8
2 Elements of Quantum Information Theory
linear combination of vectors α|a + β|b is defined as α
∗
a| + β
∗
b|, where α
∗ is
the complex conjugate of α ∈ C.
The action of a linear map φ| ∈ H
∗ on a vector |ψ ∈ H is written as a “bra-ket”:
|ψ → →φ|ψ ∈ C and defines the inner product of vectors |ψ and |φ in H. For the
definition of dual vector, it follows that: φ|ψ
∗
= =ψ|φ. Moreover, given a linear
operator A, the quantity φ|A|ψ ∈ C can be interpreted as the result of the inner
product between vectors A|ψ and |φ in H.
Two vectors are said to be orthogonal if their inner product is zero. The norm
induced by the inner product is given by: |ψ =
√
ψ|ψ. A vector |ψ is said to be
normalized, or called a unit vector, if |ψ = 1. An orthonormal set of vectors {|ψ i }
is exclusively composed of normalized and mutually orthogonal vectors: ψ i |ψ j =
δ i, j , where δ i, j is the Kronecker delta.
The Dirac notation provides a useful way to represent the action of linear operators
on H, through the outer product. The outer product of |ψ ∈ H and φ| ∈ H
∗ is
represented by |ψφ | and acts as follows on |γ ∈ H: |γ → →φ|γ |ψ. The outer
product of a vector |ψ by itself defines a linear operator that projects a vector
|φ ∈ H in the one-dimensional subspace spanned by |ψ: |ψψ ||φ = =ψ|φ|ψ.
From this definition, it immediately follows that any orthonormal basis {|b i }
d
i=1
of the d-dimensional Hilbert space H satisfies the completeness relation:
d
i=1 |b i b i | = 1, where 1 is the identity operator. With the completeness relation, it is possible to represent the action of any linear operator A in the outer product
notation:
A = 1 A 1 =
d
i, j=1
b i |A|b j |b i b j |,
(2.1)
where the element b i |A|b j can be regarded as the matrix entry in the i-th row and
j-th column of the matrix representation of A with respect to the basis {|b i }
d
i=1 .
Let A be a linear operator with domain D(A). We define the adjoint operator A
†
of A as that operator with domain D(A
†
) such that:
(ψ|A
†
|φ)
∗
= =φ|A|ψ ∀|ψ ∈ D(A), ∀ |φ ∈ D(A
†
).
(2.2)
This implies that the matrix representing A
† is obtained from that of A by applying
transposition and complex conjugation. It also follows that (|ψφ |)
†
= |φψ | and
that the dual vector of A|ψ is ψ|A
† .
The evolution of a closed quantum system is determined by a unitary operator U ,
that is an operator for which U
†
= U
−1 , where U
−1 is the inverse of U . A unitary
transformation also links any two bases {b i }
d
i=1 and {b
i }
d
i=1 in H: |b
i = U |b i . Two
bases are called mutually unbiased if b
i |b j = 1/d for every i and j.
The observable quantities in quantum mechanics are represented by self-adjoint
operators. Let us take a moment to distinguish symmetric, self-adjoint and Hermitian
operators.
2 Elements of Quantum Information Theory
linear combination of vectors α|a + β|b is defined as α
∗
a| + β
∗
b|, where α
∗ is
the complex conjugate of α ∈ C.
The action of a linear map φ| ∈ H
∗ on a vector |ψ ∈ H is written as a “bra-ket”:
|ψ → →φ|ψ ∈ C and defines the inner product of vectors |ψ and |φ in H. For the
definition of dual vector, it follows that: φ|ψ
∗
= =ψ|φ. Moreover, given a linear
operator A, the quantity φ|A|ψ ∈ C can be interpreted as the result of the inner
product between vectors A|ψ and |φ in H.
Two vectors are said to be orthogonal if their inner product is zero. The norm
induced by the inner product is given by: |ψ =
√
ψ|ψ. A vector |ψ is said to be
normalized, or called a unit vector, if |ψ = 1. An orthonormal set of vectors {|ψ i }
is exclusively composed of normalized and mutually orthogonal vectors: ψ i |ψ j =
δ i, j , where δ i, j is the Kronecker delta.
The Dirac notation provides a useful way to represent the action of linear operators
on H, through the outer product. The outer product of |ψ ∈ H and φ| ∈ H
∗ is
represented by |ψφ | and acts as follows on |γ ∈ H: |γ → →φ|γ |ψ. The outer
product of a vector |ψ by itself defines a linear operator that projects a vector
|φ ∈ H in the one-dimensional subspace spanned by |ψ: |ψψ ||φ = =ψ|φ|ψ.
From this definition, it immediately follows that any orthonormal basis {|b i }
d
i=1
of the d-dimensional Hilbert space H satisfies the completeness relation:
d
i=1 |b i b i | = 1, where 1 is the identity operator. With the completeness relation, it is possible to represent the action of any linear operator A in the outer product
notation:
A = 1 A 1 =
d
i, j=1
b i |A|b j |b i b j |,
(2.1)
where the element b i |A|b j can be regarded as the matrix entry in the i-th row and
j-th column of the matrix representation of A with respect to the basis {|b i }
d
i=1 .
Let A be a linear operator with domain D(A). We define the adjoint operator A
†
of A as that operator with domain D(A
†
) such that:
(ψ|A
†
|φ)
∗
= =φ|A|ψ ∀|ψ ∈ D(A), ∀ |φ ∈ D(A
†
).
(2.2)
This implies that the matrix representing A
† is obtained from that of A by applying
transposition and complex conjugation. It also follows that (|ψφ |)
†
= |φψ | and
that the dual vector of A|ψ is ψ|A
† .
The evolution of a closed quantum system is determined by a unitary operator U ,
that is an operator for which U
†
= U
−1 , where U
−1 is the inverse of U . A unitary
transformation also links any two bases {b i }
d
i=1 and {b
i }
d
i=1 in H: |b
i = U |b i . Two
bases are called mutually unbiased if b
i |b j = 1/d for every i and j.
The observable quantities in quantum mechanics are represented by self-adjoint
operators. Let us take a moment to distinguish symmetric, self-adjoint and Hermitian
operators.