How can one know the initial state from measurements?

[pellman]

Theoretical problems often begin with "given a system in state ψ₀" For example, the 2-slit experiment begins with the assumption of a plane wave incident on the slits. I had always understood to this mean some prior set of measurements had been made to determine the initial state. But how can this be done in principle?

For simplicity, suppose our system is such that a single observable O is sufficient for a complete description, and the eigenvalues of O have discrete spectrum. Let ϕ_k be the orthonormalized eigenstate corresponding to the kth eigenvalue of O. Then any initial state of the system has the form

ψ₀ = ∑ a_k ϕ_k

To know the initial state ψ₀ is to know all the a_k . But repeated measurements of O can only give us |a_k|². So how can we ever know the relative phases?

I expect there must be some way of making measurements so as to take advantage of "interference effects" to get the phases. Can someone else explain further?

In the case of spin 1/2, I derived a way of getting the relative phases of the up and down coefficients by making measurements of the spin along the other two axes, and using the resulting amplitudes to calculate the relative phase along the desired axis. This works. But what about the general case?

 

[A. Neumaier]

One creates the initial state by preparation, using the knowledge about which kind of experimental arrangement produces which state.

 

[Truecrimson]

What you did was akin to linear inversion, one of the methods for doing quantum tomography which is what you need in general.

The problem of determining pure states ("wave functions" as opposed to mixtures) came to be known as the Pauli problem. Pauli originally asked whether the measurements of position and momentum suffice to determine the wave function, and the answer is no. Weigert 1992 discusses the problem, for example. For finite-dimensional systems and general measurements, the problem of minimal measurements was solved by Flammia et al. (up to a set of measure zero, later addressed by Finkelstein). If you restrict yourselves to orthogonal basis measurements, I believe the current best result is that five bases are enough in an arbitrary finite dimension.

For an infinite-dimensional system, doing homodyne measurements for all phase space angles (essentially the observables $x\cos \theta + p\sin \theta$) is one way to determine the wave function.

 

[A. Neumaier]

I believe the current best result is that five bases are enough in an arbitrary finite dimension.

... but only if you know (or assume) that the system is in a pure state. To get the state of an unknown source (which is in general a mixed state) you need at least $N^2-1$ binary test statistics, and this many suffice.

 

[Truecrimson]

Yes, the last two paragraphs of my reply were all about pure states.