Quantum Computation and Communication with Qubit Systems

10 1. Basics of quantum computation and communication
1.4.1 Multiple spin systems
The description of two or more systems requires a formalism called tensor product or
more precisely Kronecker product. In quantum mechanics the Kronecker product is
useful to describe systems composed by multiple uncorrelated particles. The Hilbert
space for two qubits is then the space obtained by multiplying two Hilbert spaces, each
correspondingtoasinglequbit,followingtherulesoftheKroneckerproductdescribedin
theappendix. Inmathematical termswedenotethisnewspacewiththenotationC
2
⊗C
2
,where⊗isthesymbolcommonlyusedfortheKroneckerproduct. Inparticular,C
2
⊗C
2
is a four-dimensional space spannedby the vectors{|0⊗| 0 ,|0⊗| 1 ,|1⊗| 0 ,|1⊗| 1} .
In the same way it is possible to define an operator on multiple particles just taking
kronecker productof the matrices representing the operator in the chosen Hilbert space.
Usually asubscriptis usedto denote theparticle theoperator isacting on. For example,
σ
1
x
⊗σ
2
z
is a tensor product operator withσ
x
applied to the first particle andσ
z
applied
to the second particle. When this operator acts on a tensor product vector such as
|ψ
1
⊗|φ
2
, the operator acts on the corresponding particles 1 and 2, respectively:
(σ
1
x
⊗σ
2
z
)(|ψ
1
⊗|φ
2
)=(σ
1
x
|ψ
1
)⊗(σ
2
z
|φ
2
) (1.19)
Ingeneral, a shortcutfortheabove mentioned notation isusedtoindicate Kronecker
product of two or more pure states:
|0⊗| 0...|0=|00...0 (1.20)
Wewillseelaterthatthisoperationisnotpossiblewhentheparticlesarestatistically
correlated, or entangled. Entangled particle therefore cannot be written as Kronecker
product of two independent particles.
1.5 Tools
Here we introduce the mathematic tools used in quantum information to describe and
modify the state of a quantum system. In the previous section we have introduced the
Pauli matrices, used to measure the spin along the axes x,y and z. In the following
we will use them to do more complex operations like rotations and to introduce logic
gates, then we will see how to characterize the uncertainty about the state of a quantum
system, using the so called density matrices.
1.5.1 Rotations in the Bloch Sphere
Inquantummechanics, thepurestate spaceofatwo-level quantummechanicalsystemis
geometrically represented by a sphere, as sketched in Fig.1.21 named Bloch sphere after
the physicist Felix Bloch. In this geometrical construction the correspondence between