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2 A Little Bit of Quantum Mechanics
p(m) = Tr(M
†
m M m ρ),
and the state after the measurement is
σ =
M m ρ M
†
m
Tr(M
†
m M m ρ)
.
The operators M m ’s satisfy the completeness equation
m
M
†
m M m = I.
Postulate 2.2.4 Given any composite physical system, its state of space is the tensor
product of the state spaces of the component physical systems. Additionally, in the
case in which the systems are numbered 1 to n, and the system number i is prepared
in the state ρ i , then the join state is described by
ρ 1 ⊗ ρ 2 ⊗ . . . ⊗ ρ n .
2.3 Universal Quantum Gates
In this subsection we recall some important quantum gates, essential for quantum
mechanics. We assume the reader is familiar with theory of quantum gates (single
and multiple qubit quantum gates). For more details with respect to this topic, we
suggest the textbook [121].
2.3.1 Single Qubit Gates
The single qubit gates, as the own name says, are gates acting on one qubit state. The
quantum NOT gate X can be represented in terms of matrices as
X ≡
0 1
1 0
.
Hence, if we have a generic quantum state |v = α|0 + β|1 represented as a column
matrix
|v =
α
β
,