Talk:Quantum channel

Examples
The way this article employs circular description is far from helpful. One concrete example of a channel that transmits quantum information would be more helpful. P0M 21:12, 14 January 2006 (UTC)

Mct Mht - When I added the examples of teleportation and the fiber optic transmission of photons I was responding to the above request. These are both concrete experimental examples of quantum channels and their real life physical meaning. In other words, when experimentalists say "quantum channel" they mean it in exactly the same way as this article. Please justify your changes or revert the article so that it once again responds to P0M's request.--J S Lundeen 22:38, 23 July 2006 (UTC)

whoa, i meant no disrespect towards experimentalists, and believe what i added didn't suggest any. i didn't delete any material, simply relocated and added comments. notice the fiber optic example given describes physical transportation of photons, tell me if i am wrong. in the langauge of, say, quantum operation, such transportation is taken for granted implicitly and not considered an "operation". compare for example, your description of teleportation here and the general teleportation scheme in quantum teleportation. sending half of an entangled state to the recieving party is not considered part of the scheme in the latter. i added comments comparing this difference in usage of the term, that's all. Mct mht 02:39, 24 July 2006 (UTC)

I didn't mean you disrespect experimentalists, but you have presumed to speak for us. You are making a distinction between meanings of "quantum channel" where there is none. Moving a quantum particle is an example of a quantum channel. Hence, one talks about channel fidelity and channel capacity in quantum cryptography - here moving the particle is not taken for granted.


 * if by channel capacity you mean as defined in the article, or in some other rigorous fashion (IIRC, they are all equivalent, in that they induce the same topology on the CPTP maps), i'd be interested to see a reference from the literature. Mct mht 11:58, 24 July 2006 (UTC)

The point of my two concrete examples were to illustrate that the quantum communication refered to in the intro need not involve moving the actual particle carrying the information. Teleportation is an example of a quantum channel where the particle is not moved. Nonetheless, it is important to remember that to set up the teleportation scheme one requires a quantum channel. The transmission then later only requires a classical channel. I realize that these steps are sometimes left out of the description of teleportation but they are necessary.


 * agreed that they are necessary (see text i inserted in article). being "left out in the description" and "not an operation, by definition" are fundamentally different. Mct mht 12:16, 24 July 2006 (UTC)

Moreover, teleportation is the perfect example of the quantum and classical channels. Teleportation is only surprising because one can use a classical channel to create a quantum channel (as long as there is pre-existing shared entanglement) --J S Lundeen 11:00, 24 July 2006 (UTC)


 * there's a obvious difference, and one shouldn't confuse the two. relocating a particle is obviously not a quantum channel in the sense of being modelled by a CP map between C*-algebras. that should be pointed out, and that's what i did. not to say the experimentalists use the term strictly in that sense. applying a measurement and physically transporting a particle are not treated on equal footing in the first part of the particle. in that formulation, it is a priori assumed that you can arrange the location of various systems however you please, and it is not part of the operation. again, compare the descriptions of teleportation schemes. so one needs to distinguish, since first part of the article laid out clearly what is meant by a quantum channel, in a mathematical sense. likewise, using the terminology in the same paragraph while mixing these two interpretations is not good.Mct mht 11:29, 24 July 2006 (UTC)


 * to summarize, so there are two meanings of the term:
 * A formal operational meaning of quantum mechanical (or classical for that matter) objects and processes, defined by CP maps on C*-algebras.
 * Operations one might perform on a quantum mechanical system in practice.


 * for instance, measurement would belong in both categories. but physical transportation of a particle is 2 but not 1. Mct mht 12:31, 24 July 2006 (UTC)


 * Mct mht, The distinction you are making is not meaningful. CP maps correspond to practical operations on physical systems.


 * well, first there's no distinction and now the distinction is not meaningful. i am not at all rejecting the other view of quantum channels. if the references below are supposed to convince me that it indeed is used in that sense (2 above), well, i am already more than convinced. however there is a distinction far from unmeaningful. Mct mht 16:13, 24 July 2006 (UTC)


 * Physical transportation of a particle does fit in your first meaning as well as the second.


