Decoherence and the Density Matrix

[captain_clogs]

Hi all,

I've been reading the seminal Zurek papers on decoherence but there is one (crucial) point on which I am confused. I understand the mathematical demonstrations that over very short timescales the superpositions of states represented as off-diagonal terms in the density matrix can be shown to go to zero over very short timescales due to interaction of the apparatus/system with the environment, leaving a diagonal density matrix. However, why exactly does a diagonal density matrix mean that we can never measure a superposition of states?

Thanks for any insight!

 

[Demystifier]

To be able to measure the superposition means to be able to measure the relative PHASE between different components of the superposition. For example, the superposition
|a>+|b>
is very different from the superposition
|a>-|b>
In the first case the relative phase factor is +1, while in the second it is -1.

If you write down the density matrix for these two superpositions, you will see that their diagonal matrix elements are the same, while they differ in the off-diagonal matrix elements. In other words, the information about the relative phase is encoded in the off-diagonal matrix elements. Thus, by destroying the off-diagonal matrix elements you destroy the information about the relative phase, which implies that you cannot see the superposition. Instead of a superposition above with a well defined relative phase, you have a mixture
|a> or |b>