Operators for comparing superposition components -- definable?

[Agrippa]

Hello,

I'm wondering, is it possible to define an operator on a Hilbert space that gives information about the "distinctness" of superposition components?

As a simple example, imagine that we have two particles. Let |3> designate the state in which they are 3 meters apart, let |5> designate the state in which they are 5 meters apart, and |7> for seven metres apart. Now imagine the three possible states:

(s1) a|3> + b|3> = |3>
(s2) a|3> + b|5>
(s3) a|3> + b|7>

Where a and b are amplitudes such that |a|²+|b|²=1.
Is it possible to define an operator that gives, say, a null result for s1 (no distinctness) while giving nonzero values for s2 and s3 and ranking them so that s3 gets a higher value (more distinct?)?

If there are real quantum mechanical applications for such operators I would be very interested to learn of them. If there are no known applications even though they are nonetheless in principle definable, then I would still be very interested.

Thanks!

 

[wabbit]

I don't think so, at least if you mean an operator as used in quantum measurement : it will always pick one of the basis states, regardless of whether that was the original state or not.

However, if you can duplicate your state, then repeated measurements will give you either always the same result (no "distinctness"), or not - and furthermore this will give you an estimate of the $p_i=|q_i|^2$, your state being $\sum {q_i|i\rangle}$ .

Then for instance $\sum{p_i^2}$ is equal to one for any basis state (and only for those) and goes down to zero the more spread out the state is over the basis (if you have n basis states, the minimum value of this indicator is 1/n). This is not an operator however.

As to application I don't know, only replying regarding the math here.

 

[Agrippa]

I don't think so, at least if you mean an operator as used in quantum measurement : it will always pick one of the basis states, regardless of whether that was the original state or not.

Thanks, but I wonder if you could say a little more about what you mean? It's still not immediately clear to me why an operator could not perform this function. An operator O on some vector space maps a vector |A> in that space to some other vector O|A> in that same space. Assuming O|A> is an eigenvector of O, the corresponding eigenvalue yields information. The question then becomes, why can't we define an operator that yields (say) the following results:

O|(s1)> = 0|(s1)>
O|(s2)> = 2|(s2)>
O|(s3)> = 4|(s3)>

Here, the eigenvalues are a measure of the distinctness of the superposition components (in the original post basis); 0 for no distinctness, 2 for slightly distinct etc.

Perhaps the problem is that such operators won't be Hermitian? Eigenvectors of a Hermitian operator (that don't share the same eigenvalue) are all mutually orthogonal. That is likely to break down. But I take it that we are not forced to only use Hermitian operators in our models?

However, if you can duplicate your state, then repeated measurements will give you either always the same result (no "distinctness"), or not - and furthermore this will give you an estimate of the $p_i=|q_i|^2$, your state being $\sum {q_i|i\rangle}$ .

Then for instance $\sum{p_i^2}$ is equal to one for any basis state (and only for those) and goes down to zero the more spread out the state is over the basis (if you have n basis states, the minimum value of this indicator is 1/n). This is not an operator however.

This is a very interesting suggestion. But consider states (s1),(s2), and (s3) again. Your function F gives the right result for (s1) since F(s1) = 1 (and we can stipulate that 1 means "no distinctness"). The problem is that it gives the same result for (s2) and (s3) for if a=b then F(s2) = F(s3) = 0.25. But I need a function that ranks (s3) as having "more distinct" components than (s2).

 

[wabbit]

You are correct. If you re asking whether you can build an (Hermitian) operator that has s1, s2, s3 as eigenstates then as you say the answer is no in this case because these are not orthogonal.

The operator you defined is OK mathematically but far as you can't interpret it as a measurement operator in QM though since those are hermitian.

But it looks like I may have misunderstood your question - does the above respond to your orginal query? Otherwise, please feel free to elaborate : )

 

[Agrippa]

You didn't misunderstand my question - your reply was helpful (e.g. the non-Hermitian requirement wasn't immediately obvious to me).
I agree that the operator I have in mind won't be a (Hermitian) measurement operator.
The remaining question is just whether such an operator is going to be well-defined, particularly if we demand that all eigenvalues be real positive numbers.
For example, it seems unlikely that it can be expressed in matrix form. I can't think of any other way of defining it than in terms of a set of pairs, pairing specific state vectors (like (s1), (s2) etc.) to other vectors in the space that happen to have the right (real positive) eigenvalues.
My concern is: if that's the only way of defining the operator, then it's not a real operator?

