# An Introduction to Quantum Entanglement

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One of the most important objections to the interpretation of quantum mechanics (Copenhagen interpretation) is the EPR paradox, which was proposed by Einstein, Podolsky and Rosen. According to the EPR paradox, the description of physical reality provided by quantum mechanics is incomplete. In this article, we will partially explain entanglement and discuss why nature is ‘spooky’ in the framework of quantum mechanics.

## Classical Correlation

Before going further on quantum entanglement, it is important to understand the concept of classical correlation first. For this purpose, it is convenient to illustrate correlation as a thought experiment with the help of Alice (A) and Bob (B), the ‘most popular couple in physics’, and their friend Carol (C).

Imagine Carol has two different colored balls, white and black. She shuffles the 2 balls and gives one to Alice and one to Bob, not showing them which color they have. Now, suppose Alice gets on her shuttle and goes to Alpha Centauri while Bob goes to Palo Alto. The rule is they cannot look at which ball they are holding until they arrive at their destination.

However, their clocks are synchronized and they calculate a few seconds time dilation due to the rule of special relativity before their trips. In other words, Alice will look at her ball a few seconds before Bob did. According to that, when Alice looks at her ball, she will instantly know the color of Bob’s ball even before he looks. There is a perfect correlation between Alice and Bob’s observations. If Alice’s ball is black, that means Bob’s ball is certainly white and vice versa.

Nevertheless, our classical correlation example does not violate the principle of Einstein’s special relativity theory which states that nothing can travel faster than the speed of light in a vacuum.  Because of this, there is no way to send a signal faster than the speed of light. Alice may know the color of Bob’s ball, but she cannot tell her observation to Bob before he looks at his ball. In other words, Bob’s observation is not influenced by Alice’s. Consequently, our classical entanglement example is consistent with our grasp of the physical world.

## Spin Correlation and Entanglement

Entanglement is a way to describe particles which are associated with strange, non-classical correlation, regardless of how apart they are. In other words, it is simply quantum mechanical generalization of correlation. At this point, instead of balls, a pair of spin states can be used as quantum systems in order to describe entanglement. Now, examine the decay of neutral pi meson as an example of a maximally entangled quantum state, which is also known as a simplified version of the EPR paradox.

π0 → e- + e+.

Electron and positron occupy the singlet spin configuration and it is given as,

According to this, if the electron is in a spin-up state, the positron must be spin-down, and vice versa. In other words, our measurements are correlated and independent of the distance between electron and positron. Therefore, if you measure one of the two entangled particles, you will instantly determine the outcome of the measurement of the other one.

One can say that it is just like our classical correlation example which contains black and white balls; however, in quantum reality, our quantum mechanical balls are not only black and white colored, they can also be red and green. This means that one can measure any component of the electron’s spin and any component of the positron’s spin.

For instance, if Alice measures the x-component of the electron’s spin and Bob measures the x-component of the positron’s spin, the outcome is perfectly correlated such as above example. Furthermore, if Alice measures the x-component of spin and Bon measures the y-component, in this case there is a random correlation between two measurements.

Consequently, the outcome of the positron’s spin measurement depends on which component of electron’s spin is measured, regardless of how far apart they are.  Einstein felt uncomfortable with this idea and claimed that the quantum mechanical description of physical world was incomplete. He described this strange effect of quantum mechanics as “spooky action at distance”.

The EPR paradox relies on the assumption that information cannot travel faster than speed of light and entanglement does not seem compatible with this assumption. Therefore, one may ask: does quantum mechanics violate the principle of locality? Some people think so. However, entanglement has received increasing attention over last decade and has proved to be great source for quantum computation.

• Griffiths, David J., Introduction to Quantum Mechanics
• Sakurai, J. J. and Napolitano Jim, Modern Quantum Mechanics
• Susskind, Leonard and Friedman, Art, Quantum Mechanics

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Written by

### Kazim Gokhan Atman

Kazim Gokhan Atman is a Ph.D. student in mathematical physics. He is mainly focused on fractional calculus and possible violation of locality at a small distance in Quantum Field Theory.

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