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arXiv:2107.00803

A microscopic derivation of the quantum measurement postulates

Vyacheslav Lysov, Yasha Neiman

Published Jul 2, 2021• 0 citations
arXivHTMLPDF

Abstract

In the mid-19th century, both the laws of mechanics and thermodynamics were known, and both appeared fundamental. This was changed by Boltzmann and Gibbs, who showed that thermodynamics can be *derived*, by applying mechanics to very large systems, and making simple statistical assumptions about their behavior. Similarly, when Quantum Mechanics (QM) was first discovered, it appeared to require two sets of postulates: one about the deterministic evolution of wavefunctions, and another about the probabilistic measurement process. Here again, the latter is derivable from the former: by applying unitary evolution to large systems (apparatuses, observers and environment), and making simple assumptions about their behavior, one can derive all the features of quantum measurement. We set out to demonstrate this claim, using a simple and explicit model of a quantum experiment, which we hope will be clear and compelling to the average physicist.

DOI
10.48550/arxiv.2107.00803

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In the mid-19th century, both the laws of mechanics and thermodynamics were known, and both appeared fundamental. This was changed by Boltzmann and Gibbs, who showed that thermodynamics can be derived, by applying mechanics to very large systems, and making simple statistical assumptions about their behavior. Similarly, when Quantum Mechanics (QM) was first discovered, it appeared to require two sets of postulates: one about the deterministic evolution of wavefunctions, and another about the probabilistic measurement process. Here again, the latter is derivable from the former: by applying unitary evolution to large systems (apparatuses, observers and environment), and making simple assumptions about their behavior, one can derive all the features of quantum measurement. We set out to demonstrate this claim, using a simple and explicit model of a quantum experiment, which we hope will be clear and compelling to the average physicist.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the "measurement problem" of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.
Why yet another book on quantum mechanics? Quantum mechanics was born in the first quarter of the twentieth century and has received an enormous number of theoretical and experimental confirmations over the years. It is considered to be the fundamental physical paradigm, and has a wide range of applications, from cosmology to chemistry, and from biology to information sciences. It is one of the greatest intellectual achievements of the past century. As an effect of its invention, the very concept of physical reality was changed, and “observation,” “measurement,” “prediction,” and “state of the system” acquired a new and deeper meaning. Probability was not unknown in physics: it was introduced by Boltzmann in order to control the behavior of a system with a very large number of particles. It was the missing concept in order to understand the thermodynamics of macroscopic bodies, but the structure of the physical laws remained still deterministic. The introduction of probability was needed as a consequence of our lack of knowledge of the initial conditions of the system and of our inability to solve an enormous number of coupled non-linear differential equations. In quantum mechanics, the tune is different: if we have 10 6 radioactive atoms no intrinsic unknown variables decide which of them will decay first. What we observe experimentally seems to be an irreducible random process. The original explanation of this phenomenon in quantum mechanics was rather unexpected.
Abstract The quantum measurement postulate remains a fundamental physical contradiction, enforcing a non-unitary, observer-dependent projection that interrupts the continuous dynamical evolution governed by the Schrödinger equation. In this work, we demonstrate that the orthodox instantaneous collapse postulate is both a thermodynamic impossibility and a logical redundancy. The observer-dependent actualization of macroscopic particle ensembles would require an instantaneous deposition of latent heat ( dt→0dt \rightarrow 0dt→0 d t → 0 ), triggering an unphysical, infinite power divergence that delivers a macroscopic thermal shock, fundamentally contradicting the thermodynamic stability of the apparatus. Moreover, because an unobserved apparatus must undergo continuous open-system thermodynamic relaxation prior to conscious inspection, any subsequent mathematical projection is theoretically obsolete. Furthermore, we mathematically invalidate leading alternative theories, proving that mean-field thermodynamic ensemble models (such as the Allahverdyan, Balian, and Nieuwenhuizen formalism) fundamentally contradict their own claims: while explicitly attempting to derive the probabilistic actualization dictated by the Born rule, they inadvertently force deterministic single-run outcomes. We emphasize that our critique and the theoretical framework proposed herein are explicitly directed at the objective actualization of individual quantum systems. Consequently, the purely statistical ensemble interpretation—which conceptually bypasses the single-run measurement problem—falls outside the scope of the present analysis. To replace these untenable paradigms, we establish the Open-System Measurement Framework (OSMF), shifting the focus entirely to the continuous, objective physical evolution of the individual wavefunction. We demonstrate that the quantum system-apparatus interaction generates a transient, thermodynamically unstable macroscopic superposition. Driven by its coupling to a reservoir, this state rapidly undergoes a continuous thermodynamic relaxation into a stable local minimum of its thermodynamic potential, actualizing a definite outcome. By applying the OSMF to canonical experiments, we prove that measurement is an objective, irreversible process. Crucially, this framework seamlessly reconciles Bell-type non-local correlations (the EPR paradox) with the strict causal constraints of special relativity by providing a local, continuous physical mechanism for outcome actualization.
Quantum mechanics is usually presented starting from a series of postulates about the mathematical framework. In this work we show that those same postulates can be derived by assuming that measurements are discrete interactions: that is, that we measure at specific moments in time (as opposed to a continuous measurement that spans a long time interval) and that the system is in general affected by our measurement. We believe that this way of presenting quantum mechanics would make it easier to understand by laying out a more cohesive view of the theory and making it resonate more with our physics intuition.
In textbooks, ideal quantum measurements are described in terms of the tested system only by the collapse postulate and Born's rule. This level of description offers a rather flexible position for the interpretation of quantum mechanics. Here we analyse an ideal measurement as a process of interaction between the tested system S and an apparatus A, so as to derive the properties postulated in textbooks. We thus consider within standard quantum mechanics the measurement of a quantum spin component ŝz by an apparatus A, being a magnet coupled to a bath. We first consider the evolution of the density operator of S + A describing a large set of runs of the measurement process. The approach describes the disappearance of the off-diagonal terms ("truncation") of the density matrix as a physical effect due to A, while the registration of the outcome has classical features due to the large size of the pointer variable, the magnetization. A quantum ambiguity implies that the density matrix at the final time can be decomposed on many bases, not only the one of the measurement. This quantum oddity prevents to connect individual outcomes to measurements, a difficulty known as the "measurement problem". It is shown that it is circumvented by the apparatus as well, since the evolution in a small time interval erases all decompositions, except the one on the measurement basis. Once one can derive the outcome of individual events from quantum theory, the so-called collapse of the wavefunction or the reduction of the state appears as the result of a selection of runs among the original large set. Hence nothing more than standard quantum mechanics is needed to explain features of measurements. The employed statistical formulation is advocated for the teaching of quantum theory.