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Bloch sphere and mixed stats

  1. Sep 24, 2014 #1
    1. The problem statement, all variables and given/known data
    What is reduced density matrix ##\rho_A## and the Bloch vector representation for a state that is 50% ##|0 \rangle## and 50% ##\frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle)##


    2. Relevant equations



    3. The attempt at a solution

    I haven't seen many (any?) examples of this so I'm trying to feel my way through it. So first matrix should be
    ##\begin{bmatrix}
    1 & 0 \\
    0 & 0
    \end{bmatrix}##
    and the second:
    ##\frac{1}{2}\begin{bmatrix}
    1 & 1 \\
    1 & 1
    \end{bmatrix}##
    Both have a 50% chance so ##\frac{1}{4}\begin{bmatrix}
    1 & 1 \\
    1 & 1
    \end{bmatrix}+\frac{1}{2}\begin{bmatrix}
    1 & 0 \\
    0 & 0
    \end{bmatrix}##
    Giving:
    ##\rho_A=\begin{bmatrix}
    \frac{3}{4} & \frac{1}{4} \\
    \frac{1}{4} & \frac{1}{4}
    \end{bmatrix}##

    I think this is correct.

    I'm not sure about the Bloch sphere portion. Can anyone give me some direction?
     
  2. jcsd
  3. Sep 28, 2014 #2

    jfizzix

    User Avatar
    Science Advisor
    Gold Member

    I would look up the Bloch vector representation of spin-1/2 systems. In short, any
    Any [itex]2\otimes 2[/itex] matrix can be expressed as a sum over the 4 basis matrices [itex]I,\sigma_{x},\sigma_{y}[/itex] and [itex]\sigma_{z}[/itex]. For convenience, we'll define [itex]\sigma_{0}=I[/itex] as the [itex]2\otimes 2[/itex] identity matrix.

    These matrices form an orthogonal basis with the inner product between two matrices defined as the trace of the product of the two matrices:
    [itex]Tr[\sigma_{i}\sigma_{j}]=2\delta_{ij}[/itex]

    As an example, we can have a [itex]2\otimes 2[/itex] matrix [itex]A[/itex] expressed as
    [itex]A=a_{0} \sigma_{0} +a_{1}\sigma_{x}+a_{2}\sigma_{y}+a_{3}\sigma_{z}[/itex]

    We can use the orthogonality of the basis matrices to find [itex]a_{0}[/itex] through [itex]a_{3}[/itex].

    For example:
    [itex]a_{2}= \frac{Tr[A\sigma_{2}]}{Tr[\sigma_{2}\sigma_{2}]}[/itex]

    Now for a density matrix, [itex]A=\rho[/itex], and [itex](a_{1},a_{2},a_{3})[/itex] form a vector [itex]\vec{a}[/itex] on the Bloch sphere. Once you know the components of [itex]\vec{a}[/itex], you will have the Bloch vector of the spin-1/2 system.
     
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