How the Born rule is understood in various interpretations?

[Jarek 31]

TL;DR Summary

General discussion about Born rule in various interpretations

While intuitively we use
1) 3rd Kolmogorov axiom: probability of alternative of disjoint events is sum of their probabilities
somehow QM allows to use instead:
2) Born rule: probability of alternative of disjoint events is proportional to square of sum of their amplitudes.

This nonintuitive Born rule e.g. allows to violate Bell-like inequalities for example Mermin's Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1 "tossing 3 coins, at least 2 give the same value" - absolutely obvious, derived from just existence of joint probability distribution and 3rd axiom ... but QM formalism allows to violate it (e.g. 4.2.1 in Preskill notes http://theory.caltech.edu/~preskill/ph229/notes/chap4_01.pdf or https://arxiv.org/pdf/1212.5214 ).

From "Quantum Interpretation poll", I see there are representants of various interpretations here - I would like to propose a general discussion about Born rule.
For example:
Does Born rule have to be assumed, or can we derive it? (can be e.g. derived in Ising model, something similar is in Malus law)

Both 3rd axiom and Born rule coexist in physics - how to specify which one should be used in a given situation?
Connected: for the "tossing 3 coins" violation, there is required difference between unknown and unmeasured value - how to understand this difference?

What is interpretation of QM amplitude - how does it differ from probability, where the square comes from?

 

[Demystifier]

The Born rule in arbitrary basis can be derived from the Born rule in the position basis. See e.g. the paper linked in my signature.

 

[Jarek 31]

Thank you, I see you use amplitude in formula (12), and then get density in (13) - could you elaborate on this most crucial step?

In other words, you work on |ket>, while we also need second amplitude <bra|, multiplied in Born rule e.g. by going through some representation like $<bra|ket> = \int_k <bra|k> <k|ket>dk$ - what is the difference between <bra| and |ket> types of amplitude?
For example in scattering matrix they come from two time directions: $S_{fi} = \lim_{t_1\to-\infty} \lim_{t_2\to \infty} \langle \Phi_f |U(t_2,t_1)|\Phi_i\rangle$.

 

[Demystifier]

Thank you, I see you use amplitude in formula (12), and then get density in (13) - could you elaborate on this most crucial step?

At this point I just use standard QM, which postulates (without a derivation from something deeper) that probability is |amplitude|$^2$.

In other words, you work on |ket>, while we also need second amplitude <bra|, multiplied in Born rule e.g. by going through some representation like $<bra|ket> = \int_k <bra|k> <k|ket>dk$ - what is the difference between <bra| and |ket> types of amplitude?

Perhaps I'm stating the obvious, but $\langle\psi|=|\psi\rangle^{\dagger}$.

For example in scattering matrix they come from two time directions: $S_{fi} = \lim_{t_1\to-\infty} \lim_{t_2\to \infty} \langle \Phi_f |U(t_2,t_1)|\Phi_i\rangle$.

I don't think that it's a correct interpretation that <bra| comes from the opposite time direction. The <bra| appears because you want to get a positive real number (probability) from |ket> which, by itself, is not a real positive real number.

If you want a deeper eplanation of the Born rule from Bohmian mechanics, see the paragraph after Eq. (23) and references therein. But before trying to understand it, first make sure that you understand where do probabilities come from in classical statistical mechanics. I saw a billion times people trying to understand quantum probabilities without understanding probabilities in classical statistical physics. It never works well.

 

[Jarek 31]

Ok, so you postulate Born rule, not derive it. In this case my questions remain.

Regarding <bra|, |ket>, in scattering matrix one comes from propagation of initial conditions, second from final - but when they meet I think they indeed should be always their conjugates (consequence of Hamiltonian being self-adjoint ... or maybe there are exemptions? https://en.wikipedia.org/wiki/Non-Hermitian_quantum_mechanics ).

However, going from Feynman to Boltzmann path ensemble e.g. in Ising model (derivation), the propagator becomes transfer matrix, and the two amplitudes are its left and right eigenvectors - they are different if transfer matrix is not symmetric, what for Ising mode can be realized e.g. with gradient of magnetic field.

