NoteNextra-origin/content/Math401/Freiwald_summer/Math401_T2.md

Math401 Topic 2: Finite-dimensional Hilbert spaces

Recall the complex number is a tuple of two real numbers, z=(a,b) with addition and multiplication defined by

(a,b)+(c,d)=(a+c,b+d)

(a,b)\cdot(c,d)=(ac-bd,ad+bc)

or in polar form,

z=re^{i\theta}=r(\cos\theta+i\sin\theta)

where r=\sqrt{a^2+b^2}=\sqrt{z\overline{z}} and \theta=\tan^{-1}(b/a).

The complex conjugate of z is \overline{z}=(a,-b).

Section 1: Finite-dimensional Complex Vector Spaces

Here, we use the field \mathbb{C} of complex numbers. or the field \mathbb{R} of real numbers as the field \mathbb{F} we are going to encounter.

Definition of vector space

A vector space \mathscr{V} over a field \mathbb{F} is a set equipped with an addition and a scalar multiplication, satisfying the following axioms:

1. Addition is associative and commutative. For all u,v,w\in \mathscr{V},

Associativity:

(u+v)+w=u+(v+w)

Commutativity:

u+v=v+u

2. Additive identity: There exists an element 0\in \mathscr{V} such that v+0=v for all v\in \mathscr{V}.

3. Additive inverse: For each v\in \mathscr{V}, there exists an element -v\in \mathscr{V} such that v+(-v)=0.

4. Multiplicative identity: There exists an element 1\in \mathbb{F} such that v\cdot 1=v for all v\in \mathscr{V}.

5. Multiplicative inverse: For each v\in \mathscr{V} and c\in \mathbb{F}, there exists an element c^{-1}\in \mathbb{F} such that v\cdot c^{-1}=1.

6. Distributivity: For all u,v\in \mathscr{V} and c,d\in \mathbb{F},

c(u+v)=cu+cv

A vector is an ordered pair of elements over the field \mathbb{F}.

If we consider \mathbb{F}=\mathbb{C}^n, n\in \mathbb{N}, then u=(a_1,a_2,\cdots,a_n), v=(b_1,b_2,\cdots,b_n)\in \mathbb{C}^n are vectors.

The addition and scalar multiplication are defined by

u+v=(a_1+b_1,a_2+b_2,\cdots,a_n+b_n)

cu=(ca_1,ca_2,\cdots,ca_n)

c\in \mathbb{C}.

The matrix transpose is defined by

u^T=(a_1,a_2,\cdots,a_n)^T=\begin{pmatrix}
a_1 \\
a_2 \\
\vdots \\
a_n
\end{pmatrix}

The complex conjugate transpose is defined by

u^*=(a_1,a_2,\cdots,a_n)^*=\begin{pmatrix}
\overline{a_1} \\
\overline{a_2} \\
\vdots \\
\overline{a_n}
\end{pmatrix}

In physics, the complex conjugate is sometimes denoted by z^* instead of \overline{z}. The complex conjugate transpose is sometimes denoted by u^\dagger instead of u^*.

Hermitian inner product and norms

On \mathbb{C}^n, the Hermitian inner product is defined by

\langle u,v\rangle=\sum_{i=1}^n \overline{u_i}v_i

The norm is defined by

\|u\|=\sqrt{\langle u,u\rangle}

Definition of Inner product

Let \mathscr{H} be a complex vector space. An inner product on \mathscr{H} is a function \langle \cdot, \cdot \rangle: \mathscr{H}\times \mathscr{H}\to \mathbb{C} satisfying the following axioms:

1. For each u\in \mathscr{H}, v\mapsto \langle u,v\rangle is a linear map.

\langle u,av+bw\rangle=a\langle u,v\rangle+b\langle u,w\rangle

For all u,v,w\in \mathscr{H} and a,b\in \mathbb{C}.

2. For all u,v\in \mathscr{H}, \langle u,v\rangle=\overline{\langle v,u\rangle}.

u\mapsto \langle u,v\rangle is a conjugate linear map.

3. \langle u,u\rangle\geq 0 and \langle u,u\rangle=0 if and only if u=0.

