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State space dimensionality in short memory hidden variable theories

Alberto Montina

arXiv:1008.4415v2quant-ph

TL;DR

The paper studies whether hidden-variable theories can reduce ontological state-space dimension while retaining short-memory dynamics, a tension exposed by the economical qubit model. It refines the no-shrinking analysis using invertible processes, topology, and regularity conditions, and develops a construction method for the non-Markovian model. It concludes that Markov theories cannot shrink below the quantum state-manifold dimension, whereas the qubit model achieves half the Bloch-sphere dimension by abandoning Markovian dynamics.

  • Problem

    The paper asks whether a hidden-variable state space can be smaller than the quantum state manifold while preserving Markov dynamics, given the earlier no-shrinking result's trajectory-relaxation assumption.

    Source: §I · pp. 1–2

  • Method

    The paper introduces invertible processes and regularity properties of conditional probabilities to analyze trajectory relaxation, topology, and construction of an economical non-Markovian qubit model.

    Source: §I · p. 2, A. Conditional probabilities and… · p. 4, B. Set S(X) and its symmetry property · p. 6

  • Results

    Markov hidden-variable theories cannot have ontological dimension below the quantum state-manifold dimension, while the qubit model uses a one-dimensional space, half the Bloch-sphere dimension, with non-Markovian dynamics.

    Source: §V · pp. 8–9

  • Takeaways & Limitations

    Dimensional reduction can let a single realization carry less information than the quantum state, but in the presented qubit model this requires rejecting short-memory dynamics.

    Source: §V · p. 9

    Whether ontological shrinking is possible for a general N-dimensional Hilbert space without Markov dynamics remains an open question.

    Source: §V · p. 9

Abstract

from arXiv · show

Recently we have presented a hidden variable model of measurements for a qubit where the hidden variable state space dimension is one-half the quantum state manifold dimension. The absence of a short memory (Markov) dynamics is the price paid for this dimensional reduction. The conflict between having the Markov property and achieving the dimensional reduction was proved in [A. Montina, Phys. Rev. A, 77, 022104 (2008)] using an additional hypothesis of trajectory relaxation. Here we analyze in more detail this hypothesis introducing the concept of invertible process and report a proof that makes clearer the role played by the topology of the hidden variable space. This is accomplished by requiring suitable properties of regularity of the conditional probability governing the dynamics. In the case of minimal dimension the set of continuous hidden variables is identified with an object living an N-dimensional Hilbert space, whose dynamics is described by the Schr\"odinger equation. A method for generating the economical non-Markovian model for the qubit is also presented.

I. INTRODUCTION

The paper motivates reduced ontological spaces as a possible way to represent single quantum realizations with fewer resources, while examining their tension with Markov dynamics. It develops an invertibility-based refinement of the no-shrinking result and reviews an economical non-Markovian qubit model.

Motivation

  • Quantum-state resources grow exponentially for composite systems, motivating hidden-variable descriptions that may carry less information about a single realization.The quantum state contains complete event-probability information, whereas an actual measurement produces a single outcome.Source: §I · p. 1
  • A reduced ontological space could provide an efficient classical weak simulation by tracking single-realization dynamics rather than computing quantum probabilities.The paper distinguishes strong simulation of probability weights from weak simulation of outcomes sampled according to those weights.Source: §I · p. 1

Prior results

  • Known Markov hidden-variable theories do not achieve ontological shrinking, and the no-shrinking theorem required an additional trajectory-relaxation hypothesis.The qubit model reviewed here has a one-dimensional state space, half the Bloch-sphere dimension, but its dynamics is non-Markovian.Source: §I · p. 1

Contribution

  • The paper analyzes trajectory relaxation through invertible processes and uses regularity properties of conditional probabilities to clarify how hidden-space topology enters the theorem.It identifies a sub-region of a compact ontological space where all processes are invertible and emphasizes that dimension is topological rather than cardinality-based.Source: §I · p. 2
  • For minimal dimension, the continuous hidden variables can be identified with an object in an N-dimensional Hilbert space whose dynamics follows the Schrödinger equation.The paper also presents a systematic construction method for the economical non-Markovian qubit model.Source: §I · p. 2

II. GENERAL FRAMEWORK

The framework represents quantum states by probability distributions over continuous and discrete ontic variables, with preparation and measurement contexts allowed to affect those distributions. Ontic dynamics is modeled by conditional transition probabilities, and the short-memory assumption imposes Markovian evolution.

