## Quantum Mechanics, Probability, Covid-19 Analogy

I hesitated to include “Covid-19” in the title. This is not a commentary on the scientific, social, economic aspects of the Covid-19 crisis. This is about an analogy between the probability of catching Covid-19 and the concept of probability used in Quantum Mechanics.

The probability of catching the Sars-Cov-2 virus which causes the Covid-19 disease depends on where you go an what you do there. God forbid, if you test positive, after that moment forward, the probability of catching Covid-19 becomes irrelevant for you. The concept of probability used in Quantum Mechanics is very similar.

In Quantum Mechanics (QM), when we measure an observable (position, momentum, spin direction, etc.) we observe one of the possible outcomes of that observable. Can we predict which one? Not precisely. We can compute the probability of measuring that outcome, however. This probability is given by $|\psi|^2$. The $\psi$ is known as the quantum mechanical wavefunction. In the literature you can also find the term  “probability amplitude” referring to $\psi$. Remember, $\psi$ can be complex-valued. When we square a complex number we can get a negative number. Negative probability is not meaningful, hence the “absolute square”.

Where do we get this $\psi$ from? We have equations known as Schrodinger’s equation and Dirac’s equation. Solutions of these equations in specific configurations give us the $\psi$. How did Schrodinger come up with his equation? How did Dirac do it? Good questions! Let me answer this with another question (don’t you hate people who do that). How did Maxwell and Einstein come up with their equations? These great physicists (Maxwell, Einstein, Schrodinger, Dirac) started with data. They tried to explain the existing observations. But, they also used their intuition to expand the coverage of their equations. So, initially these equations were hypotheses. Later, more observations confirmed the validity of their equations.

Let’s come back to the analogy. Before the measurement the probability of obtaining a particular outcome is $|\psi|^2$. After the measurement $\psi$ becomes meaningless. Physicists don’t say “meaningless”, they say $\psi$ becomes a sharp distribution (delta function). In other words, after the measurement, the probability of obtaining that particular outcome is 1. In my opinion the term “meaningless” is a better description. In physicist jargon this abrupt adjustment of $\psi$ is known as the “collapse of the wavefunction”. This is very similar to the probability of catching Covid-19. Before you go for testing you can estimate a probability for positive outcome. If the test turns out to be positive, after that moment, the probability of catching the virus becomes meaningless.

Is that all?

Surely, there is more to the story.  Once you know your Covid-19 test result, if it is positive, the probability of catching the virus is irrelevant for you but you still have an effect on the system. Where you go and what you do there changes the probabilities of catching the virus for other people. Similarly, the knowledge of the outcome of a physical measurement performed on a single electron changes the probability amplitudes for the other electrons in the system. Here we differentiate $\psi_{system}$ and $\psi_{individual}$. After the measurement on the electron labeled “individual” the $\psi_{individual}$ becomes meaningless but $\psi_{system}$ is still meaningful and it has to be adjusted.

Example with qubits

Let’s consider a system consisting of entangled qubits and two observers. It is customary among physicists to name the observers Bob and Alice. When Bob measures the state of one of the qubits and obtains the result of 1, after that moment this particular qubit will stay in the (1) state until the next disturbance (measurement or thermal noise) in the system. Between the disturbances the system will evolve as if all qubits except this particular one are in 1/0 superposition.

It is possible that another observer (Alice) knows the state of another qubit (similar to another person being aware of her Covid-19 test result). If there is no communication between Bob and Alice they will assess the system differently. If they report their measurement results to each other they will re-adjust their models ( $\psi_{system}$) to make the models agree.

What is actually (physically) happening irrespective of reporting between Bob and Alice? Well….physicists have been debating this for almost 100 years now.  In my modest opinion, analogies can be helpful with conceptual understanding.

In terms of analogy, the communication between Bob and Alice is similar to reporting of Covid-19 test results on various web sites. Every day, once we know the Covid-19 test results we re-evaluate our risk (probability of catching the virus) instinctively and statistical modelers update their numbers. This is Bayesian approach to probability. This is similar to $\psi_{system}$ being adjusted after we learn more about the internal states of the system.

The crux of the analogy is this: We certainly re-evaluate our risk after learning the latest Covid-19 statistics but does this act of re-evaluation change our behavior? Human beings are semi-rational beings so they do change their behavior. Electrons are not rational beings, do they change their individual behavior based on the measurements performed on the other components of the system? Based on classical physics the answer would be no but the answer according to QM is yes. Empirical evidence confirms QM.

In a quantum mechanical system the “collective” is more than the sum total of the individual parts. Information generated from measurements is stored in the “collective” and this information influences the behavior of the individual parts even though the individuals are not in direct (causal) contact. Quantum fields are such collectives. Entangled qubits are such collectives as well. Contrast this to the understanding in classical mechanics which considers the “information generated from measurements” as completely extraneous to the physical system.

Limitations of $\psi$

1. Quantum Mechanics (QM) is a recipe to estimate the probability of a particular outcome from a measurement if the measurement is prepared in a certain way.
2. There is no disagreement on the recipe. That’s why we call it “mechanics”.  Everyone agrees on the mechanics. The 100 year discussion is on the interpretation of $\psi$.
3. Regarding the interpretation of $\psi$ this paper and its quick summary can be helpful. But, I have to admit, I am still confused by these discussions.
4. I mentioned above that “information generated from measurements” becomes part of the system. There is empirical evidence for this “information is physical” claim but it is still controversial. Scott Aaronson’s explanation of “information is physical” is interesting (the comments section is also interesting). One can also read the (Oxford) PhD thesis of C.G.Timpson.
5. $\psi$ does not reflect the conservation laws. We integrate the conservation laws into $\psi$ in an ad hoc way.
6. $\psi$ cannot model inelastic scattering (interactions where elementary particles transform into other particles).
7. We cannot model “interaction” with $\psi$. We talk about the interdependence of $\psi_{individual}$ and $\psi_{system}$ but the difficulty is a fundamental one. Quantum Mechanics is a linear theory. Interaction is a nonlinear process. We cannot model a nonlinear process with a linear theory. This is why an improved version of Quantum Mechanics was invented. That improved version is known as Quantum Field Theory.
8. Measurement is a type of interaction. Since $\psi$ cannot model interaction, it is difficult to incorporate measurement process into $\psi$.
9. “Information generated from measurements” introduces nonlinearity into the system. Measurement is a type of interaction. Interaction is nonlinear by definition. Measurement generates information. Therefore, “information generated from measurements” must be expressions of nonlinearity.
10. The next question would be “what is information?”. Questions beget questions. I am not saying this to discourage you but I have to conclude this post here.

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