The Exercise

Two systems $A$ and $B$ share a pure state $|\psi\rangle$, and $\rho_\psi$ is the state of $A$ alone. Copies of $\rho_\psi$ can be converted by local operations and classical communication (LOCC) into states close to maximally mixed ones; the distillable entanglement $D(\rho_\psi)$ is the best rate at which this can be done, per copy, in the limit of many copies. Exercise 12.23 is to show that this rate cannot exceed the von Neumann entropy $S(\rho_\psi)$.^✦

A Quantum System

A quantum system $A$ is a physical system whose state space $\mathcal{H}_A$ is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $A$ is a positive operator of unit trace on $\mathcal{H}_A$; such an operator is called a density operator.^✦ A pure state $|\psi\rangle$ of $A$ is a unit vector in $\mathcal{H}_A$, and the state it determines is the projector $|\psi\rangle\langle\psi|$ onto that vector.

The composite system $A \otimes B$ of two quantum systems $A$ and $B$ is the quantum system whose state space is $\mathcal{H}_A \otimes \mathcal{H}_B$.^✦ The reduced state on $A$ of a state $\rho$ of $A \otimes B$ is the state $\operatorname{tr}_B \rho$ of $A$ given by the partial trace over $B$.^✦ The reduced state $\rho_\psi$ on $A$ of a pure state $|\psi\rangle$ of $A \otimes B$ is $$\rho_\psi = \operatorname{tr}_B |\psi\rangle\langle\psi| .$$(1)

The von Neumann entropy of a state $\rho$ of $A$ is $$S(\rho) = -\operatorname{tr}(\rho \log \rho),$$(2)^✦ with $\log$ the natural logarithm here and below.

Many Copies

The composite system $\bigotimes_i A_i$ of a finite family of quantum systems $A_i$ is the quantum system whose state space is the tensor product of the spaces $\mathcal{H}_{A_i}$; the system $A^{\otimes m}$ of $m$ copies of $A$ is the composite of $m$ factors all equal to $A$. The product state $\bigotimes_i \rho_i$ of states $\rho_i$ of the $A_i$ is the state of $\bigotimes_i A_i$ given by the tensor product of the operators $\rho_i$.^✦ Taking all $m$ factors equal to a state $\rho$ of $A$ gives the product state $\rho^{\otimes m}$ of $A^{\otimes m}$.^✦

Distillation

A measurement on $A$ with a finite set of outcomes is a family of operators $M_j$ on $\mathcal{H}_A$, one for each outcome $j$, satisfying the completeness relation $\sum_j M_j^\dagger M_j = I$.^✦ A state $\rho$ of $A$ converts to a state $\sigma$ of $A$ by LOCC, written $\rho \rightsquigarrow \sigma$, when some measurement on $A$ leaves the system in $\sigma$ whatever its outcome: outcome $j$ occurs with a probability $p_j$, the $p_j$ being nonnegative and summing to one, and $$M_j \rho M_j^\dagger = p_j\, \sigma \qquad \text{for every outcome } j.$$(3)^✦

The trace distance between two states $\rho$ and $\sigma$ of $A$ is $$D(\rho, \sigma) = \tfrac{1}{2} \operatorname{tr} |\rho - \sigma| .$$(4)^✦

The rank of a state $\mu$ of $A$ is the dimension of the range of $\mu$.^✦ A state $\mu$ is maximally mixed on $K$ dimensions when it has rank $K$ and entropy $S(\mu) = \log K$.^✦

A real number $R$ is an achievable distillation rate for a state $\rho$ of $A$ when, for every number of copies $m$, there are a natural number $K_m$ and states $\sigma_m$ and $\mu_m$ of $A^{\otimes m}$ with $\rho^{\otimes m} \rightsquigarrow \sigma_m$ and $\mu_m$ maximally mixed on $K_m$ dimensions, such that, as $m \to \infty$, $$D(\sigma_m, \mu_m) \to 0, \qquad \frac{\log K_m}{m} \to R .$$(5)^✦ The distillable entanglement of $\rho$ is the supremum of its achievable rates, $$D(\rho) = \sup \{ R : R \text{ is an achievable distillation rate for } \rho \} .$$(6)^✦

The Claim

What Is Checked

The claim is stated over the reduced state (1) of a pure state of $A \otimes B$ and two numbers attached to it: its von Neumann entropy (2) and its distillable entanglement (6).

The library proves that the projector onto a unit vector is a density operator and that the partial trace of a density operator is again one, so $\rho_\psi$ is a state of $A$. It proves the same of a tensor product of density operators, which makes $\rho^{\otimes m}$ a state of $A^{\otimes m}$ for every state $\rho$ of $A$.