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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

    Demystifier

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    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>
     
    Last edited: Mar 30, 2012
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