The Exercise

A system $Q$ starts in a state $\rho$, purified by a reference system $R$ so that $RQ$ is in a pure state. Two quantum operations then act on $Q$ in turn, first $\mathcal{E}_1$ and then $\mathcal{E}_2$. Each couples $Q$ through a unitary to an environment of its own, $E_1$ for the first and $E_2$ for the second, prepared in a pure state, and the reference is left alone. After the first operation the system is in the state $\rho' = \mathcal{E}_1(\rho)$, after the second in $\rho'' = \mathcal{E}_2(\rho')$.

The exercise bounds the entropies left at the end against one another: those of the system, of reference and system, of reference and second environment, and the entropy exchanges of the two operations. It also bounds the coherent information of the two operations together by the entropy $S(\rho)$ of the input.

The library states the bounds for a single operation and for two in turn.

A Quantum System

A quantum system $A$ is described by its state space, a finite dimensional complex Hilbert space^✦, and the dimension $\dim A$ of $A$ is the complex dimension of that space.^✦ The composite system $A \otimes B$ of two systems $A$ and $B$ has as its state space 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 $\psi$ of $A$ is a unit vector in the state space of $A$^✦, and the state it determines is the projector $|\psi\rangle\langle\psi|$.^✦ 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 density operator is the tensor product of the two.^✦

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

An identification $e$ of a system $A$ with a system $B$ is a linear isometry of the state space of $A$ onto that of $B$.^✦ It carries a state $\rho$ of $A$ to the state $e \rho e^{-1}$ of $B$^✦ and a pure state $\psi$ of $A$ to the pure state $e\psi$ of $B$^✦, and its inverse $e^{-1}$ identifies $B$ with $A$.^✦ Two identifications reorder composites: the swap of $A \otimes B$ with $B \otimes A$, $u \otimes v \mapsto v \otimes u$^✦, and the regrouping $\alpha$ of $A \otimes (B \otimes C)$ with $(A \otimes B) \otimes C$, $u \otimes (v \otimes w) \mapsto (u \otimes v) \otimes w$.^✦

The reduced state $\rho_A = \operatorname{Tr}_B \rho$ of a state $\rho$ of $A \otimes B$ is the state of $A$ left by the partial trace over $B$^✦; the reduced state $\rho_B = \operatorname{Tr}_A \rho$ is the state of $B$ left by the partial trace over $A$.^✦

The von Neumann entropy of a state $\rho$ is $$S(\rho) = -\operatorname{tr}(\rho \log \rho),$$(1)^✦ in nats. For a state $\rho$ of $A \otimes B$ one writes $S(A,B) = S(\rho)$ and $S(B) = S(\rho_B)$. The conditional entropy of $\rho$ is $S(A|B) = S(A,B) - S(B)$^✦, and its coherent information is the negative of that, $$I(A\rangle B) = -S(A|B) = S(B) - S(A,B).$$(2)^✦

Two Operations in Turn

The dilated channel $\mathcal{E}_{U,\omega}$ on a system $Q$, built from an environment $E$, a pure state $\omega$ of $E$ and a unitary $U$ of $Q \otimes E$, sends a state $\rho$ of $Q$ to $$\mathcal{E}_{U,\omega}(\rho) = \operatorname{Tr}_E\bigl(U(\rho \otimes |\omega\rangle\langle\omega|)U^\dagger\bigr).$$(3)^✦ Every trace preserving quantum operation on $Q$ arises this way, by Stinespring’s theorem.

An operation can also act on one factor of a composite. For a state $\sigma$ of $A \otimes B$, a unitary $U$ of $B \otimes C$ and a pure state $\omega$ of an ancilla $C$, the dilated joint state $\sigma_{ABC}$ is the state of $(A \otimes B) \otimes C$ $$\sigma_{ABC} = W(\sigma \otimes |\omega\rangle\langle\omega|)W^\dagger, \qquad W = \alpha\,(1 \otimes U)\,\alpha^{-1},$$(4)^✦ in which the product unitary $1 \otimes U$ of the identity of $A$ with $U$ acts on $B$ and $C$ together, the factors regrouped by $\alpha$ before and after. Tracing out the ancilla leaves the operation run on $B$ alone, $(\mathrm{id} \otimes \mathcal{E}_{U,\omega})(\sigma) = \operatorname{Tr}_C\,\sigma_{ABC}$.^✦

