic states.This motivates comparing one qubit with two classical bits rather than with one classical bit.Source: §I · p. 20

J. Nonmaximally informative measurements

• Nonmaximally informative measurements provide partial information and can arise by coarse-graining more informative product or joint measurements.A joint example reveals relations between systems without extracting all available information.Source: J. Nonmaximally informative measurements · pp. 20–21

K. Measurement update rule

• For nonmaximally informative joint measurements, reproducibility does not determine a unique state-update rule.One natural analogue of quantum projection selects the epistemic state with highest classical fidelity to the initial state.Source: K. Measurement update rule · p. 21

A. Epistemic states

• A maximally knowledgeable triplet answers three of six canonical questions and therefore contains eight ontic states.Every pairwise marginal must also satisfy the knowledge-balance principle.Source: A. Epistemic states · p. 21
• Triplet epistemic states fall into three types: uncorrelated, correlated across one pair, or correlated across all three systems.These correspond respectively to product states, products involving a Bell state, and GHZ states.Source: A. Epistemic states · pp. 21–22

B. The monogamy of pure entanglement

• The toy theory identifies monogamy of pure entanglement with monogamy of perfect correlations.Perfect correlations are permitted between a pair, but not simultaneously between one system and two others.Source: B. The monogamy of pure entanglement · p. 22
• Three systems that are all pairwise perfectly correlated would occupy only four ontic states, violating the eight-state minimum imposed by the knowledge-balance principle.Thus the candidate state with identical ontic states for all three systems is not an allowed triplet epistemic state.Source: B. The monogamy of pure entanglement · p. 22

C. Teleportation

• The teleportation analogy reframes the protocol as transferring the applicability of a state of knowledge rather than transmitting a complete physical state.The paper notes that this transfer can occur without communication between the systems themselves when the measurement outcome is communicated only to the describer.Source: C. Teleportation · pp. 23–24
• Shared correlations, a joint measurement, and two communicated bits transfer an unknown epistemic state from Alice’s system to Bob’s.Bob applies the inverse of the measured permutation, while the original pair remains in a correlated epistemic state; the protocol also supports entanglement swapping.Source: C. Teleportation · pp. 23–24

VI. FURTHER ANALOGUES

• Convex combinations and ancilla-assisted procedures generate toy-theory analogues of unsharp measurements and irreversible transformations.These mirror quantum POVM-like measurements and non-unitary operations.Source: §VI · pp. 24–25
• The toy theory also reproduces no information gain without disturbance, no deterministic error-free discrimination of nonorthogonal states, and indeterministic unambiguous discrimination.These are listed among its analogues of quantum state-discrimination phenomena.Source: §VI · p. 25
• The theory contains analogues of locally indistinguishable and unextendible product bases, while remaining explicitly local in Bell’s sense.The paper also expects key distribution to be possible because the toy theory reproduces no-information-gain-without-disturbance.Source: §VI · p. 25
• A one-to-one correspondence between maximally informative outcomes and pure epistemic states is recovered because only uniform distributions are allowed.Allowing arbitrary distributions would produce multiple epistemic states with the same ontic base.Source: §VI · p. 25

Implications

• These failures suggest that contextuality may identify an additional conceptual ingredient beyond the principle that maximal information is incomplete.Related quantum information-processing advantages, including random access codes and communication-complexity problems, have been connected to contextuality or nonlocality.Source: §VII · pp. 25–27

Continuum of states, measurements and transformations

• The toy theory’s finite sets of states, measurements, and transformations cannot reproduce quantum theory’s continuum of each.Allowing non-uniform distributions, probabilistic measurements, and probabilistic transformations creates a continuum but still cannot reproduce quantum predictions.Source: §VII · pp. 25–26

Binary operations on epistemic states

• The toy theory’s convex and coherent binary operations are partial, unlike quantum theory’s full operations on states.Extending convex combinations alone produces variants with unique decompositions or lacking purifications, while extending both operations conceptually remains unresolved.Source: §VII · pp. 25–26

Embedding two-level systems in three-level systems

• The qutrit variant reproduces qutrit phenomena but cannot be combined straightforwardly with the original qubit theory.Two disjoint toy-qutrit epistemic states involve six ontic states, whereas two toy-qubit states involve four, unlike the corresponding quantum isomorphism.Source: §VII · pp. 25–27

Contextuality and nonlocality

• Contextuality and nonlocality cannot be reproduced because the toy theory is a local, noncontextual hidden-variable theory.The Kochen-Specker and Bell theorems rule out reproducing all quantum predictions under these assumptions.Source: §VII · pp. 25–26

