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Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW - WH) B) = (E_A - E_B) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = e^itx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, e^itx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.

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Thermal (canonical) condensed-phase systems are of considerable interest in computational science, and include for example reactions in solution. Time-independent properties of these systems include free energies and thermally averaged geometries - time-dependent properties include correlation functions and thermal reaction rates. Accounting for quantum effects in such simulations remains a considerable challenge, especially for large systems, due to the quantum nature and high dimensionality of the phase space. Additionally time-dependent properties require treatment of quantum dynamics. Most current methods rely on semi-classical trajectories, path integrals or imaginary-time propagation of wave packets. Trajectory based approaches use continuous phase-space trajectories, similar to classical molecular dynamics, but lack a direct link to a wave packet and so the time-dependent schrodinger equation. Imaginary time propagation methods retain the wave packet, however the imaginary-time trajectory cannot be used as an approximation for real-time dynamics. We present a new approach that combines aspects of both. Using a generalisation of the coherent-state basis allows for mapping of the quantum canonical statistical average onto a phase-space average of the centre and width of thawed Gaussian wave packets. An approximate phase-space density that is exact in the low-temperature harmonic limit, and is a direct function of the phase space is proposed, defining the Gaussian statistical average. A novel Nose-Hoover looped chain thermostat is developed to generate the Gaussian statistical average via the ergodic principle, in conjunction with variational thawed Gaussian wave-packet dynamics. Numerical tests are performed on simple model systems, including quartic bond stretching modes and a double well potential. The Gaussian statistical average is found to be accurate to around 10% for geometric properties at room temperature, but gives energies two to three times too large. An approach to correct the Gaussian statistical average and ensure classical statistics is retrieved at high temperature is then derived, called the switched statistical average. This involves transitioning the potential surface upon which the Gaussian wave packet propagates, and the system property being averaged. Switching functions designed to perform these tasks are derived and tested on model systems. Bond lengths and their uncertainties calculated using the switched statistical average were found to be accurate to within 1% relative to exact results, and similarly for energies. The switched statistical average, calculated with Nose- Hoover looped chain thermostatted Gaussian dynamics, forms a new platform for evaluating statistical properties of quantum condensed-phase systems using an explicit real-time wave packet, whilst retaining appealing features of trajectory based approaches.

Entanglement is an effect at the heart of quantum mechanics, which is both useful as a resource for information theoretic tasks and important in the fundamental understanding of physics. While bipartite maximal entangled states are well understood, applications as well as detection of other forms of entanglement - multipartite, mixed, bound - still provide many open questions. The fact that some bound entangled states can be used as a resource for quantum key distribution motivates the question of how such states can be distributed between distant parties. One way would be a conventional quantum repeater starting with distillable entanglement between the nodes and performing subsequent steps of distillation and entanglement swapping. It is, however, an intriguing question whether key can be obtained between distant parties if only bound entanglement is available between the nodes of the repeater. In this work, we provide upper bounds on the key obtainable from a quantum repeater that can be severely limited for bound entangled input states. Understanding the role of entanglement in macroscopic systems is an important task. A particular interesting question is whether there are connections between the entanglement in a given system and its thermodynamic variables. It has previously been shown that for some Hamiltonians all states below a certain internal energy are entangled. We extend this result to higher internal energies by also considering the entropy of the system. This allows us to theoretically certify entanglement of thermal states at higher temperatures than previous results. Another application of entanglement is as an additional resource for classical communication via a quantum channel. While the classical capacity of a channel assisted by maximal entanglement is known, it is an open question whether for any entangled state, there always exists a channel the classical capacity of which can be enhanced by using the state as an additional resource. We show that this is the case for one-way undistillable Werner states in small dimensions.

