2.2 Density Operator Formalism
11
Definition 2.1 (Density Operator) Consider a quantum system that is found in one
of the pure states {|ψ i } with probabilities p i < 1, where
i p i = 1. Then, the state
of the system is called a mixed state and is described by the density operator
ρ =
i
p i |ψ i ψ i |.
(2.10)
The matrix representation of ρ is called density matrix, which is often used to indicate
the operator itself.
If a quantum system is in a pure state |ψ with certainty, then its density operator
reads: ρ = |ψψ |. A criterion to determine whether a state ρ is pure or mixed is
given by the computation of its purity: Tr[ρ
2
]. A state is pure if Tr[ρ
2
] = 1, while
it’s mixed if Tr[ρ
2
] < 1.
Density operators offer an alternative formulation of quantum mechanics, which
is particularly useful in quantum information. Here we provide an intrinsic characterization of density operators, which allows us to abandon their interpretation in
terms of an ensemble of pure states.
Theorem 2.1 (Characterization of density operators) An operator ρ is the density
operator of a mixed state ρ =
i p i |ψ i ψ i | if and only if it is normalized (Tr[ρ] =
1) and positive.
The proof of this Theorem can be found in [2].
We can now reformulate the postulates of quantum mechanics in the density
operator picture.
Postulate 2.1 The state of a quantum system is completely determined by a normalized positive operator, denoted density operator, acting on a Hilbert space H named
the state space of the system.
Postulate 2.2 The evolution of a closed quantum system is determined by a unitary
transformation U . Specifically, the evolved state of the system ρ
is obtained from
the initial state ρ as follows:
ρ
= UρU
†
.
(2.11)
Postulate 2.3 The measurement of a quantum system is defined by a collection of
measurement operators {M m } acting on H and satisfying the completeness relation:
m M
†
m M m = 1. If ρ is the state of the system prior to measurement, the probability
of observing the measurement outcome m is given by:
Pr(m) = Tr[M
†
m M m ρ]
(2.12)
and the state of the system after the measurement reads
ρ m =
M m ρ M
†
m
Tr[M
†
m M m ρ]
.
(2.13)
11
Definition 2.1 (Density Operator) Consider a quantum system that is found in one
of the pure states {|ψ i } with probabilities p i < 1, where
i p i = 1. Then, the state
of the system is called a mixed state and is described by the density operator
ρ =
i
p i |ψ i ψ i |.
(2.10)
The matrix representation of ρ is called density matrix, which is often used to indicate
the operator itself.
If a quantum system is in a pure state |ψ with certainty, then its density operator
reads: ρ = |ψψ |. A criterion to determine whether a state ρ is pure or mixed is
given by the computation of its purity: Tr[ρ
2
]. A state is pure if Tr[ρ
2
] = 1, while
it’s mixed if Tr[ρ
2
] < 1.
Density operators offer an alternative formulation of quantum mechanics, which
is particularly useful in quantum information. Here we provide an intrinsic characterization of density operators, which allows us to abandon their interpretation in
terms of an ensemble of pure states.
Theorem 2.1 (Characterization of density operators) An operator ρ is the density
operator of a mixed state ρ =
i p i |ψ i ψ i | if and only if it is normalized (Tr[ρ] =
1) and positive.
The proof of this Theorem can be found in [2].
We can now reformulate the postulates of quantum mechanics in the density
operator picture.
Postulate 2.1 The state of a quantum system is completely determined by a normalized positive operator, denoted density operator, acting on a Hilbert space H named
the state space of the system.
Postulate 2.2 The evolution of a closed quantum system is determined by a unitary
transformation U . Specifically, the evolved state of the system ρ
is obtained from
the initial state ρ as follows:
ρ
= UρU
†
.
(2.11)
Postulate 2.3 The measurement of a quantum system is defined by a collection of
measurement operators {M m } acting on H and satisfying the completeness relation:
m M
†
m M m = 1. If ρ is the state of the system prior to measurement, the probability
of observing the measurement outcome m is given by:
Pr(m) = Tr[M
†
m M m ρ]
(2.12)
and the state of the system after the measurement reads
ρ m =
M m ρ M
†
m
Tr[M
†
m M m ρ]
.
(2.13)