 

[wabbit]

Let me stress that my understanding of QM is very limited, so take everything I say here with a grain of salt.

Mathematically, the operator you described is certainly well defined (and as for any finite dimensional linear map, you can represent it as a matrix) as long as the $|s_i \rangle$'s form a basis (which is the case for your choice). And I don't see any reason why you cannot define a (non QM) measurement procedure that takes an arbitrary state $\sum q_i |s_i\rangle$ and assigns to it the eigenvalue associated with the basis element $|s_i\rangle$ with probablity $p_i=|q_i|^2$.

The limitations I see to this are :

(a) it is unclear to me how you would build an apparatus to implement this procedure (would it not be impossible if QM correctly describes the world ? after all if that is the case any measurement can be described by a hermitian operator, no ?). Also this assumes a single measurement : with repeated measurements of a replicated state, even with the original basis you can estimate the $p_i$'s so you can caracterize the original state)

(b) whether or not it can be implemented physically, since your operator does not follow the QM framework, so you cannot assume that QM interpretation of things like interference etc will carry over (some may do, I don't know which ones). There may be extensions of QM that would allow this, but this is beyond my grasp.

 

[Agrippa]

Mathematically, the operator you described is certainly well defined (and as for any finite dimensional linear map, you can represent it as a matrix) as long as the $|s_i \rangle$'s form a basis (which is the case for your choice). And I don't see any reason why you cannot define a (non QM) measurement procedure that takes an arbitrary state $\sum q_i |s_i\rangle$ and assigns to it the eigenvalue associated with the basis element $|s_i\rangle$ with probablity $p_i=|q_i|^2$.

I see, so it's well defined because the eigenvectors are basis vectors. And perhaps the operator only needs to invoke a single basis vector V in that it takes an arbitrary state Si to iV where there is a unique eigenvalue i for each Si.

The limitations I see to this are :

(a) it is unclear to me how you would build an apparatus to implement this procedure (would it not be impossible if QM correctly describes the world ? after all if that is the case any measurement can be described by a hermitian operator, no ?). Also this assumes a single measurement : with repeated measurements of a replicated state, even with the original basis you can estimate the $p_i$'s so you can caracterize the original state)

(b) whether or not it can be implemented physically, since your operator does not follow the QM framework, so you cannot assume that QM interpretation of things like interference etc will carry over (some may do, I don't know which ones). There may be extensions of QM that would allow this, but this is beyond my grasp.

In response to both (a) and (b), quantum mechanics is incomplete (measurement problem) and its completion may require such non-Hermitian operators. The Schrödinger equation entails that measurements do not yield single definite outcomes, but result in superpositions of all possible outcomes. So one option is to supplement this equation with a collapse function. The one-particle master equation for the density matrix ρ is then:
$$\frac{d}{dt}ρ(t) = -\frac{i}{ħ}[H,ρ(t)] - T[ρ(t)]$$
Where H is the standard quantum Hamiltonian of the particle and T[] represents the effect of the spontaneous collapses on the particle's wave-function. In the position representation, this operator becomes:
$$<x|T[ρ(t)]|y> = Θ[1-e^{-(x-y)^{2}/4r^{2}}] <x|ρ(t)|y>$$
Where r sets the width of the localization/collapse process, x is position operator for the particle and y is a random variable corresponding to the localization center (i.e. the place where the jump occurs). Crucially, Θ fixes the circumstances in which collapse occurs. You might be aware of GRW - they set Θ = 10^-16sec^-1, which is interpreted as the probability per unit time for spontaneous collapse. But there are reasons to think this stipulation unsatisfactory. And here's where the operators I have in mind come in: Imagine that physical systems typically have a certain amount of a given physical quantity Q. What's different about Q is that superpositions of it are unstable, and, in particular, collapse frequency is a function of (i) level of Q in any given superposition component and (ii) variation in Q-level among superposition components. In that case the master equation becomes:
$$\frac{d}{dt}ρ(t) = -\frac{i}{ħ}[H,ρ(t)] - Q_1[ρ(t)]Q_2[ρ(t)][1-e^{-(x-y)^{2}/4r^{2}}] <x|ρ(t)|y>$$
Unfortunately I'm having difficulties properly spelling out these Q functions. But the idea is that Q[ρ(t)]₂ is a measure of the level of Q in a given quantum state or in a given superposition component of that state while Q[ρ(t)]₁ is a measure of the distinctness of the Q levels among superposition components.
That's how I see such operators playing a role in quantum theory.

