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Foundations and applications of sequential measurements

Kieran Flatt

Published Jan 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.5525/gla.thesis.74279

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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.
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.
We study sequential state discrimination measurements performed on the same qubit by subsequent observers. Specifically, we focus on the case when the observers perform a kind of a minimum-error type state discriminating measurement where the goal of the observers is to maximize their joint probability of successfully guessing the state that the qubit was initially prepared in. We call this the Joint Best Guess strategy. In this scheme, Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, and so on to altogether N consecutive receivers who all perform measurements on it. The goal for all observers is to determine which state Alice sent. In the Joint Best Guess strategy, every time a system is received the observer is required to make a guess, aided by the measurement, about its state. The price to pay for this requirement is that errors must be permitted, the guess can be correct or in error. There is a nonzero probability for all the receivers to successfully identify the initially prepared state, and we maximize this joint probability of success. This work is a step toward developing a theory of nondestructive sequential quantum measurements and could be useful in multiparty quantum communication schemes based on communicating with single qubits, particularly in schemes employing continuous variable states. It also represents a case where subsequent observers can probabilistically and optimally get around both the collapse postulate and the no-broadcasting theorem.
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf
We present a comprehensive technical review of quantum measurement theory, emphasizing the mathematical structures underlying the measurement problem and their role in modern quantum information science. Beginning with spectral theory and the Born rule, we develop the theory of positive operator-valued measures (POVMs), Kraus operators, and completely positive channels as essential generalizations of projective measurements. We discuss how decoherence theory addresses the preferred-basis and interference problems while leaving the outcome problem formally unresolved, motivating interpretational responses ranging from collapse models to many-worlds scenarios. Throughout, we emphasize the applicability of POVM theory to quantum state discrimination, quantum cryptography, and quantum error correction. The review prioritizes physical motivation and mathematical hints over detailed proofs, suitable for researchers transitioning into quantum foundations or quantum information theory.
This thesis studies the field of quantum measurements and more precisely sequential measurements. The aim is to provide answers to questions like when is performing a sequential or a repeated measurement meaningful, or even possible. This is important in order to understand what type of information can be acquired when performing measurements on quantum states. In basic quantum theory observables are identified with self-adjoint operators. We introduce the concept of a positive operator valued measure (POVM) to gain better understanding on observables. The downside to POVMs is that they don't produce a new state after the measurement, only measurement statistics. In order to perform multiple measurements on a quantum state, the concepts of measurement model and instrument are defined in this thesis. Using instruments to perform measurements is essential in order to describe a state after the initial measurement is performed. This way it's possible to make measurements sequentially. After defining the necessary tools for sequential measurements and a few different properties related to them, some applicable situations for these kind of measurements are shown. This way sequential measurements end up being somewhat meaningful in order to gain information from quantum systems.