2.2 Ensemble of Quantum States
19
Assume that a measurement described by measurement operators M m is done.
Thus, if the initial state is |φ i , the probability of having m is given by
p(m|i) = =φ i |M
†
m M m |φ i = Tr(M
†
m M m |φ i φ i |).
The density operator after obtaining the result m is
ρ m =
i
p(i|m)|φ
m
i φ
m
i | =
i
p(i|m)
M m |φ i φ i |
φ i |M
†
m M m |φ i
=
M m ρ M
†
m
Tr(M
†
m M m ρ)
,
where
|φ
m
i =
M m |φ i
φ i |M
†
m M m |φ i
is the state after getting the result m, generating therefore an ensemble of states |φ
m
i
with respective probabilities p(m|i).
Definition 2.2.2 A quantum system whose state |φ is known exactly is said to
be in a pure state; the density operator in this case is given simply by ρ = |φφ|.
Otherwise, we say that ρ is in a mixed state.
At this point, we are able to describe the postulates of quantum mechanics based
on the formalism of density operator.
Postulate 2.2.1 Associated with any isolated physical system is a complex Hilbert
space known as the state space of the system. The system is completely described
by its density operator, which is a positive operator with trace one, acting on the
state space of the system. If the system is in the state ρ i with probability p i , then the
density operator for the system is given by
i
p i ρ i .
Postulate 2.2.2 The evolution of a closed quantum system is described by means of
a unitary transformation: the state ρ at time t 1 is related to σ at time t 2 by a unitary
operator U ,
σ = UρU
†
.
The operator U depends only on the times t 1 and t 2 .
Postulate 2.2.3 The quantum measurements are described by a collection {M m },
where the M m ’s are called measurements operators. The operators M m ’s act on the
state space of the system being measured. The index m refers to the measurement
outcomes possible to occur. If the state of the system is in ρ, then the probability of
occurrence of m is equal to
19
Assume that a measurement described by measurement operators M m is done.
Thus, if the initial state is |φ i , the probability of having m is given by
p(m|i) = =φ i |M
†
m M m |φ i = Tr(M
†
m M m |φ i φ i |).
The density operator after obtaining the result m is
ρ m =
i
p(i|m)|φ
m
i φ
m
i | =
i
p(i|m)
M m |φ i φ i |
φ i |M
†
m M m |φ i
=
M m ρ M
†
m
Tr(M
†
m M m ρ)
,
where
|φ
m
i =
M m |φ i
φ i |M
†
m M m |φ i
is the state after getting the result m, generating therefore an ensemble of states |φ
m
i
with respective probabilities p(m|i).
Definition 2.2.2 A quantum system whose state |φ is known exactly is said to
be in a pure state; the density operator in this case is given simply by ρ = |φφ|.
Otherwise, we say that ρ is in a mixed state.
At this point, we are able to describe the postulates of quantum mechanics based
on the formalism of density operator.
Postulate 2.2.1 Associated with any isolated physical system is a complex Hilbert
space known as the state space of the system. The system is completely described
by its density operator, which is a positive operator with trace one, acting on the
state space of the system. If the system is in the state ρ i with probability p i , then the
density operator for the system is given by
i
p i ρ i .
Postulate 2.2.2 The evolution of a closed quantum system is described by means of
a unitary transformation: the state ρ at time t 1 is related to σ at time t 2 by a unitary
operator U ,
σ = UρU
†
.
The operator U depends only on the times t 1 and t 2 .
Postulate 2.2.3 The quantum measurements are described by a collection {M m },
where the M m ’s are called measurements operators. The operators M m ’s act on the
state space of the system being measured. The index m refers to the measurement
outcomes possible to occur. If the state of the system is in ρ, then the probability of
occurrence of m is equal to