Is there a very specific definition for all types of events where quantum decoherence occurs? Is it merely any event that is "thermodynamically irreversible" and/or "causes entropy to increase"? Is it mathematically defined?

Most importantly, is it possible to list some examples of events where quantum decoherence occurs?

  • $\begingroup$ That is a good questions and if you find a definitive answer you should tell someone... $\endgroup$ – dmckee Dec 29 '13 at 19:54
  • $\begingroup$ Are you still wondering about this? $\endgroup$ – DanielSank Mar 1 '15 at 10:23
  • $\begingroup$ @DanielSank: the current accepted answer definitely helps a bit, although more clarity about decoherence (and maybe also it's relationship to wave function collapse) would always be interesting $\endgroup$ – Silveri Mar 2 '15 at 13:09

I only yesterday saw this question, I am sorry. The answer is simple. Consider a quantum system described by a wave-function. This can be a system of ONE particle, two, three, or more, but on condition that we know exactly how it evolves, and it always remains a well-defined wave-function. (It even can be even a superposition of energy eigenstates, i.e. the energy not unique.)

Now, if this system interacts with another system, consisting in an undefined number of particles, and whose wave-function we cannot write (because we cannot follow the evolution of this system), e.g. a bath of particles, a macroscopic apparatus, the environment, etc., the system under observation becomes, for us, DECOHERED.

That means, if initially we could write it as,

$$|\Psi\rangle = \sum C_i |\Psi_i \rangle ,$$

the phases of the constants $C_i$ become undefined. The wave-function transforms into a mixture.

You may ask whether the decoherence is an effect of our inability to follow the evolution of a complex system. IT MAY BE. We cannot give a definite answer, because indeed, we cannot follow it and cannot say what would have happened if we could follow.

  • $\begingroup$ Typically, the decoherence occurs when we subject the quantum system described by the wave-function in the answer, to a measurement by a macroscopic apparatus, and the measurement is configured to decide in which one of the states Ψ_i the macroscopic apparatus "finds" our system. $\endgroup$ – Sofia Nov 21 '14 at 16:15

In principle the whole universe could be written in the quantum mechanical density formulation.

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

$$\hat\rho= \sum_ip_i|\psi_i\rangle\langle \psi_i|\,$$

By choosing an orthonormal basis, one may resolve the density operator into the density matrix, whose elements are:

$$\rho_{mn} = \sum_ip_i\langle u_{m}|\psi_i\rangle\langle \psi_i|u_n\rangle=\langle u_{m}|\hat\rho|u_n\rangle $$

This means that all the elements of the matrix are filled, the off diagonal ones carrying the phase information for all states, i.e., the coherence of the many body system:

for an operator $\hat A$ which describes an observable $A$ of the system the expectation value of $\langle A\rangle$ is given by

$$\langle A\rangle= \sum_i p_i\langle\psi_i|\hat A|\psi_i\rangle=\sum_{mn} \langle u_{m}|\hat\rho|u_n\rangle\langle u_{m}|\hat A|u_n\rangle= \sum_{mn} \rho_{mn}A_{nm}= \textrm{tr}(\rho A)\;.$$

Now if the off diagonal matrix elements for this density matrix are very very small, one has decoherence, i.e. the phases of one quantum mechanical state are no longer coherent, within measurement errors, with the phases of another one in the many body system.

This happens when dimension become large and the approximation h_bar=0 is effective, one has decoherence. In some sense it depends on the experiment and the accuracies of measurement possible, but it is mostly true once macroscopic dimensions enter the problem except in very special quantum mechanical situations ( lasing, superconductivity, superfluidity)

  • $\begingroup$ Anna, I have a small objection: also states having a well-defined wave-function can be represented by a density matrix, not only mixtures. $\endgroup$ – Sofia Nov 22 '14 at 14:55
  • $\begingroup$ Anna, your answer is beautiful. I have just a small objection: also states having a well-defined wave-function can be represented by a density matrix, not only mixtures. But now, THE REAL trouble: WHAT makes the off-diagonal elements of the density matrix to vanish? THIS is the question on which we don't have an answer. Some people say, DECOHERENCE (Zurek?) But it is not a satisfactory answer. $\endgroup$ – Sofia Nov 22 '14 at 15:05
  • $\begingroup$ @Sofia I think they do not really vanish, they are zero within our measurement possibilities. It is the transition between the h a number comparable with the elementary and atomic dimensions and h set to zero for all intents in classical physics. It is not zero, but the uncertainty is so small that the classical mechanics formulations are accurate for macroscopic purposes. $\endgroup$ – anna v Nov 22 '14 at 15:16
  • $\begingroup$ @Sofia as for the mixed states in a density matrix formulation, the expression is a quote from the wiki article. After all a mixed state is an ensemble of pure states, and if you only have one state no need for statistical formulations, I think. $\endgroup$ – anna v Nov 22 '14 at 15:22
  • $\begingroup$ Dear Anna, it is a bit late in my country, can we continue our discussion tomorrow? It is a very essential discussion. About the density matrix, I saw cases when people operate with the density matrix as a general form referring together to pure states and mixed. But the PAINFUL problem is the DECOHERENCE. And it's not matter to be discussed just in brief comments. This is why I prefer to leave it to tomorrow. $\endgroup$ – Sofia Nov 22 '14 at 18:50

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