Page 86 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 73

The control-NOT (CNOT) gate, as can be seen from Figure 2-12(a),
implements the exclusive-OR (XOR) operation. Thus, the gate inverts y ,

if x = 1, and leaves it as is if x = 0 . This operation is expressed as,

C x y = x (x + y )mod 2 . (85)
12
Applied to a pair of single product states of two qubits, the CNOT gate
produces a set of entangled qubits, i.e.,

C ( 0 + 1 )0 = ( 0 0 + 1 1 ). (86)
12 1 1 2 1 2 1 2

Similarly, since the CNOT gate is reversible, when applied to an entangled
state, it produces a set of disentangled states, i.e.,

C 12 ( 0 1 0 ± 1 1 1 2 ) ( 0 ±= 1 1 1 )0 , (87)
2
2
and

C ( 0 1 ± 1 0 ) ( 0 ±= 1 )1 . (88)
12 1 2 1 2 1 1 2
These operations are essential for quantum teleportation.
One may recall that a classical NOT gate is called universal in the sense
that any other logic gate may be created by combining several NOT gates.
Similarly, a universal quantum gate should generate all unitary
transformations of n qubits. It can be shown that such a gate is realized by
combining a pair of gates, namely, one that produces a general rotation on a
single bit, U ( φθ, ), where,
Universal

ª cos ( 2θ ) −ie − φ i sin ( 2θ )º
U ( ,φθ ) = « » , (89)
Universal φ i ( 2θ ) ( 2θ )
¬ −ie sin cos ¼

and a CNOT gate [100].

2.4.2 Quantum Teleportation

According to Bennett et al. [106], quantum teleportation is “a process that
disembodies the exact quantum state of a particle into classical data and EPR
correlations, and then uses these ingredients to reincarnate the state in