The canonical purification of a state $\rho = \sum_i p_i |i\rangle\langle i|$ of $Q$, written in an orthonormal eigenbasis, is the pure state $\sum_i \sqrt{p_i}\,|i\rangle|i_R\rangle$ of $Q \otimes R$.^✦ Here the reference $R$ is the system with state space $\mathbb{C}^d$, $d = \dim Q$, and $|i_R\rangle$ is its standard basis. The swap carries the canonical purification to the reference purification $|RQ\rangle$, a pure state of $R \otimes Q$ with the reference first.^✦

The entropy exchange $S(\rho,\mathcal{E})$ of a state $\rho$ of $Q$ under an operation $\mathcal{E}$ on $Q$ is the entropy $S(R'Q')$ of the state $\rho^{R'Q'}$ of reference and system once $\mathcal{E}$ has acted on $Q$ alone, $$S(\rho,\mathcal{E}) = S(\rho^{R'Q'}), \qquad \rho^{R'Q'} = (\mathrm{id}_R \otimes \mathcal{E})(|RQ\rangle\langle RQ|).$$(5)^✦

In the notation of (3), $\mathcal{E}_1 = \mathcal{E}_{U_1,\omega_1}$ and $\mathcal{E}_2 = \mathcal{E}_{U_2,\omega_2}$. The second operation has its own entropy exchange $S(\rho',\mathcal{E}_2)$, given by (5) at its input $\rho'$.

After the first operation reference and system are in the state $\rho^{R'Q'}$ of (5) with $\mathcal{E} = \mathcal{E}_1$, and the second, run on $Q$ alone, takes them to $\rho^{R''Q''} = (\mathrm{id}_R \otimes \mathcal{E}_2)(\rho^{R'Q'})$. The composite entropy exchange $S(R''Q'')$ of a state $\rho$ of $Q$ under $\mathcal{E}_1$ and then $\mathcal{E}_2$ is the von Neumann entropy of the final state $\rho^{R''Q''}$ of reference and system, $$S(R''Q'') = S\bigl((\mathrm{id}_R \otimes \mathcal{E}_2) (\mathrm{id}_R \otimes \mathcal{E}_1)(|RQ\rangle\langle RQ|)\bigr),$$(6) and the coherent information of the two operations together is the coherent information (2) of the same state, $I(\rho,\mathcal{E}_2 \circ \mathcal{E}_1) = I(R''\rangle Q'') = S(Q'') - S(R''Q'')$.^✦

Keeping the second environment instead of tracing it out gives the dilated joint state (4) of $\rho^{R'Q'}$ under $U_2$ and $\omega_2$, the state $\rho^{R''Q''E_2''}$ of reference, system and second environment. The swap and the regrouping bring its factors into the order $E_2$, $R$, $Q$, and the partial trace over $Q$ leaves the joint state of $E_2$ and $R$, whose entropy is the escape entropy $$S(R''E_2'') = S\bigl(\operatorname{Tr}_Q\,\rho^{R''Q''E_2''}\bigr).$$(7)^✦

The Claim

What Is Checked

The claims are stated over a state $\rho$ of an arbitrary finite dimensional system $Q$ and over operations given, as in (3), by a unitary of $Q$ with an environment and a pure state of that environment. Their quantities are von Neumann entropies (1) of states built from these: the outputs $\rho'$ and $\rho''$, the states of reference and system behind the entropy exchanges (5) and (6), and the joint state behind the escape entropy (7), together with the coherent information (2).

About these constructions the library proves that each yields a state. Conjugating a density operator by a unitary or by an identification of systems gives a density operator, and so do the tensor product of two density operators and the partial trace of one; the projector onto a unit vector is a density operator; the tensor product of two unitaries is unitary; the canonical purification has unit norm, and an identification carries a unit vector to a unit vector. Hence $\rho'$, $\rho''$, $\rho^{R'Q'}$, $\rho^{R''Q''}$ and $\rho^{R''Q''E_2''}$ are density operators.