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arXiv:quant-ph/0611203

Language Diversity of Measured Quantum Processes

Karoline Wiesner, James P. Crutchfield

Published Nov 19, 2006• 4 citations
arXivHTMLPDF

Abstract

The behavior of a quantum system depends on how it is measured. How much of what is observed comes from the structure of the quantum system itself and how much from the observer's choice of measurement? We explore these questions by analyzing the \emph{language diversity} of quantum finite-state generators. One result is a new way to distinguish quantum devices from their classical (stochastic) counterparts. While the diversity of languages generated by these two computational classes is the same in the case of periodic processes, quantum systems generally generate a wider range of languages than classical systems.

DOI
10.48550/arxiv.quant-ph/0611203

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The behavior of a quantum system depends on how it is measured. How much of what is observed comes from the structure of the quantum system itself and how much from the observer's choice of measurement? We explore these questions by analyzing the \emph{language diversity} of quantum finite-state generators. One result is a new way to distinguish quantum devices from their classical (stochastic) counterparts. While the diversity of languages generated by these two computational classes is the same in the case of periodic processes, quantum systems generally generate a wider range of languages than classical systems.
Abstract If an experimentalist observes a sequence of emitted quantum states via either projective or positive-operator-valued measurements, the outcomes form a time series. Individual time series are realizations of a stochastic process over the measurements’ classical outcomes. We recently showed that, in general, the resulting stochastic process is highly complex in two specific senses: (i) it is inherently unpredictable to varying degrees that depend on measurement choice and (ii) optimal prediction requires using an infinite number of temporal features. Here, we identify the mechanism underlying this complicatedness as generator nonunifilarity—the degeneracy between sequences of generator states and sequences of measurement outcomes. This makes it possible to quantitatively explore the influence that measurement choice has on a quantum process’ degrees of randomness and structural complexity using recently introduced methods from ergodic theory. Progress in this, though, requires quantitative measures of structure and memory in observed time series. And, success requires accurate and efficient estimation algorithms that overcome the requirement to explicitly represent an infinite set of predictive features. We provide these metrics and associated algorithms, using them to design informationally-optimal measurements of open quantum dynamical systems.
The field of classical stochastic processes forms a major branch of mathematics.Stochastic processes are, of course, also very well studied in biology, chemistry, ecology, geology, finance, physics, and many more fields of natural and social sciences.When it comes to quantum stochastic processes, however, the topic is plagued with pathological issues that have led to fierce debates amongst researchers.Recent developments have begun to untangle these issues and paved the way for generalizing the theory of classical stochastic processes to the quantum domain without ambiguities.This tutorial details the structure of quantum stochastic processes, in terms of the modern language of quantum combs, and is aimed at students in quantum physics and quantum-information theory.We begin with the basics of classical stochastic processes and generalize the same ideas to the quantum domain.Along the way, we discuss the subtle structure of quantum physics that has led to troubles in forming an overarching theory for quantum stochastic processes.We close the tutorial by laying out many exciting problems that lie ahead in this branch of science.
The thesis of this paper is that quantum measurement can be analyzed and understood within quantum mechanics itself. The statistics of the measurement process comes from the unknown details of the macroscopic measurement device. Quantum measurement is analyzed within the framework of scattering theory of quantum field theory with the aim of finding a physical rather than a metaphysical understanding. The measurement interaction is treated together with the quantum process to be measured. The evaluation of a Feynman diagram for the total process leads to one factor from the measurement interaction for each channel multiplying the basic scattering amplitudes. With increasing entanglement taken into account, these factors compete for dominance. They depend on unknown details of the measurement apparatus. The statistics of this competition is studied under the assumption that the measurement interaction does not introduce any bias. A binary quantum system is analyzed and after that the n-channel case. As a background, a couple of classical examples are shown, leading to the same selective mechanisms. The result of this analysis is that the quantum measurement process can be understood, whether a single measurement or an ensemble of measurements, as a result of an ordinary unitary time development, based on the interaction beteen the system subjet to measurement and the measurement apparatus. This result makes it possible to view the quantum-mechanical state as describing reality rather than merely a potentiality. This opens again for an ontology of physics, and hence for science in general.
Bell-CHSH-like inequalities have been very successful in benchmarking {\it spatial} quantum correlations. However, as this paper illustrates, they are in general not sufficient for benchmarking {\it temporal} quantum correlations. To show this, we parametrise classical and quantum stochastic finite state generators based on a single bit and a single qubit, respectively, and compare the temporal correlations of their output sequences using a Bell-CHSH-like inequality. We find that for sequential measurements by two observers, Alice and Bob, classical machines can exceed the Tsirelson bound of 222\sqrt{2}22​, due to their fundamental structure. However, when we consider a time delay between consecutive measurements, we find examples where the quantum machines outperform their classical counterparts by maintaining correlations longer under generally scrambling operations. Our result can be used to distinguish quantum from classical processes and to identify novel resources for quantum technology applications.