The Exercise

Nielsen’s theorem says when Alice and Bob can turn a shared pure state $|\psi\rangle$ into another pure state $|\varphi\rangle$ by local operations and classical communication: exactly when the reduced state $\rho$ of Alice’s system in $|\psi\rangle$ is majorized by her reduced state $\sigma$ in $|\varphi\rangle$. The exercise is the direction from majorization to conversion. Given that $\rho$ is majorized by $\sigma$, find a measurement on Alice’s system whose every outcome turns $\rho$ into $\sigma$, without assuming that $\rho$ is invertible.

States and Measurements

A quantum system $S$ is a physical object whose mathematical realisation is its state space, a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ A measurement $\{M_j\}$ on $S$ with a finite set of outcomes is a family of operators $M_j$ on the state space of $S$, one for each outcome $j$, satisfying the completeness relation $$\sum_j M_j^\dagger M_j = 1 .$$(1)^✦

Majorization

The spectrum $\lambda(\rho)$ of a state $\rho$ of $S$ is the list $\lambda_1, \dots, \lambda_d$ of the eigenvalues of $\rho$, each repeated according to its multiplicity, where $d$ is the dimension of the state space of $S$. Let $\lambda_1^\downarrow(\rho) \ge \dots \ge \lambda_d^\downarrow(\rho)$ be the spectrum of $\rho$ in decreasing order. A state $\rho$ of $S$ is majorized by a state $\sigma$ of $S$, written $\rho \prec \sigma$, when the spectrum of $\rho$ is majorized by the spectrum of $\sigma$: for every $k$, the $k$ largest eigenvalues of $\rho$ sum to at most the sum of the $k$ largest eigenvalues of $\sigma$, $$\sum_{i=1}^{k} \lambda_i^\downarrow(\rho) \le \sum_{i=1}^{k} \lambda_i^\downarrow(\sigma), \qquad k = 1, \dots, d .$$(2)^✦

Conversion by a Measurement

A state $\rho$ of $S$ is convertible by local operations and classical communication (LOCC) to a state $\sigma$ of $S$, written $\rho \rightsquigarrow \sigma$, when a single measurement on $S$ turns $\rho$ into $\sigma$ whatever its outcome: there are a measurement $\{M_j\}$ on $S$ and a probability distribution $p_j$ on its outcomes, $p_j \ge 0$ and $\sum_j p_j = 1$, such that $$M_j \rho M_j^\dagger = p_j \sigma \qquad \text{for every outcome } j .$$(3)^✦

The Claim

The library states the exercise’s direction for any two states of one quantum system, the measurement acting on that system.

What Is Checked

The claim’s hypothesis is majorization, (2); its conclusion is conversion by LOCC, (3), with a measurement satisfying (1).

A state is self adjoint, so its spectrum is a list of real numbers, and the library states that its entries are nonnegative and sum to one.