12
1 Some Linear Algebra
Definition 1.7.6 Let V be a complex Hilbert space (recall that we are dealing with
quantum mechanics). Assume also that T : V −→ V is a linear operator on V . Then
there exists a unique linear operator T
†
: V −→ V such that, for every |v, |w ∈
V , it follows that (|v, T |w) = (T
†
|v, |w). The operator T
† is called adjoint or
Hermitian conjugate of T .
Hermitian and normal operators play an important role in quantum computation
and quantum information.
Definition 1.7.7 Let T : V −→ V be a linear operator, where V is a Hilbert space.
If T
†
= T then T is called Hermitian or self-adjoint operator.
Definition 1.7.8 Let V be a complex Hilbert space. A positive operator T : V −→
V is a linear operator such that for every |v ∈ V , (|v, T |v) is a real, nonnegative
number. We say that T is positive definite if (|v, T |v) is a real, strictly greater than
zero number for all |v = 0.
Definition 1.7.9 Let T : V −→ V be a linear operator. Then T is called normal if
T T
†
= T
† T .
Remark 1.7.2 (1) There exists an important and well-known property of normal
operators: T : V −→ V is normal if and only if T is diagonalizable. This result is
known as Theorem of Spectral Decomposition.
(2) If T : V −→ V is Hermitian then T is normal; thus T is also diagonalizable.
Unitary operators are fundamental to quantum mechanics since they describe the
evolution of closed quantum systems.
Definition 1.7.10 Let U be a matrix with entries in C. We say that U is unitary if
U
† U = I . In terms of operators, we say that an operator U : V −→ V is unitary if
U
† U = I .
Remark 1.7.3 (1) If U : V −→ V is unitary then UU
†
= I , so U is normal.
(2) Unitary operators preserve inner products between vectors.
The next results states that Pauli matrices have nice properties.
Proposition 1.7.1 The Pauli Matrices are Hermitian and unitary, so they are also
normal operators. Therefore, the Pauli matrices have spectral decomposition.
Exercise 1.7.1 Show Proposition 1.7.1.
We now return the attention to tensor products of vector spaces. We begin by
observing that sometimes we use the notation |vw = |v ⊗ |w to denote the tensor
of |v and |w. Let V and W be two vector spaces of dimensions m and n, respectively.
Let T : V −→ V be a linear operator on V and S : W −→ W a linear operator on
W . The elements of V ⊗ W are linear combinations of tensors |v i ⊗ |w i , where
|v i ∈ V and |w i ∈ W . Then the function T ⊗ S : V ⊗ W −→ V ⊗ W defined by
1 Some Linear Algebra
Definition 1.7.6 Let V be a complex Hilbert space (recall that we are dealing with
quantum mechanics). Assume also that T : V −→ V is a linear operator on V . Then
there exists a unique linear operator T
†
: V −→ V such that, for every |v, |w ∈
V , it follows that (|v, T |w) = (T
†
|v, |w). The operator T
† is called adjoint or
Hermitian conjugate of T .
Hermitian and normal operators play an important role in quantum computation
and quantum information.
Definition 1.7.7 Let T : V −→ V be a linear operator, where V is a Hilbert space.
If T
†
= T then T is called Hermitian or self-adjoint operator.
Definition 1.7.8 Let V be a complex Hilbert space. A positive operator T : V −→
V is a linear operator such that for every |v ∈ V , (|v, T |v) is a real, nonnegative
number. We say that T is positive definite if (|v, T |v) is a real, strictly greater than
zero number for all |v = 0.
Definition 1.7.9 Let T : V −→ V be a linear operator. Then T is called normal if
T T
†
= T
† T .
Remark 1.7.2 (1) There exists an important and well-known property of normal
operators: T : V −→ V is normal if and only if T is diagonalizable. This result is
known as Theorem of Spectral Decomposition.
(2) If T : V −→ V is Hermitian then T is normal; thus T is also diagonalizable.
Unitary operators are fundamental to quantum mechanics since they describe the
evolution of closed quantum systems.
Definition 1.7.10 Let U be a matrix with entries in C. We say that U is unitary if
U
† U = I . In terms of operators, we say that an operator U : V −→ V is unitary if
U
† U = I .
Remark 1.7.3 (1) If U : V −→ V is unitary then UU
†
= I , so U is normal.
(2) Unitary operators preserve inner products between vectors.
The next results states that Pauli matrices have nice properties.
Proposition 1.7.1 The Pauli Matrices are Hermitian and unitary, so they are also
normal operators. Therefore, the Pauli matrices have spectral decomposition.
Exercise 1.7.1 Show Proposition 1.7.1.
We now return the attention to tensor products of vector spaces. We begin by
observing that sometimes we use the notation |vw = |v ⊗ |w to denote the tensor
of |v and |w. Let V and W be two vector spaces of dimensions m and n, respectively.
Let T : V −→ V be a linear operator on V and S : W −→ W a linear operator on
W . The elements of V ⊗ W are linear combinations of tensors |v i ⊗ |w i , where
|v i ∈ V and |w i ∈ W . Then the function T ⊗ S : V ⊗ W −→ V ⊗ W defined by