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arXiv:1306.2699 · Quantum Measurements and Quantum Metrology

Testing quantum mechanics: a statistical approach

Mankei Tsang

Published Dec 31, 2013• 15 citations
arXivHTMLPDF

Abstract

Abstract As experiments continue to push the quantum-classical boundary using increasingly complex dynamical systems, the interpretation of experimental data becomes more and more challenging: when the observations are noisy, indirect, and limited, how can we be sure that we are observing quantum behavior? This tutorial highlights some of the difficulties in such experimental tests of quantum mechanics, using optomechanics as the central example, and discusses how the issues can be resolved using techniques from statistics and insights from quantum information theory.

DOI
10.2478/qmetro-2013-0007

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Abstract As experiments continue to push the quantum-classical boundary using increasingly complex dynamical systems, the interpretation of experimental data becomes more and more challenging: when the observations are noisy, indirect, and limited, how can we be sure that we are observing quantum behavior? This tutorial highlights some of the difficulties in such experimental tests of quantum mechanics, using optomechanics as the central example, and discusses how the issues can be resolved using techniques from statistics and insights from quantum information theory.
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The work presented in this thesis considers statistical models of quantum statistics. It sets up a framework to analyse experimental data, which is independent of any particular theory, being careful to eliminate any bias towards the theory of quantum mechanics. This framework allows us to separate the discussion of quantum statistics, which are the predictions of quantum theory, from the theory of quantum mechanics enabling an evaluation of alternative models in reproducing the same statistics. The framework allows one to evaluate the non-classicality of a process or phenomenon, without relying on their definitions in quantum mechanics. The two main non-classical phenomenon analysed in this thesis are contextuality and the weak value. We provide completely statistical definitions for both, which is completely independent of quantum mechanics. The two are defined as statistical properties of experimental data. The presence of contextuality in the data for a set of experiments is shown to pose certain obstructions to the statistical modelling of the data, specifically it is shown to require the state of the system to be updated in a non-Markovian way. A statistical model of this type is presented as proof of existence of statistical models for contextual data. The construction of this model is used to o_er insight into the mechanisms required to construct statistical models for contextual data. Additionally an explicit connection is established between contextuality and classical simulability for any sub-theory of quantum mechanics. This connection is used to evaluate the contextuality found in the qubit-stabilizer sub-theory to calculate a lower bound for the minimum number of classical bits a model would require to simulate the statistics. The weak value was previously thought of an a quintessentially quantum phenomenon. We analyse the dynamics of the process that leads to the weak value using classical Hamiltonian mechanics. A statistical analysis of the experiment explains the physical objects associated with the real and imaginary parts of the weak value and its emergence as a result of post-selection. The weak value calculated using classical Hamiltonian dynamics matches the quantum predictions.
Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical probability and statistics. On the other hand, the unique character of quantum physics sets many of the questions addressed apart from those met classically in stochastics. Furthermore, concurrent advances in experimental techniques and in the theory of quantum computation have led to a strong interest in questions of quantum information, in particular in the sense of the amount of information about unknown parameters in given observational data or accessible through various possible types of measurements. This scenery is outlined (with an audience of statisticians and probabilists in mind).
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Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, theoretical developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates a number of new concepts for an extension of classical statistical inference to the quantum context.
This study offers a comprehensive technical analysis of the mechanisms through which classical behavior emerges from the dynamics of quantum measurement. Beginning with unitary evolution and the principles of measurement, we construct the framework of environment-induced decoherence, the selection of pointer states, and the reduction of interference. Our examination focuses on how effective classicality emerges through reduced density matrices, quantum Darwinism, and consistent histories, highlighting the importance of system-environment interactions in determining stable observables. The review investigates into mathematical constructs such as spectral theory, completely positive maps, and quantum trajectories, linking them to experimental observations in mesoscopic systems, optomechanics, and superconducting circuits. We explore the constraints of decoherence in addressing the outcome problem and review interpretational models that tackle the remaining issues. The discussion emphasizes applications in quantum information, metrology, and tests of macroscopic superpositions. The article favors physical insights and mathematical guidance over exhaustive proofs, making it suitable for researchers in quantum foundations and quantum technologies.