Lecture 7 - Elements of Group Theory and Tensor Products

Recap of Previous Material

  • Importance of basic concepts studied in previous lectures for understanding quantum computing.

Elements of Group Theory

  • Definition of a Group: A set G with a binary operation • that satisfies certain axioms:

    • Closure: For all x, y ∈ G, x • y ∈ G.

    • Associativity: For all x, y, z ∈ G, (x • y) • z = x • (y • z).

    • Identity Element: There exists an element I ∈ G such that for all x ∈ G, I • x = x • I = x.

    • Inverse Element: For every x ∈ G, there exists a unique x⁻¹ ∈ G such that x • x⁻¹ = x⁻¹ • x = I.

  • Group Notation: A group is denoted as (G, •). The operation • is termed the group law.

  • Abelian Groups: If the group operation is commutative (x • y = y • x), it is called Abelian.

    • Example groups include:

    • Integers under addition: (ℤ, +)

    • Nonzero rationals under multiplication: (ℚ \ {0}, ·)

Symmetry and Dihedral Groups

  • Symmetries of a Square: Examples of group elements include rotations and reflections.

  • Dihedral Group Dn: Represents the symmetries of a regular polygon. For example, D4 = (I, R, R², R³, H, V, D, D⁻) where R = rotation and H, V, D are reflections.

Matrix Groups

  • Matrix groups are defined under matrix addition and multiplication.

  • Examples include:

    • General linear group: GL_m(C) = {A ∈ C^{m×m} | det(A) ≠ 0}

    • Unitary group: U_m(C) = {A ∈ C^{m×m} | A^† = A⁻¹}

    • Special unitary group: SU_m(C) = {A ∈ C^{m×m} | det(A) = 1, A^† = A⁻¹}

Tensor Products

  • Definition: Tensor product spaces U ⊗ V consist of sums of vectors of the form u ⊗ v, where u ∈ U, v ∈ V.

  • Bilinear Maps: A bilinear function from U × V to W can be expressed in terms of a linear transformation from U ⊗ V to W.

  • Examples of Tensor products:

    • For vector spaces U, V \in \mathbb{R}^m, we can define a bilinear map as f(u, v) = u^Tv = tr(uv^T) = g(\phi(u,v)) = g(u⊗v) forms a new matrix.

  • Dimensionality:

    • If U has dimension m and V has dimension n, then U ⊗ V has dimension m·n.

  • Kronecker Product:

    • The Kronecker product extends the idea of tensor products and is commonly used in quantum computing. Defined mathematically as:
      x ⊗ y = \begin{pmatrix} x_1 y_1 … & x_1 y_n & x_2 y_1… & x_m y_n\end{pmatrix}

Important Properties of Kronecker Products

  • Non-commutativity: Generally, x ⊗ y ≠ y ⊗ x.

  • Associativity: (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z).

  • Distributes over Addition: x ⊗ (y + z) = x ⊗ y + x ⊗ z.

  • Mixed Product Properties:

    1. If A, B, C, and D are matrices such that products AC and BD are defined, then (A ⊗ B)(C ⊗ D) = AC ⊗ BD.

    2. If A, B are matrices acting on vectors x and y respectively, then (A ⊗ B)(x ⊗ y) = Ax ⊗ By.

Dirac Notation

  • Usage of Dirac notation in quantum mechanics simplifies expressions. For instance:

    • The tensor product can be denoted as |x\rangle⊗|y\rangle = |xy\rangle.

Summary

  • Covered concepts of group theory, tensor products, Kronecker products, and introduced the formalism needed to work within these frameworks in quantum computing contexts.