arXiv subjects

William Clark

Publications and source records attributed to William Clark.

42 records · Page 3Linked to original sources

arXiv2019

Time-minimum control of quantum purity for 2-level Lindblad equations

We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity. The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.

math.OC↗

arXiv2019

Secret key distillation across a quantum wiretap channel under restricted eavesdropping

The theory of quantum cryptography aims to guarantee unconditional information-theoretic security against an omnipotent eavesdropper. In many practical scenarios, however, the assumption of an all-powerful adversary is excessive and can be relaxed considerably. In this paper we study secret key distillation across a lossy and noisy quantum wiretap channel between Alice and Bob, with a separately parameterized realistically lossy quantum channel to the eavesdropper Eve. We show that under such restricted eavesdropping, the key rates achievable can exceed the secret key distillation capacity against an unrestricted eavesdropper in the quantum wiretap channel. Further, we show upper bounds on the key rates based on the relative entropy of entanglement. This simple restricted eavesdropping model is widely applicable, e.g., to free-space quantum optical communication, where realistic collection of light by Eve is limited by the finite size of her optical aperture. Future work will include calculating bounds on the amount of light Eve can collect under various realistic scenarios.

cs.IT↗

arXiv2018

Repeater-enhanced distributed quantum sensing based on continuous-variable multipartite entanglement

Entanglement is a unique resource for quantum-enhanced applications. When employed in sensing, shared entanglement between distributed quantum sensors enables a substantial gain in the measurement sensitivity in estimating global parameters of the quantum sensor network. Loss incurred in the distribution of entanglement, however, quickly dissipates the measurement-sensitivity advantage enjoyed by the entangled quantum sensors over sensors supplied with local quantum resources. Here, we present a viable approach to overcome the entanglement-distribution loss and show that the measurement sensitivity enabled by entangled quantum sensors beat that afforded by the optimum local resource. Our approach relies on noiseless linear amplifiers (NLAs) to serve as quantum repeaters. We show that unlike the outstanding challenge of building quantum repeaters to suppress the repeaterless bound for quantum key distribution, NLA-based quantum repeaters for distributed quantum sensing are realizable by available technology. As such, distributed quantum sensing would become the first application instance that benefits from quantum repeaters.

quant-ph↗

arXiv2018

Poincaré-Bendixson Theorem for Hybrid Systems

The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stability analysis of hybrid systems. We also prove a Poincaré-Bendixson Theorem for a class of one dimensional hybrid dynamical systems.

math.DS↗

arXiv2017

Optimal Control of Quantum Purity for $n=2$ Systems

The objective of this work is to study time-minimum and energy-minimum global optimal control for dissipative open quantum systems whose dynamics is governed by the Lindblad equation. The controls appear only in the Hamiltonian. Using recent results regarding the decoupling of such dissipative dynamics into intra- and inter-unitary orbits, we transform the control system into a bi-linear control system on the Bloch ball (the unitary sphere together with its interior). We then design a numerical algorithm to construct an optimal path to achieve a desired point given initial states close to the origin (the singular point) of the Bloch ball. This is done both for the minimum-time and minimum -energy control problems.

math.OC↗

arXiv2014

Stem Cell Transplantation As A Dynamical System: Are Clinical Outcomes Deterministic?

Outcomes in stem cell transplantation (SCT) are modeled using probability theory. However the clinical course following SCT appears to demonstrate many characteristics of dynamical systems, especially when outcomes are considered in the context of immune reconstitution. Dynamical systems tend to evolve over time according to mathematically determined rules. Characteristically, the future states of the system are predicated on the states preceding them, and there is sensitivity to initial conditions. In SCT, the interaction between donor T cells and the recipient may be considered as such a system in which, graft source, conditioning and early immunosuppression profoundly influence immune reconstitution over time. This eventually determines clinical outcomes, either the emergence of tolerance or the development of graft versus host disease. In this paper parallels between SCT and dynamical systems are explored and a conceptual framework for developing mathematical models to understand disparate transplant outcomes is proposed.

q-bio.OT↗