I'm trying to learn some basic quantum mechanics, mostly from I a mathematical perspective. I am trying to understand this with quantum states as vectors in a Hilbert space, bipartite systems, the difference between superpositions and ensembles of states, and density matrices. And it is the two last items I simply don't get. I have been reading the Wkipedia, Hugh Everett's thesis and this paper.(adsbygoogle = window.adsbygoogle || []).push({});

Following the latter paper, p. 16 ff., we assume that we have a bipartite system ##\mathcal H_A\otimes\mathcal H_B##, where ##\mathcal H_A## and ##\mathcal H_B## are finite dimensional complex Hilbert spaces. ##\{|i_A\rangle\}## and ##\{|\mu_B\rangle\}## are orthonormal bases of ##\mathcal H_A## and ##\mathcal H_B##, respectively. A pure state in ##\mathcal H_A\otimes\mathcal H_B## can then be expanded as ##\psi_{AB}=\sum_{i,\mu}a_{i\mu}|i\rangle_A\otimes|\mu\rangle_B##, with ##\sum_{i,\mu}|a_{i\mu}|^2=1##.

Let ##\mathbf M_A## be a Hermitian operator on ##\mathcal H_A## representing an observable, with eigenvalues ##\lambda_k## and an orthonormal eigenbasis ##|k\rangle_A## (for simplicity, we assume that all the eigenvalues are nondegenerate, i.e. their eigenspaces have dimension 1).

Now, a measurement of this observable upon a bipartite state ##|\psi\rangle_{AB}## is represented by the operator ##\mathbf M_A\otimes \mathbf 1_B##, where ##\mathbf 1_B## is the identity operator on ##\mathcal H_B##.

In the paper (just as Everett) the issue is then to calculate the expectation of the observable ##\mathbf M_A\otimes \mathbf 1_B##, and from this construct a density matrix/operator and conclude that this represents en ensemble of states, not a superposition. This what I don't understand, and moreover, I don't see the point with it.

All the above is based upon a calculation of the expectation of the observable. But why bother so much about the expectation? Isn't it more important to find the probabilities for the different outcomes of the observation, and the possible states after the observation, and their probabilities?

This is how I would do it: First, I would express ##|\psi\rangle_{AB}## in the basis ##\{|k\rangle_A\otimes|\mu\rangle_B\}## instead of ##\{|i\rangle_A\otimes|\mu\rangle_B\}##:

##|\psi\rangle_{AB}=\sum_{k,\mu}b_{i\mu}|k\rangle_A\otimes|\mu\rangle_B##. The ##b_{k\mu}##:s are obtained from the ##a_{i\mu}##:s by a unitarian coordinate transformation.

Then, measurement of the ##\mathbf M_A##-observable is performed by applying the ##\mathbf M_A\otimes \mathbf 1_B##-observable upon the given state. The outcome of this observation is ##\lambda_k## with probability ##\sum_\mu |b_{k\mu}|^2##, and in this case (assuming that this probability is nonzero), the state after the observation is ##|k\rangle_A\otimes\frac1{\sum_\mu |b_{k\mu}|^2 }\sum_{\mu}b_{k\mu}|\mu\rangle_B##.

So, I don't understand why density matrices/operators and ensembles are needed here...

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# I Ensembles and density matrices - I don't get it

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