State-Based Characterisation of Quantum Matter

QC 2026-05-19

Abstract

Condensed matter physics studies how many quantum particles, when brought together, conspire to give rise to collective behaviours that are far richer than those of the individual particles themselves. This is the physics of emergence: the idea that a large system can have properties that are not obvious from its microscopic building blocks alone. The work presented in this thesis explores how to characterise the properties of such many-particle states from three interconnected points of view: topology, quantum transport, and quantum information. The common idea behind these themes is that the properties of quantum matter may be accessed directly from quantum states themselves, without necessarily referring to an underlying Hamiltonian, circuit, or protocol used to generate them.

Symmetry-protected topological phases of matter arise from the impossibility of continuously deforming certain quantum states into others whilst preserving their symmetries. In fermionic quantum matter, the phases resulting from such impossibility are those of topological insulators and superconductors. While having an insulating bulk, these materials possess metallic boundary states robust to perturbations preserving the protecting symmetries of their phase. Mathematically, the topological nature of these materials is characterised by the calculation of topological invariants, integer or half-integer numbers distinguishing between different topological phases. These invariants have been traditionally formulated for crystalline quantum matter, where one may use momentum space to perform mathematical calculations. This thesis focuses instead on characterising the topological phases of noncrystalline matter, which includes disorder and amorphicity. We develop a series of mathematical tools, known as local topological markers, which reformulate traditional topological invariants in real space and are therefore applicable even in the absence of crystallinity. Local topological markers are formulated in terms of the one-particle density matrix, a correlation function that only requires the knowledge of the quantum state in question to be calculated. Thus, the framework for topological characterisation we discuss, which also extends to certain higher-order topological phases, is purely state-based: it does not need to specify any parent Hamiltonian nor protocol used to prepare the quantum state to be characterised.

Contrary to the classical picture of electricity where electrons behave as solid spheres drifting through a wire, the electronic transport through a nanostructure is shaped by quantum effects like tunnelling and interference. Quantum transport studies how these effects determine the conducting properties of mesoscopic samples---those that are neither microscopic nor fully macroscopic, but lie in an intermediate regime. In this thesis, we use quantum transport to study topological insulator nanowires, where the interplay between topology, deviations from crystallinity, and symmetry constraints shape electronic transmission. The characteristic feature of topological nanowires is the appearance of a perfectly transmitted mode when a certain magnetic flux is threaded through the wire's cross section. Varying the magnetic flux interpolates from perfect transmission to situations where electronic interference is enhanced, suppressing transmission. This is seen in the behaviour of conductance, which oscillates as a function of magnetic flux in what is known as Aharanov-Bohm oscillations. The mechanism originating the perfectly transmitted mode in a topological nanowire relies on time-reversal symmetry appearing at specific values of the magnetic flux. In amorphous nanowires,  where time-reversal symmetry may be broken at any magnetic flux, the perfectly transmitted mode loses its protection, and the characteristic conductance oscillations of a topological nanowire may be lost. We study how the perfectly transmitted mode may be protected even when time-reversal symmetry is broken, and relate its loss to a topological phase transition driven by amorphicity.

Entanglement is a distinctive feature of condensed matter systems: the possibility that quantum particles may be correlated in a way that has no classical analogue. Although inherently quantum-mechanical, entanglement alone is not a sufficient ingredient for quantum computation to outperform classical computers. The reason is that there exists a special class of quantum states, stabiliser states, that may be highly entangled whilst admitting an efficient classical representation---calculations involving stabiliser states are tractable in modern-day computers. The deviation of a quantum state from the set of stabiliser states goes by the name of quantum nonstabiliserness or magic, and, in the same way as entanglement, it is not a guarantee of quantum complexity on its own. In this thesis, we develop a method to characterise quantum states that combines entanglement and nonstabiliserness. To do so, we use the framework of the information lattice, whichsystematically decomposes the correlations in a quantum state in terms of their scale---the spatial extent of correlations. Through the information lattice, we distinguish different contributions to the nonstabiliserness content of a quantum state, which may have either a single-particle origin or come from intrinsic many-particle correlations. This framework, which is purely state-based, provides a first step to a fully scale-resolved description of nonstabiliserness and contributes to the characterisation of quantum states in terms of their complexity.

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