Spekkens’ toy model in all dimensions and its relationship with stabiliser quantum mechanics

Abstract

Spekkens' toy model is a non-contextual hidden variable model with an epistemic restriction, a constraint on what an observer can know about reality. The aim of the model, developed for continuous and discrete prime degrees of freedom, is to advocate the epistemic view of quantum theory, where quantum states are states of incomplete knowledge about a deeper underlying reality. Many aspects of quantum mechanics and protocols from quantum information can be reproduced in the model. In spite of its significance, a number of aspects of Spekkens' model remained incomplete. Formal rules for the update of states after measurement had not been written down, and the theory had only been constructed for prime-dimensional and infinite dimensional systems. In this work, we remedy this, by deriving measurement update rules and extending the framework to derive models in all dimensions, both prime and non-prime. Stabiliser quantum mechanics (SQM) is a sub-theory of quantum mechanics with restricted states, transformations and measurements. First derived for the purpose of constructing error correcting codes, it now plays a role in many areas of quantum information theory. Previously, it had been shown that Spekkens' model was operationally equivalent to SQM in the case of odd prime dimensions. Here, exploiting known results on Wigner functions, we extend this to show that Spekkens' model is equivalent to SQM in all odd dimensions, prime and nonprime. This equivalence provides new technical tools for the study of technically difficult compounddimensional SQM.

State-injection schemes of quantum computation in Spekkens' toy theory

“…Let us start by stating the list of instructions to construct the maximal Spekkens' subtheory that corresponds to a given choice of γ (in analogy with the framework of [18]). 3 We recall that the function γ uniquely specifies the function β(λ, λ ), defined such that T 2. The set of allowed states S is given by the states corresponding to common eigenstates of d n commuting observables in M.…”

“…These turn out to be exactly equivalent to Spekkens' epistemic states and measurements. The measurement update rules are consistent and positivenesspreserving, while Clifford gates are mapped into consistent symplectic affine transformations [3].…”

A Classical Formulation of Quantum Theory?

“…Spekkens’ model has previously been provided with a contextual extension, but without fully capturing quantum theory [ 21 ]. It has also been shown to hold strong similarities to stabilizer physics [ 22 , 23 , 24 ], thereby providing a link to a subtheory of quantum mechanics that plays an important role in quantum computing. Spekkens’ model is naturally set in phase space, which provides the backbone for our work.…”

A quantum prediction as a collection of epistemically restricted classical predictions

“…Moreover, part of our goal is to create a formulation of quantum mechanics that clearly displays the relationship between an epistemically restricted classical theory and the full quantum theory. The connection between different versions of Spekkens' model and the stabilizer subtheory of quantum mechanics has been thoroughly studied previously [3,8,9] The goal of expressing quantum theory in terms of actual probability distributions has motivated other authors as well. Fuchs and Schack [10] have shown how to express states, transformations, and Born's rule in terms of genuine probability distributions, not over phase space but over the set of outcomes of a symmetric informationally complete measurement.…”