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arXiv:1402.2919

Quantum-mechanical measurement apparatus as a black box

A. V. Nenashev

Published Feb 12, 2014• 0 citations
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Abstract

It is commonly believed that the most general type of a quantum-mechanical measurement is one described by a positive-operator valued measure (POVM). In the present paper, this statement is proven for any measurements on quantum systems with a finite-dimensional state space. The proof of POVM nature of an arbitrary measurement is carried out using a purely operational approach, which is fully ignorant about what is inside a measurement apparatus. The suggested approach gives also an opportunity to derive the Born rule.

DOI
10.48550/arxiv.1402.2919

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It is commonly believed that the most general type of a quantum-mechanical measurement is one described by a positive-operator valued measure (POVM). In the present paper, this statement is proven for any measurements on quantum systems with a finite-dimensional state space. The proof of POVM nature of an arbitrary measurement is carried out using a purely operational approach, which is fully ignorant about what is inside a measurement apparatus. The suggested approach gives also an opportunity to derive the Born rule.
Born's rule in its conventional textbook form applies to the small class of projective measurements only. It is well-known that a generalization of Born's rule to realistic experiments must be phrased in terms of positive operator valued measures (POVMs). This generalization accounts for things like losses, imperfect measurements, limited detection accuracy, dark detector counts, and the simultaneous measurement of position and momentum. Starting from first principles, this paper gives a self-contained, deductive introduction to quantum measurement and Born's rule, in its generalized form that applies to the results of measurements described by POVMs. It is based on a suggestive definition of what constitutes a detector, assuming an intuitive informal notion of response. The formal exposition is embedded into the context of a variaety of quotes from the literature illuminating historical aspects of the subject. The material presented suggests a new approach to introductory courses on quantum mechanics.
Quantum theory combines density matrices, Born probabilities, tensor-product composites, positive-operator-valued measures (POVMs), and quantum channels. In a finite-dimensional causal operational theory, we prove that two postulates suffice: local input-output statistics identify channels, and every state admits an equivalent-system purification, unique up to reversible dynamics. The full complex quantum formalism follows; every consistent probability rule is realized as a POVM, so measurement no-restriction is derived rather than assumed.
We show that probabilities of results of all possible measurements performing on a quantum system depend on the system's state only through its density matrix. Therefore all experimentally available information about the state contains in the density matrix. In this study, we do not postulate that measurements obey some given formalism (such as observables, positive-operator valued measures, etc.), and do not use Born rule. The process of measurement is considered in a fully operational manner---as an interaction of a measured system with some black-box apparatus. The key point of our approach is the proof that, for improper mixtures, the expected value of any measurement depends linearly on the reduced density function. Such a proof is achieved by considering appropriate thought experiments. We demonstrate that Born rule can be derived as a natural consequence of our results.
The POVM theorem is a central result in Bohmian mechanics, grounding the measurement formalism of standard quantum mechanics in a statistical analysis based on the quantum equilibrium hypothesis (the Born rule for Bohmian particle positions). It states that the outcome statistics of an experiment are described by a positive operator-valued measure (POVM) acting on the Hilbert space of the measured system. In light of recent debates about the scope and status of this result, we provide a systematic presentation of the POVM theorem and its underlying assumptions with a focus on their conceptual foundations and physical justifications. We conclude with a brief discussion of the scope of the POVM theorem-especially the sense in which it does (and does not) place limits on what is "measurable" in Bohmian mechanics.
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.