The Exercise

Alice and Bob share a pure state $|\psi\rangle$ of two systems, $A$ on her side and $B$ on his. They want to turn it into another pure state $|\varphi\rangle$ of the same two systems by local operations and classical communication (LOCC): each acts only on their own system, and they may tell each other what their measurements gave. The exercise is to show that no such protocol raises the Schmidt number: if $|\psi\rangle$ can be turned into $|\varphi\rangle$, the Schmidt number of $|\varphi\rangle$ is at most that of $|\psi\rangle$.^✦

The library states the exercise for pure states of a composite system $A \otimes B$ and carries out the conversion on Alice’s side alone, by one measurement on $A$ acting on her reduced state, the state of $A$ left after $B$ is traced out.

A Quantum System

A quantum system $A$ is a physical system whose state space is 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 quantum system $A$ is a unit vector in the state space of $A$.^✦ A state $\rho$ of $A$, pure or mixed, is a positive operator of unit trace on the state space of $A$.^✦ Each pure state $|\psi\rangle$ of $A$ determines the state $|\psi\rangle\langle\psi|$ of $A$, the rank one projector onto $|\psi\rangle$.^✦

The reduced state $\operatorname{Tr}_B(\rho)$ on $A$ of a state $\rho$ of $A \otimes B$ is the state of $A$ obtained by tracing out $B$ against the standard orthonormal basis $\{|k\rangle\}$ of the state space of $B$, $$\operatorname{Tr}_B(\rho) = \sum_k \bigl(I \otimes \langle k|\bigr)\,\rho\, \bigl(I \otimes |k\rangle\bigr).$$(1)^✦ For a pure state $|\psi\rangle$ of $A \otimes B$, Alice’s reduced state is $\rho_\psi = \operatorname{Tr}_B(|\psi\rangle\langle\psi|)$.

The Schmidt Number

The coefficient operator $C_\psi$ of a pure state $|\psi\rangle = \sum_k |v_k\rangle|k\rangle$ of $A \otimes B$, expanded in the standard orthonormal basis $\{|k\rangle\}$ of $B$ with each $|v_k\rangle$ in the state space of $A$, is the linear operator from the state space of $B$ to that of $A$ with $$C_\psi |k\rangle = |v_k\rangle .$$(2)^✦

The Schmidt number $\mathrm{Sch}(\psi)$ of a pure state $|\psi\rangle$ of $A \otimes B$ is the number of its nonzero Schmidt coefficients, which are the singular values $s_i(C_\psi)$ of its coefficient operator: $$\mathrm{Sch}(\psi) = \#\{\, i : s_i(C_\psi) \neq 0 \,\}.$$(3)^✦

Conversion by LOCC

A measurement on a quantum system $A$ with a finite set of outcomes is a family of measurement operators $M_j$ on the state space of $A$, one for each outcome $j$, that satisfies the completeness relation $$\sum_j M_j^\dagger M_j = I .$$(4)^✦

A state $\rho$ of a quantum system $A$ is convertible by LOCC to a state $\sigma$ of $A$, written $\rho \rightsquigarrow \sigma$, when a measurement $\{M_j\}$ on $A$ leaves $A$ in $\sigma$ whatever its outcome: there is a probability distribution $\{p_j\}$ on the outcomes, $p_j \ge 0$ and $\sum_j p_j = 1$, with $p_j$ the probability of outcome $j$, such that $$M_j \rho M_j^\dagger = p_j\, \sigma \quad \text{for every outcome } j .$$(5)^✦

The Claim

What Is Checked

The claim is stated over pure states of the composite system $A \otimes B$, the states $|\psi\rangle\langle\psi|$ they determine and their reduced states $\rho_\psi$ on $A$, the conversion $\rho \rightsquigarrow \sigma$ by a measurement on $A$, and the Schmidt number read from the coefficient operator $C_\psi$.

About these constructions the library proves that the projector onto a unit vector is a density operator, so that $|\psi\rangle\langle\psi|$ is a state of $A \otimes B$, and that the partial trace of a density operator is again a density operator, so that $\rho_\psi$ is a state of $A$.