Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction

I. INTRODUCTION

• ERL mechanics imposes a restriction on classical knowledge and derives preparations, measurements, and transformations equivalent to Gaussian quantum mechanics.The model provides a noncontextual, ψ-epistemic hidden-variable interpretation of Gaussian quantum mechanics.Source: §I · p. 1
• The reconstruction is an axiomatization from classical mechanics, rather than an interpretation of an already-given quantum formalism.Source: §I · p. 2
• Gaussian quantum mechanics reproduces complementary and noncommuting measurements, wavefunction collapse, no-cloning, information-processing tasks, and substantial entanglement and statistical structure.Examples include teleportation, key distribution, quantum error correction, metrological improvements, and dilation and purification structures.Source: §I · pp. 2–3
• The paper uses Gaussian quantum mechanics as a foil: phenomena it fails to reproduce can help identify principles needed to derive the full quantum theory.The authors distinguish phenomena explainable by epistemic restrictions from a second tier that may require additional principles.Source: §I · p. 3
• Its scope is limited: Gaussian quantum mechanics excludes Bell violations, Kochen–Specker contextuality, operational noncontextuality violations, exponential computational speed-up, interference, and quantization phenomena.These exclusions reflect the local, noncontextual hidden-variable model and efficient classical simulation available for the Gaussian theory.Source: §I · p. 3

A. Previous work that is relevant to this article

• The work extends an epistemic-restriction approach from discrete classical systems to continuous-variable systems, while situating quantum-state-as-knowledge views within prior research.The paper presents this success as evidence for interpreting quantum states as incomplete knowledge rather than states of reality.Source: §I · p. 1, A. Previous work that is relevant to… · p. 4

B. Structure of the paper

• The paper reviews quantum and Liouville mechanics, introduces ERL mechanics and its reproduced quantum phenomena, and proves operational equivalence with Gaussian quantum mechanics.Source: B. Structure of the paper · p. 4, §II · p. 4

A. Quantum Mechanics

• Quantum canonical operators satisfy [ẑ_i, ẑ_j] = iℏΣ_ij, yielding covariance constraints that are preserved by linear symplectic transformations.These transformations act linearly on canonical operators, means, and covariance matrices while preserving the uncertainty relation.Source: A. Quantum Mechanics · p. 5
• Quantum states are represented by density operators whose means and covariance matrices obey the general uncertainty principle derived from canonical commutation relations.The one-dimensional case reduces to the usual position–momentum uncertainty relation.Source: A. Quantum Mechanics · p. 5

B. Liouville mechanics

• Liouville mechanics represents classical knowledge with normalized nonnegative probability distributions over a symplectic phase space.Classical systems combine through Cartesian-product phase spaces, and distributions define means and covariance matrices for canonical coordinates.Source: B. Liouville mechanics · pp. 5–6
• Classical Liouville covariance matrices are merely positive semidefinite, so ordinary Liouville mechanics places no nontrivial lower bound on the product of position and momentum variances.Source: B. Liouville mechanics · p. 6
• The quantum uncertainty restriction arises from the iℏΣ term, which is absent from the classical covariance condition.Source: B. Liouville mechanics · p. 6

A. The epistemic restriction

• ERL mechanics permits an epistemic state only when its phase-space distribution satisfies both the classical uncertainty principle and maximum entropy.The restriction is introduced as a foundational postulate on allowed epistemic states.Source: A. The epistemic restriction · p. 6
• The classical uncertainty principle constrains the covariance matrix, with λ > 0 setting the minimum position–momentum variance product.The constraint concerns real variances and cross-correlations, despite its compact matrix form involving i.Source: A. The epistemic restriction · p. 6
• Maximum entropy selects the distribution with greatest phase-space entropy among distributions sharing the same mean values and covariance matrix.The paper motivates this condition by its closer analogy to quantum theory and its less restrictive measurement structure.Source: A. The epistemic restriction · pp. 6–7
• Distributions satisfying both constraints are multivariate Gaussians characterized by mean vector d and covariance matrix γ.When γ is not strictly positive-definite, the Gaussian expression uses a pseudoinverse.Source: A. The epistemic restriction · p. 7
• ERL describes a classical ontology with restrictions on possible preparations, measurements, and transformations, while treating the uncertainty relation as a knowledge constraint.This contrasts with interpretations that deny uncertainty relations as constraints on knowledge of pre-existing position and momentum values.Source: A. The epistemic restriction · p. 7
• The epistemic restriction applies to pure predictions and retrodictions but not to inferences conditioned on both pre-selection and post-selection.Under pre- and post-selection, ERL can allow learning both position and momentum of a system.Source: A. The epistemic restriction · pp. 6–7

