The Exercise

Two systems $A$ and $B$ are prepared in a joint pure state, either $|\psi\rangle$ or $|\varphi\rangle$, and the two states have identical Schmidt coefficients. The exercise is to show that a unitary acting on $A$ alone, together with a unitary acting on $B$ alone, turns $|\varphi\rangle$ into $|\psi\rangle$.

Systems and Pure States

A quantum system $A$ is a physical system with a state space, a finite dimensional complex Hilbert space.^✦ 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 $B$.^✦

A pure state $|\psi\rangle$ of a system $A$ is a vector of unit norm in the state space of $A$, $\langle \psi | \psi \rangle = 1$.^✦

Schmidt Coefficients

Two pure states $|\psi\rangle$ and $|\varphi\rangle$ of a composite system $A \otimes B$ have the same Schmidt coefficients when they have Schmidt decompositions with the same coefficients: there are positive real numbers $\lambda_1, \dots, \lambda_n$, orthonormal vectors $|a_1\rangle, \dots, |a_n\rangle$ and $|a'_1\rangle, \dots, |a'_n\rangle$ in the state space of $A$, and orthonormal vectors $|b_1\rangle, \dots, |b_n\rangle$ and $|b'_1\rangle, \dots, |b'_n\rangle$ in the state space of $B$, such that $$|\psi\rangle = \sum_{i=1}^{n} \lambda_i \, |a_i\rangle |b_i\rangle , \qquad |\varphi\rangle = \sum_{i=1}^{n} \lambda_i \, |a'_i\rangle |b'_i\rangle .$$(1)^✦

Local Unitaries

A unitary $U$ of a system $A$ is an operator on the state space of $A$ with $I$ the identity operator on that space, such that $$U^\dagger U = U U^\dagger = I ,$$(2)^✦ the evolution of $A$ as a closed system. Such a unitary takes a pure state $|\psi\rangle$ of $A$ to the pure state $U|\psi\rangle$.^✦

For a unitary $U$ of $A$ and a unitary $V$ of $B$, the product unitary $U \otimes V$ is the unitary of the composite system $A \otimes B$ whose operator is the tensor product of the two, acting on a vector $|a\rangle$ of the state space of $A$ and a vector $|b\rangle$ of the state space of $B$ as $$(U \otimes V)\, |a\rangle |b\rangle = U|a\rangle \, V|b\rangle .$$(3)^✦

The Claim

What Is Checked

The claim is stated over pure states of the composite system $A \otimes B$, the condition (1) of equal Schmidt coefficients, and the product unitary $U \otimes V$ of (3) applied to the pure state $|\varphi\rangle$.

The library proves of these constructions that the tensor product of two operators obeying (2) obeys it again on the state space of $A \otimes B$, and that a unitary preserves norms.