# How are EPR Pairs used in quantum computing?

## Context

Lately, I have been reading a scholarly paper entitled An Introduction to Quantum Computing for Non-Physicists which discusses the EPR Paradox.

The paper states that:

Einstein, Podolsky and Rosen proposed a gedanken experiment that uses entangled particles in a manner that seemed to violate fundamental principles of relativity.

It concludes that the paradox is resolved as the symmetry shown by changing observers indicates that they cannot use their EPR pair to communicate faster than the speed of light. However, the paper fails to adequately explain what an EPR pair is used for and does not even define the term. The best definition I could find is referenced to in the Wikipedia article, Bell state.

An EPR pair is a pair of qubits (or quantum bits) that are in a Bell state together.

This definition doesn't provide much detail beyond the basics of what an EPR pair is.

## The Question

How are EPR Pairs used in quantum computing?

EPR pairs are a particular case of entangled pairs of qubits.

From Wikipedia: "Quantum entanglement is a physical phenomenon which occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently of the state of the other." More to the point regarding your question, entanglement is a crucially important resource for quantum computing, see Bennet's laws on the inequivalences between bits, qubits and ebits (or entanglement bits).

Elementary cases of use of EPR pairs in quantum computing would be quantum teleportation and Superdense coding, and upon those pieces people have built more sophisticated applications. For the gory details of the original proposal for quantum teleportation, please check Bennet's Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen Channels (Phys.Rev.Lett., 1993, 70, 1895-1899).

For a more sophisticated application, please check this answer on the intuition behind the Choi-Jamiolkowski isomorphism, which also heavily relies on maximally entangled qubit pairs. In turn, the application of this isomorphism was suggested to me, here, a couple of days ago, as an answer to how to obtain information on what quantum logic a quantum black box is implementing.

• +1 for mentioning teleportation as an application. – M. Stern Apr 10 '18 at 20:27

If you were to try and imagine the simplest form of correlation, you might think of two bits that were randomly either both $0$, or both $1$.

Bell states are just this, but quantum. We have bits instead of qubits, and the randomness is due to superposition.

Since they are most conceptually simple form of entanglement, and the easiest to describe using information theoretic language, they are our first choice for anything entanglement related. If you need to explain non-locality, do it with a Bell pair. If you want to teleport something, it’s simplest both mathematically and practically if you use a Bell pair. If you want to measure how much entanglement you have why not see how many Bell pairs you could turn it into?

Typically, we use four canonical forms of the Bell pairs. They represent all the ways that we can choose to have correlations or anticorrelations between the $|0\rangle / |1\rangle$ states of the computational basis and the $|+\rangle / |-\rangle$ states of the $X$ basis.

These four states form a complete basis for two qubits. So any states, entangled or not, can be expressed as a superposition of Bell states. This can sometimes make our maths easier, which is another reason we like to use them.