I am fortunate to have many smart and knowledgeable physicist friends on facebook (and in real life). I often take advantage of this by conducting surveys in which I post a provocative question and wait to hear what my friends have to say about it. I do this not because I am looking for a definite answer to the questions, but to have a feeling for how researchers may react when confronted with unusual questions, insights or puzzles.

Since I find these surveys valuable, my guess is that so will the Internet. The poll I am presenting in this post was inspired by a common reaction to our quantum fingerprinting paper (see also this post). People are often suspicious of the claim that a protocol that uses only coherent states of light as information carriers can demonstrate truly quantum features, such as an exponential reduction in communication complexity compared to the classical case.

In order to address that suspicion, I asked my friends whether they thought that coherent states of light were classical. I was trying to understand how physicists are used to thinking about coherent states. The contributions I received strongly support the claim that many physicists have memorized the statement that coherent states are ‘classical’. However, as you will see, the conversation quickly turned from an informal discussion into a fruitful source of important insights. I hope you will enjoy the exchange of these bright minds as much as I have.

*Original post:*

**Juan Miguel Arrazola:** A survey for my physicist friends: Are coherent states of light ‘classical’?

*Comments:*

**Vadym Kliuchnikov:** I guess they are classical because their Wigner function is positive.

**Nicolas Quesada:** So are squeezed states which are not classical.

**Andres Garcia Escovar:** How about quasi classical?

**JMA:** So far only Vadym has given a definite answer! Are they classical or not?

**AGE:** http://galileo.phys.virginia.edu/…/CoherentStates.htm

Torsten Scholak: Yes, because they do not fulfil any of the established criteria that would make them qualitatively nonclassical.

**JMA:** If coherent states are classical, how come they are widely used in quantum cryptography?

**Erika Janitz:** Coherent states have a non-zero expectation value for the electric field, and have a mean photon number and variance of |alpha|^2, which is what you would expect from a classical field in the limit of large alpha @_@

**Varun Narasimhachar:** They achieve minimum uncertainty, so in this sense, they are as classical as anything can be. But turns out, the extent to which they are quantum is enough to enable Eve-busting, which is why they are useful in cryptography.

**Oleg Gittsovich:** Yes.

**Evan Meyer-Scott:** Coherent states are not widely used in quantum cryptography. Phase-randomized coherent states (or Poissonian mixtures of Fock states) are! Are they classical?

**Marco Piani:** I think you should distinguish between classicality of the single state and classicality of the set of states that are involved in whatever protocol you are considering. In general, there might be conflicting notions of classicality. Any single coherent state is the most classical possible with respect to certain criteria of classicality that have little to do with information and a lot to do with physics. When you start to consider instead sets of coherent states and the relevant feature Is something information-theoretic as distinguishability, things are different. Does your question mean that you have heard back from the journal?

**OG:** Evan, well here I think they are not. Just because if you say mixtures, you out of a sudden jump to convex sets.. Classical states do not form a convex set, if I am not mistaken.. So you can find non-classical examples of those mixtures..

**JMA:** Thanks everyone for your replies. I was trying to have a feel for just how engraved in physicists minds is the idea that coherent states are ‘classical’. For me, it is clear that coherent-states can exhibit properties with no classical analog, like non-orthogonality (would everyone in this thread also jump to say that a coherent state with alpha=1 or alpha=0 is ‘classical’?) However, most people’s impression seems to be that there can’t be anything quantum about a protocol that uses coherent states, which isn’t quite right! Of course, this just goes to show how ill-defined the conceptions of classical and quantum are in the average physicist’s mind, but it helps me understand how to best explain some aspects of our recent work.

**JMA:** Marco, we haven’t heard back from the journal but from the QIP commitee. Nevertheless, the reactions “there is nothing quantum about that protocol” and “you are using too many modes” are extremely popular amongst anyone that first encounters the results. I’m trying to best understand how to address or prevent these reactions.

**OG:** uhm… “However, most people’s impression seems to be that there can’t be anything quantum about a protocol that uses coherent states, which isn’t quite right!”

This is totally true.. I believe Mr. Glauber explains in which sense coherent states are understood to be classical quite well.

**OG:** “there is nothing quantum about that protocol”… ask referee for truly quantum protocols… this is not a scientific answer… I am sorry

**Juan Bermejo-Vega:** Woa, my favorite topic!: what is quantum and what is not? Ok, ultimately, I have no idea of whether they are quantum or not, but IF you give me one coherent state I will give you a quantum algorithm to do a hybrid phase estimation quantum algorithm: which something quite quantum in the same sense that Shor’s algorithm is something quite quantum http://arxiv.org/abs/quant-ph/0008057

**JBV:** Also, I was under the impression that coherent states + Gaussian unitaries + local measurements are enough to enable universal (measurement-based) quantum computation. If that holds, there should be at least one quantum ingredient there that makes the computation quantum. So… which one do you wanna pick?

**MP:** I have not really read it, but you might be interested in having a look at http://arxiv.org/abs/1203.2661, at least just to have an idea/example of how different definitions of classicality can conflict.

