Bell state

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 CNOT₁₂ 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.