# A new look at quantum mechanics?

1. Jun 11, 2014

### naima

In QM you often read that apparatus measures the value of an observable (energy, position and so on).
This observable may have different values which are the eigenvalues of a hermitian operator. The eigenvectors are orthogonal. We associate projectors Pi such that Pi * Pi = Pi. if i <> j Pi * Pj = 0

You will see that all that was an idealization. The Pi must be replaced by positive operators Fi which are not necesseraly projectors and orthogonal. the collapse on one of the vectors of a basis is replaced by the random choice of an operator Fi.
This does not seem to be only a change of vocabulary!

You can find that in http://www.phys.tue.nl/ktn/Wim/muynck.htm#quantum

As i discover all that, i am not yet able to ask questions! But i would be interested in your comments.

Last edited by a moderator: Jun 11, 2014
2. Jun 11, 2014

### strangerep

Well, it's been known some time that POVMs have advantages over the conventional PVM approach.

Indeed, alternate formulation of QM based on generalized coherent states offers better insight into the relationship between classical and quantum, based on the underlying dynamical group for any given physical situation. (Coherent states give a type of POVM.)

3. Jun 11, 2014

### Staff: Mentor

Its the generalised observation/measurement view.

Its been known for a while now that the so called Von Neumann measurement is simply one example of a much wider class of measurements that is represented not by a resolution of the identity, but by a POVM, which is a resolution of the identity, but with the disjoint requirement removed. You can in fact reduce all measurements to Von Neumann measurements by considering a probe and a Von Neumann observation on the probe:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

IMHO it is the correct place to start.

I have posted the following before but will do so again because it shows exactly how it forms the basis of a very elegant axiomatic treatment.

First we define a POVM. A POVM is a set of positive operators Ei ∑ Ei =1 from, for the purposes of QM, an assumed complex vector space.

Now we have the single fundamental axiom of QM.

An observation/measurement with possible outcomes i = 1, 2, 3 ..... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

From that axiom alone you can derive Born's rule via Gleason's Theorem:
http://kof.physto.se/cond_mat_page/t...ena-master.pdf [Broken]

The paper above gives two versions of the theorem.

The first version is Gleason's original version which is quite difficult but based on resolutions of the identity. The second version is much simpler but based on the stronger assumption of POVM's. But since our basic axiom uses POVM's - we are in luck and can use the much simpler second version.

Note only by Ei means regardless of what POVM the Ei belongs to the probability is the same. This is the assumption of non contextuality and the rock bottom essence of Born's rule. But the above axiom has that built in, so its not an assumption we need to make. Still its good to know that is the essence of Born's rule.

You can run through the proof in the link above. Its proof of continuity is a bit harder than it needs to be so I will give a simpler one. If E1 and E2 are positive operators define E2 < E1 as a positive operator E exists E1 = E2 + E. This means f(E2) <= f(E1). Let r1n be an increasing sequence of rational's whose limit is the irrational number c. Let r2n be a decreasing sequence of rational's whose limit is also c. If E is any positive operator r1nE < cE < r2nE. So r1n f(E) <= f(cE) <= r2n f(E). Thus by the pinching theorem f(cE) = cf(E).

Hence a positive operator P of unit trace exists such that probability Ei = Trace (PEi).

This is the Born rule and by definition P is the state of the system.

Its simply a mathematical requirement that follows from the fundamental axiom I gave.

You possibly haven't seen it in that form. To put it in a more recognisable form by definition a Von Neumann measurement is described by a resolution of the identity which is a POVM where the Ei are disjoint. Associate yi with each outcome to give O = ∑ yi Ei. O is a Hermitian operator and via the spectral theorem you can recover uniquely the yi and Ei. By definition O is called the observable associated with the measurement. The expected value of O E(O) = ∑ yi probability outcome i = ∑ yi Trace (PEi) = Trace (PO).

A state of the form |u><u| is called pure. A state that is the convex sum of pure states is called mixed. It can be shown (it's not hard) all states are either mixed or pure. For a pure state E(O) = trace (|u><u|O) = <u|O|u> which is the most common form of the Born rule.

Neat hey. QM from just one axiom. Well not really, its just that the other assumptions you need to develop it from what an observable is, and the Born rule, is very natural - you will find the detail in Ballentine. For example I have made the tacit assumption every POVM represents an actual measurement - if I recall correctly that's called the strong principle of superposition - obviously that goes into it as well. Still another is the collapse postulate for filtering type measurements following from physical continuity. Physical continuity is very reasonable, but still an assumption. Its that sort of thing - assumptions - yes - but very reasonable. The key weirdness is however contained in that single axiom.

I have to say every time I think of it it brings a smile to my face - QM from just one axiom.

Thanks
Bill

Last edited by a moderator: May 6, 2017
4. Jun 11, 2014

### Staff: Mentor

Indeed.

Just another reason POVM's are the correct starting point IMHO.

