On Friday, February 22, 2019 at 3:18:01 PM UTC-6, Brent wrote:
>
>
>
> On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
>
> This sounds almost tautological. I have not read Masanes' paper, but he
> seems to be saying the Born rule is a matter of pure logic. In some ways
> that is what Born said.
>
> The Born rule is not hard to understand. If you have a state space with
> vectors |u_i> then a quantum state can be written as sum_ic_i|u_i>. For an
> observable O with eigenvectors o_i the expectation values for that
> observable is
>
> sum_{ij}<u_j|O|u_i> = sum_{ij}<u_j|o_i|u_i> = sum_ip_io_i.
>
> So the expectations of each eigenvalue is multiple of the probability for
> the system to be found in that state. It is not hard to understand, but the
> problem is there is no general theorem and proof that the eigenvalues of an
> operator or observable are diagonal in the probabilities. In fact this has
> some subtle issues with degeneracies.
>
>
> Doesn't Gleason's theorem show that there is no other consistent way to
> assign probabilities to subspaces of a Hilbert space?
>
> Brent
>
It is close. Gleason's theorem tells us that probabilities are a
consequence of certain measurements. So for a basis Q = {q_n} then in a
span in Q = P{q_n}, for P a projection operator that a measure μ(Q} is
given by a trace over projection operators. This is close, but it does not
address the issue of eigenvalues of an operator or observable. Gleason
tried to make this work for operators, but was ultimately not able to.
Many years ago I had an idea that since the trace of a density matrix may
be thought of as constructed from projection operators with tr(ρ_n) = sum_n
|c_n|^2P_n, that observables that commute with the density matrix might
have a derived Born rule following Gleason. Further, maybe operators that
do not commute then have some dual property that still upholds Born rule. I
was not able to make this work.
LC
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