Michael James Kastoryano

A thesis submitted December, 2011 for the degree of Doctor of Philosophy and defended February 17, 2012.

The PhD School of Science
Quantum Information Group
QUANTOP - Danish National Research
Foundation Centre for Quantum Optics
Niels Bohr Institute
and Niels Bohr International Academy

Academic advisor:
Michael M. Wolf
Anders S. Sørensen

Abstract

Quantum Markov Chain Mixing and Dissipative Engineering

This thesis is the fruit of investigations on the extension of ideas of Markov chain mixing to the quantum setting, and its application to problems of dissipative engineering. A Markov chain describes a statistical process where the probability of future events depends only on the state of the system at the present point in time, but not on the history of events. Very many important processes in nature are of this type, therefore a good understanding of their behavior has turned out to be very fruitful for science. Markov chains always have a non-empty set of limiting distributions (stationary states). The aim of Markov chain mixing is to obtain (upper and/or lower) bounds on the number of steps it takes for the Markov chain to reach a stationary state. The natural quantum extensions of these notions are density matrices and quantum channels. We set out to develop a general mathematical framework for studying quantum Markov chain mixing.

We introduce two new distance measures into the quantum setting; the quantum Χ2- divergence and Hilbert's projective metric. The quantum Χ2 divergence allows us to extend the class of functional techniques called Poincaré inequalities to the quantum setting. In the process, we identify the appropriate framework for discussing notions such as detailed balance and Χ2 -mixing: monotone Riemannian metrics on matrix spaces. This insight allows us to characterize the connection between spectral properties of a (primitive) quantum channel and its mixing time. Within the same framework we also derive a restricted quantum version of the celebrated conductance bound (Cheeger's inequality). We then consider Hilbert's projective metric - a well known tool in Perron-probenius theory - in the context of quantum information theory. As a classic tool for analyzing convergence of positive maps on cones, Hilbert's projective metric is especially good at providing existence proofs of maps between two spaces (or cones). We relate this measure to distinguishability measures on restricted cones of operators, and analyze the observable loss of information after the application of a quantum channel under a restrict set of measurements. Various notions of contractivity of quantum channels are revealed through Hilbert's metric.

Still on the topic of quantum Markov chains, we introduce the notion of cutoff phenomenon to the quantum setting. The cutoff phenomenon describes the situation when a Markov chain does not converge for a potentially long time, and then at a specific point in time abruptly converges to equilibrium. In the thermodynamic limit, the convergence profile will then look like a step function. We show that this type of convergence behavior occurs, as measured in trace-norm, when a system is subject to a product channel, and the noise on each product is modeled by a one-parameter semigroup of quantum channels.

Finally, we consider three independent tasks of dissipative engineering. The first task, which is aimed at experimental realization in the near future, consists in dissipatively preparing a maximally entangled state of two atoms trapped in an optical cavity. We show that this is indeed possible with very hight fidelity, and in a short amount of time, using present day technology. We also show that the scaling of the fidelity with the quality factor of the cavity (the cooperativity) scales quadratically better in our dissipative setup than in any known coherent unitary setup; indicating that dissipative state preparation can lead to fundamental improvements over closed system protocols. The second task that we consider is dissipative preparation of graph states, where we show the this class of states can be prepared in a time scaling as log n in the number of stabilizer elements. We also show that this process exhibits a cutoff. As a third task of dissipative engineering, we revisit the dissipative quantum computing construction of [VWC09]; we rigorously prove that it is as efficient as the circuit model.