Hierarchy of Bounds on Accessible Information and Informational Power

Abstract

Quantum theory imposes fundamental limitations to the amount of information that can be carried by any quantum system. On the one hand, Holevo bound rules out the possibility to encode 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 formulae 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.

8 pages, 1 table, 1 figure, updated references, published version

References in corpus (4)

• Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases
• Optimal signal states for quantum detectors
• Maximally informative ensembles for SIC-POVMs in dimension 3
• Accessible Information and Informational Power of Quantum 2-designs

Cited by in corpus (9)

• The SIC Question: History and State of Play
• Quantum Conical Designs
• Informational power of the Hoggar SIC-POVM
• Device-independent tests of quantum measurements
• Quantum dynamical entropy, chaotic unitaries and complex Hadamard matrices
• Communication capacity of mixed quantum t designs
• Discrimination power of a quantum detector
• A classical interpretation of the Scrooge distribution
• Tradeoff relations between accessible information, informational power, and purity