Talk:Quantum state

@Roffaduft I reverted a couple of edits related to pure states. You may have a valid point but I don't think changes help the article. If you have a reference maybe we can agree on a better solution. Johnjbarton (talk) 17:33, 8 June 2024 (UTC)

Regarding pure states, please read the first answer on this stackexchange page. It’s explained in full the point I was trying to get across Roffaduft (talk) 18:57, 8 June 2024 (UTC)

Much of the content on stackexchange is excellent, but it is often obscure and difficult to confirm with reliable sources. The page you point illustrates this point. The issue of pure states is a side comment to a discussion on another topic. I don't think it help us.

I think the "Formalism in quantum physics" section of the article is poorly presented. I almost deleted several sections with zero refs. The article is confused on "pure states", referring to them in different ways, mostly unsourced. I put the Messiah ref in to at least give one ref that discusses the topic. Johnjbarton (talk) 19:24, 8 June 2024 (UTC)

I do agree that the section “Formalism..” could do with a good cleanup. Mixed states are discussed in two different subsections throughout the article for example, which makes it a bit messy overall.

I personally prefer to have a clear “mathematical definition” and “quantum physical definition” when it comes to topics in quantum mechanics. That is because quite ofter there is a lot of ambiguous use of terminology (something we’ve discussed earlier on the topic of Wave function collapse).

Regardless, all I wanted to do with the clarification was to emphasize the mathematical nuance between pure states and bound states.

If you feel it is too technical, I’m ok with removing the subsection “pure states vs bound states” all together and just include the notion of pure states being bound states in the subsection “eigenstates” Roffaduft (talk) 19:46, 8 June 2024 (UTC)

@Tercer instead of making a rude remark, you could perhaps point out where on page 204 of Messiah it is stated that “Any state that is not pure is called a mixed state”

I couldn’t find it. The reason I subsequently removed the claim is because the terminology can become quite ambigous, i.e., use of “eigenstates”, “states”, “pure states” etc.

The phrasing would suggest that any eigenstate that is not a pure state is automatically a mixed state for example, which doesn’t make much sense Roffaduft (talk) 19:36, 8 June 2024 (UTC)

I'm not going to waste my time leafing through references to find an exact quote to support the blinding obvious. The quote is true by definition.

As for "eigenstate", the author must have meant any state (pure or mixed) belonging to the (possibly degenerate) eigenspace of the measured eigenvalue. So yeah, any eigenstate which is not a pure state is automatically a mixed state. Tercer (talk) 19:55, 8 June 2024 (UTC)

so the eigenstate of the position operator; the dirac delta function, is a mixed state then? As it is clearly not in ${\displaystyle L^{2}}$, not bounded and hence not a pure state. Roffaduft (talk) 20:04, 8 June 2024 (UTC)

It is my understanding that you cannot answer the question "is the eigenstate of the position operator a mixed state?" A pure state requires all compatible observables to be eigenvalues. Pure states are common in theory and rare in experiment. The section "Eigenstates and pure states" was attempting to get that across so if there is some way to clarify that would be great.

Many presentations of QM delay pure/mixed states discussions, I suppose for these reasons of confusion. They are something of a different axis than issues about eigenstates/values. For example you can discuss energy levels and transitions with just eigenstates. To do any comparison with experiment or discuss statistics you have to move on to pure/mixed states. Johnjbarton (talk) 21:20, 8 June 2024 (UTC)

Just to be clear, it's not that I don't know what a mixed state is. The only issue I had is with the phrasing of the sentence: "Any state that is not pure is called a mixed state" in addition to this claim not being mentionned in Messiah p.204. The phrasing is just a bit too strong and context dependent to my liking.

For example, I prefer the definition used in "Hall, Brian C. (2013). Quantum theory for mathematicians"

• A density matrix ρ ∈ B(H) is a pure state if there exists a unit vector ψ ∈ H such that ρ is equal to the orthogonal projection onto the span of ψ. The density matrix ρ is called a mixed state if no such unit vector ψ exists.

It just leaves a lot less room for ambiguity.

The reason I reverted your reversion was because you summarized it as: "it has a references and page number". Yeah, I know. I read the reference, which is what led me to making the edit in the first place.