Definition of norm

Let \mathscr{H} be a complex vector space. A norm on \mathscr{H} is a function \|\cdot\|: \mathscr{H}\to \mathbb{R} satisfying the following axioms:

1. For all u\in \mathscr{H}, \|u\|\geq 0 and \|u\|=0 if and only if u=0.

2. For all u\in \mathscr{H} and c\in \mathbb{C}, \|cu\|=|c|\|u\|.

3. Triangle inequality: For all u,v\in \mathscr{H}, \|u+v\|\leq \|u\|+\|v\|.

Definition of inner product space

A complex vector space \mathscr{H} with an inner product is called a Hilbert space.

Cauchy-Schwarz inequality

For all u,v\in \mathscr{H},

|\langle u,v\rangle|\leq \|u\|\|v\|

Parallelogram law

For all u,v\in \mathscr{H},

\|u+v\|^2+\|u-v\|^2=2(\|u\|^2+\|v\|^2)

Polarization identity

For all u,v\in \mathscr{H},

\langle u,v\rangle=\frac{1}{4}(\|u+v\|^2-\|u-v\|^2+i\|u+iv\|^2-i\|u-iv\|^2)

Additional definitions

Let u,v\in \mathscr{H}.

\|v\| is the length of v.

v is a unit vector if \|v\|=1.

u,v are orthogonal if \langle u,v\rangle=0.

Definition of orthonormal basis

A set of vectors \{e_1,e_2,\cdots,e_n\} in a Hilbert space \mathscr{H} is called an orthonormal basis if

1. \langle e_i,e_j\rangle=\delta_{ij} for all i,j\in \{1,2,\cdots,n\}.

\delta_{ij}=\begin{cases}
1 & \text{if } i=j \\
0 & \text{if } i\neq j
\end{cases}

2. n=\dim \mathscr{H}.

Subspaces and orthonormal bases

Definition of subspace

A subset \mathscr{W} of a vector space \mathscr{V} is a subspace if it is closed under addition and scalar multiplication.

Definition of orthogonal complement

Let E be a subset of a Hilbert space \mathscr{H}. The orthogonal complement of E is the set of all vectors in \mathscr{H} that are orthogonal to every vector in E.

E^\perp=\{v\in \mathscr{H}: \langle v,w\rangle=0 \text{ for all } w\in E\}

Definition of orthogonal projection

Let E be a $m$-dimensional subspace of a Hilbert space \mathscr{H}. An orthogonal projection of E is a linear map P_E: \mathscr{H}\to E

P_E(v)=\sum_{i=1}^m \langle v,e_i\rangle e_i

Definition of orthonormal direct sum

A inner product space \mathscr{H} is the direct sum of E_1,E_2,\cdots,E_n if

\mathscr{H}=E_1\oplus E_2\oplus \cdots \oplus E_n

and E_i\cap E_j=\{0\} for all i\neq j.

That is, \forall v\in \mathscr{H}, there exists a unique v_i\in E_i such that v=v_1+v_2+\cdots+v_n.

Definition of meet and join of subspaces

Let E and F be two subspaces of a Hilbert space \mathscr{H}. The meet of E and F is the subspace \mathscr{H} such that

E\land F=E\cap F

The join of E and F is the subspace \mathscr{H} such that

E\lor F=\{u+v: u\in E, v\in F\}

Null space and range

Definition of null space

Let A be a linear map from a vector space \mathscr{V} to a vector space \mathscr{W}. The null space of A is the set of all vectors in \mathscr{V} that are mapped to the zero vector in \mathscr{W}.

\text{Null}(A)=\{v\in \mathscr{V}: Av=0\}

Definition of range

Let A be a linear map from a vector space \mathscr{V} to a vector space \mathscr{W}. The range of A is the set of all vectors in \mathscr{W} that are mapped from \mathscr{V}.

