The interpretation of probability

[Killtech]

Quite likely what you are looking for is the Wigner peudo-probability representation...

Thanks! Took me a little reading because Wigners pseudo-probability is quite a bit more special. However the articles on quasiprobability distribution does explain the general framework which i am looking for.

That article claims that coherent states form a overcomplete basis (of the Hilbert space) and allow to diagonalize ever DO. But seeing that the quasiprobability distribution is formulated in terms of DOs like this $\hat \rho = \int f(\alpha, \alpha^*)|\alpha\rangle \langle \alpha|d\alpha^2$, we can just take the $|\alpha\rangle \langle \alpha|$ as a basis for for DOs instead. Also note that this measure is very suspiciously close to the classic probability measure from @gentzen sources.

Going back to the simplex discussion, if we were to look what happens if we restrict the quasiprobabilities to actual probabilities, we see the Bloch sphere encompasses the resulting probability simplex in the following way:

So we can directly see that for every state on the Bloch sphere there exist a rotation of the simplex vertices (i.e. DO basis) that the state will be within the simplex. This is interesting because we can make this work with time evolution: if our classic probability space instead uses the time dependent basis $|\alpha(t)\rangle \langle \alpha(t)|$, then a DO that was initially on the simplex will at all times remain on the simplex within that moving basis. So for a given initial state we can have a low dimensional classical probability space that fully describes its behavior. But different initial states need different probability spaces.

However a good choice of the DO basis we could, we could try to make a rotating simplex that covers the whole of the Bloch sphere if not initially, then at least over time. This way, we can evolve any initial state to the point it gets into the simplex i.e. we model the system as having an unknown time shift.

That may establish a connection between the classical full p.measure $\rho_\mu = \int \mu(d\psi)|\psi\rangle\langle\psi|$ by simply averaging the integral over subspaces of states that are time shifted version of each other.

I have to find more to read on this topic and contemplate on this.

 

[A. Neumaier]

as a basis for for DOs instead.

Except that it is not a basis in the mathematical sense, but an overcomplete set.

So for a given initial state we can have a low dimensional classical probability space that fully describes its behavior.

No. You also need the time dependence of the rotation to describe the behavior. Moreover, you introduce a lot of representation ambiguity since one can start with any simplex whose vertices are on the Bloch sphere.

 

[Killtech]

Except that it is not a basis in the mathematical sense, but an overcomplete set.

overcomplete in the Hilbert space of quantum states, yes. The DOs formed from those states by $|\psi\rangle\langle\psi|$ however are in a different space which has a has a higher dimension, thus a true basis of the Hilberts space basis just isn't enough to cover it all. The fact that those states are enough to diagonalize any DO is probably enough for them (written as DOs) to be a complete basis of the space they are embedded in, but i am not sure and have to check that.

No. You also need the time dependence of the rotation to describe the behavior. Moreover, you introduce a lot of representation ambiguity since one can start with any simplex whose vertices are on the Bloch sphere.

Mixed states will always come with an ambiguity no matter what, so it's just a question where you want to put it. Hence its perfectly fine for the initial rotation matrix to be ambiguous - specifically if all ambiguity can be isolated into it.

My thinking was that the time dependence of the matrix can be removed by instead of choosing a static probability space using the space of a time dependent process for which there is no problem of representing a rotating simplex implicitly: For each $t$ we have an own probability space anyway for which we may chose its very own basis - so we may always pick the slightly time evolved DOs from the basis of the space before. Practically the time dependent rotation becomes incorporated into the process space via a choice of time dependent basis.

Unfortunately, it would mean that any random variables acting on that probability space would have to unwind that rotation themselves, so practically such a choice means we are in a Heisenberg like picture. Yeah, that might not be an ideal choice but again, i have to think this through.

 

[A. Neumaier]

overcomplete in the Hilbert space of quantum states, yes. The DOs formed from those states by $|\psi\rangle\langle\psi|$ however are in a different space which has a has a higher dimension, thus a true basis of the Hilberts space basis just isn't enough to cover it all.

This doesn't help. The dimension of the space of Hermitian matrices is the square of the dimension of the space of wave functions. Thus a basis has only finitely many terms, while the set of coherent states is infinite-dimensional.

Mixed states will always come with an ambiguity

No. Representing a mixed state by its density matrix is not ambiguos at all.

My thinking was that the time dependence of the matrix can be removed by instead of choosing a static probability space using the space of a time dependent process for which there is no problem of representing a rotating simplex implicitly: For each $t$ we have an own probability space anyway for which we may chose its very own basis

then everything depends on a time-dependent coordinate system, which is not better.
Think of representing classical motions that way...

 

[Killtech]

then everything depends on a time-dependent coordinate system, which is not better.
Think of representing classical motions that way...

I don't know. Actually now that you mention it, it reminds me quite a bit on Hamilton-Jacobi equation in classical mechanics where any complex problems is reduced to the triviality of a simple constant abstract rotation in some crazy complicated representation. I think HJE isn't employed often mostly due to the impossibility of finding a proper transformation - but here for the finite dimensional QM situation it actually looks very reliably doable. But i haven't time to think this all through yet and maybe the resemblance is misleading me and this isn't comparable.

