In 2004 Robert W. Spekkens introduced a toy theory designed to make a case for the epistemic view of quantum mechanics. But how does Spekkens’ toy model differ from quantum theory? While some differences are well-established, we attempt to approach this question from a tomographic point of view. More specifically, we provide experimentally viableprocedureswhichenablesustocompletelycharacterizethestatesandgatesthatare available in the toy model. We show that, in contrast to quantum theory, decompositions of transformations in the toy model must be done in a non-linear fashion.
In 2004 Robert W. Spekkens introduced a toy theory designed to make a case for the epistemic view of quantum mechanics. But how does Spekkens’ toy model differ from quantum theory? While some differences are well-established, we attempt to approach this question from a tomographic point of view. More specifically, we provide experimentally viableprocedureswhichenablesustocompletelycharacterizethestatesandgatesthatare available in the toy model. We show that, in contrast to quantum theory, decompositions of transformations in the toy model must be done in a non-linear fashion.
In order to better understand a complex theory like quantum mechanics, it is sometimes useful to take a step back and create alternative theories, with more intuitive foundations, and examine which features of quantum mechanics can be reproduced by such a foil theory. A prominent example is Spekkens' toy theory, which is based off a simple premise: "What if we took a common classical theory and added the uncertainty principle as a postulate?" In other words, the theory imposes an epistemic restriction on our knowledge about a physical system: only half of the variables can ever be known to an observer. Like good science fiction, from this simple principle a rich behaviour emerges, most notoriously when we compose several systems. The toy theory emulates some aspects of quantum non-locality, although crucially it is still a non-contextual model. In this pedagogical review we consolidate different approaches to Spekkens' toy theory, including the stabilizer formalism and the generalization to arbitrary dimensions, completing them with new results where necessary. In particular, we introduce a general characterization of measurements, superpositions and entanglement in the toy theory.
Abstract Information is physical(Landauer 1961IBM J. Res. Dev.5183–91), and for a physical theory to be universal, it should model observers as physical systems, with concrete memories where they store the information acquired through experiments and reasoning. Here we address these issues in Spekkens’ toy theory (Spekkens 2005Phys. Rev.A71052108), a non-contextual epistemically restricted model that partially mimics the behaviour of quantum mechanics. We propose a way to model physical implementations of agents, memories, measurements, conditional actions and information processing. We find that the actions of toy agents are severely limited: although there are non-orthogonal states in the theory, there is no way for physical agents to consciously prepare them. Their memories are also constrained: agents cannot forget in which of two arbitrary states a system is. Finally, we formalize the process of making inferences about other agents’ experiments and model multi-agent experiments like Wigner’s friend. Unlike quantum theory (Nurgalieva and del Rio Lidia 2019Electron. Proc. Theor. Comput. Sci.287267–97; Fraseret al2020 Fitch’s knowability axioms are incompatible with quantum theory arXiv: 2009.00321 ; Frauchiger and Renner 2018Nat. Commun.93711; Nurgalieva and Renner 2021Contemp. Phys.611–24; Brukner 2018Entropy20350) or box world (Vilasiniet al2019New J. Phys.21113028), in toy theory there are no inconsistencies when physical agents reason about each other’s knowledge.
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 non-prime. This equivalence provides new technical tools for the study of technically difficult compound-dimensional SQM.
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 non-prime. This equivalence provides new technical tools for the study of technically difficult compound-dimensional SQM.
We provide two mathematical descriptions of Spekkens's toy qubit theory, an inductively one in terms of a small set of generators, as well as an explicit closed form description. It is a subcategory MSpek of the category of finite sets, relations and the cartesian product. States of maximal knowledge form a subcategory Spek. This establishes the consistency of the toy theory, which has previously only been constructed for at most four systems. Our model also shows that the theory is closed under both parallel and sequential composition of operations (= symmetric monoidal structure), that it obeys map-state duality (= compact closure), and that states and effects are in bijective correspondence (= dagger structure). From the perspective of categorical quantum mechanics, this provides an interesting alternative model which enables us to describe many quantum phenomena in a discrete manner, and to which mathematical concepts such as basis structures, and complementarity thereof, still apply. Hence, the framework of categorical quantum mechanics has delivered on its promise to encompass theories other than quantum theory.