The example of the position operator eigenstate was just in response to the rather rude and dismissive remarks by @Tercer Roffaduft (talk) 04:24, 9 June 2024 (UTC)

The eigenvectors of the position operator are not normalizable, and hence they are not states at all. Not eigenstates, not pure states, and not mixed states. Tercer (talk) 22:28, 8 June 2024 (UTC)

This is just nonsense. The position operator has an eigenfunction: the Dirac delta, which is also referred to as a (generalized) eigenstate.

It sounds like "leafing throough references" would not be such a waste of your time at all. Roffaduft (talk) 03:49, 9 June 2024 (UTC)

It's not my job to teach you quantum mechanics. If you want to wallow in your ignorance, be my guest. Don't try to put it in Wikipedia, though. Tercer (talk) 06:43, 9 June 2024 (UTC)

Thank god it's not your job indeed. Given that you've contributed absolutely nothing to this discussion other than being very rude and dismissive, I don't feel I am missing out on much.

As you have no intention to engage in polite conversation, I am not going to waste my time continuing this discussion with you. Turning a talk page into a toxic environment doesn't benefit anyone.

I wish you all the best. Roffaduft (talk) 07:33, 9 June 2024 (UTC)

I think the misunderstanding here is whether you regard operators and their eigenvalues and eigenfunctions (i.e. eigenstates) as purely mathematical constructs or subject them to the axioms of quantum mechanics first.

Roughly speaking, I consider QM as an application of more general mathematical theory, with QM borrowing a lot of terminology from mathematics. That is perfectly fine, but a consequence is that certain words have a much more restrictive meaning in QM than they do in mathematics.

A strong claim like: "Any state that is not pure is called a mixed state" might make complete sense from a QM point of view, but it may also lead to ambiguity about the definition of a "state" down the line.

I used the eigenfunction of the position operator as an example, where the physicist has to jump through hoops in order to explain why the dirac delta function is "somewhat like" a normalizable eigenvector in the "continuous sense" Roffaduft (talk) 06:48, 9 June 2024 (UTC)

Perhaps we need to change the section "Eigenstates and pure states" to "Eigenstates and mixed states". The section needs to help readers relate eigenstates, the result of a measurement, into the overall "quantum state" topic. Pure states are not directly needed for that goal. Johnjbarton (talk) 15:51, 9 June 2024 (UTC)

At this point I’m honestly fine with whatever. My initial edit was primarily based on the reference not backing the “strong claim” (mathematically speaking) and only secondly the on “strong claim” itself. An extra reference has been added which resolves most of the issue I had. That is, if one wants extra context, one can look up the reference. Roffaduft (talk) 16:48, 9 June 2024 (UTC)

@PyetroPy edited the Superposition section, then I tried again. The questionable section was I think trying to do too much. My version uses more math. Alternatively we could just state that the overall phase is ambiguous and leave it at that. Johnjbarton (talk) 23:21, 4 September 2024 (UTC)

@Johnjbarton I like the final version. In my comment I was just trying to link the two previous concepts together (the multiplication with complex number and the fact that only the relative phases count) and to summarize, sorry if it sounded dismissive. PyetroPy (talk) 22:14, 7 September 2024 (UTC)

The introduction says: "Two broad categories are" ... and then mentions wave functions and vector formulations.

Can we clarify in this section (1) whether the two formulations here ("broad categories"?) are equivalent to each other, (2) whether these two formulations are the only ones possible.

It might also be useful to state explicitly whether these two formulations are those sometimes respectively attributed to Shrodinger and Heisenberg - Shrodinger's wave description and Heisenberg's matrix mechanics. I also understand, maybe wrongly, that these two formulations were shown to be equivalent to each other. But I don't know the relation between what's stated in the article, and the Shrodinger/Heisenberg formulations, and was unable to determine whether the two formulations in the article are equivalent to each other, without going elsewhere. ~2025-34023-50 (talk) 18:46, 21 November 2025 (UTC)

I think you are bringing something in that is not part of the article. The intro is simply summarizing the article content, "Wave function representations" and "Formalism" (ok that's horrible name). These are two common "notations", meaning "how the function dependency is expressed". Wave functions have explicit position and indexes to distinguish them; vectors subsume the position are named by the index.

The Schrödinger/Heisenberg split is usually called "formalism" or "picture". See Heisenberg_picture.

I think this article should be about the concept of "state" in QM as is discussed in the first sections. The most of the content in the sections "Wave function representations" and below are redundant and should be replaced with short summaries. Then the intro would not have that "two broad category" comment at all. Johnjbarton (talk) 19:38, 21 November 2025 (UTC)

In that case, what do you think of the following:

Two semantically different methodologies for quantum states that give the same predictions as each other are wave functions and vector quantum states. Historical, educational, and application-focused uses typically feature wave functions; modern professional physics typically uses vector states. [why? reference?]