 

[Demystifier]

Ok, so you postulate Born rule, not derive it. In this case my questions remain.

Perhaps you missed the last paragraph in my post above. The Born rule is derived, but just not in Eq. (13).

 

[Jarek 31]

If you want a deeper explanation of the Born rule from Bohmian mechanics, see the paragraph after Eq. (23) and references therein.

As (23) I see standard continuity equation with Madelung substitution - which assumes Born rule.

But before trying to understand it, first make sure that you understand where do probabilities come from in classical statistical mechanics. I saw a billion times people trying to understand quantum probabilities without understanding probabilities in classical statistical physics. It never works well.

Indeed, as emphasized in the first post, the main question here is understanding the difference between probability and amplitude - why do we need to square the latter, in contrast to (different intuitive) 3rd Kolmogorov's axiom?

For example in Ising model (similarly as for scattering matrix), two amplitudes correspond to propagators from two directions: one from left, second from right.
To get probability distribution in a given position, we need to meet both propagator there - and multiply their amplitudes.

 

[Demystifier]

Indeed, as emphasized in the first post, the main question here is understanding the difference between probability and amplitude - why do we need to square the latter, in contrast to (different intuitive) 3rd Kolmogorov's axiom?

Ah, now I think I understand what really bothers you. You ask why do we add probability amplitudes rather than probabilities themselves.

In the Bohmian interpretation that's related to the fact that, at the fundamental level, this interpretation is not about probabilities at all. The wave function is fundamentally interpreted as a pilot wave for particle trajectories, not as a "probability amplitude". The wave functions are added because the Schrodinger equation is linear so its solutions obey the superposition principle. A priori, this adding has nothing to do with probabilities and Kolmogorov. But the Schrodinger equation then implies a continuity equation for |wave function|$^2$, so it turns out that the continuity equation in equilibrium implies that the emergent probability is given by |wave function|$^2$.

 

[Jarek 31]

Thank you, the square still looks postulated, not derived.
Personally I also like dBB picture, e.g. intuitions from walking droplet experiments ( https://dualwalkers.com/ ) - they also had this "Wavelike statistics from pilot-wave dynamics in a circular corral" paper ( https://journals.aps.org/pre/pdf/10.1103/PhysRevE.88.011001 ), but I don't see square of Born rule mentioned (?)

Probably the simplest situation is: what stationary probability distribution $\rho$ should we expect in [0,1] range?
Standard diffusion/chaos predicts uniform $\rho = 1$.
In contrast, QM predicts very different $\rho \propto \sin^2$ localized stationary distribution ... why?

It can be explained e.g. using uniform path ensemble (~euclidean path integrals/MERW), getting the same stationary probability as QM, with clear intuition for the square - as in Ising model or scattering matrix:

 

[A. Neumaier]

Does Born rule have to be assumed, or can we derive it?

For a derivation from natural assumptions see my paper

• A. Neumaier, Born's rule and measurement, arXiv:1912.09906

 

[Jarek 31]

Thank you, I have briefly looked and mostly see generalization to POVMs - could you elaborate on derivation of Born rule?
Or why in [0,1] we should expect rho~sin^2 localized stationary probability distribution?

Regarding POVM-like measurement, could there be built e.g. POVM-polarizer (triangular lattice?) or spin measurement: distinguishing not two, but e.g. three possibilities rotated by 120deg?

 

[A. Neumaier]

could you elaborate on derivation of Born rule?

POVMs are derived in Section 2 from the new and intuitively motivated Detector response principle (DRP), and Born's rule is derived on p.16 from POVMs under the additional idealizing assumption of projectivity - a necessary condition for the Born rule to hold.

why in [0,1] we should expect rho~sin^2 localized stationary probability distribution?

Because this is what the formula (12) derived on p.16 gives in this situation.

Regarding POVM-like measurement, could there be built e.g. POVM-polarizer (triangular lattice?) or spin measurement: distinguishing not two, but e.g. three possibilities rotated by 120deg?

By combining several POVM (similar to what is done in Section 2.1) one gets new POVMs that can realize these cases. I don't know whether some simple material allows such polarizers, but it seems not inconceivable.