the reduced density matrix and invariance under "collapse"
I thought I posted something like this already but can't find it. So I'll try to write it down here. The point I want to make is that, if you have a composite system A + B, then the reduced density matrix of A doesn't change when we apply a la Copenhagen, a measurement on B.
First of all, what does it mean to have a "collapse" from the density matrix point of view ?
Let us first consider a simple 2-state system: |psi> = a |1> + b|2>. The associated density matrix of this (pure state) is: rho = |psi><psi|, or:
( a a* ; a b* )
( b a* ; b b* )
Performing a measurement in the basis {|1>, |2>} comes down to PUTTING THE NON-DIAGONAL ELEMENTS TO 0 IN THE DENSITY MATRIX IN THIS BASIS.
Indeed, after measurement, we have:
( a a* ; 0 )
( 0 ; b b* )
which comes down to a statistical mixture with probability |a|^2 to be in state |1> and a probability |b|^2 to be in state |2>, exactly what the Born rule prescribes for the state |psi>.
Important remark: this operation is of course sensibly dependent on IN WHAT BASIS we apply this zeroing ! We know indeed that the Born rule is dependent on in what basis we apply it: we're supposed to do so in the eigenbasis of the observable we're measuring.
Now, let us switch to an entangled system A + B, and an overall state |psi>.
|psi> can be written as Sum_{ij} a_{ij} |i> |j>, with |i> an orthogonal basis in the hilbert space H_A of system A, and |j> an orthogonal basis in the hilbert space H_B of system B.
The density matrix corresponding to it is then:
rho = |psi><psi| = Sum_{ij} Sum{kl} a_{ij} a*_{kl} |i>|j><k|<l|
Let us assume that we observe a property of system A, of which |i> is an eigenbasis. We have then that the probability for each |i> to be realized is given by Sum_j a_{ij}a*_{ij}. This is in fact what can be obtained if we define a LOCAL density matrix rho_A:
rho_A_{i,k} = Sum_j a_{ij} a*_{kj}, which is nothing else but the partial trace of rho: each block with given i and k is a submatrix (with indices l and j), and we take the trace of that submatrix.
In the case we perform a measurement in the basis |i>, we have, as said before, that we only keep the diagonal elements of rho_A ; if we change basis {|i>} in H_A (but keeping the basis |j> in H_B for the moment), it should be clear that this comes down to transforming the matrix rho_A{i,k} in the new basis. In THAT basis, if we perform a measurement (another observable of A), we take again the diagonal elements of that transformed matrix rho_A.
So it should be clear that rho_A is the correct density matrix resulting in the right diagonal elements (the probabilities of outcome), when written in the measurement basis of A.
One can even show that, for any observable O_A, acting only on H_A, the expectation value is given <O_A> = Tr( O_A rho_A).
But let us now see what happens to rho_A when we do a measurement on system *B*. This comes down to setting each non-diagonal element in the |j> basis to 0 in the density matrix rho. However, this DOESN'T AFFECT THE PARTIAL TRACE because that only uses the DIAGONAL elements, which are not touched upon!
Let us consider now what happens if we have another observable for B: we have to change the basis |j> (while keeping our basis |i> for A). Again, the trace is INVARIANT UNDER CHANGE OF BASIS.
So we see that the partial traces over the basis in H_B do not change, whether we perform a measurement or not, and in that case, no matter in what basis we perform a measurement.
The matrix rho_A being made up only of partial traces over B, this means that every individual element of the matrix rho_A remains intact, whether we perform a measurement or not, and in that case, no matter in what basis we perform a measurement.
cheers,
Patrick.