Quantum states may be categorized as (a) pure versus mixed, and (b) coherent versus incoherent. Specially recognized states include stationary states for time independence, and quantum vacuum states in quantum field theory.

- ATBS ~2025-34023-50 (talk) 20:07, 21 November 2025 (UTC)

These "two broad categories" are complete nonsense. I'll rewrite the lead into something reasonable. Tercer (talk) 20:15, 21 November 2025 (UTC)

The new introduction has the same problem as the rest of the article: math and physics are all mixed up. I think this is very common in discussions about QM because all of the participants understand which is which. So wave function on a chalkboard will be called a "state" as shorthand for "representation of the state". In the encyclopedia we should try to avoid this assumption.

For example "measurement of quantum state" is a shorthand. We don't measure a mathematical entity. Similarly

• Quantum states can be divided into pure or mixed quantum states. Pure quantum states are represented as a vector in a Hilbert space. Mixed states are statistical mixtures of pure states and cannot be represented as vectors on that Hilbert space, and instead are represented as density matrices.

Given that quantum states are math, then "represented as a vector" is confusing. I think this would be clearer as

• Quantum states can be divided into pure or mixed quantum states. Pure quantum states are vectors in a Hilbert space. Mixed states are statistical mixtures of pure states in the form of density matrices.

Johnjbarton (talk) 21:22, 21 November 2025 (UTC)

Regarding the current intro:

"Quantum states can be divided into pure or mixed quantum states. Pure quantum states are represented as a vector in a Hilbert space. Mixed states are statistical mixtures of pure states and cannot be represented as vectors on that Hilbert space, and instead are represented as density matrices."

The first sentence is ambiguous: (a) it doesn't make it clear whether a given state can be both pure and mixed, and (b) "divided into" could be read as "decomposed into" and I don't think that is what is meant here. As is often the case, some of these issues stem from the use of the plural when a single thing might be better.

How about :

Each single quantum state is either pure or mixed, and not both. [with hyperlinks for pure and mixed]

The next two sentences (starting with "Pure quantum states are represented as ...") have an apples-oranges problem. The first part describes in terms of vectors in a Hilbert space and the second part describes in terms of density matrixes. If Hilbert spaces are going to be mentioned here then I'd think it'd be better to explain how each of pure and mixed relate to a Hilbert space. Similarly if density matrices are going to be mentioned here then I'd think it'd be better to explain how each of pure and mixed relate to density matrices.

- ATBS ~2025-34023-50 (talk) 22:25, 21 November 2025 (UTC)

The lead cannot be too technical. I think vector, Hilbert space, and density matrix is as far as we can go.

As for representations, what I had in mind are the several possible representations a pure state can have: as a vector in a Hilbert space, as a ray, as a density matrix, as a Wigner function, as a linear functional, etc. The article does explain these things, so we shouldn't try to explain them badly in the lead. I'll just make clear that there are more possible representations. Tercer (talk) 08:22, 22 November 2025 (UTC)

OK. Can you please address the problems I mentioned - ambiguity, and apples/oranges comparison? I can review a draft if you want. - ATBS ~2025-34023-50 (talk) 09:21, 22 November 2025 (UTC)

I think they are already addressed by my edits (and Stepwise Continuous Dysfunction's). The lead is now only saying that they are common representations. I think giving more detail than that would be out of place. If you are unhappy then make a concrete suggestion. Tercer (talk) 12:53, 22 November 2025 (UTC)

A given quantum state is either pure or mixed (not both). Several representations are possible. Pure quantum states may be represented as a vector in a Hilbert space. Mixed states may be viewed as statistical mixtures of pure states and cannot be represented as vectors on that Hilbert space; they may be represented as density matrices.

Examples of quantum states are the wave functions describing the position and momentum degrees of freedom, finite-dimensional vectors describing the spin degree of freedom such as the singlet, and states describing many-body quantum systems in a Fock space.

(One thing this does is to remove lots of "common"; "common" may be seen as describing current practice, arguing from authority (something being in common use does not make it right), and not describing the underlying thing (quantum states).)

