The Exercise

Four quantum systems $A$, $B$, $C$ and $D$ share a state $\rho$. The problem asks whether strong subadditivity for $A$, $B$ and $C$ still holds when every entropy in it is taken conditional on the fourth system $D$.

The library states that the conditional inequality holds for every state of the four systems. For four random variables on a finite probability space it states a counterexample to a classical inequality of the same shape, one in which each entropy is that of $D$ given some of the other three variables.

Four Quantum Systems

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 systems $S$ and $T$ is the system whose state space is the tensor product of their state spaces.^✦ An identification $e$ of a system $S$ with a system $T$ is a unitary map from the state space of $S$ onto that of $T$.^✦ The regrouping $\alpha_{S,T,U}$ of three systems $S$, $T$ and $U$ is the identification of $S \otimes (T \otimes U)$ with $(S \otimes T) \otimes U$ that sends $u \otimes (v \otimes w)$ to $(u \otimes v) \otimes w$.^✦

A state $\rho$ of a system $S$ is a positive operator of unit trace on the state space of $S$.^✦ The relabelling of $\rho$ along an identification $e$ of $S$ with $T$ is the state $e \rho e^{-1}$ of $T$.^✦

For a state $\rho$ of $S \otimes T$, the reduced state $\rho_S = \operatorname{tr}_T \rho$ of $S$ is the partial trace of $\rho$ over $T$, computed in an orthonormal basis $\{|k\rangle\}$ of the state space of $T$: $$\operatorname{tr}_T \rho = \sum_k \bigl(I \otimes \langle k|\bigr)\, \rho\, \bigl(I \otimes |k\rangle\bigr).$$(1)^✦ Here $I$ is the identity on the state space of $S$, the map $I \otimes |k\rangle$ embeds $|x\rangle$ as $|x\rangle|k\rangle$^✦, and $I \otimes \langle k|$ sends $|x\rangle|y\rangle$ to $\langle k|y\rangle\,|x\rangle$^✦. The reduced state $\rho_T = \operatorname{tr}_S \rho$ of $T$ is the same sum with the two factors exchanged.^✦

The four systems of the problem are composed as $A \otimes ((B \otimes D) \otimes C)$, and $\rho$ is a state of this composite. Tracing out $A$ gives the reduced state $\rho_{BDC}$ of $(B \otimes D) \otimes C$; tracing $C$ out of $\rho_{BDC}$ gives $\rho_{BD}$, and tracing $B$ out of $\rho_{BD}$ gives $\rho_D$. The reduced state $\rho_{ABD}$ of $A \otimes (B \otimes D)$ is the partial trace over $C$ of the relabelling of $\rho$ along the regrouping $\alpha_{A,\,B \otimes D,\,C}$.

Entropies

The Shannon entropy of a probability distribution $q$ on a finite set is $$H(q) = -\sum_i q_i \log q_i ,$$(2)^✦ with the natural logarithm, so that entropies are measured in nats. The von Neumann entropy of a state $\rho$ is the Shannon entropy of its eigenvalues $\lambda_1, \dots, \lambda_n$, counted with multiplicity, where $n$ is the dimension of the state space: $$S(\rho) = -\operatorname{tr}(\rho \log \rho) = H(\lambda_1, \dots, \lambda_n).$$(3)^✦ The conditional entropy of a state $\sigma$ of a composite $X \otimes Y$, with reduced state $\sigma_Y$ on $Y$, is $$S(X|Y) = S(\sigma) - S(\sigma_Y).$$(4)^✦

Conditioned on $D$, the entropy of $A$, $B$ and $C$ together is $$S(A,B,C|D) = S(\rho) - S(\rho_D),$$(5)^✦ and the entropy of $B$ is the conditional entropy of $\rho_{BD}$, $$S(B|D) = S(\rho_{BD}) - S(\rho_D).$$(6)^✦ The entropy of the pair $A$, $B$ given $D$ is $$S(A,B|D) = S(\rho_{ABD}) - S(\rho_D),$$(7)^✦ and that of the pair $B$, $C$ given $D$ is $$S(B,C|D) = S(\rho_{BDC}) - S(\rho_D).$$(8)^✦

Random Variables

A finite probability space is a finite sample space $\Omega$ with probabilities $p(\omega) \ge 0$ that sum to one. A random variable $f$ on $\Omega$ is a function from $\Omega$ to a finite set, and its entropy $H(f)$ is the Shannon entropy of its distribution: $$H(f) = H(P_f), \qquad P_f(b) = \sum_{\omega :\, f(\omega) = b} p(\omega).$$(9)^✦ The conditional entropy of a random variable $f$ given a random variable $g$ is $$H(f|g) = H(f,g) - H(g),$$(10)^✦ where $(f,g)$ is the random variable $\omega \mapsto (f(\omega), g(\omega))$. Several variables after the bar stand for their tuple, as in $H(D|A,B,C)$.

The Claim

What Is Checked

The library proves that the partial trace of a state over either factor is a state of the other system, and that the relabelling of a state along an identification of systems is again a state.