The Exercise

Two quantum systems $A$ and $B$ share a pure state $|\psi\rangle$ , and their state spaces may differ in dimension: $A$ a qubit, say, and $B$ a three level system. The exercise is the Schmidt decomposition in this setting, the state written as one sum of products of orthonormal states of $A$ and of $B$ with positive weights, with no more terms than the smaller of the two dimensions.^✦ For the qubit and the three level system that means two terms at most, whatever the state.

The library states it as two theorems: the decomposition with that bound on its number of terms, and the same bound on the Schmidt number of the state, defined below.

A Composite System

A quantum system $A$ is given by its state space $\mathcal{H}_A$ , a finite dimensional complex Hilbert space, and its dimension $\dim A$ is the dimension of $\mathcal{H}_A$ as a complex vector space. A pure state $|\psi\rangle$ of $A$ is a vector in $\mathcal{H}_A$ 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 their state spaces, $\mathcal{H}_{A \otimes B} = \mathcal{H}_A \otimes \mathcal{H}_B .$ (1)^✦

The Schmidt Decomposition

Theorem 1. Let $|\psi\rangle$ be a pure state of the composite system $A \otimes B$ . Then there are a number of terms $n \le \min(\dim A, \dim B)$ , positive reals $\lambda_1, \dots, \lambda_n$ with $\sum_i \lambda_i^2 = 1$ , and orthonormal vectors $|1_A\rangle, \dots, |n_A\rangle$ in $\mathcal{H}_A$ and $|1_B\rangle, \dots, |n_B\rangle$ in $\mathcal{H}_B$ , such that $|\psi\rangle = \sum_{i=1}^{n} \lambda_i \, |i_A\rangle |i_B\rangle .$ (2)^✦

The Schmidt Number

The coefficient operator $M$ of a pure state $|\psi\rangle$ of $A \otimes B$ is the linear operator from $\mathcal{H}_B$ to $\mathcal{H}_A$ that reads $|\psi\rangle$ as a matrix in the standard orthonormal basis $\{|k\rangle\}$ of $\mathcal{H}_B$ : expanding the state over that basis, it sends each basis vector to the vector of $\mathcal{H}_A$ that multiplies it, $|\psi\rangle = \sum_k |v_k\rangle |k\rangle , \quad |v_k\rangle \in \mathcal{H}_A , \qquad M |k\rangle = |v_k\rangle .$ (3)^✦

The Schmidt number $\mathrm{Sch}(\psi)$ of a pure state $|\psi\rangle$ of $A \otimes B$ is the number of its nonzero Schmidt coefficients, the singular values $\sigma_j(M)$ of its coefficient operator: $\mathrm{Sch}(\psi) = \#\{\, j : \sigma_j(M) \neq 0 \,\} .$ (4)^✦

Theorem 2. Let $|\psi\rangle$ be a pure state of the composite system $A \otimes B$ . Then $\mathrm{Sch}(\psi) \le \min(\dim A, \dim B) .$ (5)^✦

What Is Checked

Both theorems are stated over a pure state $|\psi\rangle$ of the composite system $A \otimes B$ and over the dimensions $\dim A$ and $\dim B$ ; the second is stated also over the Schmidt number (4), which rests on the coefficient operator (3).