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if it is non-decomposable.
The concept is easy to illustrate in the case of a two qubit system. The
computational basis of such a system is
00 , 01 , 10 , 11 .
| i | i | i | i
Consider then the state
α00 +β 11 . (6.41)
| i | i
By studying the coefficients in the expansion of the product of two qubits
α 0 +β 1 α 0 +β 1 =
1 1 2 2
| i | i | i | i
(cid:0) (cid:1)(cid:0) (cid:1)
α α 00 +α β 01 +β α 10 +β β 11 (6.42)
1 2 1 2 1 2 1 2
| i | i | i | i
it is clear that there is no way to choose the coefficients α ,α ,β and β so
1 2 1 2
that α α = 0, α β = 0, β α = 0 and β β = 0 simultaneously in order to
1 2 1 2 1 2 1 2
6 6
reproduce the entangled state (6.41) as a product of single qubit states.
Theentangledstate(6.41)canbeproducedbytheapplicationoftheCNOT-
gate to a product state. The following figure illustrates an example.
|0>+|1>
|01>+|01>
|1>
Figure 6.12: Entangling a two-qubit product state using CNOT.
Working with un-normalized states for simplicity, we have the incoming
product state
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0 0 0
1 0 1
0 + 1 1 = 01 + 11 = + = .
| i | i | i | i | i 0 0 0
(cid:0) (cid:1) 0 1 1
     
Applying the CNOT gate yields
1 0 0 0 0 0 0 0
0 1 0 0 1 1 1 0
  = = + = 01 + 10 .
0 0 0 1 0 1 0 1 | i | i
0 0 1 01 0 0 0
        
ItisinterestingandimportanttonotethattheCNOTgatecannotbewritten
asa -productoftwosinglequbitgates. Thisisclearfromstudyingthematrix
⊗
elements of the product A B in equation (6.37). In fact, there is no way to
⊗
produce entanglement using only single qubit gates. This observation will be
put in context in section X.X.X on universal quantum gates.
Mathematically, entanglement is quite trivial, but the concept is far from
trivial from a physicalpoint of view, and has been a subject of discussionsince
the mid nineteen thirties. We will return briefly to this discussion in the last
chapter.
6.2.10 Some important gate constructions
We need to able to build complicated gates out of simpler ones. These simple
gatesare,apartfromthespecialdiscreteone-qubitgates,alsogeneralone-qubit
gates U and CNOT gates. We will call these basic or elementary operations.
Here follows a few useful constructions.
Decomposition of a single qubit gate into Z and Y rotations
Every unitary 2 2 matrix U can be expressed as
×
U =eiαR (β)R (γ)R (δ), (6.43)
z y z
in terms of the rotation operators 6.19 and 6.20, and where the parameters
α,β,γ and δ are real.
Proof
First note that the unitarity constraint UU = 1 on a general 2 2 complex
†
×
matrix U reduces the number of real parameters from 8 to 4. Furthermore, a
matrixisunitaryifandonlyifits rowvectorsandcolumnvectorsareorthonor-
mal. Therefore, every unitary 2 2 matrix can be expressed in terms of four
×
real parameters α,β,γ and δ as
ei(α β/2 δ/2)cosγ ei(α β/2+δ/2)sinγ
− − 2 − − 2 . (6.44)
(cid:18)ei(α+β/2 δ/2)sinγ ei(α+β/2+δ/2)cosγ (cid:19)
− 2 2
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For example, the column vectors are orthogonal,since
γ γ
ei(α β/2 δ/2)cos ) ei(α β/2+δ/2)sin +
− − ∗ −
2 − 2
(cid:0) (cid:0) (cid:1)
γ γ
ei(α+β/2 δ/2)sin ) (ei(α+β/2+δ/2)cos )=
− ∗
2 2
(cid:0)
γ γ γ γ
eδcos sin +eδsin cos =0.
− 2 2 2 2
The rest of the conditions on the matrix 6.44 can be checked similarly.
Multiplying out equation 6.43, yields exactly the matrix in 6.44.
Decomposition of a general Λ (U) gate
1
TheaboveresultallowsadecompositionofgeneralΛ (U)gateintermsofsingle