1. Reversible transformations

• General symplectic transformations preserve the uncertainty constraint, but nonlinear ones can violate the maximum-entropy condition by failing to preserve Gaussianity.A transformation that reduces uncertainty in both position and momentum is therefore not allowed within ERL mechanics.Source: §1 · pp. 7–8, Figure 1 · p. 8
• Linear symplectic transformations are the allowed reversible transformations because they preserve both the classical uncertainty principle and the maximum-entropy Gaussian form.They are generated by Hamiltonians at most quadratic in canonical coordinates and include displacements, rotations, and squeezing.Source: §1 · pp. 7–8

2. Perfect knowledge of quadrature variables

• For one degree of freedom, valid epistemic states can have perfect knowledge of one variable and no knowledge of its canonically conjugate partner.These states correspond in quantum theory to eigenstates of quadrature operators, which are linear combinations of position and momentum.Source: §2 · pp. 8–9
• Infinite squeezing along position produces a Dirac delta for position and a uniform momentum distribution, yielding the ERL analogue of a position eigenstate.The limit is obtained as s → 0 from a Gaussian state whose position uncertainty vanishes while momentum uncertainty diverges.Source: §2 · p. 8
• Any quadrature qθ = cos(θ)q + sin(θ)p can be known perfectly because linear symplectic transformations map valid epistemic states to valid states.The conjugate quadrature qθ+π then has no knowledge in the corresponding limiting state.Source: §2 · pp. 8–9
• For composite systems, ERL permits product states with arbitrary quadrature knowledge on each component as well as correlated epistemic states.The construction extends the single-degree-of-freedom argument through componentwise linear symplectic transformations.Source: §2 · p. 9

3. Correlated epistemic states

• Perfect knowledge of relative position qA − qB and total momentum pA + pB can be obtained while their conjugate variables remain unknown.This follows from a symplectic transformation acting on a valid epistemic state.Source: §3 · p. 9
• The ERL state with qA − qB = 0 and pA + pB = 0 reproduces the correlations of the maximally entangled EPR state.It is constructed as the limit of Gaussians squeezed along those two collective variables.Source: §3 · p. 9

4. Variables whose values can be jointly known

• Variables can be jointly known in ERL mechanics precisely when they commute under the Poisson bracket, paralleling joint measurability of commuting quantum observables.Canonical positions provide the basic jointly knowable set, and symplectic transformations preserve the relevant commutation structure.Source: §4 · p. 9

5. The impossibility of concentrating uncertainty in a subsystem

• For composite systems, the epistemic restriction applies to subsystem marginals, so uncertainty cannot be concentrated into one subsystem to make another perfectly known.The result follows from covariance-matrix positivity and the fact that Gaussian marginals preserve the maximum-entropy condition.Source: §5 · p. 10
• The same prohibition extends to symplectic partitions and virtual subsystems, including attempts to make qA − qB and pA − pB certain by relaxing qA + qB and pA + pB.Thus the restriction is stronger than Liouville’s theorem for isolated systems and applies beyond physically separated subsystems.Source: §5 · p. 10, Figure 3 · p. 10
• ERL mechanics strengthens Liouville’s theorem by ruling out transformations that squeeze uncertainty entirely out of one subsystem, while allowing uncertainty to be shuffled between subsystems.This boundary is illustrated by the contrast between the allowed and forbidden transformations in Figure 3.Source: Figure 3 · p. 10, §5 · p. 10

