Hierarchy of bounds on accessible information and informational power

Quantum theory imposes fundamental limitations on the amount of information that can be carried by any quantum system. On the one hand, the Holevo bound rules out the possibility of encoding more information in a quantum system than in its classical counterpart, comprised of perfectly distinguishable states. On the other hand, when states are uniformly distributed in the state space, the so-called subentropy lower bound is saturated. How uniform quantum systems are can be naturally quantified by characterizing them as $t$-designs, with $t=\infty$ corresponding to the uniform distribution. Here we show the existence of a trade-off between the uniformity of a quantum system and the amount of information it can carry. To this aim, we derive a hierarchy of informational bounds as a function of $t$ and prove their tightness for qubits and qutrits. By deriving asymptotic formulas for large dimensions, we also show that the statistics generated by any $t$-design with $t>1$ contains no more than a single bit of information, and this amount decreases with $t$. Holevo and subentropy bounds are recovered as particular cases for $t=1$ and $t=\infty$, respectively.

Figure 1

Upper bounds $W_t\left(d\right)$ on the informational power $W\left({\pi }_y\right)$ (in bits) of any quantum $t$-design POVM ${\pi }_y$ as a function of the dimension $d$ (on a log scale). From top to bottom: $t=1$ (blue line; see key), $t=2$ and 3 (red lines), $t=4$ and 5 (green lines), and $t=\infty$ (blue line), as provided by Corollaries I, II, III, IV, V, and VI, respectively. The asymptotes $W_{2,3}\left(d\right)\to 1\mathrm{bit},W_{4,5}\left(d\right)\to 0.744\mathrm{bit}$, and $W_{\infty }\left(d\right)\to 0.609\mathrm{bit}$ are depicted too (horizontal black lines).