Quantum measurement process


The impressive success story of quantum mechanics and quantum field theory is based on the fact that these theories have, up to now, worked wherever they were applied, such as to solid state physics, atomic physics, quantum chemistry, high energy physics and the early Universe.
It is surprising that its foundations are still as heavily debated as in the 1930's. This occurs because the quantum measurement process is not well understood, a fact which generated a zoo of interpretations of quantum mechanics (Copenhagen interpretation, multi-universe picture, mind-body-problem, the modal interpretation, the wavefunction is a state of knowledge or of belief, bohmian or nelsonian QM, extensions of QM are needed to describe collapse in measurements, ...).

In our group a solvable model for the quantum measurement process has been introduced, that embodies enough relevant physics, [L41]. A spin-1/2 is measured by coupling it to an apparatus, which itself consists of a magnet and a bath. The magnet itself consists of N spins, and becomes macroscopic for large N. The magnet starts out in a metastable paramagnetic state and is driven by the measurement in the up or down magnetized state, which gives a macroscopic amplification of the value s_z=1 or s_z=-1, respectively, of the tested spin. This setup is the magnetic analog of a bubble chamber, where an oversaturated liquid develops bubbles of its stable gas phase (just as in champagne), when triggered by a particle.


The measurement goes in two steps: first a fast diagonalization occurs on the basis set by the interaction Hamiltonian between system and apparatus; quickly after this, the hidden knowledge about the would-be off-diagonal terms is washed out by the bath, never to be recovered. The remaining state has classical features (no Schroedinger-cat terms), and it appears that the quantum solution for the registration of this state can be explained in classical terms, so there is a certain integration of classical and quantum measurements, in agreement with an expectation of Bohr.

It is set out from the start that the apparatus should begin in a mixed state; the pure state setup most often considered in literature is hopelessly unrealistic for measurement apparati.

Updates on our view on the problem occurred first in [C28] and more recently in [C30], [C33] and [C36]. A long article giving all details is in preparation.


A similar setup for a quantum measurement process was worked out in [P55] for a more complicated models with bosons, which start out close to an ideal Bose-Einstein transition and are driven into it by the measurement.



Disappearance of Schrodinger cats. Off-diagonal terms of the density matrix are popularly called "Schrodinger cat states", where cats are in a quantum superposition of being alive and being dead. (Nobody has ever observed such states, so this popular view has an unclear meaning, but anyhow). The von Neumann collapse postulates that these states disappear in a measurement. In our model this appears to occur due to a physical process: a rather quick dephasing, and, on a longer timescale, decoherence due to coupling to the bath. There are two main ingeredients: macroscopic size of the apparatus and mixed nature of its initial state. Such a mixed nature is the physical situation: an experimentalist waits untill his apparatus has stabilized. In theoretical approaches one mostly considers a pure initial state for the apparatus, but this is hopelessly unrealistic for macroscopic measurement apparati.

Derivation of the Born rule. Quantum probablities are given by Born's postulate: squares of wavefunctions, or, more generally, diagonal elements of the density matrix. Till now the cause of this rule has remained obscure. At various moments researchers have tried to explain this connection on the basis of some mathematical connections. This reasoning must be circular: in mathematics you only get out what you put in.
We have put forward another approach: Having the solution for the quantum measurement process at hand, we could look at the outcome for the macroscopic pointer variable alone. This is a certain formula derived from quantum theory. Now the question is what this has to do with the practice of an experimentalist. The only sensible interpretation is that it is a classical distribution of the possible pointer values, with probabilities indeed given by the Born prescription. The derivation is thus not based on mathematics but on the identification of the role of a mathematical formula in experimental practice.

Interpretation of quantum mechanics: In our approach, the apparatus is explicitly taken into account. The analytical solution of the model supports the view that the statistical interpretation or ensemble interpretation is the only meaningful way to interpret quantum mechanics. This means that any quantum state, also a pure state determined by a wavefunction, describes an ensemble of identically prepared systems. For example, in an ideal Stern-Gerlach experiment all electrons in the upper beam together are described by the wavefunction |up > or density matrix |up > < up|. A fundamental conclusion is that quantum mechanics does not describe individual events (``the world'') but only the statistics of outcomes of measurements.


Towards sub-quantum mechanics. A tempting conclusion is then: this theory is incomplete; everyone is invited to join Einstein's dream of a more complete theory. That such a theory is highly wanted is already clear for the first electron in a two-slit experiment, since we have no way to know through which slit it came, not even whether it existed before it hit the screen.... In other words: except for idealized limits, no single experiment is described by quantum theory. My adagio "It is the task of physicists to describe nature" then forces to take this problem very seriously. This view is completely opposite to the widely accepted status quo emphasized by Bohr, namely that "one should not ask such questions". For me this just says " Bohr closed the door" and threw away the key. For my own attempts to find the key, see the section Sub-quantum mechanics.


[C40] Armen E. Allahverdyan, Roger Balian and Theo M. Nieuwenhuizen,
The quantum measurement process: Lessons from an exactly solvable model, quant-ph/0702135 ,
in Beyond the Quantum, Lorentz Center Leiden, The Netherlands, 29 May - 2 June 2006.
Edited by Theo M. Nieuwenhuizen, Bahar Mehmani, Vaclav Spicka, Maryam J. Aghdami and Andrei Yu. Khrennikov. World Scientific 2007 , pp 53-65


[C36] A.E. Allahverdyan, R. Balian and Th. M. Nieuwenhuizen,
Phase Transitions and Quantum Measurements, quant-ph/0508162.
AIP Conf. Proc. 810: Quantum Theory: Reconsideration of Foundations-3, G. Adenier, A.Yu Khrennikov and Th.M. Nieuwenhuizen, eds, (Am. Inst. Phys., Melville, NY, 2006), pp 47-58.

[C33] Armen E. Allahverdyan, Roger Balian and Theo M. N., Dynamics of a quantum measurement, quant-ph/0412045

[C30] Armen E. Allahverdyan, Roger Balian and Theo. M. N., The quantum measurement process in an exactly solvable model, Proceedings `Foundations of Probability and Physics-3', Vaxjo, Sweden, cond-mat/0408316

[C28] Armen E. Allahverdyan, Roger Balian and Theo M. Nieuwenhuizen, The quantum measurement process: an exactly solvable model, Atti d. Fond. G. Ronchi, An. LVIII, N. 6, 719-727 (2003); cond-mat/0309188

[L41] Armen E. Allahverdyan, Roger Balian and Th.M. N., Curie-Weiss model of the quantum measurement process, Europhys. Lett. 61, (2003) 452-458

[P55] Armen E. Allahverdyan, Roger Balian and Theo M. N., Quantum measurement as driven phase transition: An exactly solvable model, Phys. Rev. A 64, 032108 (2001)