 * you keep claiming that, i am sry but it seems to me patently untrue in general. show me a paper where that's shown and we'll see. Mct mht 16:13, 24 July 2006 (UTC)


 * i am not claiming that such transportation is never a quantum channel in the sense of 1. but, for instance, if you call sending the half of the entangled state to Bob, in the teleportation scheme, a quantum channel in that sense, then i don't believe it's right. Mct mht 07:25, 25 July 2006 (UTC)


 * The idea of a quantum channel was pioneered by Shor, Lloyd, Bennett and Milburn. Any of their early papers should make it clear that they were first considering physical transportation of particles as a quantum channel. And moreover, they were particularly concerned with the resources required for protocols such as teleportation. This led to a categorization of resources in terms of classical and quantum channels, ebits, etc.


 * again, i ain't disputin' none of that. that doesn't imply what you're saying. Mct mht 16:13, 24 July 2006 (UTC)


 * The more general theoretical concept has evolved from those first papers.


 * not true, the formalization of quantum operation has been discussed in a different context for decades now (not sure about priority there, but that's not relevant). see Quantum Theory of Open Systems by Davies, or States, Effects and Operations: Fundamental Notions of Quantum Theory, by Kraus. Mct mht 16:13, 24 July 2006 (UTC)


 * Nonetheless, fiber communication is still a good example of a quantum channel. Here is a ref: Lloyd, S., “Capacity of the Noisy Quantum Channel,” Physical Review A 55, R1613-1622, 1997.  A quote: "This paper puts fundamental limits on the amount of quantum information that can be transmitted reliably along a noisy communication channel such as an optical fiber." Here are some more refs from Lloyd:
 * Giovannetti, V., and S. Lloyd, L. Maccone, P.W. Shor, ‘Entanglement assisted capacity of the broadband lossy channel,’ Phys. Rev. Lett. 91 047901, 2003. Giovannetti, V., and S. Guha, S. Lloyd, L. Maccone, J.H. Shapiro, H.P. Yuen, ‘Classical capacity of thelossy bosonic channel: the exact solution,’ Physical Review letters 92 (2): Art. No. 027902 (2004). Giovannetti, V., Guha, S., Lloyd, S., et al. ‘Classical capacity of free-space optical communication,’Quant. Inf. Comp. 4, 489-499 (2004)


 * What references are you basing your concept of quantum channel on? Keep in mind that quantum channel is not just a theoretical concept - it corresponds to real-life operations and processes --J S Lundeen 13:56, 24 July 2006 (UTC)


 * all i am saying is that there are fundamental and basic difference in notions assigned to the same term. they should not be confused in the article. Mct mht 16:13, 24 July 2006 (UTC)


 * At quantiki, their article already gives a optical fiber as an example of a quantum channel: http://www.quantiki.org/wiki/index.php/Channels_%28CP_maps%29


 * i believe they are confusing the meanings there, in the intro. Mct mht 02:44, 25 July 2006 (UTC)


 * Okay. It seems that we fundamentally disagree - I say an optical fiber (capable of transmitting single photons) is a physical realization (in other words, a concrete example) of a quantum channel in the sense of a CP map on a C-star algebra. You say, that it is an example of another meaning of "quantum channel".


 * here's a correct fact from the quantiki link you gave: a quantum channel in the sense of 1. above is by definition a composition of operations from the following: a) partial trace, b) tensoring with an ancilla, i.e. ρ &rarr; ρ tensor σ, for some fixed σ, and c) unitary evolution. this is an immediate consequence of Stinespring's theorem. if whatever process you are talking about (fiber optic, for example) admits such a model, then it is a channel in the sense of 1. if not, then you are using the term channel "as experimentalists might..." (from the article). Mct mht 02:44, 25 July 2006 (UTC)


 * so that's the definition. you could very well be right. if so, i'd like to know the experimental details, how it is done and how exactly can it be modelled as described above (try to keep it elementary for me though.) Mct mht 07:25, 25 July 2006 (UTC)


 * As I am an experimentalist, why don't you let me handle the concrete


 * i never intended otherwise. notice i didn't change anything you added, other than adding comments distinguishing what needs to be distinguished. in fact the experimental section could use some expanding. but if, in giving an example, one wants to use the term both in the sense of 1. and 2. (above), that needs to be clarified. Mct mht 02:44, 25 July 2006 (UTC)


