Say I build some stimulated emission-based cloning machine for photons (e.g. some active laser medium). Alternatively I could do cloning via interference, e.g. a linearly polarized photon could be made to interfere with and copy its state on a circularly polarized photon. Here, the no cloning theorem... as far as I am aware... says that we can never exactly copy the exact quantum state of a photon.
Quoting from the abstract of: Lamas-Linares, A., et. al. Experimental quantum cloning of single photons. Science 296, pp. 712 - 714 (2002): http://web.physics.ucsb.edu/~quopt/sci_clo.pdf
"Although perfect copying of unknown quantum systems is forbidden by the laws of quantum mechanics, approximate cloning is possible. A natural way of realizing quantum cloning of photons is by stimulated emission. In this context, the fundamental quantum limit to the quality of the clones is imposed by the unavoidable presence of spontaneous emission. In our experiment, a single input photon stimulates the emission of additional photons from a source on the basis of parametric down-conversion. This leads to the production of quantum clones with near-optimal fidelity. We also demonstrate universality of the copying procedure by showing that the same fidelity is achieved for arbitrary input states."
Now my question is, what are the practical consequences of "approximate" cloning with optimal fidelity (assuming optimal rather than experimental near-optimum fidelity). Does this mean that if we "clone" a polarized photon via stimulated emission, there should be some probability distribution for the cloned photon's frequency or polarization angle? If so, what shape or properties do these distributions have?
I think my confusion here is the following (I apologize for the length of this, but I want to illustrate where my ability to reason about the "no cloning" theorem is failing):
My understanding is that the "no cloning" theorem says, for example, that we cannot simply inject a single photon with unknown polarization and/or frequency into an optical cavity with an active gain medium, and then expect to be able to copy its information content perfectly, even if we somehow stop the experiment after a single stimulated emission event and block all spontaneous emission events.
Why? Because we would need a priori information to properly set up the optical cavity or gain medium in order to perform the copying procedure. It wouldn't work for just any polarization angle or frequency.
However, doesn't stimulated emission exactly copy both the polarization and frequency information of the stimulating photon? And, within reason, given the eigenstate distribution of the medium a photon is being extracted from via stimulated emission, shouldn't this process work for a wide range of photon frequencies and arbitrary polarization angles? Even if the efficiency of stimulated emission is very low, what stops us from e.g. running the experiment until a stimulated emission event occurs (as opposed to a spontaneous emission event), and then isolating and individually measuring each photon? Is it the inability to be able to tell for sure that a stimulated emission event has occurred, as opposed to a spontaneous emission event, that leads to the satisfication of the "no cloning" theorem without requiring some degree of error for the stimulated emission copying process?
If so, does the lowest achievable probability of spontaneous emission set a hard bound on how well we can simultaneously measure both the polarization and frequency (among other properties) of a photon's state vector? Also, does this tell us that there is a fixed lower-bound error rate for determining a photon's properties regardless of how many properties we measure (since we can individually measure multiple clones produced by the same photon)?