The Quantum Botanist

June 24, 2026

In quantum mechanics, there is a very well known paradox called the Einstein-Podolsky-Rosen paradox. It stems from a paper written by those three about their qualms regarding the current understandings of quantum mechanics. The paper can be found here. I’ll be going through the paper and explaining it in different terms, and then going over some implications and responses.

How to Read

Reading papers like this can be very intimidating, so I want to start with the method that I used when I first read it. I had to read it for a presentation in my first quantum mechanics class, so I was not nearly as proficient as I am now, and I definitely did not understand everything fully on my first read. To begin understanding it, though, you have to go through that first reading. For any paper that you aren’t fully confident in, the first step is to read the whole thing. If you don’t get something, that’s fine, just keep reading and finish the paper. You may have questions answered in the next section, or you can always look into it after reading. It’s important to get the big picture before getting lost in the details.

Big Picture

The big picture argument that the EPR paper asserts is that the wave function view of quantum mechanics is incomplete, meaning that it does not fully describe reality. The reasoning presented is that, if we take two quantum systems, allow them to interact for a certain amount of time, then stop them from interacting, we can get two wave functions describing system 2 at once, but only depending on what variable we measure on system 1. They assert that this would imply that either a single wave function is incomplete, as it only contains half the information of reality, or that the measured quantities cannot have simultaneous reality.

Detailed Explanation

Let’s now get into the details of how these conclusions were reached. We first need to define completeness and reality. They define completeness as “every element of the physical reality must have a counterpart in the physical theory.” What this means in less scientific terms is that, if there is some property that exists in the physical world, then we have to have some mathematical representation of that in our theories. Then, reality is a bit more complicated. There are a lot of definitions we can use for reality, but they decide to use one that they determined is reasonable. A physical quantity is considered real if we can predict with certainty its value without disturbing the system in any way.

Now we move on to the idea of state. The state of a quantum system is supposed to be the complete set of variables of the system. From this, we should be able to derive all real properties of the system. In classical mechanics, this is generally the position and momentum, as with these two values, we can derive any other values, as long as we have outside influences like force fields. In quantum mechanics, the state is generally given as a wave function $\psi$. For each observable quantity, there is a corresponding operator that can be applied to the state to measure that variable. To address the concern of reality, then, we need to see if the operator is able to measure the state without changing it. While this is possible, it is not guaranteed. An important rule in quantum systems is that, if two operators do not commute, then we cannot know both properties simultaneously, as mmeasurement of one alters the state in such a way that destroys the knowledge of the other. The conclusion here is that either the wave function description of quantum mechanics is incomplete, or that two properties with operators do not commute cannot both be real simultaneously. If the two properties existed simultaneously, then for the wave function to be complete, it would require both to be measurable, which they are not.

Now, let’s assume we have two quantum systems, 1 and 2, and we allow them to interact for some amount of time. We will define the combined state mathematically as $\Psi$. We cannot directly calculate the individual states of the two systems after they are done interacting from this combined state, we must employ the method of wave packet reduction. Let’s take some physical quantity, $A$, with corresponding eigenvalues $a_1, a_2, …$. Eigenvalues are the allowed values of the variable upon measurement. The corresponding eigenfunctions are $u_1(x_1), u_2(x_1), …$, where $x_1$ is the variable used to describe system 1, and it can represent multiple variables. In this case, we can rewrite the combined wave function as

where the $\psi_n$ are the coefficients of expansion into the orthogonal basis of the eigenfuctions, and describe the state of the second system. If we assume that we observe system 1 in state $u_k(x_1)$, then we can be certain that system 2 is in state $\psi_k(x_2)$. We’ve reduced the wave packet from an infinite series to a single term by measuring.

Now, assume we did the same thing, but instead of chosing the property $A$, we chose property $B$. This gives a different expansion and different values. We now observe the first system in state $\phi_r(x_1)$, and so we know the second system is in state $v_r(x_2)$. When doing two different measurements, we obtain two different states for the second system, which was not touched. Therefore, we conclude that both must simultaneously be true for system 2. Now, if these two different states are eigenstates of different values, say position and momentum, then we would have two values that the wave function interpretation cannot predict simultaneously existing at the same time in reality. The paper proves that this can be true, but I will leave it out here. Therefore, if there are two real properties that the wave function cannot predict simultaneously, then it must not be complete. This is where the EPR paper ends.

Responses

One response involves the idea of entanglement. When the two systems interact in such a way that their states are dependent on each other, they become entangled, and any action on one affects the other, even if they are no longer interacting. This would contest the claim that both wave functions are true for system 2 simultaneously. They only become true when system 1 is mesured. This would mean that there is no problem with both wave functions giving values that should be unknowable simultaneously, as they are not both real at the same time. Einstein later responded to claims like this stating that he did not believe in “spooky action at a distance,” as it should not be possible for two systems that are separated to interact the way entanglement would allow them to.

The proposed explanation for how the wave function cannot be complete is that there are hidden variables, which are variables that exist, but we do not know about. Bell later disproved the possibility of local hidden variables with what is known as Bell’s theorem, and it has been confirmed experimentally many times. Locality here means that the vairables cannot have an effect faster than the speed of light. For example, a force field that is local can only propagate at the speed of light. If it were to move faster, it would be considered non-local. This leads us to the conclusion that the wave function is complete, and that properties of quantum mechanical systems are not real until observed, either directly or with the help of an entangled system, or that there are non-local hidden variables.

Further Reading

The paper used for this can be found at this link: https://cds.cern.ch/record/405662/files/PhysRev.47.777.pdf
(this is the same one as the paper at the top of the post)

I would also recommend this book to learn more about entanglement: https://direct.mit.edu/books/book/4632/Quantum-Entanglement

I am planning on making a separate post detailing Bell’s theorem at some point, so you can read that when I write it if you are interested. I likely will not update this section, but if you search for it on the blog, it should be easy to find, assuming I’ve written it.