Actually the limit predicted by Shannon can be significantly beaten, because Shannon assumes gaussian noise, but if we use photon counting receivers we need to use a poisson distribution. This is the Gordon-Holevo limit.
To beat Shannon you need PPM formats and photon counters (single photon detectors).
One can do significantly better than the numbers from voyager in the article using optics even without photon cpunting. Our group has shown 1 photon/bit at 10 Gbit/s [1] but others have shown even higher sensitivity (albeit at much lower data rates).
The Deep Space Optical Communication (Dsoc) between earth and the psyche spacecraft uses large-M PPM for this reason! This mission is currently ongoing.
They send optical pulses in one of up to 128 possible time slots, thereby carrying 7 bits each. And each optical pulse on earth may only be received by 5-10 photons.
Can’t you calculate the CRLB for any given distribution if you wanted? That’s what my lab did for microscopy anyway. Saying you’re beating the Shannon limit is like saying you’re beating the second law of thermodynamics to me.. but I could be wrong.
You are correct. People often say "Shannon limit" (the general case) when they are really referring to the "Shannon-Hartley Limit" (the simplified case of an additive white Gaussian noise channel).
For example, MIMO appears to "break" the Shannon-Hartley limit because it does exceed the theoretical AWGN capacity for a simple channel. However, when you apply Shannon's theory to reformulate the problem for the case of a multipath channel with defined mutual coupling, you find that there is a higher limit you are still bounded by.
I often wondered why MIMO was such an investigated topic. It would make sense if the Shannon limit is higher for this channel. Is there a foundational paper or review that shows this?
Shannon theory assumes Gaussian noise, however in the very low power regime that's just not true. I agree it's unintuitive. Have a look at the Gordon paper I posted earlier.
Very interesting, I studied telecommunications and I thought the Shannon limit was the absolute limit. I wonder now if this Gordon Holevo limit is applicable for "traditional" telecommunications (like 5G) as opposed to photon counting a deep space probe
As also explained in the conclusion of the paper linked by you, "photon-counting" detectors are possible only when the energy of one photon is high enough, which happens only for infrared light or for higher frequencies.
"Photon-counting" methods cannot be implemented at frequencies so low as used in 5G networks or in any other traditional radio communications.
Weird, it works for me. You can use any of the other published links in the sibling comments. The name of the paper is "Quantum Limits in Optical Communications" by Banaszek
I am not sure if you can use 1 photon per bit because (as I understand) emitting and capturing photons is a probabilistic process and when you have 1 photon, there is a probability that it will not be captured by an antenna, but rather will be reflected or will turn into heat. Or am I wrong here?
In principle, you can send more than one photon per bit on average. Photons have a lot of ways they can encode additional bits, eg frequency, polarisation, timing.
As an example, assume every photon can encode 10 bits without losses, but you lose 10% of your photons. Then with a clever error correcting code you can encode just shy of 9 bits per photon.
You can think of the error correcting code 'smearing' 9 * n bits of information over 10 * n photos, and as long as you collect 0.9 * n photons, you can recover the 9 * n bits of information.
It's the same reason your CD still plays, even if you scratch it. In fact, you can glue a paper strip of about 1 cm width on the bottom of your CD, and it'll still play just fine. Go wider, and it won't, because you'll be exceeding the error correcting capacity of the code they are using for CDs.
*typo: you can send more than one bit per photon on average
I'm very curious to learn more about 1cm, what is the math behind it? Do you speak about classical music CD with ±700mb of capacity? I was always fascinating by ability of old super scratched optical disks still functioning without problems.
> I'm very curious to learn more about 1cm, what is the math behind it?
So I actually got that from a cool math talk I attended about 20 years ago. At the end the professor had a cool demonstration where he glued paper strips of various sizes radially on the CD, and exactly as the math he spend an hour explaining predicted, the CD player could cope with up to a 1cm width strip, but no more.
In a nutshell, you arrive at the 1cm like this: you can look up what proportion of 'wrong' bits the CD's coding can correct and other overhead. Then you look up the circumference of a CD (about 28 cm), then you do some multiplication, and figure out that you can lose about 1cm out of every 28cm, and still be able to correct.
Most of the interesting math happens at the first step of 'what proportion of errors can the music CD correct?' and more interestingly 'how does the CD player do that?'
> I was always fascinating by ability of old super scratched optical disks still functioning without problems.
Keep in mind that CD-ROMs have one additional layer of coding on top of what music CDs have. That's because if a bit error slips through the error correction chances are it still won't be audible to the human ear for music, but software might still crash with a single wrong bit.
> Do you speak about classical music CD with ±700mb of capacity?
