About quantum interference: space and time

[Husserliana97]

Hello to all,
Questions that I hope are not completely devoid of physical meaning.

Firstly, about space. Let be a Hilbert space, in which we can by definition establish the existence of complete and orthonormal vector bases; and a Psi vector (state) that we write as a linear combination (a "superposition") of these base vectors. Here we can define a notion of distance, because this space is provided with a scalar product; for example, the distance between two vectors in the combination; but this 'distance', if I understand correctly, is only a measure of the similarity/dissimilarity between these two vectors (and cannot therefore be interpreted as a Euclidean distance).
So my first question is: what role does this distance play in interference? Or, more specifically, is a large distance (and therefore a large dissimilarity between vectors/amplitudes) an obstruction to the fact that they interfere? Which also raises the question, I think, of whether the angle matters for interference, and not just the phase ratio.

Now to time. I'll put my question like this: is quantum interference a process that takes place over time (and is therefore a 'process' in the strict sense), or an instantaneous consequence of superposition?
A physicist (I'm not a physicist by profession, as you'll have gathered) told me, without being more specific, that interference is not an 'interaction' in the strict sense of the word, in other words in the physical sense of the term. I can only guess at what is meant by this (but perhaps you'll disabuse me of the notion): an interaction would be a process during which two 'entities' (in this case two Psi amplitudes) act on each other and therefore exchange energy, impulse and, more generally, information. So interference would be nothing of the sort. But why? I can only make the following assumptions:

1) Quantum interference does not involve the exchange of forces between probability amplitudes. The amplitudes are simply superimposed and interfere without exerting any force on each other.
2) It does not require the transmission of information between them. The interference pattern is determined by the superposition of the amplitudes themselves, not by any communication between them.
3) Finally, it can be considered an instantaneous process, occurring at the same time as the superposition of the states/amplitudes. There is no propagation of influence in time.

And it's this last point that interests me. Doesn't quantum interference emerge instantaneously (so to speak) from superposition? It is, so to speak, the logical consequence, without us being able to speak of a succession (the superposition's anteriority would just be logical, not chronological).
I have in mind an analogy with quantum entanglement. In the same way that entanglement does not imply an interraction at a distance (no transmission of a signal or information) between two quantum systems, but translates the simple fact of their non-separability (the superposition of correlations between their respective states), in the same way, there would be no interaction and therefore no spatio-temporal process involved in interference; but because at least two states are superimposed, it 'follows' (logically) that they interfere, and do so differently according to their phase ratios (and their angles, then ?).
Does this hypothesis, and the analogy with entanglement, simply make sense?

 

[anuttarasammyak]

So my first question is: what role does this distance play in interference? Or, more specifically, is a large distance (and therefore a large dissimilarity between vectors/amplitudes) an obstruction to the fact that they interfere? Which also raises the question, I think, of whether the angle matters for interference, and not just the phase ratio.

I interprete distance as ket-bra inner product. It is coefficient of basis expansion when one of bra or ket is basis. Phase is additional issue as shown in Bloch sphere https://en.wikipedia.org/wiki/Bloch_sphere .

 

[PeterDonis]

Here we can define a notion of distance, because this space is provided with a scalar product

No, a scalar product is not a distance. That's not even true in standard Euclidean space. In Euclidean space, the scalar product of two vectors is the cosine of the angle between them. In other spaces, the scalar product has an analogous meaning of some sort of "angle" or angle-like thing. It is never anything like a distance.

is quantum interference a process that takes place over time (and is therefore a 'process' in the strict sense), or an instantaneous consequence of superposition?

Neither.

Questions that I hope are not completely devoid of physical meaning.

Unfortunately, as far as I can tell, they are. See above.

 

[anuttarasammyak]

And it's this last point that interests me. Doesn't quantum interference emerge instantaneously (so to speak) from superposition? It is, so to speak, the logical consequence, without us being able to speak of a succession (the superposition's anteriority would just be logical, not chronological).

To me superposision and interference of the states are same, so I do not use the latter. As you assumed, concerns of when, what successive order, how they are prepared do not apply beween the superposition components of the state. They do not apply to entagled state neither because it is a single state describing the two remote partial states.

