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Decoherence and the Density Matrix

  1. Mar 29, 2012 #1
    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!
  2. jcsd
  3. Mar 30, 2012 #2


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    Science Advisor

    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
    is very different from the superposition
    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>
    Last edited: Mar 30, 2012
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