The measurement problem in quantum mechanics is the supposedly unresolved problem of how (or if) wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer. The wavefunction in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states, but actual measurements always find the physical system in a definite state. Any future evolution is based on the state the system was discovered to be in when the measurement was made, meaning that the measurement "did something" to the process under examination. Whatever that "something" may be does not appear to be explained by the basic theory.
The obvious conclusion of the above description is that the result of the measurement is not directly described by the wavefunction. Indeed, one has to construct the probability amplitude as a sum over all relative amplitudes that can occur and construct the squared norm in order to obtain the probability. The experiment can in fact add new information to the original problem making some quantities and types of questions well defined in the new context. This is in fact always the case and is the main reason why quantum logics must be considered when analyzing the compatibility of various propositions or questions.  To express matters differently (to paraphrase Steven Weinberg), the Schrödinger wave equation determines the wavefunction at any later time. If observers and their measuring apparatus are themselves described by a deterministic wave function, why can we not predict precise results for measurements, but only probabilities? As a general question: How can one establish a correspondence between quantum and classical reality?
Although there is a difficulty in understanding basic quantum mechanics and it may be a challenge for most undergraduate students, many scientists are accepting the fact that there is no paradox related to quantum measurement. The basic idea of how a measurement is performed and what can be measured are general knowledge. In reality some properties that can be assigned to macroscopic objects by using classical logics are ill-defined for quantum objects. Because of this, several quantities that would make physical sense for a classic physicist are not defined in reality. A simple example would be the spin of a particle and its spin projection. While the spin of a particle is a well defined quantity the spin projection implies the definition of an axis. Following the classical way of thinking if the particle has a well defined spatial orientation of its angular momentum this could be considered as a natural choice for an axis. In reality quantum mechanics restricts the possible orientations. There is no specified orientation for a particle with internal spin in the real space. Because of this logical impossibility of defining a natural preferred axis the question "what is the orientation of the spin for a particle?" is not defined unless a measuring device gives the required supplemental information (an axis).  
All results of quantum mechanics must be interpreted in a probabilistic sense. This means that one has to construct a statistics and calculate the probabilities. This implies that one implicitly deals with sets of particles prepared in the same way. By doing so one samples the full space of the given physical situation (its topology) and obtains non-local information. Nevertheless there is no non-locality in basic quantum mechanics.
Another enlightening experiment is the double slit experiment. One interpretation is that one particle exists in two positions at once in order to allow for the quantum interference. This is wrong for several reasons. First, one single particle traversing the apparatus will give its energy on the screen behind the double slit arrangement. If the detector is placed there then the logical question that one asks is "where does the particle fall on the screen?". This question has a perfectly determined answer given by the result of a single measurement. Nevertheless, a single event does not give any information about the global construction of the device. In order to obtain this information (and the only one predicted by quantum mechanics) one has to perform a statistics. Nevertheless, as long as the same question is asked, the answer will not be compatible with another question, for example "through what slit did the particle come on the screen?". In this way, the information demanded by the last question is not available neither for the set of particles that generated the pattern on the screen nor for the detector before they actually passed through one or the other of the slits. In order to calculate the right probability one has to include this uncertainty in the description of the end result (a probability distribution) and this is done using the probability amplitude (or "pre-probability") as described in  This and other examples should be able to show that there is in fact no "measurement problem" in quantum mechanics.
The best known example is the "paradox" of the Schrödinger's cat. A mechanism is arranged to kill a cat if a quantum event, such as the decay of a radioactive atom, occurs. Prior to observation, the cat is apparently evolving into a linear superposition of basis vectors that can be characterized as an "alive cat" and states that can be described as a "dead cat". Each of these possibilities is associated with a specific nonzero probability amplitude; the cat seems to be in some kind of "combination" state called a "quantum superposition". However, a single, particular observation of the cat does not measure the probabilities: it always finds either a living cat, or a dead cat. After the measurement the cat is definitively alive or dead. The question is: How are the probabilities converted into an actual, sharply well-defined outcome? The answer to this question is again given by the fact that there is no physical reality that can be associated to the wavefunction by itself. The wavefunction can best be described as a "pre-probability". Of course one can use various orthonormal bases in order to describe the combined Hilbert space of the apparatus and the cat. However, the physical properties which can be described by the resulting probabilities depend upon which basis is used and not every choice of basis at time t allows one to say whether the cat is dead or alive. The projector into the subspace of the Hilbert space of the system saying that the cat is "dead or alive" does not commute with any of the projectors that identify a specific state for the detector being triggered by a decay event or not being triggered by a decay event. In this way it is not possible to speak meaningfully about a macroscopic state of the cat "dead + alive" while considering one or another outcome for the decay. This means the superposed "cat" state does not have an independent physical interpretation. By this, the so called "Schrodinger cat's paradox" is solved. This and many other examples show that there is in fact no "measurement problem" in quantum mechanics.
Hugh Everett's many-worlds interpretation attempts to solve the problem by suggesting there is only one wavefunction, the superposition of the entire universe, and it never collapses—so there is no measurement problem. Instead, the act of measurement is simply an interaction between quantum entities, e.g. observer, measuring instrument, electron/positron etc., which entangle to form a single larger entity, for instance living cat/happy scientist. Everett also attempted to demonstrate the way that in measurements the probabilistic nature of quantum mechanics would appear; work later extended by Bryce DeWitt.
