On the EPR Paradox

Wenjie Chen

Mar. 19th, 2019

In 1935, Einstein, Podolsky and Rosen published a paper named "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" on the Physical Review journals.¹ In their paper, an imaginary experiment was proposed in order to prove the incompleteness of the quantum mechanics (described by wave functions). This imaginary experiment along with its siblings appeared later were given the name "the EPR paradox" historically. In this report, the EPR paradox will be briefly reviewed along with some personal understandings.

1. Interpretation of The Original Paper

In Einstein, Podolsky and Rosen's original paper, several very important arguments are addressed, which I will re-clarify one by one. Each of these arguments is intriguing and worths to think deeply and carefully. Please note that from now on I will simply uses "Einstein" instead of "Einstein, Podolsky and Rosen" when refering to the authors of this paper.

Q1: How to determine whether a physical theory is successful?

Einstein thinks a successful physical theory should meet the following two properties:

correctness,
completeness.

While the correctness is judged by experiment and measurement, the completeness is defined by the following statement.

Every element of the physical reality must have a counter part in the physical theory. We shall call this the condition of completeness.

Therefore as soon as we decide what are the elements of physical reality, the completeness of the physical theory can be tested.

Q2: What is physical reality?

Einstein gave out a sufficient condition of reality as the criterion of reality:

If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding lo this physical quantity.

I think this criterion is very intriguing because of the word "disturb". For example, in most cases when you measure the coordinate of a moving particle with fixed momentum $p_0$ , you can get some quantity, but that doesn't mean that the coodinate is a physical reality in this problem, because you cannot do that measurement without disturbing the particle.

Q3: What's "wrong" with quantum mechanics?

First I want to point out that Einstein never questioned the correctness of quantum mechanics (described by wave functions). He merely doubted the completeness of it and wished to find a better theory (for example, to discover some hidden variables).

In quantum mechanics, if the operators corresponding to two physical quantities do not commute, i.e.

[A,B] \neq 0,

then they can not be experimentally determined at the same time.

Then with the definitions mentioned above, Einstein immediately pointed out that

From this follows that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantifies cannot have simultaneous reality.

By introducing an imaginary experiment, Einstein showed that if we assume that wave function is a complete quantum-mechanical description of reality, then two physical quantities do not commute the two quantifies can have simultaneous reality. This paradox leads to the conclusion that the wave funcition is not a complete description.

The imaginary experiment Einstein proposed is explained as followed.

Suppose there are two systems, I & II. They interact with each other from $t = 0$ to $t = T$ , and before $t = 0$ we know the states of two systems. After $t = T$ the two systems are separated and no longer interact with each other.

Now consider: what will happen when we do measurements?

For the two systems, the whole wave function (at $t > T$ ) can be written as an expansion with eigenstates for some physical quantity $A$ of the first system:

\Psi(x_1,x_2) = \sum_{n = 1}^{\infty}\psi_n(x_2)u_n(x_1),

where $x_1$ & $x_2$ are variables to describe two systems, $u_n(x_1)$ is the eigenstate of $A$ :

A u_n(x_1) = a_nu_n(x_1),

and $\psi_n(x_2)$ is the expansion coefficient.

When we measure the first system for the quantity of $A$ , let's say the system is found to have the value of $a_k$ , then we know after the measurement the first system is left in the state given by the wave function $u_k(x_1)$ , and the second system is left in the state given by the wave function $\psi_k(x_2)$ . This process, where infinite series reduce to a single term is called the reduction of wave packet.

But what if we choose to measure another physical quantity?

The whole wave function (at $t > T$ ) can also be written as an expansion with eigenstates for another physical quantity $B$ of the first system:

\Psi(x_1,x_2) = \sum_{s = 1}^{\infty}\phi_s(x_2)v_s(x_1),

where $x_1$ & $x_2$ are variables to describe two systems, $v_s(x_1)$ is the eigenstate of $B$ :

B v_s(x_1) = b_sv_s(x_1),

and $\phi_s(x_2)$ is the expansion coefficient.

After a measurement for the quantity $B$ on the first system, the system is left in the state given by the wave function $v_r(x_1)$ , and the second system is left in the state given by the wave function $\phi_r(x_2)$ .

Einstein argued that now since the second system no longer interacts with the first system, any measurements carried on system I should not affect system II. Which means $\psi_k(x_2)$ and $\phi_r(x_2)$ describe the same reality (i.e. system II). Here we have already assumed that the wave function is a complete description of the system.

It seems strange that two wave functions are the same, but everything will be just fine if $A$ and $B$ commute. In that case they have the simultaneous eigenstates, which means $u_k(x_1)$ and $v_s(x_1)$ are same sets of wave functions.

However, Einstein proposed a special case where $A$ and $B$ do not commute (the case is about two particles' coordinations and momenta). In these measurements, we never disturb system II since the two systems do not interact with each other anymore, thus according to Einstein's definition of reality, $\psi_k(x_2)$ and $\phi_r(x_2)$ both reflex the same reality of system II. This violates the second statement:

when the operators corresponding to two physical quantities do not commute the two quantifies cannot have simultaneous reality.

Therefore a paradox rises.

2. Key Points of the EPR Paradox

There are several important facts about the EPR paradox that I would like to point out.

The paradox rises inside the frame of quantum mechanics. That is to say, Einstein managed to find inconsistency using the knowledge provided by quantum mechanics, nothing more.
An important fact is that the EPR paradox is based on the assumption of local realism, which both Einstein and Bohr² believe to be naturally true. Therefore what the EPR paradox really proved is that if the local realism is true, then the quantum mechanics is incomplete. Otherwise, if the quantum mechanics is a complete theory, then the local realism is wrong!
Note that the original EPR paradox only includes pure states. Schrödinger later termed the state of the two particle in the EPR paradox "entangled" state. In this problem where only pure states are involved, the entanglement and the inseparability is the same. However we should keep in mind that in real experiments, what we can produce are usually mixed states. In that case different entangled states can occur, such as nonlocal states, steering states and inseparable states. But we should not worry about those states and focus on the original paper.

3. How to Understand the EPR Paradox

Based on the Bell's inequality or the CHSH inequality, experiments have been carried out to decide whether the EPR paradox is correct. We now know that all experiments showed a violation of Bell's inequality, which means Einstein was wrong and more importantly, the assumption of local realism doesn't hold.

This fact is somewhat frustrating since the assumption of local realism is very intuitive. However, we also know that when Einstein proposed the theory of special relativity for the first time in history, that theory, especially the fact that the speed of light in vacuum is the same for all observers, was also counter-intuitive, as much as the incorrectness of local realism. As Einstein pointed out, the correctness of a physical theory should be judged by experiments, and the experiments told us the local realism should be abandoned, just like the electromagnetic aether one hundred years ago. "The history of science is a graveyard of dead scientific theories and abandoned theoretical posits."³ That's how science evolves.

Only this time, to abandon the local realism may turn out to be a giant leap of faith.

1 A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?, Phys. Rev. 47, 777 (1935).↩

2 N. Bohr, Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, Phys. Rev. 48, 696 (1935).↩

3 Moti Mizrahi, The History of Science as a Graveyard of Theories: A Philosophers’ Myth?, Int. Stud. Philos. 30, 263 (2017).↩