Mark S. Williamson

Unified View of Quantum and Classical Correlations

We discuss the problem of the separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states.

Quantum correlations in mixed-state metrology

We analyze the effects of quantum correlations, such as entanglement and discord, on the efficiency of phase estimation by studying four quantum circuits that can be readily implemented using NMR techniques. These circuits define a standard strategy of repeated single-qubit measurements, a classical strategy where only classical correlations are allowed, and two quantum strategies where nonclassical correlations are allowed. In addition to counting space (number of qubits) and time (number of gates) requirements, we introduce mixedness as a key constraint of the experiment. We compare the efficiency of the four strategies as a function of the mixedness parameter. We find that the quantum strategy gives square-root of N enhancement over the standard strategy for the same amount of mixedness. This result applies even for highly mixed states that have nonclassical correlations but no entanglement.

Emergent constraint on equilibrium climate sensitivity from global temperature variability

Equilibrium climate sensitivity (ECS) remains one of the most important unknowns in climate change science. ECS is defined as the global mean warming that would occur if the atmospheric carbon dioxide (CO2) concentration were instantly doubled and the climate were then brought to equilibrium with that new level of CO2. Despite its rather idealized definition, ECS has continuing relevance for international climate change agreements, which are often framed in terms of stabilization of global warming relative to the pre-industrial climate. However, the ‘likely’ range of ECS as stated by the Intergovernmental Panel on Climate Change (IPCC) has remained at 1.5–4.5 degrees Celsius for more than 25 years. The possibility of a value of ECS towards the upper end of this range reduces the feasibility of avoiding 2 degrees Celsius of global warming, as required by the Paris Agreement. Here we present a new emergent constraint on ECS that yields a central estimate of 2.8 degrees Celsius with 66 per cent confidence limits (equivalent to the IPCC ‘likely’ range) of 2.2–3.4 degrees Celsius. Our approach is to focus on the variability of temperature about long-term historical warming, rather than on the warming trend itself. We use an ensemble of climate models to define an emergent relationship between ECS and a theoretically informed metric of global temperature variability. This metric of variability can also be calculated from observational records of global warming, which enables tighter constraints to be placed on ECS, reducing the probability of ECS being less than 1.5 degrees Celsius to less than 3 per cent, and the probability of ECS exceeding 4.5 degrees Celsius to less than 1 per cent.

Topological phases and multiqubit entanglement

In this thesis we investigate geometric and topological structures in the context of entanglement and quantum computation.A parallel transport condition is introduced in the context of Franson interferometry based on the maximization of two-particle coincidence intensity. The dependence on correlations is investigated and it is found that the holonomy group is in general non-Abelian, but Abelian for uncorrelated systems. It is found that this framework contains a parallel transport condition developed by Levay in the case of two-qubit systems undergoing local SU(2) evolutions.Global phase factors of topological origin, resulting from cyclic local SU(2) evolution, called topological phases, are investigated in the context of multi-qubit systems. These phases originate from the topological structure of the local SU(2)-orbits and are an attribute of most entangled multi-qubit systems. The relation between topological phases and SLOCC-invariant polynomials is discussed. A general method to find the values of the topological phases in an n-qubit system is described.A non-adiabatic generalization of holonomic quantum computation is developed in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. It is shown how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing transitions in a generic three-level Λ configuration. The robustness of the proposed scheme to different sources of error is investigated through numerical simulation. It is found that the gates can be made robust to a variety of errors if the operation time of the gate can be made sufficiently short. This scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.

Composite geometric phase for multipartite entangled states

When an entangled state evolves under local unitaries, the entanglement in the state remains fixed. Here we show that the dynamical phase acquired by an entangled state in such a scenario can always be understood as the sum of the dynamical phases of its subsystems. In contrast, the equivalent statement for the geometric phase is not generally true unless the state is separable. For an entangled state an additional term is present, the mutual geometric phase, that measures the change the additional correlations present in the entangled state make to the geometry of the state space. For

Detection of bifurcations in noisy coupled systems from multiple time series

N

Eavesdropping on practical quantum cryptography

qubit states we find that this change can be explained solely by classical correlations for states with a Schmidt decomposition and solely by quantum correlations for

Trends in sea-ice variability on the way to an ice-free Arctic

W

Correlation-induced non-Abelian quantum holonomies

states.

Abrupt Climate Change in an Oscillating World

We generalize a method of detecting an approaching bifurcation in a time series of a noisy system from the special case of one dynamical variable to multiple dynamical variables. For a system described by a stochastic differential equation consisting of an autonomous deterministic part with one dynamical variable and an additive white noise term, small perturbations away from the systems fixed point will decay slower the closer the system is to a bifurcation. This phenomenon is known as critical slowing down and all such systems exhibit this decay-type behaviour. However, when the deterministic part has multiple coupled dynamical variables, the possible dynamics can be much richer, exhibiting oscillatory and chaotic behaviour. In our generalization to the multi-variable case, we find additional indicators to decay rate, such as frequency of oscillation. In the case of approaching a homoclinic bifurcation, there is no change in decay rate but there is a decrease in frequency of oscillations. The expanded method therefore adds extra tools to help detect and classify approaching bifurcations given multiple time series, where the underlying dynamics are not fully known. Our generalisation also allows bifurcation detection to be applied spatially if one treats each spatial location as a new dynamical variable. One may then determine the unstable spatial mode(s). This is also something that has not been possible with the single variable method. The method is applicable to any set of time series regardless of its origin, but may be particularly useful when anticipating abrupt changes in the multi-dimensional climate system.