The Structure of the Kirkwood-Dirac Distribution and its Applications in Quantum Foundations, Computation, and Information

Abstract

Quantum theory is fundamentally different from its classical counterpart. This difference has led to striking new insights in the foundations of physics and the theories of computation and information. Nevertheless, characterising the difference between classical and quantum theory has proven to be a difficult problem. In this thesis, I present my progress towards a general solution for this open problem. My main tool is the Kirkwood-Dirac (KD) quasiprobability distribution. The KD distribution provides an alternative description of quantum states, operations and measurements in terms of quasiprobability distributions. The KD quasiprobability distribution obeys all properties of probability distributions except that its values may, in general, lie in the complex unit disc instead of the interval [0,1]. For some experiments, however, all associated KD distributions are genuine probability distributions. For such `KD-positive’ states, operations and measurements, the KD distribution enables the construction of a classical model. The absence of such a construction can provide a natural criterion for the definition of the quantum-classical boundary. In the first part of my thesis, I present my results on the mathematical structure of the KD distribution. I summarise previous literature, and prove several new results, including a proof that the set of KD-positive states is almost always minimal, which was published in Physical Review A; novel bounds on the entries of the KD distribution and KD nonpositivity, a measure of how far a KD distribution is from being positive; and the construction of exotic states, mixed KD-positive states that cannot be written as the convex combination of pure KD-positive states, for Hilbert spaces of any odd dimension. These results lay the foundation for the objective of the remainder of this thesis: the application of the KD distribution in quantum foundations, computation and information. In the foundational setting, I study contextuality, a notion of nonclassicality often used in quantum foundations. An experiment is contextual if it is impossible to describe its indistinguishable procedures identically classically. I design an experiment that is contextual if and only if the input state of the experiment is not KD-positive. By analysing this connection with respect to the aforementioned exotic states, I obtain a quantum protocol that allows an experimenter to verify contextuality using a noncontextual experiment, thereby uncovering a previously unknown feature of contextuality. This section of the thesis is based on my publication in Physical Review A `Verification of contextuality by noncontextual experiments’ KDContextualityPaper. In the computational setting, I study the Delfosse-Guerin-Bian-Raussendorf (DGBR) model of quantum computation with a restricted set of input states. For certain sets of input states, this model is computationally as powerful as a universal quantum computer. However, for other sets of input states, this model is efficiently simulable on a classical computer and thus cannot be as powerful as a universal quantum computer, unless quantum computers offer no advantage over their classical counterparts. I prove that if all states in the set of input states are KD-positive, then the DGBR model is classically efficiently simulable. In this setting, some exotic states are bound-magic states, magic states that are nevertheless simulable input states in the DGBR model. I use my previous work on exotic states to construct the first bound-magic states on qubits. Furthermore, I show that KD nonpositivity is a resource monotone for quantum computation, establishing KD nonpositivity as a necessary resource for quantum computational advantage. This section of the thesis is based on my article `Kirkwood-Dirac nonpositivity is a necessary resource for quantum computing’, which has been accepted for publication in Physical Review Letters. In the information-theoretical setting, I study the uncertainty principle. In modern quantum information theory, the uncertainty principle is formulated as an `entropic lower bound’ on the sum of the entropies of the outcome distributions of two observables. I extend the definition of the standard entropic functions to the KD distribution and the Margenau-Hill (MH) distribution, which is the real part of the KD distribution. I introduce the MH Shannon entropy and the KD Rényi entropy, and establish their basic properties. For the MH Shannon entropy, I prove a generalised subadditivity relation, which yields a state-dependent entropic uncertainty relation that strengthens the previous best bound in most regimes and implies the finite-dimensional analogue of the Van Herstraeten--Cerf conjecture. For the KD Rényi entropy, I prove an entropic uncertainty relation strengthening the Maassen--Uffink bound, in which the KD nonpositivity appears as an additive correction. This yields a new interpretation of KD nonpositivity as a measure of quantum uncertainty, as well as a new entropic witness of KD positivity. Applying these results to cryptographic games, I show that KD nonpositivity is the key resource governing the optimal strategy in each case. This section is based on my article `Kirkwood-Dirac nonpositivity is sufficient for quantum uncertainty'. These results demonstrate that the KD distribution provides an effective formulation of quantum theory for studying the quantum-classical boundary in various settings. My work establishes explicit connections between the KD distribution and quantum foundations, computation, and information, opening up these fields to be studied using the extensive KD toolkit.