Geometry of contextuality from Grothendieck's coset space

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli operators [Information 5, 209 (2014); 1310.4267 and 1404.6986 (quant-ph)]. Now we find that the non-existence of a dessin for which the commutator (a, b) = a^ (--1) b^( --1) ab precisely corresponds to the commutator of quantum observables [A, B] = AB -- BA on all lines of the geometry is a signature of quantum contextuality. This occurs first at index |G : H| = 9 in Mermin's square and at index 10 in Mermin's pentagram, as expected. Commuting sets of n-qubit observables with n \textgreater{} 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.Comment: 13 pages, Quant. Inf. Pro

Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

In this paper, we show that a qutrit version of ZX-calculus, with rules significantly different from that of the qubit version, is complete for pure qutrit stabilizer quantum mechanics, where state preparations and measurements are based on the three dimensional computational basis, and unitary operations are required to be in the generalized Clifford group. This means that any equation of diagrams that holds true under the standard interpretation in Hilbert spaces can be derived diagrammatically. In contrast to the qubit case, the situation here is more complicated due to the richer structure of this qutrit ZX-calculus.Comment: In Proceedings QPL 2017, arXiv:1802.0973

Wigner function negativity and contextuality in quantum computation on rebits

We describe a universal scheme of quantum computation by state injection on rebits (states with real density matrices). For this scheme, we establish contextuality and Wigner function negativity as computational resources, extending results of [M. Howard et al., Nature 510, 351--355 (2014)] to two-level systems. For this purpose, we define a Wigner function suited to systems of $n$ rebits, and prove a corresponding discrete Hudson's theorem. We introduce contextuality witnesses for rebit states, and discuss the compatibility of our result with state-independent contextuality.Comment: 18 + 4 page

The role of cohomology in quantum computation with magic states

A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality. Here we extend this web to quantum computation with magic states. In this computational scheme, the negativity of certain quasiprobability functions is an indicator for quantumness. However, when constructing quasiprobability functions to which this statement applies, a marked difference arises between the cases of even and odd local Hilbert space dimension. At a technical level, establishing negativity as an indicator of quantumness in quantum computation with magic states relies on two properties of the Wigner function: their covariance with respect to the Clifford group and positive representation of Pauli measurements. In odd dimension, Gross' Wigner function -- an adaptation of the original Wigner function to odd-finite-dimensional Hilbert spaces -- possesses these properties. In even dimension, Gross' Wigner function doesn't exist. Here we discuss the broader class of Wigner functions that, like Gross', are obtained from operator bases. We find that such Clifford-covariant Wigner functions do not exist in any even dimension, and furthermore, Pauli measurements cannot be positively represented by them in any even dimension whenever the number of qudits is n>=2. We establish that the obstructions to the existence of such Wigner functions are cohomological.Comment: 33 page

Contextuality and Ontological Models: A Tale of Desire and Disappointment

Since being defined by Kochen and Specker, and separately by Bell, contextuality has been proposed as one of the key phenomena that distinguishing quantum theory from classical theories. However, with the rise of quantum information contextuality's position as the leading definition of the quantum/classical boundary has been called into question. This is due to the fact that a contextual explanation is required by subtheories that offer no exponential quantum computational advantage over classical computation. In this thesis, submitted in requirement for a PhD in physics with quantum information, we shall explore this unwanted contextuality, and show that generalized contextuality is more prevalent than was previously thought. First we will show that the single-qubit stabilizer subtheory, which was previously thought of as a non-contextual subtheory, requires a generalized contextual ontological model, when transformations are included in the operational description. In addition to this we show that the even smaller single-rebit stabilizer subtheory, a strict subtheory of the single-qubit stabilizer subtheory, must also admit a generalized contextual ontological model, again when transformations are included in the operational description. We then show that both these subtheories require negatively represented quasi-probability representations, re-establishing the link between contextuality and negativity for transformations. We also investigate the representation of transformations in generic quasi-probability representations, showing that under a reasonable assumption almost-all unitaries must be negatively represented by a finite quasi-probability representation. Second we will investigate the efficiently simulable n-qubit stabilizer subtheory, which exhibits all forms of contextuality and thus represents the main obstacle to identifying contextuality as a resource for quantum computation. To this end we present an attempt at constructing a model based on a frame-switching Wigner function. This leads us to constructing a contextual ψ-epistemic ontological model of the n-qubit stabilizer formalism. We shall see that this model is outcome deterministic, which is one of the core assumptions in the definition of traditional non-contextuality. This model then will lead us to a result that proves that any ontological model of the n-qubit stabilizer formalism requires at least n-1 generators to be encoded in the ontology of the model. As n-1 generators represents almost full knowledge about the stabilizer state, we therefore posit that ψ-onicity, a more controversial notion of non-classicality, is actually the resource for quantum computation

Quantum Contextuality with Stabilizer States

The Pauli groups are ubiquitous in quantum information theory because of their usefulness in describing quantum states and operations and their readily understood symmetry properties. In addition, the most well-understood quantum error correcting codes—stabilizer codes—are built using Pauli operators. The eigenstates of these operators—stabilizer states—display a structure (e.g., mutual orthogonality relationships) that has made them useful in examples of multi-qubit non-locality and contextuality. Here, we apply the graph-theoretical contextuality formalism of Cabello, Severini and Winter to sets of stabilizer states, with particular attention to the effect of generalizing two-level qubit systems to odd prime d-level qudit systems. While state-independent contextuality using two-qubit states does not generalize to qudits, we show explicitly how state-dependent contextuality associated with a Bell inequality does generalize. Along the way we note various structural properties of stabilizer states, with respect to their orthogonality relationships, which may be of independent interest