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arXiv:1704.07137

Sequential measurements: Busch-Gleason theorem and its applications

Kieran Flatt, Stephen Mark Barnett, Sarah Croke

Published Apr 24, 2017• 0 citations
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Abstract

Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to sequential measurement processes is the aim of this paper. Given only a simple set of postulates, a probability measure is derived utilising the concept of Liouville space and the most general permissible quantum channel arises in the same manner. Super-Liouville space is constructed and a Bayesian interpretation of this object is provided. An important application of the new method is demonstrated, providing an axiomatic derivation of important results of the BB84 protocol in quantum cryptography.

DOI
10.48550/arxiv.1704.07137

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•Sep 10, 2019•1 citation
Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to sequential measurement processes is the aim of this paper. Given only a simple set of postulates, a probability measure is derived utilising the concept of Liouville space and the most general permissible quantum channel arises in the same manner. Super-Liouville space is constructed and a Bayesian interpretation of this object is provided. An important application of the new method is demonstrated, providing an axiomatic derivation of important results of the BB84 protocol in quantum cryptography.
Gleason's theorem is a statement that, given some reasonable assumptions, the Born rule used to calculate probabilities in quantum mechanics is essentially unique [A. M. Gleason, Indiana Univ. Math. J. 6, 885 (1957)]. We show that Gleason's theorem contains within it also the structure of sequential measurements, and along with this the state update rule. We give a small set of axioms, which are physically motivated and analogous to those in Busch's proof of Gleason's theorem [P. Busch, Phys. Rev. Lett. 91, 120403 (2003)], from which the familiar Kraus operator form follows. An axiomatic approach has practical relevance as well as fundamental interest, in making clear those assumptions which underlie the security of quantum communication protocols. Interestingly, the two-time formalism is seen to arise naturally in this approach.
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.
Busch's theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleason's theorem. Here we show that a further generalisation is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction-retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.
It has recently been claimed by De Zela that Gleason's theorem, for probability measures on the lattice of projection operators, can be extended to qubits by adding assumptions related to continuity and the existence of 'eigenstates'. This amounts to a claim of the derivation of Born's rule for Hermitian qubit observables. I point out a simple counterexample, and the flaw in De Zela's derivation (these are equally applicable to the repetition of the derivation given in a recent Reply). I also briefly discuss a valid extension to qubits given by Busch.
Buschʼs theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleasonʼs theorem. Here we show that a further generalization is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction–retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.
The postulates of quantum theory are rather abstract in comparison with those of other physical theories such as special relativity. This thesis considers two tools for investigating this discrepancy and makes a connection between them. The first of these tools, Gleason-type theorems, illustrates the interplay between postulates concerning observables, states and probabilities of measurement outcomes, demonstrating that they need not be entirely independent. Gleason’s original and remarkable result applied to observables described by projection-valued measures; however, the theorem does not hold in dimension two. Busch generalised the idea to observables described by positive operator measures, proving a result which holds for all separable Hilbert spaces. We show that Busch’s assumptions may be weakened without affecting the result. The manner in which we weaken the assumptions brings them closer to Gleason’s original treatment of projection-valued measures. We will then demonstrate the connection between Gleason-type theorems and Cauchy’s functional equation, a connection which yields an alternative proof of Busch’s result. The second tool we consider is the family of general probabilistic theories which offers a means of comparing quantum theory with reasonable alternatives. We identify a general probabilistic theory which reproduces the set of non-local correlations achievable in quantum theory, a property often thought to be particular to quantum theory. Finally, we connect these two tools by determining the class of general probabilistic theories which admit a Gleason-type theorem.