 

[wabbit]

Saying that quantum mechanics is incomplete because of the measurement problem is however I believe more a philosophical stand than a proven assertion. Myself, I find the assumption than QM is complete, as presented in Relational Quantum Mechanics, far more convincing - though unproven of course, it is my understanding that there is so far no observation that shows it to be incorrect, or in other words that QM is a complete theory of the observations made so far.

This does not imply of course that it will not be superseded in the future, just that there is no observational evidence yet that makes such an extension necessary.

Regarding non-unitary evolution (which is in any case part of QM, otherwise we would be speaking of the strange theory of QM without measurement - but is understood in RQM as it relates to unitary evolution from a different perspective), the approach proposed in The Montevideo Interpretation looks very interesting to me, though its technical content is beyond my level of competence.

 

[Agrippa]

I'm more interested in the mathematics here, than the philosophical issues.

But just briefly, RQM is not complete and is based on a misunderstanding of what the measurement problem is, in particular, when I measure a quantum system I am still given no instruction as to when exactly I should invoke wave-function collapse in my description. The idea that an outside observer will use a different (no-collapse) description, treating me as part of a closed system governed by the usual no-collapse Schrödinger dynamics, in no way helps solve the problem even if the idea is true. On top of that, RQM is highly unmotivated. Rovelli says on the first page: "This conclusion derives from the observation that the experimental evidence at the basis of quantum mechanics forces us to accept that distinct observers give different descriptions of the same events." My equations above refute this claim since they (i) are consistent with our experimental evidence and (ii) entail that distinct observers give identical descriptions of events. On top of all that, RQM seems inconsistent with the rest of science, particularly all of the historical sciences (cosmology, astronomy, geology, evolutionary biology, etc) which require there to be states of the universe that exist prior to any observers. It's therefore not surprising that (from what I can tell) Rovelli is the only one is actually publishing defenses of RQM!

The equations I've offered yield unique and distinctive predictions. That alone gives us reason to take the theory seriously. If we can experimentally confirm collapse in small system with high Q while simultaneously confirming no collapse in small systems with low Q then the theory I've outlined will replace quantum mechanics.

(p.s. I wasn't aware of the Montevideo Interpretation - I'll take a look!)

 

[wabbit]

This is a very interesting discussion but perhaps it should be moved to another part of the forum as it has little to do with Linear Algebra. I was just commenting on your argument that QM is incomplete, and here is not for me the place to argue about which interpretation is or is not based on a misunderstanding.

What is your mathematical question then, leaving aside questions of QM?

 

[Agrippa]

Yes I agree.

My main concern is that the equation is not rigorous - but I don't have enough mathematical knowledge to really determine this. The equation is:
$$\frac{d}{dt}ρ(t) = -\frac{i}{ħ}[H,ρ(t)] - Q_1[ρ(t)]Q_2[ρ(t)][1-e^{-(x-y)^{2}/4r^{2}}] <x|ρ(t)|y>$$
The two Q functions are to be understood as distributing the collapses in time like a Poissonian process. Frequency is raised when (i) Q-level gets higher either for whole system or for any superposition component and (ii) when superposition components exhibit more distinct levels of Q.
My concern is that simply putting the quantum state in brackets and attaching 'Q' to the left of those brackets is not an adequate mathematical expression of the idea. But I struggle to make it more precise and am looking for suggestions.

 

[wabbit]

I'm afraid I don't have a suggestion here - perhaps others can help here or in the QM section - as stated, it's kind of a mixed question as it seems to be about how best to represent a certain physical idea mathematically. It might be out of bound in PF though if it relates more to theory development than to discussion of an established theory.

 

[Agrippa]

Fair enough. Thanks for your help - the point I learned about non-hermitian operators was useful.