6. Measurements of canonical variables

• Only sets of canonical variables that commute under the Poisson bracket can be jointly measured in ERL mechanics.Perfectly correlated systems transmit the inferred variables to the unmeasured partner; noncommuting inferences would violate the epistemic restriction.Source: §6 · p. 11
• The measurement discussion initially treats canonical-variable measurements and leaves nonlinear variables and probabilistic-outcome measurements for later characterization.These are identified as analogues of projective measurements and POVMs, respectively.Source: §6 · p. 11

7. Transformations induced by measurements of canonical variables

• A reproducible measurement of one quadrature leaves that quadrature perfectly known and makes its conjugate quadrature completely unknown.The update combines Bayesian learning of the measured value with a uniform random shift in the conjugate variable.Source: §7 · p. 12
• Unknown conjugate disturbance makes consecutive position and momentum statistics depend on measurement order, reproducing quantum noncommutativity.A prior position measurement is deterministic when repeated first but becomes probabilistic after an intervening momentum measurement.Source: §7 · p. 12

8. Modeling measurements with deterministic dynamics: the motility of the cut

• The measurement disturbance can arise from deterministic dynamics: a probe interaction transfers its initially unknown momentum uncertainty to the measured system.The system’s final momentum is fixed by the probe’s initial momentum, but remains unknown because that probe momentum is completely uncertain.Source: §8 · p. 13
• ERL predictions are insensitive to where the measurement cut is placed between system and apparatus, paralleling von Neumann’s motility-of-the-cut result.The probe-based account treats measurement as an internal dynamical process rather than an external stochastic intervention.Source: §8 · p. 13

C. Some quantum phenomena reproduced in ERL mechanics

• The reproduced quantum phenomena concern selected qualitative features rather than necessarily all quantitative predictions of quantum theory.The paper explicitly limits its claim to aspects previously taken to rule out classical explanations.Source: C. Some quantum phenomena reproduced… · p. 13

1. The EPR thought experiment

• For the EPR correlations considered, ERL explains remote-state updating as knowledge refinement about pre-existing correlated variables, without changing particle B’s ontic state.The model reproduces the stated EPR correlations but not Bell-inequality violations, which are outside its locally causal construction.Source: §1 · pp. 13–14

2. The no-cloning theorem

• In ERL mechanics, a pair of epistemic states can be cloned if and only if their distributions are disjoint or identical.Classical fidelity cannot decrease under processing, whereas cloning would require reduced fidelity for nondisjoint states.Source: §2 · pp. 14–15
• The no-cloning analogy follows by replacing quantum inner-product overlap with classical fidelity between probability distributions.Quantum cloning preserves inner products and therefore permits cloning only for orthogonal or identical states.Source: §2 · pp. 14–15
• Only ERL mechanics reproduces no-cloning for sets of pure quantum states, because its pure epistemic states can overlap.Without the epistemic restriction, pure Liouville states are point distributions and can be cloned.Source: §2 · p. 15

3. Teleportation

• ERL teleportation transfers Victor’s knowledge of an unknown particle to Bob’s particle after Alice communicates only two real parameters.Alice measures relative position and total momentum, then Bob applies corresponding phase-space shifts; the original particle is disturbed, preventing cloning.Source: §3 · p. 16
• The same ERL protocol transfers correlations with a third particle, providing a model of entanglement swapping for Gaussian states.The transferred knowledge applies to Bob’s particle even when Victor’s original state was correlated with another system.Source: §3 · p. 16

IV. OPERATIONAL EQUIVALENCE OF ERL MECHANICS AND GAUSSIAN QUANTUM MECHANICS

• Gaussian quantum mechanics is operationally equivalent to ERL mechanics with λ = ℏ.Operational equivalence means corresponding preparations, transformations, and measurements produce precisely the same statistics for every experiment.Source: §IV · p. 17
• Gaussian quantum mechanics is the subtheory whose allowed preparations, measurements, and transformations have Gaussian Wigner representations.This operational restriction supplies the quantum theory being matched to ERL mechanics.Source: §IV · p. 17

