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arXiv:1409.0341 · Physical Review A

Accessible information and informational power of quantum 2-designs

Michele Dall’Arno

Published Nov 10, 2014• 16 citations
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Abstract

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.

DOI
10.1103/physreva.90.052311

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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.
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 current study aims to examine uncertainty relations for quantum measurements assigned to a projective two-design. Complete sets of mutually unbiased bases and symmetric informationally complete measurements are important cases of such measurements. To characterize the amount of uncertainty, we use the Tsallis and Rényi entropies as well as the probabilities of separate outcomes. The obtained results are based on an estimation of the index of coincidence. They improve some uncertainty relations given in the literature.
The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.
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.
Fuchs and Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in ddd-dimensional Hilbert space, minimum quantumness is 2d+1\frac{2}{d+1}d+12​, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than d2d^2d2 states can achieve this bound. Recently, in a different context, Scott introduced a concept of generalized ttt-design in \cite{GenSphet}, which is a natural generalization of spherical ttt-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to d2d^2d2, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases. Additionally, it provides a natural way to define two more families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.