Suppose we are given an ensemble of systems which are believed to contain coherent superposition of the metric. How would we confirm this?
Well, in order to verify that an arbitrary system is in a coherent superposition, which is always relative to a preferred basis, it’s well known that we need to make measurements with respect to (at least?) two non-commuting bases. If we can make measurement M we expect it to be possible to make measurement M` = RM for some symmetry R.
I consider essentially two types of Hilbert spaces: the infinite-dimensional space associated with position, and the finite-dimensional space associated with spin. They have a very different relationship with the fundamental symmetries of spacetime.
For spin, an arbitrary rotation in space is represented by a unitary which can produce proper superpositions. Rotating 90 degrees about the y axis takes a z-up eigenstate to an equal superposition of z-up and z-down. The rotation takes one basis to another with which it does not commute.
In contrast, for position, the unitary representing spatial translation is essentially just a permutation on the space of position eigenstates. It does not produce superpositions from non-superpositions with respect to this basis.
You might think things are different when you consider more realistic measurements with respect to the over-complete basis of wavepackets. (Not surprisingly, the issue is one of preferred basis!) If you imagine the wavepackets as discretely tiling space, it’s tempting to think that translating a single wavepacket a half-integer number of tile spacing will yield an approximate superposition of two wavepackets. But the wavepackets are of course not discrete, and a POVM measurement of “fuzzy” position (for any degree of fuzziness σ) is invariant under spatial translations.
Furthermore, it’s surprisingly difficult to make a measurement of anything besides position or, less commonly, momentum. (I am shaky on this, but I believe it can essentially be traced back to the fact that the interaction term in a QFT lagrangian is usually diagonal in position or, less commonly, momentum in the form of a first derivative.) Even simple momentum measurements (i.e. “speed meters”) are usually unfeasible even though they are greatly desired because they would avoid the test-mass Standard Quantum Limit (SQL).
Although the above isn’t a water-tight argument, it does suggest that the class of measurements on the Hilbert space of spatial wavefunctions (rather than of spin) available to us is very restricted.
After direct measurements in non-commuting bases, the next most direct way to verify a superposition is to allow the unitary self-evolution of the system to connect an eigenstate in the preferred basis to a proper superposition of them. This, I now claim, is the most illuminating way to think about the design of a single-photon 2-slit experiment.
Take the initial single slit (which creates spatial coherence across the two slits) and the double slit (which prepares the superposition) as a “black box” preparation device. Now, we wish to confirm that this device is indeed producing a superposition of L and R. How do we do this? Well, first we simply move our (position) detector very close to the opening of our black box, so the slits have no chance to interfere. This is a measurement in the L/R basis, and we get a 50-50 split between the two outcomes as expected. Now, as described above, we do not have a good way of directly measuring in a non-commuting basis like L+R/L-R. However, by moving our detector the screen back away from the slits, we can allow the natural, unitary evolution of the photon to carry it from an eigenstate of L+R (which is part of a basis we cannot measure) to an eigenstate of position (which is part of a basis we can measure).
In this way, we are able to effectively make two, non-commuting measurements of the photon’s wavefunction and thereby verify that the black box is preparing the photons in a coherent superposition rather than a mixture. This is not quite as direct as truly measuring in the L+R/L-R basis, but it’s pretty good.
Even more indirect would be to allow some intermediate probe to interact with the photon and measure the probe. In order for this to do us any good, the probe much become entangled with the photon in the L+R/L-R basis (i.e. this is the Schmidt basis for the probe-photon decomposition).
[This post was prompted by conversation with Max Tegmark. See also this later post.]
