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 in a vector space is said to be linearly independent if no vector in the set can be written as a linear combination of the others:

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 on a vector space satisfies:

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

3. Inner Products

An inner product defines a notion of length and angle in vector spaces. It satisfies:

1. Conjugate symmetry:
2. Linearity:
3. Positivity: with equality only if .

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

4. Eigenvalues and Eigenvectors

For a linear operator , an eigenvector satisfies:

where is the eigenvalue. The set of all eigenvalues of is called the spectrum of .

5. Adjoints and Hermitian Operators

The adjoint (or Hermitian conjugate) of a matrix is defined such that:

A matrix is Hermitian if , meaning its eigenvalues are always real.

6. Tensor (Outer) Products

The tensor product of two vectors and produces a larger-dimensional vector:

This is fundamental in quantum mechanics for describing composite systems.

6.1 Spectral Decomposition Theorem

Any Hermitian operator can be decomposed as:

where are the eigenvalues, and are the corresponding eigenvectors.

7. Operator Functions

Functions of an operator can be defined using its spectral decomposition:

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:

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

The anti-commutator is defined as:

9. Polar and Singular Value Decomposition

• Polar decomposition: Any operator can be written as:

  where is unitary and is positive semi-definite.

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

  where are unitary and is diagonal with nonnegative entries.

Definitions of Terms

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

• Unitary Operator: An operator is unitary if:

  Unitary matrices preserve inner products and describe quantum evolution.

• Completeness Relation: A set of orthonormal vectors satisfies:

  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.