Ontic representation

  • A hidden-variable theory maps each quantum state to a probability distribution over an ontological space of continuous variables and a possible discrete index.The ontological variables are intended to contain the information about a single realization.Source: §II · p. 2

Preparation

  • Preparation-context parameters η allow the ontic distribution to depend on how the same quantum state was physically prepared.The framework introduces these parameters because distinct preparation procedures can produce the same quantum state.Source: §II · p. 2

Measurement

  • Measurement probabilities are represented by a conditional probability for an event given the ontic state, with possible dependence on measurement-context parameters τ.This conditional probability must reproduce the corresponding quantum probability.Source: §II · p. 2

Dynamics

  • The framework assumes Markovian ontic dynamics, in which a transition probability K(X2,t2|X1,t1) governs evolution between times and satisfies the Chapman–Kolmogorov equation.At equal times, the transition kernel is a delta distribution.Source: §II · p. 2
  • For time-homogeneous processes, the transition probability depends only on the time difference, while transformation contexts χ may distinguish physically different implementations of the same unitary operator.Consequently, even an identity unitary need not correspond to a delta transition for every implementation context.Source: §II · p. 2

Unitary processes

  • Quantum unitary evolution is organized as trajectories generated by Hamiltonians, with experimentally available building-block processes concatenated to realize other evolutions.The framework links conditional transition probabilities to the corresponding unitary operators.Source: §II · p. 2

III. ECONOMICAL ONTOLOGICAL MODEL OF MEASUREMENTS

The paper presents a one-dimensional hidden-variable model for qubit measurements, extending it across the Bloch sphere with finite discrete information. The model achieves ontological shrinking, but its dynamics is non-Markovian.

Ontological shrinking

  • The construction is an explicit example of ontological shrinking: a single ontic realization contains less information than the quantum state, while the full state is encoded in ρ.For a qubit, the ontic space is smaller than the two-dimensional Bloch sphere.Source: §III · p. 3

Model construction

  • A one-dimensional ontic space with continuous x and binary n represents qubit states through distributions determined by the Bloch-vector angles.The continuous variable records a zenith or azimuth angle according to n, producing two delta peaks for each prepared state.Source: §III · p. 3, Figure 1 · p. 3

Measurement probabilities

  • The probability functions satisfy the required measurement condition and define event probabilities through the conditional distributions P(w|x,n).Events with negative w_z are represented through the complementary relation P(−w|x,n)=1−P(w|x,n).Source: §III · p. 3

Model construction

  • The model initially works only inside a Bloch-sphere cone, then covers the whole sphere using 12 coordinate patches labeled by a discrete index.The cone has aperture 2θ0, with θ0 ≃53.13 degrees.Source: §III · p. 3

Non-Markovian dynamics

  • The dimensional reduction requires abandoning a short-memory description because no positive conditional probability can represent every unitary evolution.For σ_y evolution, the relevant equation cannot be satisfied because its two sides depend differently on the azimuthal angle ϕ.Source: §III · p. 3
  • The model’s non-Markovian character is demonstrated by the failure of any function K to satisfy the required dynamical equation for the distribution.The left-hand side depends on both θ and ϕ, whereas the right-hand side depends only on θ.Source: §III · p. 4

IV. ONTOLOGICAL SHRINKING AND MARKOV PROCESSES

The paper examines why ontological shrinking conflicts with Markov dynamics and frames the proof around regularity, invertibility, topology, and minimal-dimensional consequences.

