Foundations and applications of sequential measurements
Kieran Flatt
When a system is measured, its state is changed. A mathematical consequence of this statement is that scenarios in which a quantum system is measured repeatedly, or the same system is used to measure many others, require the use of Kraus’s formalism. Three projects which fall into this category are discussed in this thesis. One is of foundational interest and two are more oriented towards experiment. The first piece of work is an analysis of the uniqueness of each of Kraus's formulae for joint and conditional probabilities. Gleason, Busch and others were interested in whether the probability rules of quantum mechanics were constructed ad hoc or whether they had deeper significance. They showed that the Born rule was the only way of calculating quantum probabilities consistent with some basic assumptions about the nature of a physical theory. I extend this work to the sequential measurement case and show that no further assumptions are required for joint, over single, probabilities. A mathematical technique, the use of operator space, from that work is then developed, in my second reported piece of work, for use as a tool in quantum cryptanalysis. I show that calculations of the best eavesdropping strategies for quantum key distribution protocols can be done in a straightforward manner. I rediscover optimal strategies for BB84 and B92, two of the most commonly discussed protocols, and report a new attack for PBC00. Multiple-copy state discriminators look for methods of distinguishing states given a number of systems all in that state. An open question is whether a quantum memory, a device which interacts with other systems and does not decohere, aids this problem. In the third piece of work reported here, I compare the ability of two schemes, one which uses a quantum memory and one which does not, for performing multiple-copy state discrimination. One surprising result is that the scheme that uses quantum memory always performs worse than the one which does not. Another is that both schemes tend to the same limit in the case that the resource is an infinite number of copies. This suggests that a quantum memory may not be helpful.
•Jan 1, 2019•0 citations