## Entanglement

The Bell states are the 4 most simple states that a 2 Qbit (Quantum bit) quantum system can be in and be considered entangled. For the uninitiated this means that the probability of a bit being in given state is related to the probability of another bit being in a given state. For example, you could know that both bits are the same, or both are different, but not know anything more until you measure one of them.

## Formal definition

The Bell states are:

- |Ψ
^{+}> = 1/√2 ( |01> + |10> )

- |Ψ
^{-}> = 1/√2 ( |01> - |10> ) - |Φ
^{+}> = 1/√2 ( |00> + |11> ) - |Φ
^{-}> = 1/√2 ( |00> - |11> )

The first two states describe the Qbits being in different, unknown states whereas the bottom two are when both Qbits are in the same unknown state.

## Creating the states

The states are very simple to create from two unentangled Qbits using the 1 bit Hadamard gate (which is actually a generalisation of the Fourier transform and the conditional NOT gate (apply NOT to bit 2 if bit 1 is 1).

Applying Hadamard to the first Qbit then CNOT<sub>12</sub> creates the state |Φ^{+}> above. This can then be transformed into any of the others by applying one of three 2 Qbit unitary transforms described by the tensor product of a Pauli matrix and the identity.

For example, the Pauli matrix X (which happens to be analgous to traditional, boolean NOT) is:

0 1

1 0

The tensor product of X and I gives:

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

The bell states I've given above are in Dirac's bra-ket notation, but they can equally written as column vectors, in a fashion that should be familiar to anyone that's used binary. The numbers inside the kets above indicate their position in the column, missing terms are 0. So, |Φ^{+}> is equivalent to:

1/√2

0

0

1/√2

If we multiply the 4x4 matrix above with the vector Φ^{+ }the result is

0

1/√2

1/√2

0

or, Ψ^{+}.

## Practical use

The Bell states are a fundamental part of quantum dense coding, whereby two people who each have a Qbit can follow a protocol to allow one to transmit two Cbits (classical bit) to the other party by transmitting their single Qbit.