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

Spekkens’ toymodel is a non-contextual hidden variablemodel with an epistemic restriction, a constraint onwhat an observer can know about reality. The aimof themodel, 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 quantummechanics and protocols fromquantum information can be reproduced in themodel. In spite of its significance, a number of aspects of Spekkens’model remained incomplete. Formal rules for the update of states aftermeasurement had not beenwritten down, and the theory had only been constructed for prime-dimensional and infinite dimensional systems. In this work, we remedy this, by derivingmeasurement update rules and extending the framework to derivemodels in all dimensions, both prime and non-prime. Stabiliser quantummechanics (SQM) is a sub-theory of quantummechanics with restricted states, transformations andmeasurements. First derived for the purpose of constructing error correcting codes, it nowplays a role inmany 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 onWigner 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.