\text{Range}(A)=\{w\in \mathscr{W}: \exists v\in \mathscr{V}, Av=w\}

Dual spaces and adjoints of linear maps

Definition of linear map

A linear map T: \mathscr{V}\to \mathscr{W} is a function that satisfies the following axioms:

1. Additivity: For all u,v\in \mathscr{V} and a,b\in \mathbb{F},

T(au+bv)=aT(u)+bT(v)

2. Homogeneity: For all u\in \mathscr{V} and a\in \mathbb{F},

T(au)=aT(u)

Definition of linear functionals

A linear functional f: \mathscr{V}\to \mathbb{F} is a linear map from \mathscr{V} to \mathbb{F}.

Here, \mathbb{F} is the field of complex numbers.

Definition of dual space

Let \mathscr{V} be a vector space over a field \mathbb{F}. The dual space of \mathscr{V} is the set of all linear functionals on \mathscr{V}.

\mathscr{V}^*=\{f:\mathscr{V}\to \mathbb{F}: f\text{ is linear}\}

If \mathscr{H} is a finite-dimensional Hilbert space, then \mathscr{H}^* is isomorphic to \mathscr{H}.

Note v\in \mathscr{H}\mapsto \langle v,\cdot\rangle\in \mathscr{H}^* is a conjugate linear isomorphism.

Definition of adjoint of a linear map

Let T: \mathscr{V}\to \mathscr{W} be a linear map. The adjoint of T is the linear map T^*: \mathscr{W}\to \mathscr{V} such that

\langle Tv,w\rangle=\langle v,T^*w\rangle

for all v\in \mathscr{V} and w\in \mathscr{W}.

Definition of self-adjoint operators

A linear operator T: \mathscr{V}\to \mathscr{V} is self-adjoint if T^*=T.

Definition of unitary operators

A linear map T: \mathscr{V}\to \mathscr{V} is unitary if T^*T=TT^*=I.

Dirac's bra-ket notation

Definition of bra and ket

Let \mathscr{H} be a Hilbert space. The bra-ket notation is a notation for vectors in \mathscr{H}.

\langle v|w\rangle

is the inner product of v and w. That is, \langle v|w\rangle: \mathscr{H}\to \mathbb{C} is a linear functional satisfying the property of inner product.

|v\rangle

is the vector (or linear map) v.

|u\rangle\langle v|

is a linear map from \mathscr{H} to \mathscr{H}.

The spectral theorem for self-adjoint operators

Spectral theorem for self-adjoint operators

Definition of spectral theorem

Let \mathscr{H} be a Hilbert space. A self-adjoint operator T: \mathscr{H}\to \mathscr{H} is a linear operator that is equal to its adjoint.

Then all the eigenvalues of T are real and there exists an orthonormal basis of \mathscr{H} consisting of eigenvectors of T.

Definition of spectrum

The spectrum of a linear operator on finite-dimensional Hilbert space T: \mathscr{H}\to \mathscr{H} is the set of all distinct eigenvalues of T.

\operatorname{sp}(T)=\{\lambda: \lambda\text{ is an eigenvalue of } T\}\subset \mathbb{C}

Definition of Eigenspace

If \lambda is an eigenvalue of T, the eigenspace of T corresponding to \lambda is the set of all eigenvectors of T corresponding to \lambda.

E_\lambda(T)=\{v\in \mathscr{H}: Tv=\lambda v\}

We denote P_\lambda(T):\mathscr{H}\to E_\lambda(T) the orthogonal projection onto E_\lambda(T).

Definition of Operator norm

The operator norm of a linear operator T: \mathscr{H}\to \mathscr{H} is the largest eigenvalue of T.

\|T\|=\max_{\|v\|=1} \|Tv\|

We say T is bounded if \|T\|<\infty.

We denote B(\mathscr{H}) the set of all bounded linear operators on \mathscr{H}.

Partial trace

Definition of trace

Let T be a linear operator on \mathscr{H}, (e_1,e_2,\cdots,e_n) be a basis of \mathscr{H} and (\epsilon_1,\epsilon_2,\cdots,\epsilon_n) be a basis of dual space \mathscr{H}^*. Then the trace of T is defined by

\operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle

This is equivalent to the sum of the diagonal elements of T.

Note, I changed the order of the definitions for the trace to pack similar concepts together. Check the rest of the section defining the partial trace by viewing the tensor product section first, and return to this section after reading the tensor product of linear operators.

