The Exercise

A pure state $|\psi\rangle$ of two systems $A$ and $B$ has a Schmidt decomposition $|\psi\rangle = \sum_i \lambda_i |i_A\rangle|i_B\rangle$ , with orthonormal states $|i_A\rangle$ of $A$ , orthonormal states $|i_B\rangle$ of $B$ and nonnegative real coefficients $\lambda_i$ . The number of nonzero $\lambda_i$ is the Schmidt number $\mathrm{Sch}(\psi)$ .^✦ Problem 2.2 asks for three of its properties. It equals the rank of the reduced density operator $\rho^A = \operatorname{tr}_B(|\psi\rangle\langle\psi|)$ .^✦ It is at most the number of terms in any expansion $|\psi\rangle = \sum_j |a_j\rangle|b_j\rangle$ with vectors $|a_j\rangle$ of $A$ and $|b_j\rangle$ of $B$ , normalized or not.^✦ And when $|\psi\rangle = \alpha|\varphi\rangle + \beta|\gamma\rangle$ is a superposition of two pure states, $\mathrm{Sch}(\psi) \ge |\mathrm{Sch}(\varphi) - \mathrm{Sch}(\gamma)|$ .^✦

Two Systems and Their States

A quantum system $S$ is a physical system whose state space $\mathcal{H}_S$ is a finite dimensional complex Hilbert space.^✦ The composite system $AB$ of two quantum systems $A$ and $B$ is the quantum system whose state space is the tensor product $\mathcal{H}_A \otimes \mathcal{H}_B$ .^✦

A pure state $|\psi\rangle$ of $S$ is a unit vector in $\mathcal{H}_S$ .^✦ A state $\rho$ of $S$ is a positive operator of unit trace on $\mathcal{H}_S$ .^✦ The projector $|\psi\rangle\langle\psi|$ onto a pure state $|\psi\rangle$ of $S$ is the operator on $\mathcal{H}_S$ that takes $|\phi\rangle$ to $\langle\psi|\phi\rangle|\psi\rangle$ . The rank of a state $\rho$ of $S$ is the dimension of the range of $\rho$ , the number of its nonzero eigenvalues counted with multiplicity: $\operatorname{rank}\rho = \dim \rho(\mathcal{H}_S).$ (1)^✦

The reduced state $\rho^A$ of a state $\rho$ of $AB$ is the state of $A$ obtained from $\rho$ by the partial trace over $B$ , taken against an orthonormal basis $\{|k\rangle\}$ of $\mathcal{H}_B$ : $\rho^A = \operatorname{tr}_B \rho = \sum_k (I_A \otimes \langle k|)\,\rho\,(I_A \otimes |k\rangle).$ (2)^✦

The Schmidt Number

In the same basis a pure state of $AB$ expands as $|\psi\rangle = \sum_k |v_k\rangle|k\rangle$ , with vectors $|v_k\rangle$ in $\mathcal{H}_A$ . The coefficient operator $C_\psi$ of $|\psi\rangle$ is the operator from $\mathcal{H}_B$ to $\mathcal{H}_A$ that takes each $|k\rangle$ to $|v_k\rangle$ : $C_\psi = \sum_k |v_k\rangle\langle k|.$ (3)^✦

The Schmidt number $\mathrm{Sch}(\psi)$ of a pure state $|\psi\rangle$ of $AB$ is the number of nonzero singular values $s_i(C_\psi)$ of its coefficient operator (3), counted with multiplicity: $\mathrm{Sch}(\psi) = \#\{\, i : s_i(C_\psi) \neq 0 \,\}.$ (4)^✦ These nonzero singular values are the nonzero coefficients $\lambda_i$ of the exercise.

The Claim

Theorem 1. Let $|\psi\rangle$ be a pure state of $AB$ . Then $\mathrm{Sch}(\psi) = \operatorname{rank}\, \operatorname{tr}_B(|\psi\rangle\langle\psi|).$ (5)^✦

Theorem 2. Let $|\psi\rangle$ be a pure state of $AB$ , and let $|\psi\rangle = \sum_{j=1}^{m} |a_j\rangle|b_j\rangle$ for vectors $|a_j\rangle$ in $\mathcal{H}_A$ and $|b_j\rangle$ in $\mathcal{H}_B$ , not necessarily normalized. Then $\mathrm{Sch}(\psi) \le m.$ (6)^✦

Theorem 3. Let $|\varphi\rangle$ , $|\gamma\rangle$ and $|\psi\rangle$ be pure states of $AB$ , and let $\alpha$ and $\beta$ be nonzero complex numbers with $|\psi\rangle = \alpha|\varphi\rangle + \beta|\gamma\rangle$ . Then $|\mathrm{Sch}(\varphi) - \mathrm{Sch}(\gamma)| \le \mathrm{Sch}(\psi).$ (7)^✦

What Is Checked

The claims (5), (6) and (7) are stated over pure states of the composite system $AB$ and their Schmidt numbers (4); the first is stated also over the rank (1) of the reduced state (2) of $|\psi\rangle\langle\psi|$ .

The projector onto a unit vector is a state, and the partial trace of a state of $AB$ is a state of $A$ , so $\operatorname{tr}_B(|\psi\rangle\langle\psi|)$ is a state of $A$ .