Plan for the 10th and the 17th
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Our overall objective is to use the Density Matrix, Purity, and the Partial Trace
in order to find out "how entangled" a composite system is.

Here's a brief outline:

(0) Pure and Mixed States (Review)

    - The state of any given system is a pure state.
      We may or may not know what that state is.

    - If we have an ensemble of systems (like a bunch of bits) either they
      are all in the same state, or they are in a mixture of different states.

    - If we take one bit out of a pure ensemble, then it clearly has a pure state.

    - If we take one bit out of mixed ensemble, that one bit has a pure state.
      But if we don't know what the state is, then the best we can do is to
      represent that bit by the mixed state of the ensemble.

(1) The Density Matrix

    - How to calculate the DM of a state.

    - How to calculate the DM of a mixture of states.

    - Notice that the DM is different for pure states and mixtures.

    - But you can't tell much just by looking at it.

(2) Purity

    - How to calculate Purity.

    - Purity tells you "how mixed" the state is.

(3) The Partial Trace

    - Review the tensor product of two matrices.

    - The TP definition shows how to trace out either of the separate matrices.

    - We can do this even if the 2-bit matrix was not a product to begin with.

(4) Quantifying Entanglement

    - When we recover a DM via partial trace, its "mixedness" is really a
      measure of the entanglement of the composite system.

    - Note that we use the density matrix for two different purposes:

      (a) To represent an actual mixture of different states, or a single
          system drawn from such a mixture. In this case, a "mixed" state
          represents our lack of knowledge of the specific state.

      (b) To represent an individual system which been separated out from a
          composite system by using the partial trace. In this case the system
          may not have an individual state (if the composite was entangled).
          The "mixed" state then represents the degree of entanglement of the
          composite system.

    - Work some problems.