Measurements play a central role in quantum information. This thesis looksat two types: contextual measurements and symmetric measurements. Contextualityoriginates from the Kochen-Specker theorem about hidden variablemodels and has recently undergone a subtle shift in its manifestation. Symmetricmeasurements are characterised by the regular polytopes they formin Bloch space (the vector space containing all density matrices) and are thesubject of several investigations into their existence in all dimensions.We often describe measurements by the vectors in Hilbert space ontowhich our operators project. In this sense, both contextual and symmetricmeasurements are connected to special sets of vectors. These vectors areoften special for another reason: they form congurations in a given incidencegeometry.In this thesis, we aim to show various connections between congurationsand measurements in quantum information. The congurations discussedhere would have been well-known to 19th and 20th century geometers andwe show they are relevant for advances in quantum theory today. Specically,the Hesse and Reye congurations provide proofs of measurement contextuality,both in its original form and its newer guise. The Hesse congurationalso ties together dierent types of symmetric measurements in dimension3called SICs and MUBswhile giving insights into the group theoreticalproperties of higher dimensional symmetric measurements.
Measurements play a central role in quantum information. This thesis looksat two types: contextual measurements and symmetric measurements. Contextualityoriginates from the Kochen-Specker theorem about hidden variablemodels and has recently undergone a subtle shift in its manifestation. Symmetricmeasurements are characterised by the regular polytopes they formin Bloch space (the vector space containing all density matrices) and are thesubject of several investigations into their existence in all dimensions.We often describe measurements by the vectors in Hilbert space ontowhich our operators project. In this sense, both contextual and symmetricmeasurements are connected to special sets of vectors. These vectors areoften special for another reason: they form congurations in a given incidencegeometry.In this thesis, we aim to show various connections between congurationsand measurements in quantum information. The congurations discussedhere would have been well-known to 19th and 20th century geometers andwe show they are relevant for advances in quantum theory today. Specically,the Hesse and Reye congurations provide proofs of measurement contextuality,both in its original form and its newer guise. The Hesse congurationalso ties together dierent types of symmetric measurements in dimension3called SICs and MUBswhile giving insights into the group theoreticalproperties of higher dimensional symmetric measurements.
This thesis is centered around mathematical structures of fundamental importance to quantum foundations and quantum information. Its introductory part acquaints the reader with the basics of the general ideas of frames and simplices, as well as with some central quantum mechanical concepts like density matrices and Hilbert spaces. Article I deals with the question of contextuality in relation to quantum measurement. Using a specific strategy of assigning truth values to state vectors, some aspects of the Kochen-Specker theorem are explored. In Article II an object called SIC-POVM is studied, more specifically by investigating the behaviour of a function used to measure the closeness of an arbitrary vector set to such a configuration. In addition to putting the results of these two papers into context, the aim of this thesis is to offer a discussion of the search for connections between the SIC-POVM and so-called MUBs in odd prime dimensions, as well as a brief comment on a structure here introduced as a Kochen-Specker frame; it is shown that the latter plays a part in the proof of Hardy’s paradox as well as for an inequality for testing contextuality due to Klyachko et al.
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
Quantum theory departs from classical probabilistic theories in foundational ways. These departures--termed quantumness here--power quantum information and computation. This thesis charts the role of discrete structures in assessing quantumness, synthesizing elements of my postdoctoral research through this lens. After an introduction to the necessary background concepts, I present my work under three broad categories. First, I present work on contextuality that extensively relies on (undirected) graphs and hypergraphs as the discrete structures of interest; more specifically, it relies on invariants associated with them. This work includes Kochen-Specker (KS) contextuality and its operationalization to generalized contextuality, expressed via (hyper)graph-theoretic frameworks. I also present work on KS-contextuality in multiqubit systems and an application of generalized contextuality to a one-shot communication task, both of which rely on hypergraphs. Second, I present work on causality, where the discrete structures of interest are directed graphs. This includes work on indefinite causal order, specifically its connections to the gap between local operations and classical communication (LOCC) and separable operations (SEP), and a device-independent notion of nonclassicality--termed antinomicity--that generalizes Bell nonlocality without global causal assumptions. Finally, I present work on the incompatibility of quantum measurements, its connection to Bell nonlocality, and its role in discriminating between quantum and almost quantum correlations in the single-system setting. The discrete structures of interest here are hypergraphs that model joint measurability relations between quantum measurements. I conclude with a summary and an overview of work that is not covered in this thesis.
The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden-variable theory. It is shown that such a hidden-variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidden state is altered through interaction with the measuring device, thereby giving rise to a possibly different final hidden-variable state from which the measurement outcome is obtained. In light of this observation, a less restrictive and, arguably, more reasonable definition of noncontextuality is suggested. To prove that such a description is possible, an explicit and, in this sense, noncontextual hidden-variable model is constructed, which reproduces all quantum theoretic predictions for the Mermin-Peres square. A critical analysis of some recent and proposed experimental tests of contextuality is also provided. Although the discussion is restricted to a four-dimensional Hilbert space, the approach and conclusions are expected to generalize to any Hilbert space.
A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here, compatible measurements are those that can be performed simultaneously or in any order without disturbance. This conflict is generically called quantum contextuality. In this article, we present an introduction to this subject and its current status. We review several proofs of the Kochen-Specker theorem and different notions of contextuality. We explain how to experimentally test some of these notions and discuss connections between contextuality and nonlocality or graph theory. Finally, we review some applications of contextuality in quantum information processing.
Quantum contextuality plays a significant role in supporting quantum computation and quantum information theory. The key tools for this are the Kochen-Specker and non-Kochen-Specker contextual sets. Traditionally, their representation has been predominantly operator-based, mainly focusing on specific constructs in dimensions ranging from three to eight. However, nearly all of these constructs can be represented as low-dimensional hypergraphs. This study demonstrates how to generate contextual hypergraphs in any dimension using various methods, particularly those that do not scale in complexity with increasing dimensions. Furthermore, we introduce innovative examples of hypergraphs extending to dimension 32. Our methodology reveals the intricate structural properties of hypergraphs, enabling precise quantifications of contextuality. Additionally, we investigate several promising applications of hypergraphs in quantum communication and quantum computation, paving the way for future breakthroughs in the field.