### Mixed states II: decoherence; important measures of purity and entropy

Decoherence

At the end of this section, you should be able to:
• appreciate why the density matrix is really a great way of expressing states, even for pure states (they uniquely determine the dynamics of the system, without any "overall phase", etc.)
• develop an intuition for measurement, even "inadvertent" measurement
• understand on a somewhat high level how classical physics arises as a limit of quantum physics
• hang out with Wigner's friend
• admit that complex phases matter in quantum mechanics and link them to interference

Suppose we have a system that we wish to measure it under an operator whose eigenvectors are $|0\rangle_A$ and $|1\rangle_B$. The idea is that we have some measurement apparatus, and their original combined state evolves from something like:

$$|\psi\rangle_{AB}=(\lambda|0\rangle_A+\mu|1\rangle_B)\otimes|0\rangle_B$$
To the entangled state:

$$|\psi\rangle_{AB} = \lambda|0\rangle_A\otimes|0\rangle_B+\mu|1\rangle_A\otimes|1\rangle_B$$
Then observing the apparatus is sufficient to observe the system. The idea is that ultimately, the observer himself (or his "knowledge") are the apparatus, and the he entangles with the system to measure it.

Well, we know that often, we end up seeing things we didn't really want to. After all, physics does not care about your wants and preferences. In fact, in pretty much any situation, information about the system will leak out into the surroundings in some specific way. For example, Schrodinger's cat leaks information about the life of the cat by making the environment smelly, i.e. the state evolves from:

$$|\psi\rangle_{AB}=(\lambda|\mathrm{alive}\rangle+\mu|\mathrm{dead}\rangle)\otimes|\mathrm{clean}\rangle$$
To the entangled state:

$$|\psi\rangle_{AB}=\lambda|\mathrm{alive}\rangle\otimes|\mathrm{clean}\rangle+\mu|\mathrm{dead}\rangle\otimes|\mathrm{smelly}\rangle$$
What this means is that the density matrix of the cat evolves as:

$$\left[ {\begin{array}{*{20}{c}}{{{\left| \lambda \right|}^2}}&{\lambda \bar \mu }\\{\mu \bar \lambda }&{{{\left| \mu \right|}^2}}\end{array}} \right] \mapsto \left[ {\begin{array}{*{20}{c}}{{{\left| \lambda \right|}^2}}&0\\0&{{{\left| \mu \right|}^2}}\end{array}} \right]$$
(Check that I got the right transpose.) OK, what happened here?

Recall that the probabilities of collapsing to $|0\rangle$ and $|1\rangle$ are determined purely by the elements on the diagonal -- the off-diagonal elements, or the coherences, are only relevant for collapsing on to some combination of $|0\rangle$ and $|1\rangle$. What's going on here is that when the environment entangles with the system, it has "kinda" already observed it -- like your Wigner's friend. It "knows" that the system isn't in $|0\rangle+|1\rangle$, and even though you haven't observed the environment yet (you haven't smelled it), you know how the combined state has evolved, and the probability has become a classical probability, because the quantum stuff has already been observed -- by the environment.

The idea behind decoherence is the same idea that ensures that the Wigner's friend scenario is consistent.

"Eventually", "all" the information about the system will leak into the environment -- i.e. in principle, we should be able to determine anything about the system from measuring the environment, and our uncertainty about the system arises entirely from our completely classical uncertainty about the environment -- so the density matrix becomes a classical one, i.e. a diagonal one (the off-diagonal terms go to zero).

What basis is it diagonal in? In the basis corresponding to the states of the environment -- i.e. if the environment can be in states $|0\rangle_B$ and $|1\rangle_B$, then the states of the system that precisely induce these states of the environment form the preferred basis. These are often called the "environmentally selected basis".

This process is called decoherence. You may also hear the terms pointer states (for the preferred basis), einselection (environmentally induced selection of the preferred basis), or Quantum Darwinism (what the heck?) -- but they're really synonymous. We'll just use the fancy words when they're grammatically useful.

Well, the following may not be completely clear, but you should at least be able to appreciate that it is true: the off-diagonal terms approach zero, rather than hit it. Why? Although the system leaks information into the surroundings, we aren't really certain about what we're inferring about the system from the environment -- a live cat may be smelly too, etc. So the pointer states are not exactly orthogonal, either.

The precise behavior of decoherence depends on the Hamiltonian of the system -- e.g. predicting the generation of the smelliness of the air from the state of the cat based on what's going on microscopically is something that could be done in principle by solving a really complicated Schrodinger equation. You can, given a Hamiltonian, at least make order-of-magnitude estimates of at how much time and at how macroscopic a scale (i.e. with how many degrees of freedom) does the system begin to behave in a way that can be described as classical.

Decoherence does not remove the need for wavefunction collapse -- one still needs the observer to note an observation, collapsing the system.

TBC: purity, entropy, correlation functions