2.2 Density Operator Formalism
13
Often, a projective measurement is equivalently defined as the measurement of an
observable M, meaning that the projectors are those appearing in its spectral decomposition (2.16). If the projectors are all rank-one P m = |mm |, the measurement is
called a von Neumann measurement.
Identifying a projective measurement with the observable M is useful when, for
instance, one wants to compute the average outcome, since it can be directly written
in terms of the observable M:
M :=
m
mPr(m) =
m
m Tr[P m ρ] = Tr[Mρ].
(2.17)
Finally we remark that, although projective measurements are particular cases
of POVMs, the statistics of any POVM on a d-dimensional Hilbert space can be
reproduced by combining a projective measurement on a Hilbert space of dimension
d
≥ d with a unitary operation. This result is known as the Naimark theorem [6, 7].
2.3 Qubits and Pauli Operators
In many quantum information applications, the fundamental quantum system is a
two-level system called quantum bit or qubit. Physical realizations of qubits are, for
example: a photon that can be found in one of two distinct paths, two orthogonal
polarizations of a photon, the spin state of spin1
2
particles, or the two lowest energy
levels of an electron orbiting a nucleus.
The state space of a qubit is a two-dimensional Hilbert space, H 2 . The commonly used basis for H 2 is the computational basis {|0, |1}. Thus, any pure
qubit state is represented by a superposition of the form: |ψ = α|0 + β|1, where
|α|
2
+ |β|
2
= 1.
Conversely, any mixed qubit state is represented by a density operator ρ acting
on H 2 and can be expressed as a combination of the identity operator 1 and the Pauli
operators σ x , σ y and σ z [8]:
ρ =
1 + r · σ
2
, r ∈ R
3
: r ≤ 1,
(2.18)
where σ = (σ x , σ y , σ z )
T . The matrix representation of the Pauli operators with
respect to the computational basis reads:
σ x =
0 1
1 0
; σ y =
0 −i
i 0
; σ z =
1 0
0 −1
(2.19)
Note that, depending on the context, we also indicate the Pauli operators as σ x =
σ 1 = X , σ y = σ 2 = Y and σ z = σ 3 = Z . The Pauli operators are Hermitian with
eigenvalues ±1, traceless, and satisfy the following relation:
13
Often, a projective measurement is equivalently defined as the measurement of an
observable M, meaning that the projectors are those appearing in its spectral decomposition (2.16). If the projectors are all rank-one P m = |mm |, the measurement is
called a von Neumann measurement.
Identifying a projective measurement with the observable M is useful when, for
instance, one wants to compute the average outcome, since it can be directly written
in terms of the observable M:
M :=
m
mPr(m) =
m
m Tr[P m ρ] = Tr[Mρ].
(2.17)
Finally we remark that, although projective measurements are particular cases
of POVMs, the statistics of any POVM on a d-dimensional Hilbert space can be
reproduced by combining a projective measurement on a Hilbert space of dimension
d
≥ d with a unitary operation. This result is known as the Naimark theorem [6, 7].
2.3 Qubits and Pauli Operators
In many quantum information applications, the fundamental quantum system is a
two-level system called quantum bit or qubit. Physical realizations of qubits are, for
example: a photon that can be found in one of two distinct paths, two orthogonal
polarizations of a photon, the spin state of spin1
2
particles, or the two lowest energy
levels of an electron orbiting a nucleus.
The state space of a qubit is a two-dimensional Hilbert space, H 2 . The commonly used basis for H 2 is the computational basis {|0, |1}. Thus, any pure
qubit state is represented by a superposition of the form: |ψ = α|0 + β|1, where
|α|
2
+ |β|
2
= 1.
Conversely, any mixed qubit state is represented by a density operator ρ acting
on H 2 and can be expressed as a combination of the identity operator 1 and the Pauli
operators σ x , σ y and σ z [8]:
ρ =
1 + r · σ
2
, r ∈ R
3
: r ≤ 1,
(2.18)
where σ = (σ x , σ y , σ z )
T . The matrix representation of the Pauli operators with
respect to the computational basis reads:
σ x =
0 1
1 0
; σ y =
0 −i
i 0
; σ z =
1 0
0 −1
(2.19)
Note that, depending on the context, we also indicate the Pauli operators as σ x =
σ 1 = X , σ y = σ 2 = Y and σ z = σ 3 = Z . The Pauli operators are Hermitian with
eigenvalues ±1, traceless, and satisfy the following relation: