Forming a unitary operator from measurement operators

[Danny Boy]

If we consider a measurement of a two level quantum system made by using a probe system followed then by a von Neumann measurement on the probe, how could we determine the unitary operator that must be applied to this system (and probe) to accomplish the given measurement operators.

 

[jambaugh]

You simply need to couple the system to probe system in such a way that the probe evolves distinctly for each level.
Supposing the probe is a quantum particle B's position and the system A's observable is X, you can exponentiate:
$$U(t) = \exp( iX_A \otimes P_B t )$$
which will unitarily translate the probe particle in proportion to the value of the system's observable. You'll have (assuming an initially sharply defined system) a superposition of system in each measured mode $x_k$ with the probe translated over a period of time $t$ by a correlated distance. The system and probe are of course "entangled" (correlated). Then the measurement of the probe gives the system's measured value.

Note this is exactly what happens with an S-G magnet's measurement of electron spin. The electron's position is the probe system for the electron's intrinsic spin.

[EDIT] I should mention that the probe needs to have prepared initial state (almost fixed momentum) as it will reciprocally affect the system to be measured in proportion to IT's observable's value. Since that observable (momentum) is dual to the position recording the registration you will need a very large scale of evolution so that the position uncertainty can be large enough for the momentum to be close to a fixed value prior to the interaction, yet still be measurably distinct after the coupled evolution occurs. In the electron spin measurement example the transverse momentum is close to zero and the transverse position is localized only to the beam width. The system must evolve enough so that the beam deflection is significantly larger than the beam width so you see separate beams (and the beam's transverse momentum needs to be pretty close to zero so the reverse coupling doesn't change the spins being measured... thus subsequent measurement of the same observable will give identical outcomes.)

 

[jambaugh]

I would add to this topic a few point I partially mentioned in the edit.

• Critical to this format of the measurement process is the preparation of the registry system the system which will record the observable.
  
• The system needs to be high dimensional so the observable values are easily distinguishable without bumping against the uncertainty principle. [See Below]
  
• The systems prior state must be carefully "refrigerated" to a sharp mode so that it acts on the system being observed uniformly, thus preserving the observable within the observed system.
  
• This is necessary so that we can say we are observing something physical, we must be able to look again and confirm we get the same measurements twice.
• The reason the uncertainty principle is important is that the observable for the registry system which couples to the observed system must "move the needle" i.e. does not commute with the registry observable which records the datum. And that coupled observable is also the one which must be zeroed out before the measuring coupling occurs. Contrawise the non-commuting recording registry observable will be later read (and re-read) as a classical record of the measured value and thus the precision issue with the uncertainty principle mentioned above.

Measurement is a thermodynamic process, amplifying the very small signal of the quantum to a loud enough signal for us to treat classically. I have not mentioned the additional step in the above which is the stabilization of the record which amounts to placing the registry system into a dynamic where the transition probabilities between the distinct records is infinitesimal and the system is stable under perturbations (due to repeated readings of the record). Picture say a double well potential with coupling to a cold entropy dump.

Think about the chemistry of photography (which is after all measuring the position of groups of photons). There is the exposure to the signal then a developing state where the effect of the observable on the registry is amplified and stabilized.