De Broglie–Bohm theory tries to solve the measurement problem very differently: this interpretation contains not only the wavefunction, but also the information about the position of the particle(s). The role of the wavefunction is to generate the velocity field for the particles. These velocities are such that the probability distribution for the particle remains consistent with the predictions of the orthodox quantum mechanics. According to de Broglie–Bohm theory, interaction with the environment during a measurement procedure separates the wave packets in configuration space which is where apparent wavefunction collapse comes from even though there is no actual collapse.
Erich Joos and Heinz-Dieter Zeh claim that the latter approach was put on firm ground in the 1980s by the phenomenon of quantum decoherence. Zeh further claims that decoherence makes it possible to identify the fuzzy boundary between the quantum microworld and the world where the classical intuition is applicable. Quantum decoherence was proposed in the context of the many-worlds interpretation, but it has also become an important part of some modern updates of the Copenhagen interpretation based on consistent histories. Quantum decoherence does not describe the actual process of the wavefunction collapse, but it explains the conversion of the quantum probabilities (that exhibit interference effects) to the ordinary classical probabilities. See, for example, Zurek, Zeh and Schlosshauer.
The present situation is slowly clarifying, as described in a recent paper by Schlosshauer as follows:
- Several decoherence-unrelated proposals have been put forward in the past to elucidate the meaning of probabilities and arrive at the Born rule ... It is fair to say that no decisive conclusion appears to have been reached as to the success of these derivations. ...
- As it is well known, [many papers by Bohr insist upon] the fundamental role of classical concepts. The experimental evidence for superpositions of macroscopically distinct states on increasingly large length scales counters such a dictum. Superpositions appear to be novel and individually existing states, often without any classical counterparts. Only the physical interactions between systems then determine a particular decomposition into classical states from the view of each particular system. Thus classical concepts are to be understood as locally emergent in a relative-state sense and should no longer claim a fundamental role in the physical theory.
References and notes
- The Logic of Quantum Mechanics, Garrett Birkhoff and John Von Neumann, The Annals of Mathematics, Second Series, Vol. 37, No. 4 (Oct., 1936), pp. 823-843
- Steven Weinberg (1998). The Oxford History of the Twentieth Century (Michael Howard & William Roger Louis, editors ed.). Oxford University Press. p. 26. ISBN 0-19-820428-0.
- Steven Weinberg: Einstein's Mistakes in Physics Today (2005); see subsection "Contra quantum mechanics"
- Wojciech Hubert Zurek Decoherence, einselection, and the quantum origins of the classical Reviews of Modern Physics, Vol. 75, July 2003
- Robert B. Griffiths "Consistent Quantum Theory, Cambridge University Press"
- R. P. Feynman "Space-Time Approach to Non-Relativistic Quantum Mechanics, Review of Modern Physics, Vol. 20, Number 2, April, 1948"
- Joos, E., and H. D. Zeh, "The emergence of classical properties through interaction with the environment" (1985), Z. Phys. B 59, 223.
- H D Zeh in E. Joos .... (2003). Decoherence and the Appearance of a Classical World in Quantum Theory (2nd Edition; Erich Joos, H. D. Zeh, C. Kiefer, Domenico Giulini, J. Kupsch, I. O. Stamatescu (editors) ed.). Springer-Verlag. Chapter 2. ISBN 3-540-00390-8.
- V. P. Belavkin (1994). "Nondemolition principle of quantum measurement theory". Foundations of Physics 24 (5): 685–714. arXiv:quant-ph/0512188. Bibcode:1994FoPh...24..685B. doi:10.1007/BF02054669.
- V. P. Belavkin (2001). "Quantum noise, bits and jumps: uncertainties, decoherence, measurements and filtering". Progress in Quantum Electronics 25 (1): 1–53. arXiv:quant-ph/0512208. Bibcode:2001PQE....25....1B. doi:10.1016/S0079-6727(00)00011-2.
- Maximilian Schlosshauer (2005). "Decoherence, the measurement problem, and interpretations of quantum mechanics". Rev. Mod. Phys. 76 (4): 1267–1305. arXiv:quant-ph/0312059. Bibcode:2004RvMP...76.1267S. doi:10.1103/RevModPhys.76.1267.
- Maximilian Schlosshauer (January 2006). "Experimental motivation and empirical consistency in minimal no-collapse quantum mechanics". Annals of Physics 321 (1): 112–149. arXiv:quant-ph/0506199. Bibcode:2006AnPhy.321..112S. doi:10.1016/j.aop.2005.10.004.
- R. Buniy, S. Hsu and A. Zee On the origin of probability in quantum mechanics (2006)
- Walker, Evan (2000). The Physics of Consciousness. Cambridge: Perseus. ISBN 0-7382-0436-6. The book's author makes leading contributions to the measurement problem, observer theory, and consciousness.
- Measurement in quantum mechanics
- Philosophy of physics
- Quantum Zeno effect
- Objective collapse theory
- Wave function collapse
- Ghirardi-Rimini-Weber theory
- Quantum decoherence
- Born rule
- EPR paradox
- Quantum pseudo-telepathy
- Absolute time and space
- The Quantum Measurement Problem Two presentations: a non-technical and a more technical presentation.