Fredrik said:
But you specifically mention reduced density matrices, and those can't even be defined without a tensor product decomposition.
Yes of course you are right, but my point with the passage you quoted was that
even if we can decompose a system into subsystems (with tensor product) and look at the reduced density matrix of the particular subsystem we are interested in, you
still do not necessarily get irreversible loss of coherence! Sure, the reduced density matrix will generally be of the mixed kind but will it irreversibly go towards a total loss of coherence?
This depends on what the other subsystem is. Mostly when people talk about "environment" I believe it is implied that it consists of an infinite amount of degrees of freedom (i.e. it is macroscopic). In other words, the macroscopic nature is at least as important to the concept of decoherence as the decomposition into subsystems. My general point is that you can actually remove the decomposition as a necessity.
Let me try one last time to convince you that you can get a loss of coherence even without the decomposition feature by returning to my original example of Schrödingers cat.
Let us write the general microscopic state of the macroscopic system (cat) as:
<br />
|\psi\rangle = \sum_i c_i(t)|i\rangle<br />
where i here denotes a set of labels completely characterizing the microscopic state of all particles the cat consists of (clearly a huge amount). Let us assume that we have chosen a basis in such a way that we can clearly distinguish for which microscopic state |i\rangle the cat is either dead or alive. We can formally define a set of projection operators:
<br />
\hat{P}_\text{alive}=\sum_{i\in alive}|i\rangle\langle i|, \quad \hat{P}_\text{dead}=\sum_{i\in dead}|i\rangle\langle i|<br />
We can then associate the state \hat{P}_\text{alive}|\psi\rangle=\sum_{i\in \text{alive}}c_i(t)|i\rangle with a macroscopic state of the cat being alive and vice versa with the macroscopic "dead state". If we, for a moment, trust the conventional probability rule we have:
<br />
\text{Prob. alive}=\langle \psi|\hat{P}_\text{alive}|\psi\rangle=\sum_{i\in \text{alive}}|c_i(t)|^2, \quad \text{Prob. dead}=\langle \psi|\hat{P}_\text{dead}|\psi\rangle=\sum_{i\in \text{dead}}|c_i(t)|^2<br />
So far so good, but is it possible to define a "relative phase" between the two macroscopic states "dead" and "alive"? In principle it should be clear already here that such a feat is difficult and designing an interference experiment without knowing the microscopic configuration of the cat is pretty much impossible. However, we could analyze the operator
<br />
\hat{P}_\text{a-d}=\sum_{i\in \text{alive}}\sum_{j\in \text{dead}}|i\rangle\langle j|<br />
which connects the dead and alive subspaces and thus the object
<br />
\langle \psi|\hat{P}_\text{a-d}|\psi\rangle=\sum_{i\in \text{alive}}\sum_{j\in \text{dead}}c_i^*(t)c_j(t)<br />
is directly related to the "coherence". Now there are two issues here: 1) The c_i's are determined by the exact microscopic state (determined by initial conditions and exact many-particle hamiltonian) of which we are clearly ignorant. 2) the time scale of variation of the c_i(t) is very short compared to macroscopic time scales. We thus expect this object to fluctuate wildly (both statistically and temporally) i.e. it is essentially a chaotic variable. Performing an average (coarse graining) over this object will average it out. The probabilities on the other hand must of course add up to unity and, while it may fluctuate depending on initial conditions and as a consequence of time dependence of c_i(t), because it is always a positive quantity between 0 and 1 it will average to some constant. What this is supposed to illustrate is that the off-diagonal elements will vanish upon statistical and temporal averaging, and it is effectively impossible to create an interference experiment that may be used to observe the relative phase between the macroscopic states "dead" and "alive".
I have started reading a pdf version of Schlosshauer (I'm still buying the real one) and in the intro, he describes decoherence as the system getting more and more entangled with the environment. Everything I have seen indicates that you need to consider at least two component subsystems: "the system" and "the environment".
Yes he seems to credit the non-locality of quantum mechanics, which of course is related to system+environment. In fact most textbooks seem to use this notion to describe decoherence. However, I feel like this is only one specific type of decoherence (environmentally induced decoherence) and that decoherence in general can be described without the use of an external environment.EDIT: Sorry everyone for the long post..hope I didn't derail the discussion too much.
/Jens