IV. ONTOLOGICAL SHRINKING AND MARKOV PROCESSES

  • The section proves that ontological shrinking contradicts Markov dynamics under a regularity-based treatment of conditional probabilities and invertible processes.It also connects minimal dimension with continuous hidden variables evolving as an N-dimensional Hilbert-space object.Source: §IV · p. 4

A. Conditional probabilities and associated unitary operators

This section develops regularity properties for Markov transition probabilities and uses them to associate ontic processes with manifolds of unitary transformations. The argument depends essentially on the local topology of the ontological space.

Regularity assumptions

  • The proof assumes a weak regularity condition: part of the support of K changes smoothly with the initial ontic state and evolution time.This condition covers Markov processes involving drift, diffusion, jumps, and combinations of these behaviors.Source: A. Conditional probabilities and… · p. 4
  • Property 1 tracks infinitesimal changes in the initial state and time through a matrix λ and vector α that shift the destination ontic state.For deterministic processes these quantities are unique, while stochastic processes may permit multiple choices.Source: A. Conditional probabilities and… · p. 4

Topological role

  • The topology of the ontological space is essential because the proof uses its local Euclidean structure rather than merely the cardinality of the state space.This is what allows dimension to constrain the possible Markov dynamics.Source: A. Conditional probabilities and… · p. 4

Unitary processes

  • Unitary evolutions can be generated from physically attainable building-block processes and concatenated with auxiliary transformations to represent general generators.The parameter t in the resulting conditional probability need not be the duration of a purely time-homogeneous process.Source: A. Conditional probabilities and… · pp. 4–5

Shift property

  • Property 2 propagates nonzero transition support under infinitesimal changes of the unitary parameters, with a shift matrix η describing the corresponding ontic displacement.The property extends by induction to concatenations of processes.Source: A. Conditional probabilities and… · p. 5

Dimensional consequence

  • A nonzero transition process is associated with a unitary submanifold of dimension Ds satisfying Ds ≥ D−M, where M is the ontological-space dimension.The actual dimension is D−NI, with NI the number of independent constraints imposed by ηδt=0.Source: A. Conditional probabilities and… · p. 5

B. Set S(X) and its symmetry property

This section studies the set S(X) of quantum states compatible with an ontic state and uses invertibility and symmetry to constrain its evolution. The resulting symmetry bound supports the no-shrinking argument.

Definition of S(X)

  • S(X) contains exactly the Hilbert-space vectors whose preparations can assign nonzero probability to the ontic state X.It therefore records which quantum states are compatible with a given ontic state.Source: B. Set S(X) and its symmetry property · pp. 5–6

Process action

  • A nonzero process X→Y associated with U implies U S(X)⊆S(Y), so the compatible-state set cannot lose vectors along the process.The reverse inclusion does not follow automatically and would require invertibility.Source: B. Set S(X) and its symmetry property · p. 6

Invertible processes

  • Invertibility is introduced by requiring a reverse nonzero-probability process, and every process is assumed invertible after insignificant transient states are discarded.Compactness is used to argue that sufficiently long process sequences approach invertibility.Source: B. Set S(X) and its symmetry property · p. 6
  • Under invertibility, a process maps compatible-state sets exactly, yielding U S(X)=S(Y) rather than only inclusion.The proof uses repeated applications of U approaching the inverse transformation.Source: B. Set S(X) and its symmetry property · p. 6

Symmetry restriction

  • S(X) cannot contain every Hilbert-space vector because that would make orthogonal quantum states overlap on the same ontic state, contradicting perfect discrimination.Equivalently, S(X) is not invariant under the full group SU(N).Source: B. Set S(X) and its symmetry property · pp. 6–7
  • Any invariant set under a Lie subgroup of dimension greater than (N−1)^2 must contain every Hilbert-space vector, including the special N=4 case considered through Sp(2).Thus a proper compatible-state set cannot have symmetry larger than the stated bound.Source: B. Set S(X) and its symmetry property · p. 7

Dimensional consequence

  • The orientation of a nontrivially symmetric set S requires at least 2N−2 parameters, except for a completely symmetric set with no orientation.This parameter bound is linked to the symmetry group SU(N−1)×U(1).Source: B. Set S(X) and its symmetry property · p. 7

C. No-shrinking theorem

Under regularity and invertibility conditions, the no-shrinking theorem shows that a Markov ontological space cannot have lower dimension than the quantum state manifold.

