Uniform Additivity in Classical and Quantum Information

Information theory quantifies the optimal rates of resource interconversions, usually in terms of entropies. However, nonadditivity often makes evaluating entropic formulas intractable. In a few auspicious cases, additivity allows a full characterization of optimal rates. We study uniform additivity of formulas, which is easily evaluated and captures all known additive quantum formulas. Our complete characterization of uniform additivity exposes an intriguing new additive quantity and identifies a remarkable coincidence—the classical and quantum uniformly additive functions with one auxiliary variable are identical.

Figure 1

Using a quantum channel to generate a quantum state. A noisy quantum channel from input $A$ to output $B$ can always be thought of as a unitary interaction of the input with some inaccessible environment $E$. We can generate a quantum state from this interaction by creating ${\varphi }_{V_1\dots V_{n}A}$ and acting on $A$ with $U_{\mathcal{N}}$, leading to the state ${\rho }_{V_1\dots V_{n}BE}=I\otimes U_{\mathcal{N}}{\varphi }_{V_1\dots V_{n}A}I\otimes U_{\mathcal{N}}^{†}$.

Figure 2

Decoupling is the process of mapping one state that can be acted on by two channels into two separate states, each of which can be acted on by a single channel use. It maps a state ${\varphi }_{V_1\dots V_n{A}_1{A}_2}$ to two states, ${\tilde{\varphi }}_{{\tilde{V}}_1\dots {\tilde{V}}_n{A}_1}$ and ${\widehat{\varphi }}_{{\widehat{V}}_1\dots {\widehat{V}}_n{A}_2}$. Here, $A_1$ and $A_2$ are the input spaces to $\mathcal{N}$ and $\mathcal{M}$, so that $U_{\mathcal{N}}\otimes U_{\mathcal{M}}$ can be applied to ${\varphi }_{V_1\dots V_n{A}_1{A}_2}$ to make ${\rho }_{V_1\dots V_n{B}_1{E}_1{B}_2{E}_2}$, while $U_{\mathcal{N}}$ acts on ${\tilde{\varphi }}_{{\tilde{V}}_1\dots {\tilde{V}}_n{A}_1}$ to make ${\tilde{\rho }}_{{\tilde{V}}_1\dots {\tilde{V}}_n{B}_1{E}_1}$, and $U_{\mathcal{M}}$ acts on ${\widehat{\varphi }}_{{\widehat{V}}_1\dots {\widehat{V}}_n{A}_2}$ to make ${\widehat{\rho }}_{{\widehat{V}}_1\dots {\widehat{V}}_n{B}_2{E}_2}$. For a standard decoupling, the states ${\tilde{\varphi }}_{{\tilde{V}}_1\dots {\tilde{V}}_n{A}_1}$ and ${\widehat{\varphi }}_{{\widehat{V}}_1\dots {\widehat{V}}_n{A}_2}$ are constructed from ${\varphi }_{V_1\dots V_n{A}_1{A}_2}$ as follows. To obtain ${\tilde{\varphi }}_{{\tilde{V}}_1\dots {\tilde{V}}_n{A}_1}$, we first apply $U_{\mathcal{M}}$ to make ${\varphi }_{V_1\dots V_n{A}_1{B}_2{E}_2}$. Given ${\varphi }_{V_1\dots V_n{A}_1{B}_2{E}_2}$, we define ${\tilde{V}}_i$ to contain $V_i$. $B_2$ and $E_2$ are each either assigned to one of the ${\tilde{V}}_i$ (or perhaps traced out) to generate ${\tilde{\varphi }}_{{\tilde{V}}_1\dots {\tilde{V}}_n{A}_1}$. We define ${\widehat{\varphi }}_{{\widehat{V}}_1\dots {\widehat{V}}_n{A}_2}$ similarly.

Figure 3

(Left) Quantum Entropy Cone for two systems. The entropies of a bipartite quantum state ${\rho }_{BE}$ form a vector $\mathbf{(}H(B),H(E),H(BE)\mathbf{)}$. The vectors of entropies that can be realized by a quantum state lie in a cone. For two systems, the faces of this cone are implied by strong subadditivity. This is also true for $n=3$ systems, but for $n\ge 4$, we do not know whether the quantum entropy cone lies strictly inside the cone implied by strong subadditivity. (Right) Additivity cone. Fixing a decoupling gives an entropy inequality that implies additivity. We check whether this inequality is satisfied by using known entropy inequalites, as expressed by the quantum entropy cone. We find a cone of coefficients defining the entropy formulas that are uniformly additive with respect to the fixed decoupling. The cone above is the additive cone for zero auxiliary variable formulas.