Entanglement (I): how it all began, the EPR Paradox.

Being this the first article of this blog, I was torn as to where to begin. This is mainly because the fields of study of Quantum Information and Quantum Computation (QI and QC) are very wide and can be introduced in many ways.

After sleeping on this issue for a couple of days, I decided to go for the historical approach, starting all the way back in 1935 with the famous Einstein-Podolski-Rosen (EPR) paper: “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?“. The main reason I chose to begin with a paper that was written a couple of decades before QI and QC were born is that if one wishes to study QI and QC, one has to become familiar with the notion of quantum entanglement. And this paper was to first put the spotlight on quantum entangled states.

Without going into too much detail (don’t worry, we’ll get there), we can introduce quantum entanglement as the quantum correlations that express the non-locality inherent to quantum mechanics. But let’s not get ahead of ourselves…

In May 1935, Albert Einstein and two of his postdoctoral research associates Boris Podolsky and Nathan Rosen published a paper with a thought experiment that was meant to prove that Quantum Mechanics showed internal contradictions in it’s formulation (as we all know, Einstein wasn’t very fond of this new physical theory).

In a nutshell, the authors claimed that if the description of physical reality given by the wave function is complete, then two quantities described by non-commuting operators could have simultaneous real values. Therefore concluding that quantum-mechanical’s wave function description of reality is incomplete.

Since the main goal of this article is to introduce the notion of entangled states, I will not go into detail as to how those assertions are proved (although I highly recommend the reader to check the EPR paper), instead I’ll present the thought experiment that Einstein, Podolski and Rosen used.

The tough experiment: entangled states

Two quantum systems, I and II, interact from times $t=0$ to $t=T$ , after which time there is no longer any interaction among the two parts. With the help of Schrödinger’s equation, we can calculate the state $|\Psi\rangle$ that represents the combined system I+II at $t>T$ . Given any physical observable quantity A, we can define an operator $\hat{A}$ whose eigenvalues and eigenfunctions are given by the set $\{a_\lambda, |u_\lambda\rangle\}$ , with $\lambda=1,2,3...$ . Then, $|\Psi\rangle$ can be expressed as

$|\Psi\rangle=\sum_\lambda |u_\lambda\rangle |\psi_\lambda\rangle$ , (1)

where $|\psi_\lambda\rangle$ are the coefficients of the expansion of $|\Psi\rangle$ into a series of orthogonal functions $|u_\lambda\rangle$ . If we measure the quantity A and the result is $a_m$ , the wave packet is reduced to a single term $|u_m\rangle |\psi_m\rangle$ .

It is important to notice that the set $\{a_\lambda, |u_\lambda\rangle\}$ is determined by the choice of the physical quantity A. If we would have chosen a quantity B, whose corresponding operator has the set of eigenvalues and eigenfuntions $\{b_\sigma,|v_\sigma\rangle\}$ , instead of Eq. (1), we would get the expansion

$|\Psi\rangle=\sum_\sigma |v_\sigma\rangle |\chi_\sigma\rangle$ , (2)

where $|\chi_\sigma\rangle$ are the new expansion coefficients. Same as before, if we measure the quantity B and the outcome of said measurement is $b_n$ , then the state is reduced to $|v_n\rangle|\chi_n\rangle$ .

Given this result, Einstein, Podolski and Rosen argue that “as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two different wave functions […] to the same reality (the second system after the interaction with the first). Now, it may happen that the two wave functions, $|\psi\rangle$ and $|\chi\rangle$ , are eigenfunctions of two non-commuting operators corresponding to some physical quantities P and Q, respectively. […] by measuring either A or B we are in a position to predict with certainty, and without in any way disturbing the second system, either the value of the quantity P (that is $p_m$ ) or the value of the quantity Q (that is $q_n$ ).“

Therefore, it would seem that the results of a measurement performed on a state of an entangled system can have an instantaneous effect on another state, and could allow us to obtain information about the second state without measuring it.

At the time, Einstein labeled this a “spooky action at a distance”, and it was Austrian physicist E. Scrödinger who originally proposed the name “entanglemet” (Verschränkung).

Is Quantum-Mechanical description of reality really incomplete?

Before you get all worked up and start accusing you physics professor of teaching you a theory that cannot completely describe physical reality, you have to know that the EPR paradox was solved by CERN physicist John Bell in 1964.

One of the key concepts on which Einstein, Podolski and Rosen base their argument is that of locality. These authors assume that if two systems are sufficiently apart, a measurement performed over one of the systems will not directly affect the reality of the other (unmeasured) system. However, Bell proved that quantum theory is non-local and that the EPR “paradox” is no more than a mere example of how Quantum Mechanics cannot be understood in classical physics terms. I will make an article about Bell’s work and Bell’s inequalities in the future.

Also, it’s important to remember that in order to reach that conclusion, Einstein et al. assumed that for two quantities described by non-commuting operators to have simultaneous real values, we should be able to predict with certainty (i. e. with probability equal to unity) the value of said physical quantities (this is the EPR Criterion of Reality). We could, by measuring the quantity A, infer that the system II is described by the state $|\psi_m\rangle$ . Similarly, by measuring the quantity B we can infer that the system II is described by the state $|\chi_n\rangle$ . Taking the example EPR proposed, let us suppose that the two systems are two particles such that

$\Psi(\mathbf{x_1},\mathbf{x_2})=\int_{-\infty}^{\infty}e^{(2\pi \imath /\hbar)(x_1-x_2+x_0)p}dp$ . (3)

Then, by measuring the momentum on the first particle we could predict the momentum of the second. And by measuring the coordinate of the first, we can predict the coordinate of the second. Aldo it is true that by choosing which quantity to measure we can predict the state that represents the other system, it does not suffice to be able to choose which quantity to measure, we would need to actually be able to measure them both at once. Measuring the coordinate of one particle certainly does now allow us to predict the momentum of the other particle. This is one of the arguments that Bohr presented on his response to the EPR paradox.

Not only his paper rapidly became the center of a heated discussion between physicists on the interpretation of Quantum Mechanics (leading Bohr and Einstein to discus this issue for years), but it was the first time that physicists turned their attention to entangled states and their “spooky” properties. This would be the cornerstone that would eventually lead to Quantum Information and Quantum Computation fields of study, and this is the beginning of our journey into tho fascinating world of QI and QC.

For further reading about the EPR paradox, I recommend this article. I also wrote an article explaining Bohm’s version of the EPR’s paradox, read it here.

Ref:

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).

N. Bohr, Phys. Rev. 48, 696 (1935).

Entanglement (I): how it all began, the EPR Paradox. by Marco Vinicio Sebastian Cerezo de la Roca is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.