C. No-shrinking theorem

  • The theorem establishes M ≥ 2N −2 for Markov ontological theories satisfying Property 1 with invertible processes.Source: C. No-shrinking theorem · p. 7
  • Assuming M < 2N −2 yields a unitary-evolution manifold with dimension Ds > (N−1)^2 associated with a nonzero-probability process.Source: C. No-shrinking theorem · p. 7
  • The resulting symmetry forces S(X) to be invariant under SU(N), contradicting lemma 3 and ruling out a smaller ontological dimension.Source: C. No-shrinking theorem · p. 7

D. Minimal dimension and Schr¨odinger equation

At minimal ontological dimension, continuous hidden variables correspond to a Hilbert-space vector whose rigid evolution obeys the Schrödinger equation, with a possible discrete index completing the state description.

D. Minimal dimension and Schr¨odinger equation

E. Resource cost with round off error

With fixed round-off error, discretizing M continuous ontic variables yields an error that decreases exponentially with information at a rate inversely proportional to M, implying exponential resource growth for composite systems.

E. Resource cost with round off error

V. CONCLUSIONS

The paper clarifies that Markov hidden-variable theories cannot achieve ontological shrinking, while a one-dimensional qubit model achieves it only through non-Markovian dynamics.

V. CONCLUSIONS

  • Short-memory hidden-variable theories require an ontological dimension at least as large as the quantum state manifold dimension, preserving exponential information growth.Source: §V · p. 8
  • A compact ontological space contains trajectories converging to a region where all processes are invertible, strengthening the relaxation justification.Source: §V · p. 8
  • The qubit measurement model uses a one-dimensional ontological space, half the Bloch-sphere dimension, but its dynamics is non-Markovian.Source: §V · p. 9
  • The general N-dimensional non-Markovian ontological-shrinking problem remains open, while dropping causality is discussed as a related possibility.Source: §V · p. 9

Appendix A: Deriving the economical model for a qubit

The appendix systematically constructs an economical qubit hidden-variable model using a continuous variable, a binary index, and two-point-support distributions. It derives a bijective Bloch-sphere parametrization, conditional event probabilities, and coordinate-system freedom, while noting that the construction initially covers only a subset of preparation states.

Model construction

  • The model represents each prepared Bloch vector with two ontic points, whose positions and relative weight provide three real parameters covering the Bloch sphere.The discrete index selects one of two continuous-variable values with probabilities r(0|v) and r(1|v).Source: Appendix A · p. 9

Conditional probabilities

  • The conditional probabilities are constructed from the event Bloch vector w and the model’s coordinate functions so that the Born probability is recovered in Bloch-vector form.The event |φ⟩ is represented by w, while the preparation |ψ⟩ is represented by v(x0, x1).Source: Appendix A · pp. 9, 11, 12
  • The construction determines conditional event probabilities by solving four linear relations for the four functions associated with the two ontic indices.The relations use the four pairs of continuous-variable values formed from two alternative preparations.Source: Appendix A · p. 9

Scope and extension

  • The constructed model works only for a subset of preparation states, and outside that subset, H(w) is not necessarily given by Eqs. (A23,A24).Source: Appendix A · p. 10

Bloch-sphere parametrization

  • A vector-valued mapping parametrizes the Bloch sphere through coordinates x0 and x1, with the inverse functions fn(v) recovering those coordinates.The mapping is locally invertible only in the case where the relevant four-vectors span orthogonal two-dimensional subspaces.Source: Appendix A · pp. 10–11

Coordinate systems

  • The resulting mapping is bijective after removing the poles x1 = 0 and x1 = π, and its coordinate systems are related by Möbius transformations.The parameter θ0 controls the angular distance 2θ0 between the two poles; θ0 = π/2 gives spherical coordinates.Source: Appendix A · p. 11
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