QT – A view from the inside Part V – The Born rule?

by aotell

Now that we have established that external interactions can cause an observed jump in the state evolution, we are going to construct a case where this actually happens and we can study the properties of this transition.

Generally, the system containing the observer and everything he interacts with is very large. We cannot discuss all the details of such a system, therefore we will use a minimal version consisting of just the necessary elements. And probably to your surprise, the observer is not contained in this simplified system. We do not actually need him, because we already now how he would reconstruct the system’s dynamic.

Even our entire universe will be very simple. It just consists of a single qubit and the photon field. The photons are assumed to be inaccessible locally, because they come in or leave at the speed of light. So it is just the qubit that we regard as the local system a hypothetical observer would reconstruct.

The process we are going to analyze is a simple scattering and absorption process on the qubit, which we may imagine to be realized as some two level system with two energy eigenstates \left|0\right\rangle and \left|1\right\rangle. The incoming single photon \left|\rightsquigarrow\right\rangle carries a two-dimensional polarization state spanned by \left|\circlearrowleft\right\rangle and \left|\circlearrowright\right\rangle. Before the qubit and the photon interact, their states are independent and represented as a product state.
The actual dynamics of the process is not really important to us, the output state of the scattering event is already perfectly sufficient for the demonstration. Hence we can simply define a unitary map between the input and output states. Unitarity is guaranteed by mapping an orthonormal basis of the input space to an orthonormal basis of the output space. We have a 4-dimensional input space, 2 qubit dimensions times 2 photon polarization dimensions, so we also get a 4-dimensional output space. And the whole process can be written in the form of 4 mapping rules.
The first rule is very simple, as the photon just passes through the qubit:

\left|\rightsquigarrow\circlearrowleft\right\rangle \left|0\right\rangle \mapsto \left|0\right\rangle \left|\rightsquigarrow\circlearrowleft\right\rangle

The same thing happens if we flip both the polarization and the qubit:

\left|\rightsquigarrow\circlearrowright\right\rangle \left|1\right\rangle \mapsto \left|1\right\rangle \left|\rightsquigarrow\circlearrowright\right\rangle

But in the two other cases we get a nontrivial interaction, resulting in a superposition of a scattering and an absorption or stimulated emission outcome:

\left|\rightsquigarrow\circlearrowright\right\rangle \left|0\right\rangle \mapsto \left( \left|0\right\rangle\left|\leftrightsquigarrow\circlearrowright\right\rangle + \left|1\right\rangle\left|\cdot\right\rangle \right)\frac{1}{\sqrt{2}}

\left|\rightsquigarrow\circlearrowleft\right\rangle \left|1\right\rangle \mapsto \left( \left|1\right\rangle\left|\leftrightsquigarrow\circlearrowleft\right\rangle + \left|0\right\rangle\left|\rightsquigarrow\circlearrowleft,\rightsquigarrow\circlearrowleft\right\rangle \right)\frac{1}{\sqrt{2}}

Here, \left|\leftrightsquigarrow\right\rangle is the omnidirectionally scattered photon, \left|\cdot\right\rangle the vacuum after the photon has been absorbed, and \left|\rightsquigarrow,\rightsquigarrow\right\rangle denotes the two photon state after stimulated emission.

If we apply the process to an input state \left( \alpha \left|\rightsquigarrow\circlearrowleft\right\rangle + \beta \left|\rightsquigarrow\circlearrowright\right\rangle \right) \left( a \left|0\right\rangle + b \left|1\right\rangle \right) and then take the trace over the inaccessible photon states, we get the state operator of the output state:

\rho = \left( \left|\alpha\right|^2 \left|a\right|^2 + \frac{1}{2}\left( \left|\alpha\right|^2 \left|b\right|^2 + \left|\beta\right|^2 \left|a\right|^2 \right) \right) \left| 0 \right\rangle \left\langle 0 \right|
+ \left( \left|\beta\right|^2 \left|b\right|^2 + \frac{1}{2}\left( \left|\alpha\right|^2 \left|b\right|^2 + \left|\beta\right|^2 \left|a\right|^2 \right) \right) \left| 1 \right\rangle \left\langle 1 \right|

Conveniently, this result is already decomposed into its eigenbasis. If we apply what we have learned about the locally observed state, we have to find the eigenvector with the greatest eigenvalue of this state operator. The result of which can easily be read of the representation, and we get \left|0\right\rangle if \left|\alpha\right|^2 \left|a\right|^2 > \left|\beta\right|^2 \left|b\right|^2 and \left|1\right\rangle otherwise. But the observer at the qubit does not know the state of the incoming photon, so \alpha and \beta are unknown to him and will introduce random information to the result. Since the observer does not know better, he has to assume that all photon polarizations are equally likely, which means the assumed random distribution of the photon state must invariant under a unitary transform of the photon state. With this assumption, we can calculate the probability of the observed process outcome to be \left|0\right\rangle or \left|1\right\rangle.

I cannot show the calculation here, but please see my next post for a reference to a paper on arxiv where it is actually performed. Of course, I will still give you the result. The probability of seeing an outcome \left|0\right\rangle is precisely p_0 = \frac{\left|a\right|^2}{\left|a\right|^2+\left|b\right|^2}, and the complementary event of observing \left|1\right\rangle is p_1 = \frac{\left|b\right|^2}{\left|a\right|^2+\left|b\right|^2}.

You have certainly noticed, that our result is precisely the probability as postulated by the Born rule! This means we have not only derived, that the observer will see a discontinuous jump between pure states, but also that the statistics of that jump exactly match the established statistics of performing a quantum measurement, at least for a simple interaction with an incoming photon. We have therefore deduced the full measurement postulate of the Copenhagen interpretation just by assuming the position of an internal observer, without any additional requirements, and also without having to introduce arbitrarily many quantum worlds.

My next post will contain a few final thoughts and a pointer to more technical and rigorous derivations of all results presented here and more.