Definition of partial trace

Let T be a linear operator on \mathscr{H}=\mathscr{A}\otimes \mathscr{B}, where \mathscr{A} and \mathscr{B} are finite-dimensional Hilbert spaces.

An operator T on \mathscr{H}=\mathscr{A}\otimes \mathscr{B} can be written as (by the definition of tensor product of linear operators)

T=\sum_{i=1}^n a_i A_i\otimes B_i

where A_i is a linear operator on \mathscr{A} and B_i is a linear operator on \mathscr{B}.

The $\mathscr{B}$-partial trace of T (\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})) is the linear operator on \mathscr{A} defined by

\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i

Or we can define the map L_v: \mathscr{A}\to \mathscr{A}\otimes \mathscr{B} by

L_v(u)=u\otimes v

Note that \langle u,L_v^*(u')\otimes v'\rangle=\langle u,u'\rangle \langle v,v'\rangle=\langle u\otimes v,u'\otimes v'\rangle=\langle L_v(u),u'\otimes v'\rangle.

Therefore, L_v^*\sum_{j} u_j\otimes v_j=\sum_{j} \langle v,v_j\rangle u_j.

Then the partial trace of T can also be defined by

Let \{v_j\} be a set of orthonormal basis of \mathscr{B}.

\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{j} L^*_{v_j}(T)L_{v_j}

Definition of partial trace with respect to a state

Let T be a linear operator on \mathscr{H}=\mathscr{A}\otimes \mathscr{B}, where \mathscr{A} and \mathscr{B} are finite-dimensional Hilbert spaces.

Let \rho be a state on \mathscr{B} consisting of orthonormal basis \{v_j\} and eigenvalue \{\lambda_j\}.

The partial trace of T with respect to \rho is the linear operator on \mathscr{A} defined by

\operatorname{Tr}_{\mathscr{A}}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j}

Space of Bounded Linear Operators

Recall the trace of a matrix is the sum of its diagonal elements.

Hilbert-Schmidt inner product

Let T,S\in B(\mathscr{H}). The Hilbert-Schmidt inner product of T and S is defined by

\langle T,S\rangle=\operatorname{Tr}(T^*S)

Note here, T^* is the complex conjugate transpose of T.

If we introduce the basis \{e_i\} in \mathscr{H}, then we can write the the space of bounded linear operators as n\times n complex-valued matrices M_n(\mathbb{C}).

For T=(a_{ij}), S=(b_{ij}), we have

\operatorname{Tr}(A^*B)=\sum_{i=1}^n \sum_{j=1}^n \overline{a_{ij}}b_{ij}

The inner product is the standard Hermitian inner product in \mathbb{C}^{n\times n}.

Definition of Hilbert-Schmidt norm (also called Frobenius norm)

The Hilbert-Schmidt norm of a linear operator T: \mathscr{H}\to \mathscr{H} is defined by

\|T\|=\sqrt{\sum_{i=1}^n \sum_{j=1}^n |a_{ij}|^2}

The trace of operator does not depend on the basis.

Tensor products of finite-dimensional Hilbert spaces

Let X=X_1\times X_2\times \cdots \times X_n be a Cartesian product of n sets.

Let x=(x_1,x_2,\cdots,x_n) be a vector in X. x_j\in X_j for j=1,2,\cdots,n.

Let a\in X_j for j=1,2,\cdots,n.

Let's denote the space of all functions from X to \mathbb{C} by \mathscr{H} and the space of all functions from X_j to \mathbb{C} by \mathscr{H}_j.

\epsilon_{a}^{(j)}(x_j)=\begin{cases}
1 & \text{if } x_j=a \\
0 & \text{if } x_j\neq a
\end{cases}

Then we can define a basis of \mathscr{H}_j by \{\epsilon_{a}^{(j)}(x_j)\}_{a\in X_j}.

Any function f:X_j\to \mathbb{C} can be written as a linear combination of the basis vectors.

f(x_j)=\sum_{a\in X_j} f(a)\epsilon_{a}^{(j)}(x_j)

Proof

Note that a function is a map for all elements in the domain.

