Transformation & uncertainty : some thoughts on quantum probability theory, quantum statistics, and natural bundles

Abstract

This PHD thesis is concerned partly with uncertainty relations in quantum probability theory, partly with state estimation in quantum stochastics, and partly with natural bundles in differential geometry. The laws of quantum mechanics impose severe restrictions on the performance of measurement. Among these are the Heisenberg principle, the necessity of decoherence and the impossibility of perfect joint measurement. After some introductory comments on the nature and necessity of decoherence in open systems and its absence in closed ones, we prove sharp, state-independent inequalities reflecting the above measurement restrictions. Because of state-independence, we can use them to judge how far a measurement procedure is removed from the bounds imposed on it by quantum mechanics. Relying heavily on the machinery of quantum stochastic calculus, we elaborate this for a two-level atom (qubit) interacting with the quantized electromagnetic field. Information on two qubit observables leaks into the field, which is continually probed using homodyne detection. We determine the quality of this joint measurement procedure. Turning to quantum statistics, we then propose a two-step strategy to determine the (possibly mixed) state of n identically prepared qubits, and prove that it is asymptotically optimal in a local minimax sense. We use a rough estimator to localize the state within a (large) neighbourhood, and then reduce the problem to a Gaussian one using `Quantum Local Asymptotic Normality' for qubits. The key result here is the size of the neighbourhoods on which QLAN is shown to hold. We also propose a physical implementation, based on interaction with the quantized electromagnetic field. In differential geometry, a bundle is called `natural' if diffeomorphisms of the base lift to automorphisms of the bundle in a functorial fashion. We slightly extend the notion of a natural bundle to that of an `infinitesimally natural' one by requiring diffeomorphisms of the base to lift only infinitesimally, i.e. on the level of vector fields. We then classify the infinitesimally natural principal fibre bundles. Physical fields that transform under (infinitesimal) space-time transformations must be described in terms of (infinitesimally) natural bundles. It turns out that all spin structures are infinitesimally natural. Our framework is therefore sufficiently general to encompass Fermionic fields, for which the notion of a natural bundle is too restrictive. Interestingly, generalized spin structures (e.g. spin-c structures) are not always infinitesimally natural. We classify the ones that are. Depending on the gauge group at hand, this can significantly reduce the number of allowed space-time topologies.