State space dimensionality in short memory hidden variable theories

Abstract

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 Schrödinger equation. A method for generating the economical non-Markovian model for the qubit is also presented.

Added a lemma on the dynamical properties of the set S that solves some bugs in the proof of the theorem in section IV

References in corpus (8)

• In defense of the epistemic view of quantum states: a toy theory
• Contextuality for preparations, transformations, and unsharp measurements
• Matchgates and classical simulation of quantum circuits
• Exponential complexity and ontological theories of quantum mechanics
• A condition for any realistic theory of quantum systems
• Efficient Hidden-Variable Simulation of Measurements in Quantum Experiments
• Economical ontological models for discrete quantum systems
• Compressing the hidden variable space of a qubit

Cited by in corpus (18)

• On the reality of the quantum state
• Is the quantum state real? An extended review of $\psi$-ontology theorems
• Is a system's wave function in one-to-one correspondence with its elements of reality?
• Are quantum states real?
• Distinct Quantum States Can Be Compatible with a Single State of Reality
• Implications of the Pusey-Barrett-Rudolph quantum no-go theorem
• Quantum Theory and Determinism
• No Return to Classical Reality
• No-go theorems for ψ-epistemic models based on a continuity assumption
• Dynamics of a qubit as a classical stochastic process with time-correlated noise: minimal measurement invasiveness
• Epistemic view of quantum states and communication complexity of quantum channels
• Communication complexity and the reality of the wave-function
• Communication cost of classically simulating a quantum channel with subsequent rank-1 projective measurement
• Onthological models predictively inequivalent to quantum theory
• Measurement contextuality is implied by macroscopic realism
• Exponential communication gap between weak and strong classical simulations of quantum communication
• Stochastic Mechanics Without Ad Hoc Quantization: Theory And Applications To Semiclassical Gravity
• A Simplified Basis for Bell-Kochen-Specker Theorems