Spekkens' toy theory is a noncontextual hidden variable model with an epistemic restriction, a constraint on what the observer can know about the reality. In reproducing many features of quantum mechanics in an essentially classical model, it clarified our understanding of what behavior can be truly considered intrinsically quantum. In this work, we show that Spekkens' theory can be also used to help understanding aspects of quantum computation---in particular, an important subroutine in fault-tolerant quantum computation called state injection. State injection promotes fault-tolerant quantum circuits, which are usually limited to the classically efficiently simulatable stabilizer operations, to full universal quantum computation. We show that the limited set of fault-tolerant operations used in standard state-injection circuits can be realized within Spekkens' theory, and that state injection leads to nonclassicality in the form of contextuality. To achieve this, we extend prior work connecting Spekkens' theory and stabilizer quantum mechanics, showing that subtheories of the latter can be represented within Spekkens' theory, in spite of the contextuality in qubit stabilizer quantum mechanics. The work shines light on the relationship between quantum computation and contextuality.
Spekkens' toy theory is a noncontextual hidden variable model with an epistemic restriction, a constraint on what the observer can know about the reality. In reproducing many features of quantum mechanics in an essentially classical model, it clarified our understanding of what behavior can be truly considered intrinsically quantum. In this work, we show that Spekkens' theory can be also used to help understanding aspects of quantum computation---in particular, an important subroutine in fault-tolerant quantum computation called state injection. State injection promotes fault-tolerant quantum circuits, which are usually limited to the classically efficiently simulatable stabilizer operations, to full universal quantum computation. We show that the limited set of fault-tolerant operations used in standard state-injection circuits can be realized within Spekkens' theory, and that state injection leads to nonclassicality in the form of contextuality. To achieve this, we extend prior work connecting Spekkens' theory and stabilizer quantum mechanics, showing that subtheories of the latter can be represented within Spekkens' theory, in spite of the contextuality in qubit stabilizer quantum mechanics. The work shines light on the relationship between quantum computation and contextuality.
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.
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.
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.
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.
We give a complete characterization of the (non)classicality of all stabilizer subtheories. First, we prove that there is a unique nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory in all odd dimensions, namely Gross's discrete Wigner function. This representation is equivalent to Spekkens' epistemically restricted toy theory, which is consequently singled out as the unique noncontextual ontological model for the stabilizer subtheory. Strikingly, the principle of noncontextuality is powerful enough (at least in this setting) to single out one particular classical realist interpretation. Our result explains the practical utility of Gross's representation by showing that (in the setting of the stabilizer subtheory) negativity in this particular representation implies generalized contextuality. Since negativity of this particular representation is a necessary resource for universal quantum computation in the state injection model, it follows that generalized contextuality is also a necessary resource for universal quantum computation in this model. In all even dimensions, we prove that there does not exist any nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory, and, hence, that the stabilizer subtheory is contextual in all even dimensions.