For each a\in X_j, \epsilon_{a}^{(j)}(x_j)=1 if x_j=a and 0 otherwise. So

f(x_j)=\sum_{a\in X_j} f(a)\epsilon_{a}^{(j)}(x_j)=f(x_j)

Now, let a=(a_1,a_2,\cdots,a_n) be a vector in X, and x=(x_1,x_2,\cdots,x_n) be a vector in X. Note that a_j,x_j\in X_j for j=1,2,\cdots,n.

Define

\epsilon_a(x)=\prod_{j=1}^n \epsilon_{a_j}^{(j)}(x_j)=\begin{cases}
1 & \text{if } a_j=x_j \text{ for all } j=1,2,\cdots,n \\
0 & \text{otherwise}
\end{cases}

Then we can define a basis of \mathscr{H} by \{\epsilon_a\}_{a\in X}.

Any function f:X\to \mathbb{C} can be written as a linear combination of the basis vectors.

f(x)=\sum_{a\in X} f(a)\epsilon_a(x)

Proof

This basically follows the same rascal as the previous proof. This time, the epsilon function only returns 1 when x_j=a_j for all j=1,2,\cdots,n.

f(x)=\sum_{a\in X} f(a)\epsilon_a(x)=f(x)

Definition of tensor product of basis elements

The tensor product of basis elements is defined by

\epsilon_a\coloneqq\epsilon_{a_1}^{(1)}\otimes \epsilon_{a_2}^{(2)}\otimes \cdots \otimes \epsilon_{a_n}^{(n)}

This is a basis of \mathscr{H}, here \mathscr{H} is the set of all functions from X=X_1\times X_2\times \cdots \times X_n to \mathbb{C}.

Definition of tensor product of two finite-dimensional Hilbert spaces

The tensor product of two finite-dimensional Hilbert spaces (in \mathscr{H}) is defined by

Let \mathscr{H}_1 and \mathscr{H}_2 be two finite dimensional Hilbert spaces. Let u_1\in \mathscr{H}_1 and v_1\in \mathscr{H}_2.

u_1\otimes v_1

is a bi-anti-linear map from \mathscr{H}_1\times \mathscr{H}_2 (the Cartesian product of \mathscr{H}_1 and \mathscr{H}_2, a tuple of two elements where first element is in \mathscr{H}_1 and second element is in \mathscr{H}_2) to \mathbb{F} (in this case, \mathbb{C}). And \forall u\in \mathscr{H}_1, v\in \mathscr{H}_2,

(u_1\otimes v_1)(u, v)=\langle u,u_1\rangle \langle v,v_1\rangle

We call such forms decomposable. The tensor product of two finite-dimensional Hilbert spaces, denoted by \mathscr{H}_1\otimes \mathscr{H}_2, is the set of all linear combinations of decomposable forms. Represented by the following:

\left(\sum_{i=1}^n a_i u_i\otimes v_i\right)(u, v) \coloneqq \sum_{i=1}^n a_j(u_j\otimes v_j)(u,v)=\sum_{i=1}^n a_i \langle v,u_i\rangle \langle v_i,u\rangle

Note that a_i\in \mathbb{C} for complex-vector spaces.

This is a linear space of dimension \dim \mathscr{H}_1\times \dim \mathscr{H}_2.

We define the inner product of two elements of \mathscr{H}_1\otimes \mathscr{H}_2 (u_1\otimes v_1:(\mathscr{H}_1\otimes \mathscr{H}_2)\to \mathbb{C}, u_2\otimes v_2:(\mathscr{H}_1\otimes \mathscr{H}_2)\to \mathbb{C} \in \mathscr{H}_1\otimes \mathscr{H}_2) by

\langle u_1\otimes v_1, u_2\otimes v_2\rangle\coloneqq\langle u_1,u_2\rangle \langle v_1,v_2\rangle=(u_1\otimes v_1)(u_2,v_2)

Tensor products of linear operators

Let T_1 be a linear operator on \mathscr{H}_1 and T_2 be a linear operator on \mathscr{H}_2, where \mathscr{H}_1 and \mathscr{H}_2 are finite-dimensional Hilbert spaces. The tensor product of T_1 and T_2 (denoted by T_1\otimes T_2) on \mathscr{H}_1\otimes \mathscr{H}_2, such that on decomposable elements is defined by

(T_1\otimes T_2)(v_1\otimes v_2)=T_1(v_1)\otimes T_2(v_2)=\langle v_1,T_1(v_1)\rangle \langle v_2,T_2(v_2)\rangle

for all v_1\in \mathscr{H}_1 and v_2\in \mathscr{H}_2.