 

[PeterDonis]

To me superposision and interference of the states are same

No, superposition and interference are not the same. For one thing, superposition is basis dependent but interference is not.

 

[PeterDonis]

I interprete distance as ket-bra inner product.

No, the inner product is not a distance. See post #3.

 

[Husserliana97]

No, a scalar product is not a distance. That's not even true in standard Euclidean space. In Euclidean space, the scalar product of two vectors is the cosine of the angle between them. In other spaces, the scalar product has an analogous meaning of some sort of "angle" or angle-like thing. It is never anything like a distance.

I'm sorry, but I don't think I said that scalar product = distance, you're saying things that aren't mine. I said that the possibility of operating a scalar product in Hilbert space made it possible to define a distance, which is not the same thing. Because it can be used to define a norm, which in turn can be used to calculate a distance.

Neither.

So what is it, if it is neither an interaction (a process taking place over time, an exchange of energy/information) nor a simple consequence of superposition (amplitudes interfering as soon as their associated states are superimposed)?

 

[PeterDonis]

I said that the possibility of operating a scalar product in Hilbert space made it possible to define a distance

Which it doesn't. There is no such thing as a distance between vectors. The scalar product of two vectors has nothing whatever to do with any distance.

So what is it, if it is neither an interaction (a process taking place over time, an exchange of energy/information) nor a simple consequence of superposition (amplitudes interfering as soon as their associated states are superimposed)?

What quantum interference "is" is interpretation dependent; there is no single "right" answer. The only thing you can say about it independent of any interpretation is to describe the observations that show it, and the math of QM that predicts those observations.

 

[PeroK]

Which it doesn't. There is no such thing as a distance between vectors. The scalar product of two vectors has nothing whatever to do with any distance.

It's not used often, if at all in QM, but the inner product defines a norm, which in turn defines a metric (as in a metric space), which in pure mathematical terms is a "distance" function. In that sense, the distance between two vectors is well defined.

 

[Husserliana97]

Which it doesn't. There is no such thing as a distance between vectors. The scalar product of two vectors has nothing whatever to do with any distance.

I
don't understand... since Hilbert space is normed, we can define a distance in it, can't we? That's why it's a metric space...
This distance measures the degree of difference between the two quantum states represented by their respective vectors. The greater the distance, the more different the states. If the distance is zero, this means that the two vectors are identical (or proportional in the case of vectors of different phase). At least, that's how I understood it.

What quantum interference "is" is interpretation dependent; there is no single "right" answer. The only thing you can say about it independent of any interpretation is to describe the observations that show it, and the math of QM that predicts those observations.

So let's put ourselves on the side of Psi-ontic interpreters for whom superposition is a real thing. What about the status of interference? Is there only one interpretation that sees it as a physical process? Or do they always result ‘without delay’ from the superposition of states?

 

[PeterDonis]

the inner product defines a norm, which in turn defines a metric (as in a metric space), which in pure mathematical terms is a "distance" function. In that sense, the distance between two vectors is well defined.

No, it isn't. A metric, when used as a distance function, is applied to differentials and integrated along a curve. It is not applied to vectors. When applied to vectors, the best physical interpretation of the inner product is an angle, not a distance.

 

[PeterDonis]

since Hilbert space is normed, we can define a distance in it, can't we? That's why it's a metric space...
This distance measures the degree of difference between the two quantum states represented by their respective vectors.

No, it doesn't. Such a thing is an angle, not a distance. See post #11.

Mathematically, I suppose you could do the "apply the metric to differentials and integrate along a curve" thing in a Hilbert space, but I am not aware of any meaningful physical interpretation of such a thing, nor is such a thing done in QM.

 

[PeroK]

No, it isn't. A metric, when used as a distance function, is applied to differentials and integrated along a curve. It is not applied to vectors. When applied to vectors, the best physical interpretation of the inner product is an angle, not a distance.

This is simply wrong.

 

[PeterDonis]

let's put ourselves on the side of Psi-ontic interpreters for whom superposition is a real thing.

No, it's not. Psi-ontic means the wave function or quantum state, considered as an object in its own right, is a real thing. But that does not mean basis dependent properties such as "superposition" are real.

