If you are satisfied with the traditional foundations of quantum mechanics where a mysterious measurement process (taken as fundamental) magically assigns numbers (called measurement results) to a small quantum system following Born's rule, this paper is not for you.
If however, you want to know how it is possible that one can read off from a quantum device called a measurement apparatus numbers (called measurement results) that give information about the state of a small quantum system, and why these numbers follow (under appropriate conditions) Born's rule, then this article is a must read!!
[paragraph added May 28, 2017:] It specifically addresses the question how, in a slightly idealized measurement process of the von Neumann kind (a binary measurement resulting in a state vector reduction to an eigenstate), the information about the state is transferred from the system to the measurement device, and how the system responds to the measurement. This process is usually skipped in foundational work, being assumed to happen instantaneously and without analysis. In their model, the dynamics of the measurement happens on several different time scales; the corresponding processes are referred to as (i) collapse, (ii) decoherence, and (iii) registration). The fastest, collapse phase is governed primarily by the macroscopic nature of the measurement device and hence has a characteristic time scaling like $N^{-1/2}$, where $N$ is the number of particles in the device. This phase is in principle reversible but the revivals are suppressed by the second phase, decoherence, on a characteristic time scale scaling like $N^{-1/4}$. Both terms vanish in the thermodynamic limit; thus the latter is responsible for the near-instantaneous nature of the measurement. Registration is the final and slowest phase; it takes time even in the thermodynamic limit and is responsible for the flow of information from the measured system to the measurement device through the restauration of equilibrium. The binary nature of the measurement is due to the fact that the measurement device is prepared in a metastable equilibrium state that decays after activation into one of two symmetrical stable equilibrium states.
The paper was published as
[1]
A.E. Allahverdyan, R. Balian and T.M. Nieuwenhuizen,
Curie-Weiss model of the quantum measurement process,
Europhysics Letters 61 (2003), 452-458.
http://iopscience.iop.org/article/10.1209/epl/i2003-00150-y/pdf
https://arxiv.org/abs/cond-mat/0203460
It is a short introduction (published 2003) to important work by the authors culminating in the following 200 page treatise (published 2013) containing all the details:
[2]
A.E. Allahverdyan, R. Balian and T.M. Nieuwenhuizen,
Understanding quantum measurement from the solution of dynamical models,
Physics Reports 525 (2013), 1-166.
https://arxiv.org/abs/1107.2138
Other, related papers by the authors on the same topic include
[3]
T.M. Nieuwenhuizen, M. Perarnau-Llobet and R. Balian,
Lectures on dynamical models for quantum measurements,
International Journal of Modern Physics B 28.21 (2014): 1430014.
https://arxiv.org/abs/1406.5178
[4]
R. Balian, A.E. Allahverdyan and T.M. Nieuwenhuizen,
Statistical theory of ideal quantum measurement processes,
Manuscript (2015).
https://hal-cea.archives-ouvertes.fr/cea-01108360
[5]
A.E. Allahverdyan, R. Balian and T.M. Nieuwenhuizen,
A sub-ensemble theory of ideal quantum measurement processes,
Annals of Physics 376 (2017): 324-352.
https://arxiv.org/abs/1303.7257
[6]
R. Balian,
The Theory of Measurement,
pp. 251-278 in:
The Quantum World. Philosophical Debates on Quantum Physics
(B. d'Espagnat, H. Zwirn eds.)
Springer 2017.
Real measurement processes must be described by complex multiparticle quantum processes whose natural description and analysis is in terms of quantum statistical mechanics. The art of understanding consists of finding model problems that capture all essential aspects of the measurement process while still being sufficiently simple that an analysis is mathematically feasible and physically illuminating.
In the course of time quite a number of statistical mechanics models have been proposed and analyzed for this purpose, matching one or more of the many features that a model of a measurement process should faithfully represent. In [1], the authors start by listing six relevant such features (including Born's rule) and then outline how their model (a paramagnetic metastable system coupled to a phonon bath measuring a single spin) satisfies all of them. The main findings are plausible but given without proof. The system considered is claimed to be exactly solvable in the combined thermodynamic and Markovian limit;so everything should follow without any further approximation. For details the reader must turn to [2].
Instead, the authors discuss in [1] the impact of the findings for their model problem on the microscopic interpretation of the measurement process in its various stages. Before decoherence has a chance to set in, the macroscopic size of the apparatus already induces a permanent decay of important off-diagonal entries of the marginal density matrix of the measured system. On the other hand, the measurement result becomes
visible on the measurement apparatus (via the registration in a ferromagnetic state) on a time scale much slower than the decoherence time. Thus decoherence plays a far less pronounced role in measurement than usually assumed.
Of course any derivation rests on assumptions, and this work is no exception. Although the authors frame their contribution in a setting they call Bayesian, no use is made of specific Bayesian (subjective probability) arguments. Everything is equally valid in any setting where classical probabilities are subject to Kolmogorov's axioms and describe the behavior of averages and correlations of multiple (real or fictitious) realizations of random variables.
The machinery of statistical mechanics is assumed to be valid, without any discussion of its foundations. These foundations are traditionally phrased in terms of the standard foundations of quantum mechanics
(which assume Born's rule); so the subsequent derivation of Born's rule through the analysis of the model seems to be somewhat circular. However, as shown in Chapters 8 and 10 of my online book, statistical mechanics can be founded independent of Born's rule, taking as basis only the notion of abstract expectation values $\bar A=\langle A\rangle :=\mbox{trace}\rho A$ for a system with a given density operator $\rho$, together with the uncertainty principle in the form
a Hermitian quantity $A$ whose uncertainty $\sigma_A:=\sqrt{\langle(A-\bar A)\rangle}$ is much less than $|\bar A|$ has this value within this uncertainty.
This much weaker and unproblematic rule for the interpretation of the formalism of statistical mechanics makes neither a reference to the (complex) measurement process nor to a mysterious notion of collapse, and is the basis of my thermal interpretation of quantum mechanics.
From this perspective, the models considered in [1-6] naturally complement the thermal interpretation of quantum mechanics by presenting measurement as a derived rather than fundamental theoretical concept.
[paragraph added May 28, 2017:] Actually, the authors ultimately saw the need to justify the foundations they are using and discussed them extensively in [5]. Unlike many other foundational papers, they are quite specific about what they assume and why. In particular, as in my thermal interpretation, they renounce the traditional meaning of expectation values as a mean over observations (to emphasize this they call them q-expectation values) and they assert as their Interpretative Principle I the uncertainty principle mentioned above. (In some other respects they deviate from the thermal interpretation.)
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