 * (i.e. physical) examples. I will let you, as a theorist, handle the mathematical definitions and examples? --J S Lundeen 18:23, 24 July 2006 (UTC)

let me also say i understand how someone working in experiments might take quantum channel to mean whatever process used to effect changes in a quantum system in a lab, or larger settings. that's perfectly valid. naturally such a person would expect that, whatever the formal definition of a quantum channel might be, it is simply a model of the laboratory procedures he has in mind. but it is not so in general. it's perhaps unfortunate, this abuse of language. on the mathematical side, i believe they were called quantum operations (this term is still used). more recently in quantum information theory, "operation" is sometimes replaced by "channel", with very similar meaning (someone who knows more about the history can correct me if i am wrong). thus the confusion and/or the dual meaning of the term. Mct mht 05:45, 25 July 2006 (UTC)

Mct mht, three points.
 * 1) Lloyd, Shor, Bennett are all theorists and they considered fiber optic transmission as an example of a quantum channel. So this has nothing to do with the misunderstandings of experimentalists.


 * i never said there was misunderstandings of experimentalists. Mct mht 04:50, 27 July 2006 (UTC)


 * 1) Quantum channel and quantum operation are synonomous - "quantum channel" is usually used in scenerios where information is transmitted, and "quantum operation" is more often used for general transformations such gates, measurements, etc.


 * that description is not right. yes they are synonymous. the difference is quantum channel is slightly more general (but not really) in that "hybrid" observable algebras are admitted as input and output algebras (see the example of quantum instrument in the article), while quantum operation usually means a map between purely quantum algebras. to imply information is sometimes transmitted and sometimes not is wrong, for a quantum channel in first sense(from above) always transmits information, where the term information is also used in the mathematical sense. for instance, measurement, in the Schrodinger picture, is a quantum channel with quantum input and classical output. and gates have quantum input and quantum output. Mct mht 05:12, 27 July 2006 (UTC)


 * 1) Any process in the lab can be modelled as a quantum operation and hence as a quantum channel.
 * 2) Transmission through a fiber is described by a quantum operation just as any process in the lab can be. In the case of a real fiber there is always depolarization and loss, which can be described by a set of Kraus operators or, equivalently, by a superoperator.


 * Re Transmission through a fiber ...can be described by a set of Kraus operators..., as i said above, can you provided some details and a reference? if that's indeed correct, then you would be right and we'll modify the article accordingly. Mct mht 04:50, 27 July 2006 (UTC)

In conclusion, there is no dual meaning in the use of quantum channel.--J S Lundeen 22:49, 26 July 2006 (UTC)

Mct mht, well lets consider fiber transmission with depolarization. The main (with the largest weight) Kraus operator will be the identity - the photon is transmitted undisturbed. The other Kraus operators will just be the Sx,Sy,Sz operators in an equal sum (assuming the depolarization is isotropic).


 * according to above, the channel is the map ρ &rarr; 1/8(ρ + Sx ρ Sx + Sy ρ Sy + Sz ρ Sz), that what you mean? Mct mht 16:17, 27 July 2006 (UTC)


 * if that's the case, can you supply some details as to what the experimental setup is and how exactly it admits such a description in the article? it should be included. Mct mht 01:06, 1 August 2006 (UTC)

Here is a reference where they represent fiber transmission by a superoperator (considering changing reference frames as opposed to depolarization): Phys. Rev. Lett. 91, 027901 (2003).

And Mct mht, here is another reference where two of the founders of quantum information (Shor - useful QComp, and Bennet - QCrypt) say: Any means, such as an optical fiber, for delivering quantum systems more or less intact from one place to another, may be viewed as a quantum channel. IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998.


