The Exercise

Let $X$, $Y$, $Z$ be random variables with finitely many values. Knowing $Z$, the pair $X,Y$ is never less uncertain than $X$ alone: $H(X,Y|Z) \ge H(X|Z)$.^✦ The quantum analogue for three systems $A$, $B$, $C$, with the von Neumann entropy in place of the Shannon entropy, fails already for qubits: $S(A,B|C)$ can be smaller than $S(A|C)$.^✦ It can also be smaller than $S(A|C) - S(B|C)$, so the triangle inequality for the entropies of the pair $A,B$ and of $A$ and $B$ alone, $S(A,B) = S(\rho)$, $S(A) = S(\rho_A)$, and $S(B) = S(\rho_B)$, fails once both sides are conditioned on $C$.

Classical Conditional Entropy

The Shannon entropy of a distribution $p$ on a finite set is $$H(p) = -\sum_i p(i) \log p(i),$$(1)^✦ with the natural logarithm, so that entropies are measured in nats. For a distribution $p(x,y)$ on a product of two finite sets, the marginal of the second variable is $p_Y(y) = \sum_x p(x,y)$^✦, and the conditional entropy of $X$ given $Y$ is $$H(X|Y) = H(p) - H(p_Y).$$(2)^✦

For $p(x,y,z)$ on a product of three finite sets, the marginal of the first and last variables is $p_{XZ}(x,z) = \sum_y p(x,y,z)$^✦, and $H(X|Z)$ is the conditional entropy (2) of $p_{XZ}$. With the marginal $p_Z(z) = \sum_{x,y} p(x,y,z)$, the entropy of the pair given $Z$ is $$H(X,Y|Z) = H(p) - H(p_Z).$$(3)^✦

Three Quantum Systems

A quantum system $S$ is described by its state space, a finite dimensional complex Hilbert space^✦, and every such space $H$ is the state space of a system.^✦ The qubit is the system with state space $\mathbb{C}^2$.^✦ Two systems $S$ and $T$ form the composite system $S \otimes T$, whose state space is the tensor product of theirs.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦

An identification $e$ of a system $S$ with a system $T$ is a linear isometric bijection from the state space of $S$ onto that of $T$^✦; it carries a state $\rho$ of $S$ to the operator $e \rho e^{-1}$ on the state space of $T$.^✦ For three systems, the regrouping $\alpha$ identifies $A \otimes (C \otimes B)$ with $(A \otimes C) \otimes B$ by $u \otimes (v \otimes w) \mapsto (u \otimes v) \otimes w$.^✦

Let $|j\rangle$ and $|k\rangle$ be orthonormal bases of the state spaces of $S$ and $T$. The partial traces of an operator $M$ on the state space of $S \otimes T$ are the operators $$\operatorname{Tr}_T M = \sum_k (I \otimes \langle k|)\, M\, (I \otimes |k\rangle), \qquad \operatorname{Tr}_S M = \sum_j (\langle j| \otimes I)\, M\, (|j\rangle \otimes I)$$(4) on the state spaces of $S$ and of $T$. The maps $I \otimes |k\rangle$ and $|j\rangle \otimes I$ are the embeddings $x \mapsto x \otimes |k\rangle$ and $y \mapsto |j\rangle \otimes y$, and $I \otimes \langle k|$ and $\langle j| \otimes I$ are the restrictions $x \otimes y \mapsto \langle k|y\rangle\, x$ and $x \otimes y \mapsto \langle j|x\rangle\, y$. The reduced state of a state $\rho$ of $S \otimes T$ on $S$ is $\rho_S = \operatorname{Tr}_T \rho$^✦, and its reduced state on $T$ is $\rho_T = \operatorname{Tr}_S \rho$.^✦

Quantum Conditional Entropy

The von Neumann entropy of a self adjoint operator is the Shannon entropy (1) of its eigenvalues^✦, and that of a state $\rho$ is the entropy of its operator, $S(\rho) = -\operatorname{tr}(\rho \log \rho)$.^✦ For a state $\rho$ of a composite system $A \otimes B$, the conditional entropy of $A$ given $B$ is $$S(A|B) = S(\rho) - S(\rho_B).$$(5)^✦

Let $\rho$ be a state of $A \otimes (C \otimes B)$ for three systems $A$, $B$, $C$. Its reduced states are $\rho_{CB} = \operatorname{Tr}_A \rho$ on $C \otimes B$, $\rho_C = \operatorname{Tr}_B \rho_{CB}$ on $C$, and $\rho_{AC} = \operatorname{Tr}_B (\alpha \rho \alpha^{-1})$ on $A \otimes C$, the state regrouped by $\alpha$ with $B$ traced out. The entropy of the pair $A,B$ given $C$ is $$S(A,B|C) = S(\rho) - S(\rho_C).$$(6)^✦ The entropy of $A$ given $C$ is the conditional entropy (5) of $\rho_{AC}$, $$S(A|C) = S(\rho_{AC}) - S(\rho_C),$$(7)^✦ and the entropy of $B$ given $C$ is $$S(B|C) = S(\rho_{CB}) - S(\rho_C).$$(8)^✦

The Claim

What Is Checked

The classical inequality (9) is stated over the Shannon entropy (1) and the marginals $p_{XZ}$ and $p_Z$. The quantum statements are stated over states of three qubits, through the conditional entropies (6), (7) and (8), which are built from the von Neumann entropy, the partial traces (4) and the regrouping $\alpha$.

About these constructions the library proves that a partial trace of a positive operator of unit trace is again positive of unit trace, so that $\rho_S$ and $\rho_T$ are states, and that conjugation by an isometry preserves positivity and trace, so that a state carried along an identification, $\alpha \rho \alpha^{-1}$ among them, is again a state.