# B Is it a myth that serves as motivation for decoherence?

1. Oct 20, 2016

### zonde

This paper about Quantum Darwinism https://arxiv.org/pdf/0903.5082v1.pdf starts with statement:
"The quantum principle of superposition implies that any combination of quantum states is also a legal state. This seems to be in conflict with everyday reality: States we encounter are localized. Classical objects can be either here or there, but never both here and there. Yet, the principle of superposition says that localization should be a rare exception and not a rule for quantum systems."

This sums up nicely motivation behind decoherence approach to measurement problem.
But there is no reference for these statements so I would like to test my suspicions that this is just QM folklore rather than solid predictions of QM.

First statement: "The quantum principle of superposition implies that any combination of quantum states is also a legal state."
Isn't it required that both states are described using the same Hamiltonian? So that this statement might not be true in general?

Second: "Classical objects can be either here or there, but never both here and there." implying that quantum objects can be both here and there.
As I see this comes from explanation of double slit experiment that photon (or other quantum particle, say buckyball) goes by both paths and interferes with itself but not with other particles in the beam.
This interpretation of double slit experiment is certainly falsified by experiments that create interference between beams of two phase locked lasers.

2. Oct 21, 2016

### Staff: Mentor

The first statement is a solid prediction of QM. The second is a reasonable conclusion based on our observations; the question is how best to model it in the framework of QM.

What does this mean? States aren't described by Hamiltonians; they're described by vectors in a Hilbert space (or density matrices, depending on what formalism you are using). The superposition principle just reflects the fact that Hilbert space is a vector space, so adding any two vectors in the space gives another vector that is also in the space.

The paper is using "both here and there" as a shorthand for "in a superposition of here and there". Certainly quantum objects have been experimentally placed in such superpositions.

I don't see this anywhere in the paper. The paper is simply observing that nobody has ever done an experiment that shows a classical object being in a superposition; yet if classical objects are built from quantum building blocks, it should be possible to put them in superpositions, just as you can with quantum objects. The rest of the paper tries to explain how it can be that we never observe classical objects to be in superpositions, even though QM seems to say that should be possible.

3. Oct 21, 2016

### zonde

But Hamiltonian determines Hilbert space, right? In order to write Schrödinger equation and speak about it's solutions (and superpositions of it's solutions) you need Hamiltonian, right?

4. Oct 21, 2016

### zonde

Can you give some reference for experimental observation of "superposition of here and there" (probably observation of interference effect from such superposition)?

5. Oct 21, 2016

### StevieTNZ

6. Oct 21, 2016

### zonde

7. Oct 21, 2016

### A. Neumaier

Experimentally observed are always numbers, not state vectors. And they are always observed here (where the detector is), and never there (everywhere else).

It is impossible to observe a state vector. The best one can do is to infer from many observations of an ensemble of events that the experimental setting was prepared to produce an ensemble whose state is given by a superposition of here and there. What this implies about the corresponding quantum objects remains a matter of speculation (i.e., interpretative assumptions).

8. Oct 21, 2016

### kith

As usual, I'm having a bit trouble to understand what exactly you are after because of your unusual terminology. There is a sense in which what you write makes sense to me, so I'll comment.

If you have an electron in a state |electron> and a proton in a state |proton>, the state |electron>+|proton> cannot be reached because all the known interactions conserve the number of leptons and baryons. This is a very handwaving explanation of the phenomenon of superselection, which strictly excludes some states as unphysical.

What Zurek has shown is that if we have prepared a quantum system in a superposition of eigenstates of a certain observable, the interaction with a sufficiently large environment in a thermal state leads to a rapid decay of the interference properties of the system state. So we cannot describe the system state as a superposition anymore (interestingly, the decay of coherence is the only effect on this time scale; the probabilities to obtain the different eigenvalues in a measurement of our observable don't change). Since the time-scale of this effect is so short, it can be viewed as a dynmical kind of superselection for which Zurek coined the term "environment-induced superselection".

Last edited: Oct 21, 2016
9. Oct 21, 2016

### stevendaryl

Staff Emeritus
No, the Hilbert space is prior to any particular Hamiltonian. For nonrelativistic quantum mechanics of a single spinless particle, the Hilbert space is the set of all square-integrable functions on 3D space. That's the same for every Hamiltonian. The Hamiltonian is an operator on Hilbert space.

10. Oct 21, 2016

### stevendaryl

Staff Emeritus
Any time you use a wave function, you are using a superposition of "here" and "there" (unless the wave function is a delta function, which really isn't possible, since that's not square-integrable).

11. Oct 21, 2016

### A. Neumaier

Not if, as usually the case in quantum information theory, the Hilbert space is finite-dimensional only.

12. Oct 21, 2016

### stevendaryl

Staff Emeritus
I tend to use the word "wave function" only when we are talking about the Hilbert space of square-integrable functions on $R^3$, but I guess people might use it for other types of Hilbert spaces, as well.