The Exercise

A copying machine takes a system $S$ in some state, together with a blank system $B$ prepared in a fixed state $\sigma$, and returns two systems of the kind of $S$, each in the state $S$ came in with. The machine is linear: fed a mixture of two inputs, it returns the same mixture of the two outputs. Suppose it copies two different states $\rho_1$ and $\rho_2$ of $S$. Show that it copies no mixture of $\rho_1$ and $\rho_2$ in which each occurs with a probability strictly between $0$ and $1$.^✦

Systems and States

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The composite system $S \otimes T$ of two quantum systems $S$ and $T$ is the quantum system whose state space is the tensor product of the state spaces of $S$ and $T$.^✦

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, a density operator.^✦

The mixture of two states $\rho$ and $\rho'$ of $S$ with weight $p$, a real number with $0 \le p \le 1$, is the state of $S$ with density operator $$p\rho + (1-p)\rho',$$(1)^✦ the state prepared by choosing $\rho$ with probability $p$ and $\rho'$ otherwise. The product state $\rho \otimes \sigma$ of a state $\rho$ of $S$ and a state $\sigma$ of $T$ is the state of $S \otimes T$ whose density operator is the tensor product of the two, acting as $\rho$ on the first factor and as $\sigma$ on the second.^✦

The Claim

The machine is a map $E$ taking each state of $S \otimes B$ to a state of $S \otimes S$. The map $E$ respects mixtures if, for every weight $q$ with $0 \le q \le 1$ and all states $\tau_1$ and $\tau_2$ of $S \otimes B$, $$E\bigl(q\tau_1 + (1-q)\tau_2\bigr) = q\,E(\tau_1) + (1-q)\,E(\tau_2).$$(2) Let $\sigma$ be a state of $B$, the blank state. The map $E$ copies a state $\rho$ of $S$ if $$E(\rho \otimes \sigma) = \rho \otimes \rho.$$(3)

What Is Checked

The claim is stated over the states of the composite systems $S \otimes B$ and $S \otimes S$: every input and every target in it is a product state.

About these constructions the library proves that a convex combination of two density operators with weights $p$ and $1-p$ is again positive with trace one, and that so is the tensor product of two density operators.