The tensor product of two linear operators T_1 and T_2 is a linear combination in the form as follows:

\sum_{i=1}^n a_i T_1(u_i)\otimes T_2(v_i)

for all u_i\in \mathscr{H}_1 and v_i\in \mathscr{H}_2.

Such tensor product of linear operators is well defined.

Proof

If \sum_{i=1}^n a_i u_i\otimes v_i=\sum_{j=1}^m b_j u_j\otimes v_j, then a_i=b_j for all i=1,2,\cdots,n and j=1,2,\cdots,m.

Then \sum_{i=1}^n a_i T_1(u_i)\otimes T_2(v_i)=\sum_{j=1}^m b_j T_1(u_j)\otimes T_2(v_j).

An example of

Tensor product of linear operators on Hilbert spaces

Let T_1 be a linear operator on \mathscr{H}_1 and T_2 be a linear operator on \mathscr{H}_2, where \mathscr{H}_1 and \mathscr{H}_2 are finite-dimensional Hilbert spaces. The tensor product of T_1 and T_2 (denoted by T_1\otimes T_2) on \mathscr{H}_1\otimes \mathscr{H}_2, such that on decomposable elements is defined by

(T_1\otimes T_2)(v_1\otimes v_2)=T_1(v_1)\otimes T_2(v_2)=\langle v_1,T_1(v_1)\rangle \langle v_2,T_2(v_2)\rangle

Extended Dirac notation

Suppose \mathscr{H}=\mathbb{C}^n with the standard basis \{e_i\}.

e_j=|j\rangle and

|j_1\dots j_n\rangle=e_{j_1}\otimes e_{j_2}\otimes \cdots \otimes e_{j_n}=

The Hadamard Transform

Let \mathscr{H}=\mathbb{C}^2 with the standard basis \{e_1,e_2\}=\{|0\rangle,|1\rangle\}.

The linear operator H_2 is defined by

H_2=\frac{1}{\sqrt{2}}\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}=\frac{1}{\sqrt{2}}(|0\rangle\langle 0|+|1\rangle\langle 0|+|0\rangle\langle 1|-|1\rangle\langle 1|)

The Hadamard transform is the linear operator H_2 on \mathbb{C}^2.

Singular value and Schmidt decomposition

Definition of SVD (Singular Value Decomposition)

Let T:\mathscr{U}\to \mathscr{V} be a linear operator between two finite-dimensional Hilbert spaces \mathscr{U} and \mathscr{V}.

We denote the inner product of \mathscr{U} and \mathscr{V} by \langle \cdot, \cdot \rangle.

Then there exists a decomposition of T

T=d_1 T_1+d_2 T_2+\cdots +d_n T_n

with d_1>d_2>\cdots >d_n>0 and T_i:\mathscr{U}\to \mathscr{V} such that:

1. T_iT_j^*=0, T_i^*T_j=0 for $i\neq j$(
2. T_i|_{\mathscr{R}(T_i^*)}:\mathscr{R}(T_i^*)\to \mathscr{R}(T_i) is an isomorphism with inverse T_i^* where \mathscr{R}(\cdot) is the range of the operator.

The d_i are called the singular values of T.

Gram-Schmidt Decomposition

Basic Group Theory

Finite groups

Definition of group

A group is a set G with a binary operation \cdot that satisfies the following axioms:

1. Closure: For all a,b\in G, a\cdot b\in G.
2. Associativity: For all a,b,c\in G, (a\cdot b)\cdot c=a\cdot (b\cdot c).
3. Identity: There exists an element e\in G such that for all a\in G, a\cdot e=e\cdot a=a.
4. Inverses: For all a\in G, there exists an element b\in G such that a\cdot b=b\cdot a=e.

Symmetric group S_n

The symmetric group S_n is the group of all permutations of n elements.

