14
2 Elements of Quantum Information Theory
σ i σ j = δ i, j 1 +
3
k=1
ε i jk σ k ,
(2.20)
where ε i jk is the Levi-Civita symbol, which is equal to +1 (or −1) if the triple (i, j, k)
is a cyclic (or anti-cyclic) permutation of (1, 2, 3), and zero if any two indices are
repeated.
The representation (2.18) of a qubit state, together with (2.20), is particularly
useful in many computations. For instance, the purity of a qubit state can be readily computed as: Tr[ρ
2
] = (1 + r
2
)/2. Thus, the norm of the vector r indicates
whether the state is pure (r = 1) or mixed (r < 1). When r = 0, the state is
said to be maximally mixed.
Moreover, the vector r individuates a point inside a unit sphere, called the Bloch
sphere, where the three Cartesian coordinates are associated with the eigenstates of
the Pauli operators. Often, in the quantum information jargon one can measure “in
the z direction of the Bloch sphere”, meaning that one is performing a projective
measurement in the eigenbasis of σ z , which is conventionally associated with the
computational basis.
Finally, the Pauli operators have a prominent role in quantum error correction,
since they represent all the possible errors that can occur when processing a qubit. In
particular, σ x produces bit flips, σ z yields phase flips and σ y both phase and bit flips:
σ x |a = |¯ a
(2.21)
σ z |a = (−1)
a
|a
(2.22)
σ y |a = i(−1)
a
| ¯
a, a = 0, 1
(2.23)
where ¯
a = 1 − a.
2.4 Composite Systems and Entanglement
Postulate 2.4 allows us to introduce one of the most astonishing features of quantum mechanics, entanglement, which plays a crucial role in quantum information
protocols.
Suppose that two parties, Alice and Bob, locally prepare their own quantum system
in the pure states |ψ A and |ψ B , respectively. Then, by Postulate 2.4, the state of
the composite quantum system is pure: | = |ψ A ⊗ |ψ B and is called a product
state. Note that a compact notation for |ψ A ⊗ |ψ B is |ψ A , ψ B or |ψ A ψ B . If the
pure state of a composite system is not a product state, then it is called entangled.
More generally, Alice and Bob could agree on locally preparing the states |ψ
i
A
and |ψ
i
B according to the value of a shared random variable with distribution { p i }.
This task only requires local operations and classical communication (LOCC). In
this case, the state of the composite system is described by the mixed state:
2 Elements of Quantum Information Theory
σ i σ j = δ i, j 1 +
3
k=1
ε i jk σ k ,
(2.20)
where ε i jk is the Levi-Civita symbol, which is equal to +1 (or −1) if the triple (i, j, k)
is a cyclic (or anti-cyclic) permutation of (1, 2, 3), and zero if any two indices are
repeated.
The representation (2.18) of a qubit state, together with (2.20), is particularly
useful in many computations. For instance, the purity of a qubit state can be readily computed as: Tr[ρ
2
] = (1 + r
2
)/2. Thus, the norm of the vector r indicates
whether the state is pure (r = 1) or mixed (r < 1). When r = 0, the state is
said to be maximally mixed.
Moreover, the vector r individuates a point inside a unit sphere, called the Bloch
sphere, where the three Cartesian coordinates are associated with the eigenstates of
the Pauli operators. Often, in the quantum information jargon one can measure “in
the z direction of the Bloch sphere”, meaning that one is performing a projective
measurement in the eigenbasis of σ z , which is conventionally associated with the
computational basis.
Finally, the Pauli operators have a prominent role in quantum error correction,
since they represent all the possible errors that can occur when processing a qubit. In
particular, σ x produces bit flips, σ z yields phase flips and σ y both phase and bit flips:
σ x |a = |¯ a
(2.21)
σ z |a = (−1)
a
|a
(2.22)
σ y |a = i(−1)
a
| ¯
a, a = 0, 1
(2.23)
where ¯
a = 1 − a.
2.4 Composite Systems and Entanglement
Postulate 2.4 allows us to introduce one of the most astonishing features of quantum mechanics, entanglement, which plays a crucial role in quantum information
protocols.
Suppose that two parties, Alice and Bob, locally prepare their own quantum system
in the pure states |ψ A and |ψ B , respectively. Then, by Postulate 2.4, the state of
the composite quantum system is pure: | = |ψ A ⊗ |ψ B and is called a product
state. Note that a compact notation for |ψ A ⊗ |ψ B is |ψ A , ψ B or |ψ A ψ B . If the
pure state of a composite system is not a product state, then it is called entangled.
More generally, Alice and Bob could agree on locally preparing the states |ψ
i
A
and |ψ
i
B according to the value of a shared random variable with distribution { p i }.
This task only requires local operations and classical communication (LOCC). In
this case, the state of the composite system is described by the mixed state: