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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)?

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It's likely bad taste to answer your own question, however, here I am only pointing out the argument of another in the literature. I also won't accept this answer to allow others to respond.


Surprisingly, my above guess that perfect copying is only limited by the probability of spontaneous emission... might actually be correct.

Quoting from [Milonni, P. W., Hardies, M. L. Photons cannot always be replicated. Physics Letters 92(A) no. 7, pp. 321 - 322 (1982).]:

"The probability of spontaneous $(n_s = 0)$ generation of signal photons is equal to the probability of stimulated generation when there is initially one signal photon already present."

"This result is traditionally expressed in the context of ordinary stimulated emission by an initially excited atom. The point here is that this fundamental law applies to all stimulated processes. Whenever there is a possibility of stimulated generation of photons, there is also the possibility of spontaneous generation. And when there is initially one signal photon, the rate of spontaneous generation is precisely equal to the rate of stimulated generation. (More generally, the rates are equal when the expectation value of the signal-mode photon number is unity.) Of course, the absolute rates are different for different purposes..."

"...It should be mentioned that Wooters and Zurek [7] have independently reached a similar conclusion, that no device is able to clone with certainty photons in an arbitrary state of polarization. Their agreement is general: The linearity of quantum mechanics forbids the existence of such a device. The argument of Wooters and Zurek does not rule out the amplification of any given state by a device designed specifically for that state, but it does rule out the existence of a device capable of amplifying an arbitrary state. However, if one designs a device to amplify a specific state then he is assuming a priori knowledge of the state."


Said differently: the ability to achieve stimulated emission without spontaneous emission, or more generally, the existence of quantum amplifiers capable of operating below a critical noise threshold, would mean the absurdity of allowing for superluminal information transfer via a FLASH-like mechanism: http://link.springer.com/article/10.1007%2FBF00729622.


I suppose this quote also have relevance to another question on this site: Why doesn't the no-cloning theorem make lasers impossible?

Specifically, we can say that any gain medium in the optical cavity of a laser will have some probability of spontaneous emission, and thus, rigging the gain medium so that the properties (or subset thereof) of this spontaneously emitted photon match that of the photon one is attempting to clone, means knowing a priori knowing the photons state vector. Therefore, "no cloning" does not preclude the existence of lasers or require that stimulated emission have some degree of "sloppiness".

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