Thanks
Bill

5. Jun 11, 2014

### Staff: Mentor

Gave it a quick squiz - from the link

Axiomatic approach of quantization
Quantum mechanics has been developed as a theory describing microscopic (atomic and subatomic) processes. In textbooks it is sometimes presented as being derivable from classical mechanics by a substitution -referred to as quantization- like
p → −iħ∂/∂q, ħ = h/2π,
in which h = Planck's constant (compare). However, in such a "derivation" there always is a certain arbitrariness: there are many different ways to quantize classical quantities. A quantization scheme can be justified only by the fact that its result is yielding a useful description of microscopic reality. For this reason it seems better to introduce quantum mechanics in an axiomatic way, as is done in the following, letting experiment decide about its applicability.

Note - the above does not apply to the approach to the dynamics taken by Ballentine based on symmetries. That is the correct approach IMHO. The issue of course comes when you apply it. Ballentine derived the equations from symmetry considerations and showed the energy and momentum operators have exactly the same form as classical mechanics. However given a classical system we make the assumption the Hamiltonian of the classical system translates to the quantum Hamiltonian. That's very reasonable - but is an assumption.

Thanks
Bill

6. Jun 12, 2014

### strangerep

It's less of an assumption if one's quantization procedure is based on the full dynamical group of a system, not just the kinematical symmetry. (Ballentine takes Galilean symmetry as a given, and then finds reasonably general forms for possible interaction terms in the Hamiltonian such that Galilean symmetry is still respected overall. But he stops short of a full dynamical group treatment.)

One can instead start with the dynamical group of the classical system. Recognizing that equations of motion are an indirect expression of Casimirs (in a particular representation), it becomes obvious that quadratic and higher products of operators in the Hamiltonian, etc, should be symmetrized in order to get a quantum version -- since Casimir operators always consist of sums of fully symmetrized products of the basic generators.

The generators which are transformed among themselves under commutation with the Hamiltonian then become the raw material for constructing generalized coherent states, which give POVMs.

7. Jun 13, 2014

### naima

I was surprised to see that De Muynck highlights POVMs but avoids Kraus operators. Not a word about them.
If we have to caculate the result of merging we have to add these Ki Kraus operators and not to add the $Fi=Ki^\dagger Ki$.
Look at http://arxiv.org/abs/quant-ph/0408011 where calculus is done.
Why has he to hide that?

8. Jun 19, 2014

### naima

Can we say that if we have a SIC POVM, it provides a basis of effects Ei on which any POVM output can be read as $\Sigma p(i) Ei$?
Fuchs writes that a state is a probability distribution on thes Ei full stop.

9. Jun 19, 2014

### Staff: Mentor

What do you mean by basis of effects?

I can't recall Fuchs saying that:
http://arxiv.org/abs/quant-ph/0205039

He chooses POVM's because he finds it simplified things removing the disjoint requirement of resolutions of the identity.

'I try to make this point dramatic in my lectures by exhibiting a transparency of the table above. On the left-hand side there is a list of various properties for the standard notion of a quantum measurement. On the right-hand side, there is an almost identical list of properties for the POVMs. The only diﬀerence between the two columns is that the right-hand one is missing the orthonormality condition required of a standard measurement. The question I ask the audience is this: Does the addition of that one extra assumption really make the process of measurement any less mysterious? Indeed, I imagine myself teaching quantum mechanics for the ﬁrst time and taking a vote with the best audience of all, the students. “Which set of postulates for quantum measurement would you prefer?” I am quite sure they would respond with a blank stare. But that is the point! It would make no diﬀerence to them, and it should make no diﬀerence to us. The only issue worth debating is which notion of measurement will allow us to see more deeply into quantum mechanics.'

Are you claiming that a resolution of the identity cant describe an observation? Since its a special case of a POVM obviously it can.

What I am saying is dead simple. A POVM results from a Von Neumann observation, which is a special case of a POVM, when analysing a probe interacting with a system and doing a Von Neumann observation on the probe. There is noting inconsistent with anything I have said and that.

Thanks
Bill

Last edited: Jun 19, 2014
10. Jun 19, 2014

### naima

11. Jun 19, 2014

### Staff: Mentor

So?

It certainly didn't answer my query of you:

So far I am finding your posts too vague to get much of a grip on.

Thanks
Bill

12. Jun 20, 2014

### naima

I may try to be less vague:

A povm is a set of effects (positive hermitian operators less than identity)
With a d dimentional Hilbert space there is a SIC POVM. it is a set of $d^2$ effects.
They form a basis for $\mathscr{L} (H_d )$
When you associate a probability to each effect the associate density operator is a linear combination of these effects with coefficients in [0 1]

I hope there is no problem with that
BUT my question was about something else:

In the De Muynck paper, he uses POVM and effects but he says nothing about Kraus operators or amplitudes.
I felt it was bizarre.
I found another link where Fuchs writes:

So my question is not technical: Is it possible to get interferometry results by avoiding amplitudes of probability (or Kaus operators)?

13. Jun 20, 2014

### Staff: Mentor

Don't know - have to leave that up to someone else.

Thanks
Bill

14. Jun 23, 2014

### naima

I looked for qbist calculus in interferometry avoiding probability amplitudes but i found none.
It seems to be only blah blah.