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arXiv:1504.04429 · Physical Review A

Hierarchy of bounds on accessible information and informational power

Michele Dall’Arno

Published Jul 27, 2015• 12 citations
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Abstract

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 $t=\ensuremath{\infty}$ 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 $t=\ensuremath{\infty}$, respectively.

DOI
10.1103/physreva.92.012328

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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 ttt-designs, with t=\ensuremath∞t=\ensuremath{\infty}t=\ensuremath∞ 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 ttt and prove their tightness for qubits and qutrits. By deriving asymptotic formulas for large dimensions, we also show that the statistics generated by any ttt-design with t>1t>1t>1 contains no more than a single bit of information, and this amount decreases with ttt. Holevo and subentropy bounds are recovered as particular cases for t=1t=1t=1 and t=\ensuremath∞t=\ensuremath{\infty}t=\ensuremath∞, respectively.
The accessible information quantifies the amount of classical information that can be extracted from an ensemble of quantum states. Analogously, the informational power quantifies the amount of classical information that can be extracted by a quantum measurement. For both quantities, we provide upper and lower bounds that depend only on the dimension of the system, and we prove their tightness. In the case of symmetric informationally complete (SIC) ensembles and measurements, stronger bounds are provided and their tightness proved for qubits and qutrits. From our upper bounds, we notice, perhaps surprisingly, that the statistics generated by SIC ensembles or measurements in arbitrary dimension, though optimal for tomographic purposes, in fact never contain more than just one bit of information, the rest being constituted by completely random bits. On the other hand, from our lower bounds, we obtain an explicit strategy beating the so-called pretty-good one for the extraction of mutual information in the case of SIC ensembles and measurements.
The accessible information and the informational power quantify the amount of information extractable from a quantum ensemble and by a quantum measurement, respectively. So-called spherical quantum 2-designs constitute a class of ensembles and measurements relevant in testing entropic uncertainty relations, quantum cryptography, and quantum tomography. We provide lower bounds on the entropy of 2-design ensembles and measurements, from which upper bounds on their accessible information and informational power follow, as a function of the dimension only. We show that the statistics generated by 2-designs, although optimal for the above-mentioned protocols, never contains more than 1 bit of information. Finally, we specialize our results to the relevant cases of symmetric informationally complete sets and maximal sets of mutually unbiased bases, and we generalize them to the arbitrary-rank case.
I derive a universal upper bound on the capacity of any communication channel between two distant systems. The Holevo quantity, and hence the mutual information, is at most of order EΔt/ℏ, where E is the average energy of the signal, and Δt is the amount of time for which detectors operate. The bound does not depend on the size or mass of the emitting and receiving systems, nor on the nature of the signal. No restrictions on preparing and processing the signal are imposed. As an example, I consider the encoding of information in the transverse or angular position of a signal emitted and received by systems of arbitrarily large cross section. In the limit of a large message space, quantum effects become important even if individual signals are classical, and the bound is upheld.
The accessible information and the informational power quantify the maximum amount of information that can be extracted from a quantum ensemble and by a quantum measurement, respectively. Here, we investigate the tradeoff between the accessible information (informational power, respectively) and the purity of the states of the ensemble (the elements of the measurement, respectively). Under any given lower bound on the purity, 1) we compute the minimum informational power and show that it is attained by the depolarized uniformly-distributed measurement; and 2) we give a lower bound on the accessible information. Under any given upper bound on the purity, 1) we compute the maximum accessible information and show that it is attained by an ensemble of pairwise commuting states with at most two distinct non-null eigenvalues; and 2) we give a lower bound on the maximum informational power. The present results provide, as a corollary, novel sufficient conditions for the tightness of the Jozsa-Robb-Wootters lower bound to the accessible information.
The description of the state of an n-bit quantum system requires 2n-1 complex numbers. This exponentially large information capacity of quantum states has been exploited in recent results showing both exponential speed-up, and exponential savings in communication costs in solving certain problems using quantum computers. In this dissertation, we establish limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed. More specifically, we give tight bounds for random access codes, which allow us to encode classical information using quantum bits such that only a small portion of the encoded information may be recovered via a measurement. This also sheds light on the power of computing with a finite number of quantum bits—using these techniques, we show an exponential size lower bound for quantum finite automata for a problem which can be solved on a linear size classical automaton. We then consider the complexity of solving certain problems in the quantum black-box model, an information theoretic model that has been a rich source of insights into the nature of quantum computation. We derive nearly optimal lower bounds for several problems in this model, including that of approximating the median. We also give new, optimal algorithms for approximate medians and other order statistics.
The intrinsic information of quantum systems refers to the information required to define a quantum state, and may reveal how the nature stores and processes microscopic information. However, there is an evident paradox due to the "\textit{information scale contrast}": Existing analytical results on the information bounds of quantum systems show that the information-carrying capacity of an nnn-qubit system is only O(n)O(n)O(n). Nonetheless, the intrinsic information content (estimated by, e.g., the number of parameters or the complexity of ontic embedding) indicates that defining an nnn-qubit system often requires information of the order O(2n)O(2^n)O(2n). In this paper, we aim to clarify the upper bound of intrinsic information in quantum systems, as well as explain and resolve the aforementioned paradox. Starting with an analysis of the dependence between the Bloch parameters, we take the structural constraints of the quantum state space as a prior and derive the posterior information content of a quantum state through the process of Maximum A Posteriori (MAP) estimation. We analytically prove that the upper bound of the posterior information content of a 2-qubit system is exactly equal to 2. Furthermore, we conjecturally generalized this result to nnn-qubit systems via a convex optimization process and numerical experiments. In summary, our main theoretical observation is that the tight structural constraints among the parameters of a quantum state make them highly interdependent, and thus unable to freely encode information. It turns out that the intrinsic information of an nnn-qubit system defined by posterior information is bounded by nnn classical bits.