State space dimensionality in short memory hidden variable theories

TL;DR

This paper explores a hidden variable model for qubits with reduced state space dimensionality, analyzing the role of topology and regularity in non-Markovian dynamics, and presents a method to generate such models.

Contribution

It provides a detailed analysis of the topology and regularity conditions enabling a minimal-dimensional hidden variable model for qubits, including a new proof and a generation method.

Findings

Hidden variable space can be of minimal dimension, equivalent to an N-dimensional Hilbert space.

The Schrödinger equation describes the dynamics of the hidden variables in the minimal case.

A method to generate non-Markovian models with reduced dimensionality is proposed.

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 Schr\"odinger equation. A method for generating the economical non-Markovian model for the qubit is also presented.