This dissertation is about two aspects of quantum thermodynamics, quantum equilibration and thermal machines. First, we investigate equilibration of quantum systems with regards to typical measurements. Considering any Hamiltonian, any initial state, and measurements with a small number of outcomes compared to the dimension, we show that most measurements are already equilibrated. When the initial state is an eigenstate of the observable, most observables are initially out of equilibrium yet equilibrate more rapidly than would be physically reasonable. In search for more physical equilibration times, we turn to two practical scenario: a quantum particle in a one-dimensional box, as observed by a coarse grained position measurement; and a subsystem interacting with a highly mixed environment. We show that equilibration in both of these contexts indeed takes place and does so in very reasonable time scales. Back to a more general context, we present a theory independent definition of equilibration, and show that equilibration of pure states is objectively easier for quantum systems than for classical systems. This shows that quantum equilibration is a fundamental aspect of physical systems, while classical equilibration relies on experimental ignorance. In the subject of thermal machines, we show that a quantum system (the clock) can be used to exactly implement any energy-conserving unitary operation on an engine. When the engine includes a quantum work storage device we can approximately perform completely general unitaries. This can be used to carry out arbitrary transformations of a system without external control. We then show that autonomous thermal machines suffer no intrinsic thermodynamic cost compared to externally controlled ones. Finally, we further improve this construction by showing that the results still hold if the clock and the work storage device are given a more physical Hamiltonian.

The exploitation of quantum mechanics for new technologies promises a revolution in computing, simulations, communication and metrology. In particular, quantum computation and simulation harness properties peculiar to quantum systems to process information using algorithms that run exponentially faster than their classical counterparts. While the Fourier Transform (FT) is a cornerstone tool for signal processing in science, engineering, and computing, the quantum version of the Fourier transform (QFT) is a key component of many quantum algorithms. And just as the FT has a generalisation known as the complex Hadamard, so the QFT is a special case of the quantum complex Hadamard (QCH) operator. This thesis reports implementations of the QFT and QCH, with photons in linear optical circuits, as core components for quantum information processing, quantum computation, and quantum simulations. The iterative QFT is implemented in 8hor's quantum factoring algorithm to demonstrate how qubits can be recycled to provide a resource saving. This included the first demonstration of two in-series photonic controlled logic gates. The four dimensional QCH was constructed with an internal phase implemented as a tuneable Berry phase to control quantum interference between two photons. A generator for this QCH, described as a highly connected graph, was simulated as a Hamiltonian, with full time evolution implemented in a fully reconfigurable optical network. Quantum simulations for the evolution of phonons in a model chain of atoms were performed, where QFT operators prepare and measure in a basis corresponding to localised excitations. Quantised vibrational wave packets in molecules, described as phonons in the harmonic approximation, were simulated

The spectral statistical and entanglement within the eigenstates of generic spin chain Hamiltonians are analysed. A class of random matrix ensembles is defined which include the most general nearestneighbour qubit chain Hamiltonians. For these ensembles, and their generalisations, it is seen that the long chain limiting spectral density is a Gaussian and that this convergence holds on the level of individual Hamiltonians. The rate of th is convergence is numerically seen to be slow. Higher eigenvalue correlation statistics are also considered, the canonical nearest-neighbour level spacing statistics being numerically observed and linked with ensemble symmetries. A heuristic argument is given for a conjectured form of the full joint probability density function for the eigenvalues of a wide class of such ensembles. This is numerically ver ified in a particular case. For many trans lat ion ally-invariant nearest-neighbour qubit Hamiltonians it is shown that there exists a complete orthonormal set of eigenstates for which the entanglement present in a generic member, between a fixed length block of qubits and the rest of the chain , approaches its maximal value as the chain length increases. Many such Hamiltonians are seen to exhibit a simple spectrum so that their eigenstates are unique up to phase. The entanglement within the eigenstates contrasts the spectral density for such Hamiltonians , which is that seen for a non-interacting chain of qubits. For such noninteracting chains, their always ex ists a basis of eigenstate;; for which there is no entanglement present .

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