Bell state

The Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. They are named after John S. Bell, as they are the subject of his famous Bell inequality. A Bell pair is a pair of qubits which jointly are in a Bell state, that is, entangled with each other. Unlike classical phenomena such as the nuclear, electromagnetic, and gravitational fields, entanglement is invariant under distance of separation and is not subject to relativistic limitations such as speed of light.

The Bell states

A Bell state is defined as a maximally entangled quantum state of two qubits. The qubits are usually thought to be spatially separated (held by Alice and Bob, respectively, to use quantum cryptography terms). Nevertheless they exhibit perfect correlations which cannot be explained without quantum mechanics.

To explain, let us first look at the Bell state $|Phi^+\; angle$:

$|Phi^+\; angle\; =\; frac\{1\}\{sqrt\{2\; (|0\; angle\_A\; otimes\; |0\; angle\_B\; +\; |1\; angle\_A\; otimes\; |1\; angle\_B)$

This expression means the following: The qubit held by Alice (subscript "A") can be 0 as well as 1. If Alice measured her qubit the outcome would be perfectly random, either possibility having probability 1/2. But if Bob then measured his qubit, the outcome would be the same as the one Alice got. So, if Bob measured, he would also get a random outcome on first sight, but if Alice and Bob communicated they would find out that, although the outcomes seemed random, they are correlated.

So far, this is nothing special: Maybe the two particles "agreed" in advance, when the pair was created (before the qubits were separated), which outcome they would show in case of a measurement.

Hence, followed Einstein, Podolsky, and Rosen in 1935 in their famous "EPR paper", there is something missing in the description of the qubit pair given above — namely this "agreement", called more formally a hidden variable.

But quantum mechanics allows qubits to be in quantum superposition — i.e. in 0 and 1 simultaneously, e.g. in either of the states $|+\; angle\; =\; frac\{1\}\{sqrt\{2(|0\; angle\; +\; |1\; angle)$ or $|-\; angle\; =\; frac\{1\}\{sqrt\{2(|0\; angle\; -\; |1\; angle)$. If Alice and Bob chose to measure in this basis, i.e. check whether their qubit were $|+\; angle$ or $|-\; angle$, they will find the same correlations as above. That is because the Bell state can be formally rewritten as follows:

$|Phi^+\; angle\; =\; frac\{1\}\{sqrt\{2\; (|+\; angle\_A\; otimes\; |+\; angle\_B\; +\; |-\; angle\_A\; otimes\; |-\; angle\_B)$

Note that this is still the "same" state.

John S. Bell showed in his famous paper of 1964 by using simple probability theory arguments that these correlations cannot be perfect in case of "pre-agreement" stored in some hidden variables — but that quantum mechanics predict perfect correlations. In a more formal and refined formulation known as the Bell-CHSH inequality, this would be stated such that a certain correlation measure cannot exceed the value 2 according to reasoning assuming local "hidden variable" theory (sort of common-sense) physics, but quantum mechanics predicts $2sqrt\{2\}$.

There are three other states of two qubits which lead to this maximal value of $2sqrt\{2\}$ and the four are known as the four "maximally entangled two-qubit states" or "Bell states":

$|Phi^+\; angle\; =\; frac\{1\}\{sqrt\{2\; (|0\; angle\_A\; otimes\; |0\; angle\_B\; +\; |1\; angle\_A\; otimes\; |1\; angle\_B)$

$|Phi^-\; angle\; =\; frac\{1\}\{sqrt\{2\; (|0\; angle\_A\; otimes\; |0\; angle\_B\; -\; |1\; angle\_A\; otimes\; |1\; angle\_B)$

$|Psi^+\; angle\; =\; frac\{1\}\{sqrt\{2\; (|0\; angle\_A\; otimes\; |1\; angle\_B\; +\; |1\; angle\_A\; otimes\; |0\; angle\_B)$

$|Psi^-\; angle\; =\; frac\{1\}\{sqrt\{2\; (|0\; angle\_A\; otimes\; |1\; angle\_B\; -\; |1\; angle\_A\; otimes\; |0\; angle\_B)$

Bell state measurement

The Bell measurement is an important concept in quantum information science: It is a joint quantum-mechanical measurement of two qubits that determines in which of the four Bell states the two qubits are in.

If the qubits were not in a Bell state before, they get projected into a Bell state (according to the projection rule of quantum measurements), and as Bell states are entangled, a Bell measurement is an entangling operation.

It is the crucial part of quantum teleportation (see there).

In current experiments which use photons as qubits, Bell measurements can only be partly realized as two of the Bell states cannot be distinguished with optical techniques. In Bell measurements of ion qubits in ion trap experiments, the distinction of all four states is possible.