Primer on Linear Algebra for Quantum

This primer is based on Chapter 2 of Nielsen and Chuang's book Quantum Computation and Quantum Information. It covers concepts in linear algebra that are essential for understanding quantum mechanics and quantum computing.

1. Bases and Linear Independence

A set of vectors $\{ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \}$ in a vector space $V$ is said to be linearly independent if no vector in the set can be written as a linear combination of the others:

$$c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_n \mathbf{v}_n = 0 \quad \Rightarrow \quad c_i = 0 \text{ for all } i.$$

A basis is a linearly independent set of vectors that spans the entire vector space. Every vector in the space can be uniquely expressed as a linear combination of basis vectors.

2. Linear Operators and Matrices

A linear operator $A$ on a vector space satisfies:

$$A(\alpha \mathbf{v} + \beta \mathbf{w}) = \alpha A\mathbf{v} + \beta A\mathbf{w}, \quad \forall \mathbf{v}, \mathbf{w} \in V, \alpha, \beta \in \mathbb{C}.$$

In finite dimensions, linear operators can be represented as matrices acting on column vectors.

3. Inner Products

An inner product $\langle \mathbf{v} | \mathbf{w} \rangle$ defines a notion of length and angle in vector spaces. It satisfies:

1. Conjugate symmetry: $\langle \mathbf{v} | \mathbf{w} \rangle = \overline{\langle \mathbf{w} | \mathbf{v} \rangle}$
2. Linearity: $\langle \mathbf{v} | (a \mathbf{w} + b \mathbf{u}) \rangle = a \langle \mathbf{v} | \mathbf{w} \rangle + b \langle \mathbf{v} | \mathbf{u} \rangle$
3. Positivity: $\langle \mathbf{v} | \mathbf{v} \rangle \geq 0$ with equality only if $\mathbf{v} = 0$.

For quantum mechanics, the standard inner product in Hilbert space is:

$$\langle \psi | \phi \rangle = \sum_i \overline{\psi_i} \phi_i.$$

4. Eigenvalues and Eigenvectors

For a linear operator $A$, an eigenvector $\mathbf{v}$ satisfies:

$$A \mathbf{v} = \lambda \mathbf{v},$$

where $\lambda$ is the eigenvalue. The set of all eigenvalues of $A$ is called the spectrum of $A$.

5. Adjoints and Hermitian Operators

The adjoint (or Hermitian conjugate) $A^\dagger$ of a matrix $A$ is defined such that:

$$\langle A \mathbf{v} | \mathbf{w} \rangle = \langle \mathbf{v} | A^\dagger \mathbf{w} \rangle.$$

A matrix is Hermitian if $A = A^\dagger$, meaning its eigenvalues are always real.

6. Tensor (Outer) Products

The tensor product of two vectors $\mathbf{v}$ and $\mathbf{w}$ produces a larger-dimensional vector:

$$|\psi\rangle \otimes |\phi\rangle = \begin{bmatrix} \psi_1 \phi_1 \\ \psi_1 \phi_2 \\ \vdots \\ \psi_m \phi_n \end{bmatrix}.$$

This is fundamental in quantum mechanics for describing composite systems.

6.1 Spectral Decomposition Theorem

Any Hermitian operator $A$ can be decomposed as:

$$A = \sum_i \lambda_i |\psi_i\rangle \langle \psi_i |,$$

where $\lambda_i$ are the eigenvalues, and $|\psi_i\rangle$ are the corresponding eigenvectors.

7. Operator Functions

Functions of an operator $f(A)$ can be defined using its spectral decomposition:

$$f(A) = \sum_i f(\lambda_i) |\psi_i\rangle \langle \psi_i |.$$

This is useful for defining matrix exponentials and other functions in quantum mechanics.

8. Commutator and Anti-Commutator

The commutator measures the non-commutativity of two operators:

$$[A, B] = AB - BA.$$

In quantum mechanics, nonzero commutators indicate fundamental uncertainty relationships (e.g., position and momentum).

The anti-commutator is defined as:

$$\{A, B\} = AB + BA.$$

9. Polar and Singular Value Decomposition

• Polar decomposition: Any operator $A$ can be written as:

  $$A = UP,$$

  where $U$ is unitary and $P$ is positive semi-definite.

• Singular Value Decomposition (SVD): Any matrix $A$ can be expressed as:

  $$A = U \Sigma V^\dagger,$$

  where $U, V$ are unitary and $\Sigma$ is diagonal with nonnegative entries.

Definitions of Terms

• Adjoint / Hermitian Conjugate / Conjugate Transpose: The adjoint $A^\dagger$ is given by taking the complex conjugate and transposing a matrix.

• Unitary Operator: An operator $U$ is unitary if:

  $$U^\dagger U = U U^\dagger = I.$$

  Unitary matrices preserve inner products and describe quantum evolution.

• Completeness Relation: A set of orthonormal vectors $\{ |\psi_i\rangle \}$ satisfies:

  $$\sum_i |\psi_i\rangle \langle \psi_i | = I.$$

  This ensures that any vector in the space can be expanded in this basis.

This primer provides a foundation for understanding the mathematical framework behind quantum computing and quantum mechanics.