Continuum of states, measurements and transformations

• A continuum of ontic states would require a continuum of distinct subsets to represent pure quantum states, and therefore a very large or continuous ontic state space.Hardy’s argument establishes this from the distinguishability structure of pure states.Source: §VII · p. 26

Embedding two-level systems in three-level systems

• The original toy theory and a tertiary-question variant treat pairs of qubits differently from fundamentally four-level systems.The original theory imposes knowledge balance both globally and on individual systems, whereas the tertiary-question variant lacks the latter constraint.Source: §VII · p. 27

Exponential speed-ups in computation

• The toy theory is efficiently classically simulable, so it cannot reproduce a possible exponential quantum-computational speed-up.This conclusion is conditional because the relevant problems have not been proved difficult for classical computers.Source: §VII · pp. 25, 27

VIII. RELATED WORK

• The toy theory differs from related models by combining a knowledge-balance foundation with transformations and multiple-system structure.Kirkpatrick’s model captures noncommutativity and interference but omits transformations and multiple systems; Hardy’s theory is less analogous for multiple systems, though sufficient to demonstrate local teleportation.Source: §VIII · pp. 27–28
• Smolin’s lockbox models reproduce several information-theoretic phenomena, and a reformulation without unique pair labels yields a variant of the toy theory.The reformulated model permits correlations and makes bit-commitment impossibility depend on transformations between the pair’s two epistemic states.Source: §VIII · p. 28
• Zeilinger’s operational principle resembles the toy theory’s information balance but assumes the quantum measurement structure rather than deriving it.The toy theory instead fixes the ratio of known to unknown propositions through a balance between knowledge and ignorance.Source: §VIII · p. 28

IX. CONCLUSIONS

• The toy theory’s broad reproduction of quantum-state phenomena strongly suggests interpreting quantum states as states of incomplete knowledge.The conclusion lists analogous decompositions, transformations, inversion, updating, steering, cloning, broadcasting, and teleportation.Source: §IX · p. 28
• Because the toy theory contains almost no physics, its success suggests that many reproduced phenomena concern information manipulation rather than specifically physical dynamics.Atomic spectra are not reproduced, whereas no-cloning and teleportation may not depend on such real physics.Source: §IX · pp. 28–29
• Future work asks how to justify the knowledge-balance principle physically, identify the ontic states involved, and formulate a second principle for missing phenomena.Contextuality and nonlocality require modifying the classical conception of reality, but need not force an interpretation based only on future measurement outcomes.Source: §IX · p. 29
• The knowledge-incompleteness principle may support a simple axiomatization of quantum theory but is insufficient to derive quantum theory by itself.The authors propose seeking a second principle that reproduces contextuality and nonlocality through deliberately incomplete and possibly unphysical theories.Source: §IX · p. 29

APPENDIX A: DIFFERENCES TO RESTRICTED QUANTUM

• The toy theory is not simply a restricted version of quantum theory for N qubits because its correlations, coherent binary operations, and transformations differ.Its transformations include permutations analogous to anti-unitary maps, and its coherent binary operations are not precisely quantum coherent superpositions.Source: Appendix A · pp. 29–30
• The toy theory’s mutually unbiased correlations differ from quantum correlations: its correlated epistemic states always yield an even number of anti-correlations, whereas quantum states cannot.The comparison uses Bell states and analogous correlated epistemic states across mutually unbiased measurements.Source: Appendix A · pp. 29–30, Table II · p. 30, Table III · p. 30

APPENDIX B: RELEVANCE TO QUANTUM AXIOMATICS

• The toy theory shows that many information-theoretic effects found in quantum theory are not unique to quantum theory, affecting proposed information-theoretic axiomatizations.The appendix discusses implications for key distribution, bit commitment, and the CBH characterization theorem.Source: Appendix B · p. 30
• The toy theory likely permits key distribution and likely prohibits arbitrarily concealing and arbitrarily binding bit commitment, but these claims are not rigorously established.Remote steering suggests that an analogue of BB84-based bit commitment would not be secure against Alice.Source: Appendix B · p. 30
• The toy theory does not challenge the CBH theorem because it lies outside the required C∗-algebraic framework, where convex combination is not an arbitrary binary operation.The authors suggest that this framework may either exclude physically reasonable theories or identify why the toy theory is not physically reasonable.Source: Appendix B · p. 30
• Derivation from the knowledge balance principle and the toy theory’s closeness to quantum theory suggest that predominantly information-theoretic axiomatizations remain promising.This conclusion is presented despite the toy theory’s challenge to assumptions such as the C∗-algebraic framework.Source: Appendix B · pp. 30–31