----------------------------------------------- Does decoherence solve the measurement problem? ----------------------------------------------- Many physicist nowadays think that decoherence provides a fully satisfying answer to the measurement problem. But this is an illusion. Decoherence is the (experimentally verified) decay of off-diagonal contributions in a density matrix (written in a preferred basis), when information dissipates into unobservable degrees of freedom in the environment of a system. In particular, decoherence reduces a pure state to a _mixture_ of eigenstates. This is enough to induce classical features in many large quantum systems, characterized by a lack of interference terms. Thus decoherence is very valuable in understanding the classical features of a world that is fundamentally quantum. On the other hand, the 'collapse of the wave function' selects _one_ of the eigenstates as the observed one. This ''problem of definite outcomes'' is part of the measurement problem. It is still a riddle, and not explained by decoherence. More precisely, decoherence explains the dynamical decay of off-diagonal entries in a density matrix rho, thus reducing a nondiagonal density matrix (e.g., one corresponding to a pure state psi via rho = psi psi^*) to a diagonal one, usually one with all diagonal elements occupied. In particular, this turns pure states into a mixture. On the other hand, the collapse turns a pure state psi into another pure state, obtained by projecting psi to the eigenspace corresponding to a measurement result. In terms of density matrices, and assuming that the eigenspace is 1-dimensional, a collapse turns a density matrix rho into a diagonal matrix with a single diagonal entry. This is not explained at all by decoherence. A thorough discussion is given in the excellent survey article M. Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics, Rev. Mod. Phys. 76 (2005), 1267-1305. quant-ph/0312059 More recently, Schlosshauer published a book on decoherence; see his home page http://www.nbi.dk/~schlossh/, where one can also find four reviews of the content. All reviews highly recommend the book; two reviews are wholly favorable. The review by Zeilinger (in Nature) explains why the arguments given there against the Copenhagen interpretation are not convincing, and that by Landsman (in Stud. Hist. Phil. Mod. Phys.) emphasizes conceptual shortcomings, and refers (among others) to http:/to.stanford.edu/archives/win2004/entries/qm-decoherence for a more balanced discussion of the merits of decoherence. The champions of the decoherence approach are (not always but at least sometimes) quite careful to delineate what decoherence can do and what it leaves open. For example, Erich Joos, coauthor of the nice book 'Decoherence and the Appearance of a Classical World in Quantum Theory', http://www.iworld.de/~ej/book.html explicitly states in the last paragraph of p.3 in quant-ph/9908008 that (and why) decoherence does not resolve the measurement problem. The prize-winning book by J.Bub, Interpreting the Quantum World, http://de.wikipedia.org/wiki/Lakatos_Award also emphasizes this point. If the big crowd has a cruder point of view, it means nothing but lack of familiarity with the details. If the quantum mechanical state is taken only as a description of a large ensemble, as in the Statistical Interpretation (see next question), there is no problem. But the riddle is present if one insists that the quantum mechanical state describes a single quantum system (as seems to be required for today's experiments on single atoms in a ion trap), which makes the collapse a necessity. In spite of all results about decoherence, Wigner's mathematically rigorous analysis of the incompatibility of unrestricted unitarity, the unrestricted superposition principle and collapse, Chapter II.2 in: J.A. Wheeler and W. H. Zurek (eds.), Quantum theory and measurement. Princeton Univ. Press, Princeton 1983, in particular pp. 285-288, is unassailable. In a nutshell, Wigner's argument goes as follows: If a measurement of 'up' turns the complete system (including the measured system, the detector, and the environment) into the state psi_1 = |up> tensor |up-detected> tensor |env_1> and a measurement of 'down' turns it into psi_2 = |down> tensor |down-detected> tensor |env_2> and the projections of these states are stable under repetition of the measurement (but possibly with different |env> parts>) then, by linearity, measuring the state |left> = (|up> + |down>)/sqrt(2) necessarily places the whole system into the superposition (psi_1 + psi_2)/sqrt(2) of such states and _not_ (as would be needed to account for the experimental observations) into a state of the form as psi_1 or psi_2, depending on the result of the measurement. Wigner's reasoning implies that a resolution of the measurement problem requires giving up one of the two holy tenets of traditional quantum mechanics: unrestricted unitarity or the unrestricted superposition principle. Von Neumann and with him most textbook authors opted for giving up unitarity by introducing collapse as a process independent of the Schroedinger equation. This is no longer adequate since we now know that there is no dividing line between classical and quantum, so that a measurement can no longer be idealized in the traditional fashion. But then there is no longer a clear place for when the collapse happens, and more specific solutions are called for. My paper A. Neumaier, Collapse challenge for interpretations of quantum mechanics quant-ph/0505172 (see also http://www.mat.univie.ac.at/~neum/collapse.html) contains a collapse challenge for interpretations of quantum mechanics that brings to a focus the requirements for a good solution of the measurement problem. In my opinion, the collapse is no fundamental principle but the result of _approximating_ the entangled dynamics of a system with its environment by a Markovian dynamics for the system itself, resulting in a dissipative master equation of Lindblad type. The latter have a built in collapse. The validity of the Markov approximation is an _additional_ assumption beyond decoherence, which is responsible for the collapse. Its nature is similar to that of the socalled Stosszahlansatz in the derivation of the Boltzmann equation. Quantum optics and hence all high quality experiments for the foundations of quantum mechanics are unthinkable without the Markov approximation.