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.
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.
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.
Abstract Quantum theory has an epistemic horizon, i.e. exact values cannot be assigned simultaneously to incompatible physical quantities. As shown by Spekkens’ toy theory, positing an epistemic horizon akin to Heisenberg’s uncertainty principle in a classical mechanical setting also leads to a plethora of quantum phenomena. We introduce a deterministic theory—nomic toy theory—in which information gathering agents are explicitly modelled as physical systems. Our main result shows the presence of an epistemic horizon for such agents. They can only simultaneously learn the values of observables whose Poisson bracket vanishes. Therefore, nomic toy theory has incompatible measurements and the complete state of a physical system cannot be known. The best description of a system by an agent is via an epistemic state of Spekkens’ toy theory. Our result reconciles us to measurement uncertainty as an aspect of the inseparability of subjects and objects. Significantly, the claims follow even though nomic toy theory is essentially classical. This work invites further investigations of epistemic horizons, such as the one of (full) quantum theory.
One of the main questions in the field of quantum computation is where the quantum computational speed-up comes from. Recent studies in the field of quantum foundations have suggested which are the features to be considered as inherently non-classical. One of the major contributions in this direction comes from a result known as Spekkens' toy theory, which is a model built to reproduce quantum theory as a classical phase-space-inspired theory with restrictions on what an observer can know about reality. The model reproduces many of the features of quantum mechanics, but it does not reproduce non-locality and contextuality. In this thesis we first complete Spekkens' toy theory with measurement update rules and a mathematical framework that generalises it to systems of any finite dimensions (prime and non-prime). We also extend the operational equivalence between the toy theory and stabilizer quantum mechanics to all odd dimensions via Gross' Wigner functions. We then use the toy theory to represent the non-contextual and classically simulatable part of the computation in state-injection schemes of quantum computation where contextuality is a resource. In the case of qubits, we show that the subtheories of quantum mechanics represented in the toy model can achieve the full stabilizer theory via state-injection and we associate different proofs of contextuality to different injection processes. Stepping back from Spekkens' toy theory, we conclude by focusing on single system protocols that compute non-linear functions (similarly to the popular CHSH game) which show quantum advantages even in absence of non-locality and contextuality (in its standard notions). We analyse their performances (formalised in Bell's and Tsirelson's bounds) in relation to Landauer's principle, which associates entropic costs to irreversible computations, and to a new notion of contextuality for sequences of transformations.
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 (SM) is a non-contextual hidden-variable model made to support the epistemic view of quantum theory, where quantum states are states of partial knowledge about a deeper underlying reality. Despite being a classical model, it has reproduced many features of quantum theory (entanglement, teleportation, . . .): (almost) everything but contextuality, which therefore seems to be the inherent quantum feature. In addition to the importance in foundation of quantum theory, the notion of contextuality seems to be a crucial resource for quantum computation. In particular it has been proven that, in the case of odd prime discrete dimensional systems, contextuality is necessary for universal quantum computation in state-injection schemes of computation based on stabilizer quantum mechanics (SQM). The latter is a subtheory of quantum mechanics which is very popular in the field of quantum computation and quantum error correction. State-injection schemes consist of a classically-simulable part (like SQM) and a resource state that boosts the computation to a quantum improvement. In the odd-dimensional case, SM is operationally equivalent to SQM. In the even-dimensional case, the equivalence only holds in terms of structure, not in terms of statistical predictions. This because qubit-SQM shows contextuality, while qudit(odd dimensions)-SQM does not. We believe that SM can be a valid tool to study contextuality as a resource in the field of quantum computation. Restricted versions of SM compatible with quantum mechanics (QM) can be used as the non-contextual classically-simulable part of state-injection schemes thus opening other scenarios where studying if contextuality is necessary for quantum computational speed-up.