Remaining things that should be addressed:

1) in the first paragraph, "quantum system" is undefined

2) "degree of freedom" is undefined

3) representation is used in two ways; one in regards to a model "representing" a pure state (say), and one in regards to the quantum state "representing" a system (in the first paragraph).

Also, the use of "representation" in the first paragraph seems problematic to me. Maybe it solves a lot of problems to leave "representation" out of the first paragraph. We could say that a quantum state is a mathematical model of a quantum system, and leave it at that. (We also need to define what a quantum system is.)

-ATBS ~2025-34023-50 (talk) 21:23, 22 November 2025 (UTC)

"(not both)" is redundant, that is poor writing. "common" only means that it is common, don't overthink it. we should talk about the simplest representations in the lead, that the reader is most likely to have encountered, not exotic ones. and we should definitely say that they are common or simple or usual. this is valuable information.

1) I don't think there is any useful definition of "quantum system" we can give, it is just a system described by quantum mechanics.

2) "degree of freedom" can be deleted instead of defined.

3) a quantum state is not just a mathematical model of a quantum system, it is crucial that it is supposed to represent it. Tercer (talk) 21:37, 22 November 2025 (UTC)

In quantum physics, a quantum state is a mathematical entity that can represent a physical system at a given point in time. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. Knowledge of the quantum state at a given point in time, and the rules for the state's evolution in time, exhausts all that can be known about a physical system. [reference needed]

The pure quantum states may be viewed as building blocks for all quantum states. The quantum states that are not pure are mixed. A mixed state may be viewed as a probabilistic mixture of pure states. One way to represent a pure state is as a vector in a Hilbert space. A mixed state cannot be represented in this way. The density matrix representation can be used for both pure and mixed states.

Some examples of quantum states are the wave function describing position and momentum, finite-dimensional vectors such as the singlet state, and a Fock space used to describe a many-body quantum system.

(Apologies if I added some wrong, especially "in time". I added this with the idea of it leading to something more precise perhaps after further modifications.)

- ATBS ~2025-34023-50 (talk) 22:12, 22 November 2025 (UTC)

In quantum physics, a quantum state is a mathematical entity that can represent a physical system at a given point in time. It does represent, saying it can represent is misleading. Adding "at a given point in time" is incorrect, and unnecessary even if it were true. [reference needed] No, it's not needed, and furthermore poor form. The lead should summarize the body of the article, references go there. You shouldn't add information to the lead that is not discussed in the article. One way to represent a pure state is as a vector in a Hilbert space. A mixed state cannot be represented in this way. That's incorrect, the set of Hermitian matrices do form a Hilbert space (with the Hilbert-Schmidt inner product), and a density matrix is a vector in that Hilbert space. it's just not the same Hilbert space as the usual vector pure states. Tercer (talk) 11:03, 23 November 2025 (UTC)

Thanks for the feedback, Tercer. About some of this.. (I'll try to get to the rest in the next few days.)

Some of my motivation: a mathematical structure differs from a particular use that is made of that structure. One could say that a rectangle represents a window in a building. And that's not wrong. Somewhere, at some time, someone has modeled a window in a building by using a mathematical concept of a rectangle. So it's correct to say "a rectangle represents a window." This is true even though rectangles are used for other purposes besides representing windows. On the other hand, it remains that there is a distinction between the mathematical object "rectangle", and using the rectangle to model the window. Wikipedia's entries for circle, rectangle, riemannian geometry, all start with a mathematical description of the object, and (should, and seem to at first glance) clearly differentiate uses of the object from its definition or properties. If we are going to say that a quantum state is a mathematical structure, should we introduce it that way and reinforce the distinction between the mathematical structure and one of its possible uses?

Maybe you're more interested in the following - what do you think of: A quantum state is a mathematical entity that is used in quantum physics to model a physical system.

The term "model" differs from "represent" and I'm going to guess that you might prefer "represent" or say that "model" is not correct or have some good motivation for using "represent". If so, can you explain to me what you want to capture by "represent"? The number 3 could be used to represent a physical system but that would usually not be as useful as using a quantum state to represent a physical system.

-ATBS ~2025-34023-50 (talk) 06:09, 24 November 2025 (UTC)

The sentence already starts with In quantum physics, and in quantum physics the quantum state does represent a quantum system. And it is a representation, as opposed to merely a model, because in this theory that's really all that a quantum system is, there is nothing else. Models are usually simplifications, that only focus on one aspect of the object. Tercer (talk) 10:15, 24 November 2025 (UTC)