What about the status of interference?

Interference is not basis dependent, but it's also not a property of the wave function or quantum state. It's a property of particular observations that are made. That property can be described in terms that are independent of interpretation, but of course those terms don't tell you "what's really going on". All they tell you is what is observed.

 

[PeterDonis]

This is simply wrong.

Why?

Consider two vectors in Euclidean space. Take their inner product. What does it tell you? Here, I'll even write it down:

$$
\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos \theta
$$

 

[PeroK]

Why?

Consider two vectors in Euclidean space. Take their inner product. What does it tell you? Here, I'll even write it down:

$$
\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos \theta
$$

See under "definition".

https://en.m.wikipedia.org/wiki/Hilbert_space

"A Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.[4]"

 

[PeterDonis]

See under "definition".

https://en.m.wikipedia.org/wiki/Hilbert_space

"A Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.[4]"

Sure, that's the definition. But the question is what the "distance function" actually means.

Unfortunately, Euclidean space (and Minkowski spacetime in the context of relativity) has a special property that is very unfortunate in this context: there is a one-to-one correspondence between vectors and "spatial points" (meaning interpreting points in the space as actual locations in a physical space like the one we move around in) once we pick an "origin" point. That means you can get away with interpreting the norm of a vector as the "distance" between the tail and the head of an arrow between two "spatial points".

However, as soon as you try to operate in any other space, even though it might still be a metric space, this interpretation no longer works. Instead you have to do what is standardly done in GR, which is to interpret vectors as objects in the tangent space at a point, and draw a distinction between using the norm/metric in the tangent space to get things like angles (since the "length" of a vector in the tangent space does not correspond to anything physical, so only unit vectors in the tangent space are used) and using it to integrate differentials along curves to get arc lengths. (I say "arc lengths" instead of "distances" because if the metric is not flat, there is no unique "distance" between two different points; arc length is path dependent.)

AFAIK all the latter machinery is not standardly used in QM, because there is no point: even if you did the mathematical operation "integrate differentials along a curve to get an arc length", it would have no useful physical meaning, so nobody bothers doing it. The only viable physical interpretation of the Hilbert space inner product is as something like an angle, giving the degree to which two vectors are "parallel" (maximal value of inner product) vs. "orthogonal" (zero inner product). Which corresponds to interpreting the Hilbert space as a tangent space in GR. Indeed, state vectors in Hilbert space are typically restricted to have unit norm for this very reason: the norm, by itself, has no physical meaning, only the inner product "angle" between two vectors does.

 

[PeroK]

In an inner product space, we can define a metric for vectors $x,y$:
$$d(x,y) = \sqrt {\langle x-y, x-y\rangle}$$This is called the metric induced by the inner product and satisfies, for example, the triangle inequality. Note that this definition of metric is not to be confused with the metric in differential geometry. This is discussed here:

https://math.stackexchange.com/ques...efinition-of-metric-as-defined-in-differentia

 

[PeterDonis]

this definition of metric is not to be confused with the metric in differential geometry.

Not to be confused, sure; they are mathematically distinct concepts. But AFAIK, the only cases in which the inner product space definition of "metric" has a useful interpretation as an actual "distance" are cases, like Euclidean space, where the two definitions give you the same thing.

Indeed, the inner product space definition itself, as the Wikipedia page on Hilbert spaces notes, is connected with the triangle inequality, which is a special case of the Cauchy-Schwarz inequality--which says, briefly, that the inner product of two vectors must be less than or equal to the product of their norms, with equality holding only when the two vectors are linearly dependent, i.e., parallel. Which is a statement about the "angle" interpretation of the inner product--"linearly dependent" corresponds to "parallel", i.e., maximum inner product, zero "angle". And that is the interpretation that comes into play in QM.

 

[PeroK]

Not to be confused, sure; they are mathematically distinct concepts. But AFAIK, the only cases in which the inner product space definition of "metric" has a useful interpretation as an actual "distance" are cases, like Euclidean space, where the two definitions give you the same thing.

