M1L7c.txt

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There was a question in the last class
about if you have a density matrix,
if any arbitrary density matrix can
be regarded as the partial trace of a pure state.
I was not immediately clear.
I thought it was, but I wasn't sure.
But the proof is so simple that it fits on three lines.
You can simply take a density matrix.
You can double up the Hilbert space
and define our pure state in a Hilbert
space of twice the dimension.
And you immediately see that this density matrix
is the partial trace of a pure state.
So therefore, if you want, you can always
say the density matrix is sort of entangled.
You are entitled to the opinion that the density matrix always
originated into a pure state, but the entanglement points
to the other subsystem have been broken.
And I showed you last class that if you're
one of the Bell states, the most entangled states in a fully
entangled state between two particles,
but you just look at one particle,
one particle is in just the random state of the density
matrix, which is the unity matrix.