Then again, i actually never though of that until you made me. Indeed the quantities related to the constants of motion in HJE like the periods of cyclic coordinates translate directly into the eigenvalues of the corresponding Markov process (its transition/rate matrix/kernel) which represents the same (deterministic) dynamics but for any random initial distribution. Thanks for bringing this up.

No. Representing a mixed state by its density matrix is not ambiguos at all.

Hmm, in a sense you are right, but this isn't what i had in mind.

The ambiguity in this approach is not simply about representation, since apparently the valid representations are additionally constrained in a complicated manner. One could bring that up as an argument against such an idea. But I am trying to understand if this may have something to do with further hidden physical information inherent in the system instead.

If we only have to deal with such systems purely in the isolated case, i would agree with you. But if you consider what happens if we couple it to another quantum system, things get interesting: The dimension of the DO space grows faster then one would normally expect by looking at the DO spaces of the individual systems. Higher dimension means the system stores more information. Classically one would interpret this as the complex interaction uncovering additional information from the individual systems that is physically inaccessible/not differentiated while each is in isolation.

So upon combining two quantum systems with each other knowing the density matrices each system was in exactly, i think it is insufficient to uniquely determine the density matrix of the combined system, is it? The individual DOs seem to lack some information. For the initial state, that may not make any difference if measured instantly, but once the system was evolved by a Hamiltonian producing some entangling between the coupled systems, mixed states will show more diverse behavior making use of the additional dimensions. Interestingly, the classical measure with its massive ambiguity in the isolated case never runs into this problem and uniquely determines a mixed state regardless of what other systems it is coupled to.

 

[A. Neumaier]

If we only have to deal with such systems purely in the isolated case, i would agree with you. But if you consider what happens if we couple it to another quantum system, things get interesting: The dimension of the DO space grows faster then one would normally expect by looking at the DO spaces of the individual systems. Higher dimension means the system stores more information.

The dimension is squared. Density matrices are routinely used for open systems (Lindblad equations and generalizations). They indeed contain much more information than the pure case, which is completely useless for open systems because it cannot represent any realistic system (except at very low temperature).

So upon combining two quantum systems with each other knowing the density matrices each system was in exactly, i think it is insufficient to uniquely determine the density matrix of the combined system, is it?

Of course, since the information about correlations is missing.

But since classical random systems already have the same sort of incompleteness, your demand is unreasonable.

 

[Killtech]

The dimension is squared. Density matrices are routinely used for open systems (Lindblad equations and generalizations). They indeed contain much more information than the pure case, which is completely useless for open systems because it cannot represent any realistic system (except at very low temperature).

Is the dimension squared though? i forgot to account for the monogamy of entanglement and while there are sure many Hermitian matrices, shouldn't the monogamy butcher their dimension in order to produce valid DOs?

Of course, since the information about correlations is missing.

Okay, sorry i forgot to mention i was thinking of a scenario where two system start independently in an isolated state each and are then brought into interaction. In that case there should be no kind of entanglement initially.

In terms of pure states the standard procedure/assumption is to build the initial states of the combined system as products of the individual pure states of the subsystems. This construction uniquely determines the state of the new system hence there is no incompleteness here. But i am not sure how this works for mixed states and their DOs. Maybe i was mistaken and it works similar after all? Is the combined DO given by the Kronecker product of the individual DOs? At least in terms of size this would fit.

But since classical random systems already have the same sort of incompleteness, your demand is unreasonable.

Actually, no. At very least i don't know what you are referring to. All the probability models i know always are complete in the set of quantities they model. An incompleteness implies the indecision of some statements and for example it would mean one could not determine if a a specific expectation or correlation would have a value or if they are conditioning on an impossibility and hence are not even defined. Not being able to differentiate between two such possible extremes would produce a model that is mathematically challenging but has little practical purpose. Given the history of the axiom of choice it should be clear how much complexity this entails, which is why models are usually setup so they don't run into such issues.

 

[A. Neumaier]

Is the dimension squared though?

The space of Hermitian operators on a Hilbert space of complex dimension $d$ has real dimension $d^2$. The semidefinite ones form a full-dimensional cone, Normalization intersects this cone with an affine hyperplane, hence produces a $d^2-1$ dimensional compact manifold. All these are physically resomable density operators.

i was thinking of a scenario where two system start independently in an isolated state each and are then brought into interaction. In that case there should be no kind of entanglement initially.

In this case the starting density opertor is the tensor product of the pieces. But interaction adds immediately terms that destroy the tensor product structure, Thus after the first picosecond you are on the full-dimensional manifold of density operators.

Is the combined DO given by the Kronecker product of the individual DOs? At least in terms of size this would fit.

Yes, Kronecker product = tensor product.

Given the history of the axiom of choice

The axiom of choice is irrelevant in all that.