The Theorem

Two quantum systems $A$ and $B$ are in a joint pure state $|\psi\rangle$ . In orthonormal bases $|j\rangle$ of $A$ and $|k\rangle$ of $B$ the state is a double sum $\sum_{j,k} c_{jk}\,|j\rangle|k\rangle$ with complex coefficients $c_{jk}$ , in which any vector of one basis can occur with any vector of the other. Theorem 2.7, the Schmidt decomposition, says that the double sum can be replaced by a single sum of paired terms.

Two Quantum Systems

A quantum system $S$ is a physical system whose state space $\mathcal{H}_S$ is a finite dimensional complex Hilbert space.^✦ A pure state $|\psi\rangle$ of a quantum system $S$ is a vector in its state space $\mathcal{H}_S$ of unit norm, $\langle\psi|\psi\rangle = 1$ .^✦

The composite system $A \otimes B$ of two quantum systems $A$ and $B$ is the quantum system whose state space is the tensor product of the state spaces of $A$ and $B$ , $\mathcal{H}_{A \otimes B} = \mathcal{H}_A \otimes \mathcal{H}_B .$ (1)^✦ For vectors $|a\rangle$ in $\mathcal{H}_A$ and $|b\rangle$ in $\mathcal{H}_B$ we write $|a\rangle|b\rangle$ for the product vector $|a\rangle \otimes |b\rangle$ in the space (1).

The Claim

Theorem 1. Let $A$ and $B$ be quantum systems and let $|\psi\rangle$ be a pure state of the composite system $A \otimes B$ . Then there exist a natural number $n$ , positive real numbers $\lambda_1, \dots, \lambda_n$ with $\sum_i \lambda_i^2 = 1$ , orthonormal vectors $|a_1\rangle, \dots, |a_n\rangle$ in $\mathcal{H}_A$ and orthonormal vectors $|b_1\rangle, \dots, |b_n\rangle$ in $\mathcal{H}_B$ such that $|\psi\rangle = \sum_{i=1}^{n} \lambda_i \, |a_i\rangle |b_i\rangle .$ (2)^✦

What Is Checked

The claim is stated over a pure state $|\psi\rangle$ of the composite system $A \otimes B$ , for any two quantum systems $A$ and $B$ .

About these constructions, the library proves that the tensor product $\mathcal{H}_A \otimes \mathcal{H}_B$ of two finite dimensional complex Hilbert spaces is again one, so that the composite of two quantum systems is itself a quantum system.