Let's say that "decoherence" is that transition from a pure quantum state to a mixed state due to interactions with the environment. (A reasonable definition?)
Mixed states are NOT decohered states, they are states where the phases of the wavefunctions are well defined, just not in an eigenstate that will give a unique eigen value at measurement.
How is that compatible with the Copenhagen interpretation of quantum mechanics -- specifically with the role of the observer?
In the quantum regime "observation" is interchangeable with "interaction", it is not necessary to have a human observer. An electron scattering off an atom "observes" the atom.
By the above definition, observation is an example of decoherence; the observer in the Copemhagen interpertation is part of the environment and causes the wavefunction to collapse.
The "collapse" language is a fancy way of saying that an instance was picked from a probability distribution, which is what the state function gives us where composed of pure states or mixed. The square of the state function gives us a probability distribution and an observation gives an instance in that distribution. It is as ridiculous as saying that when getting a five in a throw of the dice, the probability distribution of the dice throws has collapsed.
However, decoherence can also happen by (often-unwanted) coupling of the quantum system to a heat bath.
Now yes, a heat bath leads to decoherence. The easiest way to think of coherence and decoherence is in the density matrix formalism.
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.
A density matrix has rows and columns of all the pure state functions comprising an ensemble of paticles. Off diagonal elements carry the phase information between two pure wave functions.
In theory, the whole universe could be described by one density matrix. h_bar though being a very small number and the macroscopic dimensions of a heat bath together with the enormous number of particles ( 10^23 per mole) reduce the density matrix to its diagonal elements. That is when a system of particles is decohered, when the phase information is lost.
Does that mean that the heat bath is an "observer"? If not, what causes the apparent** wave function collapse when there is no human observer?
The multitude of interactions in a heat bath and the macroscopic dimensions ensure decoherence, the loss of phases between state functions because they cannot be measured.
Having worked with crummy superconducting qubits, I'm aware that they can decohere very quickly and apparently without human observation. E.g. suppose I initialize a qubit in a pure state. If I measure it after 10 ns, the qubit still appears to be in a pure state. If I measure the qubit after 1ms, it appears to have decohered into a mixed state.
a mixed state is not decohered. still there exists a density matrix with non zero off diagonal elements
(I could determine that by trying to perform some quantum operation on the qubit.) Since I did not measure or interfere with the qubit in the intervening time, it does not seem that I could have caused the decoherence ("collapse").
All matter emits black body radiation according to the temperature, i.e. photons impinged on your qubit. Obviously some of them interacted and changed the state function describing the setup. On a large scale this is the heat bath.