Metric spaces allow the notion of limits and continuity to be generalised. A metric is literally a "distance function". That's the motivation for defining the metric the way it is. That it generalises the concept of distance.

https://en.m.wikipedia.org/wiki/Metric_space

This goes far beyond Euclidean distances. Any Hilbert space allows the notion of continuous functions to be defined on it. Using the usual definition of continuity with the metric replacing the Euclidean modulus.

This is the starting point for functional analysis. See here if you are interested:

https://en.m.wikipedia.org/wiki/Banach_space

 

[PeterDonis]

@PeroK the question is what role, if any, the "distance" computed with the metric in a Hilbert space plays in QM. AFAIK the answer to that is "none".

 

[PeroK]

@PeroK the question is what role, if any, the "distance" computed with the metric in a Hilbert space plays in QM. AFAIK the answer to that is "none".

QM doesn't seem to make much use of the analytic properties of the Hilbert Space. All that functional analysis lying about unused!

That said, I'm sure I've seen more mathematically advanced approaches delve into C*-Algebras and such like.

 

[Husserliana97]

@PeroK the question is what role, if any, the "distance" computed with the metric in a Hilbert space plays in QM. AFAIK the answer to that is "none".

And what about fidelity and Bures Distance, in quantum information theory ?

 

[PeterDonis]

And what about fidelity and Bures Distance, in quantum information theory ?

Do you have a reference?

 

[PeterDonis]

I'm sure I've seen more mathematically advanced approaches delve into C*-Algebras and such like.

Algebraic quantum field theory does do this, but from what I've read (which is not a lot, the main text I'm familiar with is Wald's 1993 monograph Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics), it still doesn't really make any use of the concept of "distance".

 

[PeroK]

Algebraic quantum field theory does do this, but from what I've read (which is not a lot, the main text I'm familiar with is Wald's 1993 monograph Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics), it still doesn't really make any use of the concept of "distance".

The concept of distance is fundamental in defining continuity and completeness etc. A Hilbert Space, for example, is a complete inner product space. Where "complete" means every Cauchy sequence has a limit. So, you can't even talk about a Hilbert space in the first place without the concept of a distance between its elements.

Also, representing a state vector by an infinite series of eigenvectors requires the notion of convergence of the series. And, convergence requires the notion of distance.

Although physics textbooks only touch on this, the underlying metric is fundamental.

The fact that physics can skate over the mathematical details doesn't mean that the underlying mathematics is not there.

 

[Husserliana97]

Do you have a reference?

Here

https://arxiv.org/abs/2310.09231

 

[PeterDonis]

The fact that physics can skate over the mathematical details doesn't mean that the underlying mathematics is not there.

Yes, fair point.

 

[PeterDonis]

Here

https://arxiv.org/abs/2310.09231

Thanks! I was not previously aware of this work. It looks very interesting. You've added another item to my already way too long reading list.

 

[Husserliana97]

Thanks! I was not previously aware of this work. It looks very interesting. You've added another item to my already way too long reading list.

You're welcome ! Glad to contribute something for once :)

 

[Fra]

These are interesting questions, which can either be both of mathematical foundation interest or interest to those trying to understand the conceptual foundations of QM, not sure where you come from.

From the conceptual perspective I think the of significance of the distance metrics in the abstrac spaces as beeing (not equal to but) related to what one may call "a priori transition probabilites". The the bures measures measure, beeing a generalisation from the hilbert metric to one on density operators, is related to fidelty which is associated to transition probabilities. https://arxiv.org/abs/1106.0979

But It seem your question was something else, if this "distance" has an meaning in the "interference", and if the "interference" has a process or more a logical consequence.

The way I interpret your thinking, I would prefer to think of "quantum interference" first of all as a logical consequence of how the initial conditions, where we "know" values of non-commuting observables. But that doesn't mean it's "instant" whatever happens, has some dynamics, but the "phenomeman" of quantum interference i view as a logical consequence. To question further would mean to ask "why" does systems seem to exist in stable states specified by non-commuting information. This gets interpretational I think.

I do not see a clear connection between the a priori the transition probabilities between two states and their interference, as I think interference is between PARTS of ONE system, not between two different hypothetical states that don't exist at the same time?

Not sure if I missed your point

/Fredrik