The Exercise

A source prepares a quantum system $Q$ in one of two pure states $|\psi_1\rangle$ and $|\psi_2\rangle$, each with probability one half; the two are distinct but not orthogonal. A cloning device would take the system, together with a second system of the same kind in a fixed blank state $|s\rangle$, and return the pair in the state $|\psi_i\rangle|\psi_i\rangle$, whichever $i$ the source chose. Fed by the source, it would produce a new ensemble of two states of the pair, again with probability one half each. The exercise compares the Holevo quantity $\chi$, the entropy of an ensemble’s mixture less the average entropy of its members, for the two ensembles, finds the second larger, and shows that no device clones both states.

A Quantum System

A quantum system $Q$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ The composite system $Q \otimes R$ of two quantum systems $Q$ and $R$ is the quantum system whose state space is the tensor product of their state spaces.^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on the state space of $Q$.^✦

A pure state $|\psi\rangle$ of $Q$ is a unit vector in the state space of $Q$, physically determined up to a global phase.^✦ A pure state $|\psi\rangle$ of $Q$ determines the state $|\psi\rangle\langle\psi|$ of $Q$, the projector onto its vector.^✦ The product of a pure state $|\psi\rangle$ of $Q$ and a pure state $|\varphi\rangle$ of $R$ is the pure state $|\psi\rangle \otimes |\varphi\rangle$ of $Q \otimes R$ whose vector is the tensor product of the two; two copies of one state are written $|\psi\psi\rangle = |\psi\rangle \otimes |\psi\rangle$. The product state $\rho \otimes \sigma$ of a state $\rho$ of $Q$ and a state $\sigma$ of $R$ is the state of $Q \otimes R$ whose density operator is the tensor product of those of $\rho$ and $\sigma$.^✦

Evolution and Channels

A unitary $U$ of $Q$ is an operator on the state space of $Q$ with $U^\dagger U = U U^\dagger = 1$, the evolution of $Q$ as a closed system.^✦ The evolution of a state $\rho$ of $Q$ under a unitary $U$ of $Q$ is the state $U \rho U^\dagger$ of $Q$.^✦ The partial trace $\operatorname{Tr}_2$ of a state $\rho$ of $Q \otimes R$ is the reduced state of $Q$ obtained by tracing out the second factor.^✦

The dilated channel $\mathcal{E}_{U,\omega}$ on $Q$, built from an ancilla $C$ in a pure state $|\omega\rangle$ and a unitary $U$ of $Q \otimes C$, sends a state $\rho$ of $Q$ to $$\mathcal{E}_{U,\omega}(\rho) = \operatorname{Tr}_2\bigl(U\,(\rho \otimes |\omega\rangle\langle\omega|)\,U^\dagger\bigr).$$(1)^✦

Entropy and Fidelity

The fidelity of two states $\rho$ and $\sigma$ of $Q$ is the nonnegative real number $$F(\rho,\sigma) = \operatorname{tr}\sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}},$$(2)^✦ each square root of a positive operator taken eigenvalue by eigenvalue. For pure states, $F(\psi_1,\psi_2)$ stands for the fidelity of $|\psi_1\rangle\langle\psi_1|$ and $|\psi_2\rangle\langle\psi_2|$.

The mixture of states $\rho_i$ of $Q$, indexed by a finite set, with probabilities $p_i$ that are nonnegative and sum to one, is the state $\sum_i p_i \rho_i$ of $Q$.^✦ The von Neumann entropy of a state $\rho$ of $Q$ is $S(\rho) = -\operatorname{tr}(\rho \log \rho)$, measured in nats.^✦ The Holevo quantity of an ensemble $\{p_i, \rho_i\}$ of states of $Q$ is the entropy of its mixture less the average entropy of its members, $$\chi\bigl(\{p_i, \rho_i\}\bigr) = S\Bigl(\sum_i p_i \rho_i\Bigr) - \sum_i p_i\, S(\rho_i).$$(3)^✦

The Claim

In the fidelity, the two conditions on the states of the exercise read $0 < F(\psi_1,\psi_2) < 1$.

The cloning device acts on the pair $Q \otimes Q$, so it is a dilated channel on $Q \otimes Q$, with its unitary acting on $(Q \otimes Q) \otimes C$.

What Is Checked

The first claim is stated over the Holevo quantity (3) of two ensembles with weights one half, built from the mixture and the von Neumann entropy, over the states $|\psi_i\rangle\langle\psi_i|$ and the products $|\psi_i\psi_i\rangle$, and over the fidelity (2). The second is stated over the dilated channel (1) on $Q \otimes Q$, applied to the product state $|\psi_i\rangle\langle\psi_i| \otimes |s\rangle\langle s|$.

The library proves that each of these constructions gives a state. The projector onto a unit vector is a density operator, and so are the tensor product of two density operators and a convex combination of density operators; the tensor product of two unit vectors is a unit vector. Conjugation by a unitary and the partial trace take density operators to density operators, so the dilated channel sends states of $Q \otimes Q$ to states.