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What is a pure quantum state and a mixed quantum state?

  1. Feb 21, 2013 #1
    what is a pure quantum state and a mixed quantum state? I looked up the internet but I did not quite understand. For pure quantum state I have read that they can't be written down as a mixture of other states??? are entangled states pure states?
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  3. Feb 21, 2013 #2


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    The distinction between 'pure' and 'mixed' applies to ensembles of quantum systems, not the possible states of individual quantum systems.

    Consider the following examples. You have N spin 1/2 systems:

    1. All of your spin systems are identically prepared in a S_z= +1/2 eigenstate. This is a pure ensemble.

    2. All of your spins are identically prepared in the same linear combination of + and - Sz eigenstates. This is also a pure ensemble.

    3. 25% percent of your spin systems are prepared like in 1 above. 75% are prepared in the state described in 2. This is a mixed ensemble.

    Mathematically the difference is easily seen in terms of the density matrix [itex]\hat{\rho}[/itex] of the ensemble if you are familiar with that idea. For a pure ensemble, the trace of the square of the density operator is a maximum at one: [itex]Tr(\hat{\rho}^2)=1[/itex]. For any mixed ensemble the trace of [itex]\hat{\rho}^2[/itex] is less than 1. Also, for a pure state, the density operator is indempotent, i.e. [itex]\hat{\rho}^2=\hat{\rho}[/itex] So, if you are able to write down the density matrix of the system, you have an easy way of determining whether a state is pure or mixed.
    Last edited: Feb 21, 2013
  4. Feb 22, 2013 #3


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    The "ensemble" can also consist of the systems that participate in the experiment when you run it multiple times. If you use a preparation procedure that puts the spin of a silver atom in the pure state ##|+\rangle## 1/3 of the time and in the pure state ##|-\rangle## 2/3 of the time, the spin of a single silver atom that's been subjected to that procedure is in a mixed state.
  5. Feb 22, 2013 #4


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    A state in general is defined as a positive operator of unit trace.

    By definition a pure state is defined as an operator of the form |u><u|. It can be shown that any state can be written as a convex sum of pure states - but not uniquely. Non pure states are called mixed.

    Check out;

  6. Feb 22, 2013 #5
    Thank you all! I understand it now.
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