S_n=\{f: \{1,2,\cdots,n\}\to \{1,2,\cdots,n\} \text{ is a bijection}\}

Unitary group U(n)

The unitary group U(n) is the group of all n\times n unitary matrices.

Such that A^*=A, where A^* is the complex conjugate transpose of A. A^*=(\overline{A})^T.

Cyclic group \mathbb{Z}_n

The cyclic group \mathbb{Z}_n is the group of all integers modulo n.

\mathbb{Z}_n=\{0,1,2,\cdots,n-1\}

Definition of group homomorphism

A group homomorphism is a function \varPhi:G\to H between two groups G and H that satisfies the following axiom:

\varPhi(a\cdot b)=\varPhi(a)\cdot \varPhi(b)

A bijective group homomorphism is called group isomorphism.

Homomorphism sends identity to identity, inverses to inverses

Let \varPhi:G\to H be a group homomorphism. e_G and e_H are the identity elements of G and H respectively. Then

1. \varPhi(e_G)=e_H
2. \varPhi(a^{-1})=\varPhi(a)^{-1}. \forall a\in G

More on the symmetric group

General linear group over \mathbb{C}

The general linear group over \mathbb{C} is the group of all n\times n invertible complex matrices.

GL(n,\mathbb{C})=\{A\in M_n(\mathbb{C}) \text{ is invertible}\}

The map T: S_n\to GL(n,\mathbb{C}) is a group homomorphism.

Definition of sign of a permutation

Let T:S_n\to GL(n,\mathbb{C}) be the group homomorphism. The sign of a permutation \sigma\in S_n is defined by

\operatorname{sgn}(\sigma)=\det(T(\sigma))

We say \sigma is even if \operatorname{sgn}(\sigma)=1 and odd if \operatorname{sgn}(\sigma)=-1.

Fourier Transform in \mathbb{Z}_N.

The vector space L^2(\mathbb{Z}_N) is the set of all complex-valued functions on \mathbb{Z}_N with the inner product

\langle f,g\rangle=\sum_{k=0}^{N-1} \overline{f(k)}g(k)

An orthonormal basis of L^2(\mathbb{Z}_N) is given by \delta_y,y\in \mathbb{Z}_N.

\delta_y(k)=\begin{cases}
1 & \text{if } k=y \\
0 & \text{otherwise}
\end{cases}

in Dirac notation, we have

\delta_y=|y\rangle=|y+N\rangle

Definition of Fourier transform

Define \varphi_k(x)=\frac{1}{\sqrt{N}}e^{2\pi i kx/N} for k\in \mathbb{Z}_N. \varphi_k:\mathbb{Z}\to \mathbb{C} is a function.

The Fourier transform of a function F\in L^2(\mathbb{Z}_N) such that (Ff)(k)=\langle \varphi_k,f\rangle is defined by

F=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1} \sum_{k=0}^{N-1} e^{2\pi i kj/N}|k\rangle\langle j|

Symmetric and anti-symmetric tensors

Let \mathscr{H}^{\otimes n} be the $n$-fold tensor product of a Hilbert space \mathscr{H}.

We define the S_n on \mathscr{H}^{\otimes n} by

Let \eta\in S_n be a permutation.

\prod(\eta)v_1\otimes v_2\otimes \cdots \otimes v_n=v_{\eta^{-1}(1)}\otimes v_{\eta^{-1}(2)}\otimes \cdots \otimes v_{\eta^{-1}(n)}

And extend to \mathscr{H}^{\otimes n} by linearity.

This gives the property that \zeta,\eta\in S_n, \prod(\zeta\eta)=\prod(\zeta)\prod(\eta).

Definition of symmetric and anti-symmetric tensors

Let \mathscr{H} be a finite-dimensional Hilbert space.

An element in \mathscr{H}^{\otimes n} is called symmetric if it is invariant under the action of S_n. Let \alpha\in \mathscr{H}^{\otimes n}

\prod(\eta)\alpha=\alpha \text{ for all } \eta\in S_n.

It is called anti-symmetric if

\prod(\eta)\alpha=\operatorname{sgn}(\eta)\alpha \text{ for all } \eta\in S_n.