Bell Basis for Two Qubits
We come to the point of covering what is sensationally known as "teleportation". In the previous page, we have defined two body states | A > and | B > constructed from two arrows 1 and 2. we now add two more states | C > and | D > to them;
|
|A> = | u >1 | d >2, |B> = | d >1 | u >2, |C> = | u >1 | u >2, |D> = | d >1 | d >2
|
|
These four clearly exaust all possibilities of configuration of two arrows in Up-Down direction, and thus form a two arrow state basis. We can form an alternative basis set defined below, made of of entangled states using the combination of | A >, | B >, | C > and | D >;
|
|X+> = 1/\sqrt2| u >1 | d >2 + 1/\sqrt2| d >1 | u >2,
|X- > = 1/\sqrt2| u >1 | d >2 - 1/\sqrt2| d >1 | u >2,
|Y+> = 1/\sqrt2| u >1 | u >2 + 1/\sqrt2| d >1 | d >2,
|Y- > = 1/\sqrt2| u >1 | u >2 - 1/\sqrt2| d >1 | d >2.
|
|
The fact that these four states | X+ >, | X- >, | Y+ > and | Y- > form a babsis is proved by their mutual orthogonality. This entangled basis is celled Bell basis. It is noticed that | X+ > is identical to previously defined | S >. All the states in this basis is producible and detectable since that is a basic reqirement of a set being a basis.
Three Arrow Quantum State
Let us assume that Alice places a quantum arrow A in the angle \theta, namely in the state
|
| \theta >A = cos[\theta/2] | u >A + sin[\theta/2] | d >A
|
|
She wants to send this state to Bob who is located in some distance from her. This time she will have to get a collaboration of Eve. Eve prepares an entangled state of two arrows E and B in the form
|
| \Phi >EB = 1/\sqrt2 | u >E | d >B - 1/\sqrt2 | d >E | u >B
|
|
and send the arrow E to Alice, and the arrow B to Bob. We write the total quantum state of arrows A, E and B as | \Psi >AEB which is given by
|
| \Psi >AEB = | \theta >A | \Phi >EB
|
|
|
Go To:
ResearchPage