Yes, that's because that's what I heard the talk about. I am sure more modern formats also have interesting error correction, but I don't know what they use and how much you could cover up.
Is there some fundamental limit to the number of bits per photon that can be communicated via EM radiation? I think it does not exist, because photons aren't all equal, we can use very high frequency and X-ray quantum can probably carry much more information than RF quantum.
this is called the Shannon limit. To discern signal from noise, a minimum sample rate of 2x the frequency of the signal is required. A signal is something that can be turned on or off to send a bit.
Higher frequencies can carry more data as you infer but the engineering challenges of designing transmitters and receivers create tradeoffs in practical systems.
Can individual photons be measured for polarization and phase or is there a similar limit that requires more than one photon to do so? I suppose both are relative to some previous polarization or phase?
Polarization can be measured using polarization filter and light detector, but it is destructive in the usual sense of quantum theory. That is, if the detector after polarization filter clicks, we know the EM field had non-zero component in the direction of the filter, but we do not find out the other components it had before entering the filter.
I guess not without a minimum bound on the communication speed.
If you have a way to reliably transmit N bits in time T using P photons, you can transmit N+1 bits in time 2 * T using also P photons. What you would do to transmit X0,X1,...Xn is:
- During the first time slot of duration T, transmit X1,... Xn if X0 = 0 and 0 otherwise (assuming absence of photons is one of the symbols, which we can label 0)
- During the second time slot of duration T, transmit 0 if X0=0 and X1,... Xn otherwise
This only uses P photons to transmit one more bit, but it takes twice as long. So if you're allowed to take all the time that you want to transmit, and have really good clocks, I guess that theoretically this is unbounded.
Send three photons A B C. They arrive at times ta, tb, tc. Compute fraction (tc - tb) / (tb - ta). This can encode any positive real number with arbitrary precision. But clearly you need either very precise measurements or send the photons at a very slow rate.
Special relativity will give limits on how aligned the clocks can be (which should be a function of the distance between the clocks). So there will be precision limits.
Unless the clocks are accelerating, that shouldn't change the ratio between the two time differences, right? If the two ends have a constant velocity relative to each other, then each should perceive the other as being off by a fixed rate.
I don't think compressed sensing is really extracting more information than Shannon, it simply exploits the fact that the signal we are interested in is sparse so we don't need to sample "everything". But this is somewhat outside my area of expertise so my understanding could be wrong.
Maybe I’m mixing Shannon’s limit with the sampling rate imposed by the Nyquist-Shannon Sampling theorem
> Around 2004, Emmanuel Candès, Justin Romberg, Terence Tao, and David Donoho proved that given knowledge about a signal's sparsity, the signal may be reconstructed with even fewer samples than the sampling theorem requires.[4][5] This idea is the basis of compressed sensing
…
> However, if further restrictions are imposed on the signal, then the Nyquist criterion may no longer be a necessary condition.
A non-trivial example of exploiting extra assumptions about the signal is given by the recent field of compressed sensing, which allows for full reconstruction with a sub-Nyquist sampling rate. Specifically, this applies to signals that are sparse (or compressible) in some domain
In so many words, Shannon gave a proof showing that in general the sample rate of a digital sensor puts an upper bound on the frequency of any signal that sensor is able to detect.
Unlike the Nyquist-Shannon theory, compressed sensing is not generally applicable: it requires a sparse signal.
As with many other optimization techniques, it’s a trade off between soundness and completeness.
that is not correct. digital sensors detect frequencies above the nyquist limit all the time, which is why they need an analog antialiasing filter in front of them. what they can't do is distinguish them from baseband aliases
you could just as correctly say 'nyquist-shannon theory is not generally applicable; it requires a bandlimited signal' (which is why compressed sensing doesn't violate it)
Thank you for the clarification, great point about the importance of distinguishing the acts of "detecting" and "making sense of" some signal/data/information
Consider the simple example of frequency aliasing. If you sample a 3.2MHz sine wave at a 1MHz sample rate, it looks the same as a 0.2MHz wave. But if you know a priori that the signal only has frequency components between 3 and 3.5 MHz, then you know the 0.2MHz you are measuring is actually 3.2MHz - you can fully reconstruct the original signal even though you are not sampling it fast enough.
Interestingly, in a philosophical way, you might never be able to know the “original signal”, since any signal can also technically be the alias of an infinite number of other signals, including the one used for sampling
To beat Shannon you need PPM formats and photon counters (single photon detectors).
One can do significantly better than the numbers from voyager in the article using optics even without photon cpunting. Our group has shown 1 photon/bit at 10 Gbit/s [1] but others have shown even higher sensitivity (albeit at much lower data rates).
[1] https://www.nature.com/articles/s41377-020-00389-2