As the rebuttal or at least the discussion of the criticism goes:

1) I suggest that you check the literature (in particular, the quantum optics literature) and check why coherent states are considered “classical”. If I am not wrong you will find that the reasons why they are considered classical apply anyway best when the amplitude is very high. I.e., coherent states are the closest possible approximation to classical states of light, but even that best approximation is only really good in absolute terms when the amplitude is big

2) Add to that the issue of orthogonality and distinguishability for sets of states

3) As the number of modes goes, you must be very careful, and explain why, from your perspective, it is not the right thing to look at and such.

The point in the discussions about quantum vs classical is that one often tries to compare apples and pears. The taste of some apples might be more similar to that of pears, even if the shape of those apples is less similar to that of pears than the shape of other apples that in turn may taste very different from pears. So, the answer to the question of which apples are more similar to pears depends on what you care about. In this sense, you must first of all address the issue of what are the relevant properties. I think you have increasingly done so in subsequent presentations/explanations, and that is good. Unfortunately you can only do so much, simply because, to complicate the issue, different people care about different aspects of quantum vs classical. What I can suggest is that you may want to avoid (strong) claims that may trigger a very skeptical attitude. It is a difficult balance, because then you also water down your result, or at least the selling of your results. I can only wish you good luck and remain available in case you want to discuss in person. Cheers.

Deny Hamel: I guess one way you can think about is whether or not you can explain your scheme without quantizing the light field and with some kind of threshold detector.

**JMA:** Evan: What about an implementation of B92 with two non-orthogonal coherent states?

**JMA:** Deny: Help me out: Would two classical fields of very low amplitude entering a beam-splitter interfere in the same way as two coherent states?

**DH:** Yes it’s essentially the same. If I recall correctly for your protocal you just want the field to exit the BS one way if the phases are the same and the other way if they are opposite; classical waves will do that.

Actually, in a completly classical case, woudn’t it be always possible to measure the intensity of every pusle no matter how attenuated they are? In that case you could always just send the whole strings. I would say that the “constant energy” critetion only makes sense once you have some discretization, or at lease a minimum measurable energy.

**JMA:** Deny, I think you nailed it. My argument so far has been to say that you just can’t go to arbitrarily low amplitudes without encountering quantum effects. But you make me realize that this is crucial: If the energy of the electromagnetic field were continuous, you could always tell whether each output port of the beam splitter had light there or not. However, in our universe, you cannot! The discrete properties of light limit the amount of information that can be gained from a measurement of a very weak pulse. In fact this is intimately related to the non-orthogonality of the incoming coherent states, which is arguably the important quantum property at play.

**MP:** Let me say that classical information processing with continuous variables is a dangerous business, and always requires taking into account imperfections. Otherwise one ends up claiming that infinite data can be stored in the position of an infinitesimal dent on a ruler, and with all powerful analog computers.

**EMS:** I didn’t know the B92 protocol was originally for coherent states! Cool. I’ve never heard of an implementation though. Also I really like how Bennett’s optical circuits are done in ASCII art!

Regarding your response to Deny’s comment: what if you measure in the coherent state basis? Then you can get information even from arbitrarily small pulses (barring imperfections Marco). So the quantum nature doesn’t arise until you use single photon detectors. I would argue it is the discrete properties of single photon detectors that limit the information, not of light (in this case).

**MP:** Notice that as energy goes, you might always go down with the frequency, and I do not think frequency directly enters into your calculations. But with lower frequency should come probably longer pulses (larger time bins). There should be a trade-off somewhere.

**JMA:** Yes Marco, the ‘constant-energy’ adjective can be deceiving, as it really implies ‘constant mean photon number’, but it sounds nicer! Having said that, the trade-off between photon number and number of modes in quantum is something I would like to quantify and understand better. For example, in principle it is possible to communicate any number of bits with just one time-bin mode: just encode information in the number of photons in the mode!

**JMA:** Evan: the problem with measuring in the (overcomplete) coherent state basis is that you could not distinguish perfectly between the vacuum and a state of low amplitude, since they are not orthogonal. Thus, you could not tell reliably what output port of the beam splitter was a vacuum and which one was a low amplitude state. So you still have limited information.

**JMA:** I also just have to say that no matter what impact this work has, it has proved to be a catalyst for amazing discussions.

**MP:** You might be onto (or dealing with) something important and fundamental. People have tried to come up with information principles for quantum mechanics (or, sometimes, just for correlations in quantum mechanics). Maybe as relativity comes form a limit on the speed at which information can be transferred, similarly quantum mechanics can be deduced by postulating that there is some finite maximal rate (per unit of time x unit of energy) for the transmission of information? Probably people have already thought in this direction but one never knows (btw we are doing science in the open, which is quite nice 🙂 )

**JMA:** I think Seth Lloyd made a connection between general relativity and quantum mechanics by postulating a minimum unit of information per discrete unit of spacetime (or something like this). It was a talk he gave at QCMC 2012, so indeed there might be something fundamental at play, in the sense that there is a limit to the information-carrying capacity of physical systems. Once I’m done with the QFT assignment, I will think about this a bit more!

**JMA:** If someone accuses me of procrastination for being on facebook I will reply: Hey, I’m working on open science!

**EMS:** Re non-orthoganality: yeah, of course! I understand now and agree.

Note: Featured image is a sample of Kim Keever’s abstract liquid art (which I found thanks to my love Aleksandra Ignjatovic.)