A. Operational formulation of ERL mechanics

• ERL mechanics specifies valid preparations as phase-space distributions satisfying the epistemic restriction, then characterizes the compatible transformations and measurements.The operational formulation focuses on allowed procedures and outcome probabilities without requiring an ontological description.Source: A. Operational formulation of ERL… · p. 17, §IV · p. 17

1. General measurements

• Measurements need not determine outcomes from the system’s ontic state alone, because unknown ancillary degrees of freedom can make outcomes probabilistic.Quadratic interactions with ancillas can yield partial information about conjugate variables, while ignored environments produce effective noisy measurements.Source: §1 · p. 17
• A general measurement is a family of indicator functions ξ_y(z), where ξ_y(z)dy gives the probability of outcome y for ontic state z.The indicators are nonnegative and normalized so that some outcome occurs with certainty.Source: §1 · p. 17
• An indicator function is valid exactly when its normalized form satisfies the epistemic constraint and preserves validity on correlated systems.The condition is derived by requiring measurements on one subsystem to leave every allowed state of the other subsystem valid.Source: §1 · p. 17
• Valid indicator functions can be organized into normalized outcome sets, including phase-space displacement families labeled by points in the outcome space.A translated valid indicator function yields a valid measurement set when the functions integrate to one for every ontic state.Source: §1 · p. 18

2. General transformations

• A valid transformation must be completely validity-preserving and must supervene on linear symplectic dynamics with valid ancillary systems and marginalization.Validity preservation alone is insufficient, as illustrated by momentum reversal, which is not symplectic and cannot be implemented under the restriction.Source: §2 · p. 19
• Every completely validity-preserving transformation satisfies ontic supervenience, and valid transformations can be characterized by acting on half of a perfectly correlated state.This gives the transformation criterion in terms of the resulting bipartite epistemic state.Source: §2 · p. 19
• Valid transfer functionals are in one-to-one correspondence with valid bipartite epistemic states having a uniform marginal, analogous to the Choi isomorphism.The correspondence underlies the equivalence between complete validity preservation and the bipartite-state characterization.Source: §2 · pp. 19–20
• Ontically realizable transformations act on covariance matrices through Gaussian channel parameters X and Y constrained by an uncertainty relation.They arise by coupling the system to a valid environment through a joint linear symplectic transformation and then marginalizing.Source: §2 · p. 20
• The covariance-matrix characterization is equivalent to validity of the bipartite state produced by acting on half of the perfectly correlated state.This establishes the link between ancilla-based classical realizations and the bipartite-state representation.Source: §2 · pp. 21–22
• Measurement updates need not follow a unique projection-like rule because the update depends on how the measurement is physically implemented.Measurements are modeled by apparatus degrees of freedom, symplectic coupling, and acquisition of apparatus information.Source: §2 · p. 22

• The section introduces the Wigner representation and then proceeds to define Gaussian quantum mechanics.Further reading is provided in Refs..Source: B. Operational formulation of Gaussian… · p. 22

1. The Wigner representation

• Quantum states are represented by real-valued phase-space functions normalized to unity.This representation is introduced for systems with n canonical degrees of freedom.Source: §1 · p. 22
• The Wigner representation of a Hermitian observable is a real-valued function, and quantum expectation values are recovered by the Euclidean inner product of operator and state representations.Source: §1 · p. 22
• POVM measurements become phase-space functions whose Euclidean inner product with the state representation recovers the Born-rule outcome probability.The measurement functions sum to the uniform phase-space measure.Source: §1 · pp. 22–23
• A completely-positive trace-nonincreasing map is represented by a real-valued function over two phase spaces, extending the Wigner representation to general transformations.The paper notes that this representation had not previously been made explicit in the Wigner-representation literature.Source: §1 · p. 23

