Page 88 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 75

sender performs a joint (XOR) measurement between particles 1 and 2. As
we saw previously, the outcome of measuring a pair of single product states
of two qubits, such as that of particles 1 and 2, has four possible outcomes

( 0 0 ± 1 1 )
Ψ ± = 1 2 1 2 , (92)
12 2

and

( 0 1 ± 1 0 )
Φ ± = 1 2 1 2 . (93)
12 2

Taking this into account, the direct product state ψ may be expanded in
123
terms of these four outcomes and rewritten as,

( 000a + a 011 + b 100 + b 111 )
ψ = 123 123 123 123
123
2
= 1 Φ + ( 0 + b 1 )+ 1 Φ −
a
2 12 3 3 2 12 . (94)
⋅ ( a− 0 + b 1 )+ 1 Ψ + ( 0 + a 1 )
b
3 3 2 12 3 3
+ 1 Ψ − ( b− 0 + a 1 )
2 12 3 3

The result of performing the XOR between particles 1 and 2 will be the
collapse or projection of the global tensor product state ψ along one of
123
the four vector states Ψ ± and Φ ± with equal probability, namely, ¼.
12 12
Notice that this will leave a new global state consisting of the tensor product
of one of the vectors Ψ ± and Φ ± , at the sender, and a modified
12 12
qubit 3, at the receiver. One possible result might be,

Ψ + ( 0b + a 1 ). (95)
12 3 3

If these were the case then, to complete the teleportation process the sender
has to communicate to the receiver, using classical message, that the global