Number of photons required for communication On one hand, the amount of information I can transmit is proportional to the bandwidth. The higher the frequency, the more information I can transmit. On the other hand, the number of photons is reverse proportional to the frequency. I cannot possibly transmit more information than the number of photons I send. Therefore, it appears, that at low intensity levels, a higher frequency signal may contain less information than a lower frequency signal.
For example, consider a camera sensor in a high amplification mode (known in digital photography as "high ISO"). Provided the light intensity is uniform by color, blue sensor pixels would receive fewer photons than red pixels. The photon noise in modern sensors is one of the main quality limitations. Thus, in low light conditions, blue images would be grainier than red images meaning that the amount of the transmitted information is reverse proportional to the frequency.
Considering these two competing trends, an optimum must exist. 
Is there a known formula or estimate for the optimal frequency to transmit the highest amount of information for a given received power? Or, stating this in reverse, is there a formula for the minimum received power required to avoid the photon quantization noise at a given frequency?
 A: The optimal distribution of photon frequencies for sending messages, assuming no noise but quantum shot noise, is indistinguishable from thermal (blackbody) radiation at a given temperature. So find the temperature for thermal radiation corresponding to your desired power, find its entropy, convert that to bits, and you have the theoretical maximum amount of information for a given power. 
Why is this true? I'll give a brief sketch of a proof. The Holevo formula for the quantum information that can be sent over a quantum channel ${\cal N}$ at a given power is 
$$
\max_{\{ p_i, |\psi_i\rangle \langle \psi_i | \}} S\left({\cal N}\left({\sum_i p_i |\psi_i\rangle \langle \psi_i|}\right)\right) - \sum_i p_i S\left(\cal N( | \psi_i \rangle \langle \psi_i | )\right), $$
where the maximization is over all probability distributions of input states to the channel with the desired power constraint, and $S$ is entropy. In words, this is the entropy of the average output less the average entropy of the output. 
If the channel is noiseless, then the second term on the right-hand-side is 0, and you just need to maximize the entropy of the average channel output. This maximization is the same as the maximization for determining a thermal state of the channel output given a fixed power. 
Of course when you try to do this, you may run into problems, like discovering that the spatial width of your channel (which you haven't specified) makes a difference.  I suspect that you would need to specify your problem quite a bit more before you could get a definite numerical answer. 
If you're trying to send a signal through an environment with noise, the theorem that the optimal distribution of photon frequencies is a thermal state no longer holds, and things get much more complicated. But I think that for your question, you wanted the assumption of no noise.