 * at first glance, you seem to be right. i am going to revert to the last version before i added those comments. Mct mht 16:17, 27 July 2006 (UTC)

Besides, as I said "any process in the lab can be modelled as a quantum operation." Because fiber transmission is a process we can set up in the lab it is obviously a quantum operation. I suggest you confer with your colleagues about this. Perhaps they can clear up your misconceptions about the connection between quantum operations the physics they represent. I think I have done the best I can do here.--J S Lundeen 11:39, 27 July 2006 (UTC)

Unbounded CB norm
Near the top of this article, it states: "We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite dimensional." However, later on in the CB norm section it says:

"To make the operator norm even a more undesirable candidate, the quantity


 * $$\| \Phi \otimes I_n \|$$

may increase without bound as $$n \rightarrow \infty.$$"

This claim is only true in infinite-dimensional spaces (that is, all linear maps on finite-dimensional spaces are completely bounded). Either this comment should be removed or at least reworded accordingly. JokeySmurf (talk) 19:42, 28 February 2009 (UTC)

Forms of communication devices
I'm not sure if this is already there or not, but could a section on the forms a quantum-communication device could take (EG twin photons, electrons) be created? UNIT A4B1 (talk) 20:47, 28 October 2009 (UTC)

Link to quantum operation
I just made it clear in the leader paragraph that a quantum channel is just a quantum operation viewed as a communication channel. This should be emphasized more in the text. I don't think that this article should be merged with the one on quantum operations, but we should make sure that this articles doesn't repeat the description of quantum operation unnecessarily. —Preceding unsigned comment added by Njerseyguy (talk • contribs) 14:43, 23 March 2010 (UTC)

TODO list
Some things that could be improved: All the above reflects on this article .. User:Linas (talk) 05:49, 30 November 2013 (UTC)
 * The section on instrument should be harmonized with quantum instrument, which needs expansion.
 * The section on effects should link to an article on quantum effect or someting similar. Some authors seem to use 'effect' as a synonym for 'bra', others seem to use it as a synonym for a conjugate of a mixed state, some seem to imply its an observable ... all this could be clarified.
 * The section treating classical information has several minor errors, and is otherwise quite unsatisfying. My understanding is that classical info is best described by forbenius algebras, but this would need to be fully developed,
 * The article POVM is a walking disaster zone, and needs a total overhaul. Linking to that form this is bound to lead to confusion.

Quantum channel noise
This article doesn't appear to mention the noise of a quantum channel. But I've seen several articles that refer to "noisy quantum channels" and the correction of such noise (see Quantum error correction and Entanglement distillation and see also Fidelity of quantum states). The specific article on Quantum noise appears to refer to measurements (the classical domain), not superpositions.

The present article needs an explanation of "noisy quantum channels". Are quantum states inherently noisy (I think the answer is no, since uncertainty/nondeterminism is only significant when there is a measurement)? Then, how can quantum channels possibly be noisy? In exactly what basic way do the measurements required for communications generate noise? Can a classical channel be used to send a mixed quantum state from one macroscopic location to another (I think the answer is no)? This article should answer these questions and more, not just give mathematics, IMO. David Spector (talk) 19:35, 23 November 2019 (UTC)


 * I would say the whole article is about noisy quantum channels, since all channels that are not unitary describe some sort of noise or error process. The "ideal quantum channel" is the "identity channel" that sends and quantum state $$|\psi\rangle$$ unchanged from the sender (Alice) to the receiver (Bob): $$\Phi_{id}(\rho) = \rho$$. (the unitary channel and the isometry channel $$\Phi_{V}(\rho) = V\rho V^\dagger$$, where $$V$$ is a unitary or isometry are also noise-free channels. All other completely-positive maps as discussed in the article describe information transmission in the presence of some noise process (=coupling to the environment). I think it would be useful to explain this. All error processes arise from coupling to some other system (="environment") and a basic tenet of quantum mechanics is that if we include all the systems making up the environment, then the total evolution is unitary, i.e., the quantum channel from A to B+environment is noise-free. The "noise" arises from the fact that the environment is not accessible to B ("has to be traced out", in math terminology) which leads to the completely positive map mentioned in the 2nd sentence. The noisyness of the channels is reflected in the fact that their quantum capacity is not maximal: the identity channel for a qubit has one qubit capacity (obviously); all the noisy qubit channels have a smaller quantum capacity. --Qcomp (talk) 00:33, 25 November 2019 (UTC)