The Exercise

Theorem 11.15 gathers three inequalities for the entropies of a state shared by quantum systems $A$, $B$ and $C$. Conditioning reduces entropy. Write $S(X)$ for the entropy of the system $X$ in the shared state, with $S(A,B,C)$ for the three systems together; the entropy of $A$ given both $B$ and $C$, $S(A|B,C) = S(A,B,C) - S(B,C)$, is at most its entropy given $B$ alone, $S(A|B) = S(A,B) - S(B)$. Discarding a system never raises mutual information, so $A$ shares no more with $B$ than with $B$ and $C$ together, $S(A:B) \le S(A:B,C)$.^✦ Nor does an operation acting on $B$ alone: if it turns $A$ and $B$ into $A'$ and $B'$, then $S(A':B') \le S(A:B)$.^✦

Systems and States

A quantum system $A$ is a physical object described by its state space, a finite dimensional complex Hilbert space.^✦ The composite system $A \otimes B$ of two systems $A$ and $B$ is the system whose state space is the tensor product of their state spaces.^✦

A state $\rho$ of $A$ is a positive operator of unit trace on the state space of $A$.^✦ A pure state $|\psi\rangle$ of $A$ is a unit vector in the state space of $A$, fixed physically up to a global phase^✦; it determines the state $|\psi\rangle\langle\psi|$ of $A$, the projector onto that vector.^✦ The product state $\rho \otimes \sigma$ of a state $\rho$ of $A$ and a state $\sigma$ of $B$ is the state of $A \otimes B$ whose operator, its density operator, is the tensor product of the operators of $\rho$ and $\sigma$.^✦

The reduced state $\rho_A$ of a state $\rho$ of $A \otimes B$ is the state $\operatorname{Tr}_B(\rho)$ of $A$, the partial trace of $\rho$ over the second factor^✦; the reduced state $\rho_B = \operatorname{Tr}_A(\rho)$ of $B$ is its partial trace over the first.^✦

An identification $e$ of a system $A$ with a system $B$ is a linear isometry of the state space of $A$ onto that of $B$^✦; its inverse $e^{-1}$ identifies $B$ with $A$^✦, and along $e$ a state $\rho$ of $A$ is relabelled as the state $e \rho e^{-1}$ of $B$.^✦

Three systems compose in two bracketings. The regrouping $\alpha_{A,B,C}$ is the identification of $A \otimes (B \otimes C)$ with $(A \otimes B) \otimes C$ that moves the brackets, $$\alpha_{A,B,C} \colon |u\rangle \otimes (|v\rangle \otimes |w\rangle) \mapsto (|u\rangle \otimes |v\rangle) \otimes |w\rangle .$$(1)^✦

A state $\rho$ of $A \otimes (B \otimes C)$ has reduced states $\rho_A$ of $A$ and $\rho_{BC}$ of $B \otimes C$, the partial traces over the other factor. The reduced state $\rho_B$ of $B$ is the reduced state of $\rho_{BC}$ on $B$, and the reduced state $\rho_{AB}$ of $A \otimes B$ is the partial trace over $C$ of $\rho$ relabelled along $\alpha_{A,B,C}$ as a state of $(A \otimes B) \otimes C$.

Entropy and Mutual Information

The von Neumann entropy of a state $\rho$ of $A$ is the Shannon entropy of the eigenvalues of $\rho$, measured in nats: $$S(\rho) = -\operatorname{tr}(\rho \log \rho) .$$(2)^✦ The mutual information of a state $\rho$ of $A \otimes B$ is $$S(A:B)_\rho = S(\rho_A) + S(\rho_B) - S(\rho) .$$(3)^✦ For a state $\rho$ of $A \otimes (B \otimes C)$ the second system is $B \otimes C$, and the mutual information is written $S(A:B,C)_\rho$.

An Operation on One System

A unitary $U$ of a system $A$ is an operator on the state space of $A$ with $U^\dagger U = U U^\dagger = 1$^✦; under it a state $\rho$ of $A$ evolves to the state $U \rho U^\dagger$.^✦ The identity unitary $1_A$ of $A$ is the identity operator.^✦ For a unitary $U$ of $A$ and a unitary $V$ of $B$, the product unitary $U \otimes V$ of $A \otimes B$ is the tensor product of the two operators.^✦

A dilated channel $\mathcal{E}_{U,\omega}$ on a system $B$ is an operation taking states of $B$ to states of $B$, given by an ancilla, a quantum system $C$ in a pure state $|\omega\rangle$, and a unitary $U$ of $B \otimes C$. Write $\alpha = \alpha_{A,B,C}$. On a state $\rho$ of $A \otimes B$ the operation adjoins the ancilla in the product state $\rho \otimes |\omega\rangle\langle\omega|$ of $(A \otimes B) \otimes C$, relabels it along $\alpha^{-1}$ as a state of $A \otimes (B \otimes C)$, evolves it by the product unitary $1_A \otimes U$, relabels it back along $\alpha$ and traces out $C$: $$\rho' = (\mathrm{id} \otimes \mathcal{E}_{U,\omega})(\rho) = \operatorname{Tr}_C \bigl[ \alpha (1_A \otimes U) \alpha^{-1} (\rho \otimes |\omega\rangle\langle\omega|) \alpha (1_A \otimes U)^\dagger \alpha^{-1} \bigr] .$$(4)^✦ The systems $A$ and $B$ of $\rho'$ are the book’s $A'$ and $B'$.

The Claim

What Is Checked

The three inequalities are stated over states of composite systems and their partial traces, the regrouping $\alpha_{A,B,C}$ with the relabelling of states along it and its inverse, the von Neumann entropy and the mutual information, and, for the third, the operation (4) built from a pure ancilla, the product unitary $1_A \otimes U$ and a partial trace.

Each of these constructions returns a state. The tensor product of two density operators is a density operator, and so is the projector $|\omega\rangle\langle\omega|$ onto a unit vector; conjugating a density operator by a unitary or by an isometry gives a density operator, and so does taking a partial trace; the tensor product of two unitaries is a unitary.