2. Gaussian quantum mechanics

• Gaussian states are defined as quantum states whose Wigner representations have Gaussian form, equivalently to having Gaussian Wigner-characteristic functions.Their covariance matrix is γ and their mean vector is d.Source: §2 · p. 23
• Gaussian measurements are those that preserve Gaussianity on the unmeasured part of a bipartite Gaussian state, and their Wigner representations are Gaussian.Source: §2 · p. 23
• Gaussian operations are those that map Gaussian states to Gaussian states, with transformations represented by Gaussian functions over paired phase spaces.The paired variable is denoted z′′ ≡ (z, z′).Source: §2 · p. 23

C. Proof of Equivalence

• Gaussian Wigner functions are positive and normalized, so they can be interpreted as probability distributions over phase space.Source: C. Proof of Equivalence · p. 23
• Gaussian POVM representations are valid conditional outcome probabilities because they are positive and normalize over measurement outcomes.Source: C. Proof of Equivalence · pp. 23–24
• For Gaussian states and POVMs, measurement outcome probabilities equal the phase-space distribution of the state weighted by the conditional outcome density given the ontic state.Preparation is interpreted as sampling an unknown phase-space point, while measurement reveals information about that point.Source: C. Proof of Equivalence · p. 23
• Gaussian trace-preserving maps have positive, normalized Wigner representations that act as conditional probabilities mapping an initial phase-space point z′ to a final point z.The final distribution is obtained by integrating the transition function against the initial distribution.Source: C. Proof of Equivalence · pp. 23–24
• The proof concludes that Gaussian quantum mechanics yields the same preparation, measurement, and transformation descriptions as Liouville mechanics under the epistemic restriction.The equivalence is established by matching the uncertainty conditions with λ replaced by ℏ.Source: C. Proof of Equivalence · p. 24
• First moments and covariance matrices of quantum states coincide with those computed from their Wigner functions, transferring the quantum uncertainty relation to the phase-space representation.The Wigner representations therefore satisfy the classical uncertainty principle with λ = ℏ.Source: C. Proof of Equivalence · p. 24

• Gaussian quantum mechanics includes no-cloning, teleportation, dense coding, quantum key distribution, quantum error correction, and many aspects of entanglement theory.Source: §V · p. 25
• The framework applies to canonical fields and quantum optics, where Gaussian states, optical transformations, and homodyne-based measurements receive an epistemic-restriction interpretation.Direct photodetection is not included.Source: §V · p. 25
• The explanatory scope of ERL mechanics supports interpreting quantum states as representations of agents’ incomplete knowledge rather than reality itself.The reconstruction begins from classical ontology plus an epistemic restriction and derives Gaussian quantum mechanics.Source: §V · p. 25
• ERL mechanics cannot reproduce Bell’s theorem or the Kochen-Specker theorem, identifying these phenomena as targets for additional conceptual innovations toward full quantum theory.Source: §V · p. 25
• The discrete analogue extends the correspondence to stabilizer theory for qutrits, whereas the Spekkens Toy Theory is not operationally equivalent to qubit stabilizer theory because Bell and Kochen-Specker phenomena arise there.Source: §V · p. 25
• No quantum subtheory lies between Gaussian quantum mechanics and the full theory: adding one non-Gaussian unitary and closing under composition yields all unitaries.The analogous stabilizer-theory claim for qudits is widely believed but not rigorously proven.Source: §V · p. 26

• Without the max-ent condition, valid preparations could include non-Gaussian distributions that satisfy the classical uncertainty principle but contain near-perfect correlations between two systems.Source: Appendix A · p. 26
• Such correlated distributions would allow a Gaussian indicator measurement on A to infer both qB and pB at B with arbitrarily high accuracy.Source: Appendix A · p. 26
• This inference would violate the uncertainty component of the epistemic restriction, so every such indicator function must be ruled out.Source: Appendix A · pp. 26–27