10
2 Elements of Quantum Information Theory
basis {|a i }
d
i=1 , called the eigenbasis of A. Then, the operator A can be written in its
spectral decomposition as follows
1 :
A =
d
i=1
a i |a i a i |,
(2.7)
where |a i a i | are rank-one projectors on the corresponding eigenvectors |a i .
We define the trace of an operator A on H as follows:
Tr[A] =
d
i=1
b i |A|b i ,
(2.8)
where {|b i }
d
i=1 is any orthonormal basis for H. We remark that the trace definition is
independent of the chosen basis thanks to the cyclic property of the trace Tr[AB] =
Tr[B A] and to the fact that a change of orthonormal basis is represented by a unitary
operator.
At last, an operator A is said to be positive (A ≥ 0) if its expectation value on
any vector is non-negative: v|A|v ≥ 0 for all |v ∈ H. Notably, an operator A is
positive if and only if it is Hermitian ( A = A
† ) with non-negative eigenvalues.
2.2 Density Operator Formalism
We have so far identified the state of a quantum system by its wave function |ψ,
implicitly assuming that it can be completely determined. However, from a practical
point of view, this is not always feasible. Consider, for instance, an electron-target
scattering experiment where the electron beam is prepared without the use of polarizers. The electron spin will probably be oriented in a random direction for each
electron of the beam. Thus, the spin of the beam cannot be described by a pure state
of the form:
|ψ = α| ↑↑ z + β| ↓↓ z ,
(2.9)
since the latter describes a spin oriented in a specific direction, fixed by the polar
angles θ = 2 arccos |α| and ϕ = arg β − arg α. Rather, the state of the beam spin
is described by an ensemble of spins oriented in all directions, weighted by their
probability of occurrence: a mixed state.
In cases like this, where the lack of information on an ensemble of single states
prevents us from describing them one by one completely, we can still study such a
collection of states statistically by means of the density operator, introduced by von
Neumann in 1927.
1 Note that the spectral theorem also holds for self-adjoint operators, but does not hold for merely
symmetric operators.
2 Elements of Quantum Information Theory
basis {|a i }
d
i=1 , called the eigenbasis of A. Then, the operator A can be written in its
spectral decomposition as follows
1 :
A =
d
i=1
a i |a i a i |,
(2.7)
where |a i a i | are rank-one projectors on the corresponding eigenvectors |a i .
We define the trace of an operator A on H as follows:
Tr[A] =
d
i=1
b i |A|b i ,
(2.8)
where {|b i }
d
i=1 is any orthonormal basis for H. We remark that the trace definition is
independent of the chosen basis thanks to the cyclic property of the trace Tr[AB] =
Tr[B A] and to the fact that a change of orthonormal basis is represented by a unitary
operator.
At last, an operator A is said to be positive (A ≥ 0) if its expectation value on
any vector is non-negative: v|A|v ≥ 0 for all |v ∈ H. Notably, an operator A is
positive if and only if it is Hermitian ( A = A
† ) with non-negative eigenvalues.
2.2 Density Operator Formalism
We have so far identified the state of a quantum system by its wave function |ψ,
implicitly assuming that it can be completely determined. However, from a practical
point of view, this is not always feasible. Consider, for instance, an electron-target
scattering experiment where the electron beam is prepared without the use of polarizers. The electron spin will probably be oriented in a random direction for each
electron of the beam. Thus, the spin of the beam cannot be described by a pure state
of the form:
|ψ = α| ↑↑ z + β| ↓↓ z ,
(2.9)
since the latter describes a spin oriented in a specific direction, fixed by the polar
angles θ = 2 arccos |α| and ϕ = arg β − arg α. Rather, the state of the beam spin
is described by an ensemble of spins oriented in all directions, weighted by their
probability of occurrence: a mixed state.
In cases like this, where the lack of information on an ensemble of single states
prevents us from describing them one by one completely, we can still study such a
collection of states statistically by means of the density operator, introduced by von
Neumann in 1927.
1 Note that the spectral theorem also holds for self-adjoint operators, but does not hold for merely
symmetric operators.