12
2 Elements of Quantum Information Theory
Postulate 2.4 The state space H of a composite quantum system, with subsystems
numbered from 1 to N , is given by the tensor product of the state spaces H i composing
the system: H = H 1 ⊗ H 2 ⊗ · · · ⊗ H N .
Thanks to Postulates 2.1 and 2.4, we can describe the state of a composite system
commonly encountered in quantum information. Consider a quantum system Q
whose state depends on the value x of a classical random variable X , with probability
distribution Pr(x). For an observer who ignores the value of X , the global state of
the quantum system and of the classical variable is given by:
ρ X Q =
x
Pr(x) |xx | X ⊗ ρ
x
Q ,
(2.14)
where the random variable X is represented by orthogonal pure states |x, since its
classical outcomes can be perfectly distinguished. The quantum system is instead
found in one of the conditional states ρ
x
Q . Moreover, we say that ρ X Q is classical on
X or is a classical-quantum (c.q) state if it can be written in the form (2.14).
2.2.1 POVMs and Projective Measurements
Postulate 2.3 provides the most general description of a quantum measurement. There
are two special cases of quantum measurements which are of particular interest in
quantum information. The first one is the positive operator-valued measure (POVM),
which simplifies the formalism when only the measurement statistics matters.
Definition 2.2 (POVM) A POVM is defined by a set of positive operators {E m },
the POVM elements, acting on the state space, such that
m E m = 1. Then the
probability of obtaining outcome m when measuring the system in state ρ is given
by:
Pr(m) = Tr[E m ρ].
(2.15)
One can readily see that POVMs are a special case of Postulate 2.3, when the measurement operators are given by M m =
√
E m , which implies M
†
m M m = E m .
The only case in which the measurement operators and the POVM elements
coincide is for projective measurements, i.e. when they are orthogonal projectors:
E m = M m = P m . One can verify that a set of orthogonal projectors {P m }, satisfying
the completeness relation:
m P m = 1, is composed of mutually orthogonal projectors: P m P n = δ m,n P n [5]. From the measurement outcomes and the projectors it is
possible to define an Hermitian operator M, called observable, through its spectral
decomposition:
M =
m
m P m .
(2.16)
2 Elements of Quantum Information Theory
Postulate 2.4 The state space H of a composite quantum system, with subsystems
numbered from 1 to N , is given by the tensor product of the state spaces H i composing
the system: H = H 1 ⊗ H 2 ⊗ · · · ⊗ H N .
Thanks to Postulates 2.1 and 2.4, we can describe the state of a composite system
commonly encountered in quantum information. Consider a quantum system Q
whose state depends on the value x of a classical random variable X , with probability
distribution Pr(x). For an observer who ignores the value of X , the global state of
the quantum system and of the classical variable is given by:
ρ X Q =
x
Pr(x) |xx | X ⊗ ρ
x
Q ,
(2.14)
where the random variable X is represented by orthogonal pure states |x, since its
classical outcomes can be perfectly distinguished. The quantum system is instead
found in one of the conditional states ρ
x
Q . Moreover, we say that ρ X Q is classical on
X or is a classical-quantum (c.q) state if it can be written in the form (2.14).
2.2.1 POVMs and Projective Measurements
Postulate 2.3 provides the most general description of a quantum measurement. There
are two special cases of quantum measurements which are of particular interest in
quantum information. The first one is the positive operator-valued measure (POVM),
which simplifies the formalism when only the measurement statistics matters.
Definition 2.2 (POVM) A POVM is defined by a set of positive operators {E m },
the POVM elements, acting on the state space, such that
m E m = 1. Then the
probability of obtaining outcome m when measuring the system in state ρ is given
by:
Pr(m) = Tr[E m ρ].
(2.15)
One can readily see that POVMs are a special case of Postulate 2.3, when the measurement operators are given by M m =
√
E m , which implies M
†
m M m = E m .
The only case in which the measurement operators and the POVM elements
coincide is for projective measurements, i.e. when they are orthogonal projectors:
E m = M m = P m . One can verify that a set of orthogonal projectors {P m }, satisfying
the completeness relation:
m P m = 1, is composed of mutually orthogonal projectors: P m P n = δ m,n P n [5]. From the measurement outcomes and the projectors it is
possible to define an Hermitian operator M, called observable, through its spectral
decomposition:
M =
m
m P m .
(2.16)