Smooth entropies characterize basic information-theoretic properties of random variables, such as the number of bits required to store them or the amount of uniform randomness that can be extracted from them (possibly with respect to side information). In this paper, explicit and almost tight bounds on the smooth entropies of n-fold product distributions, Pn, are derived. These bounds are expressed in terms of the Shannon entropy of a single distribution, P . The results can be seen as an extension of the asymptotic equipartition property (AEP).
We introduce smooth entropy as a measure for the number of almost uniform random bits that can be extracted from a source by probabilistic algorithms. The extraction process should be universal in the sense that it does not require the distribution of the source to be known. Rather, it should work for all sources with some structural property. Smooth entropy unifies previous work on privacy amplification and entropy smoothing. It enables a systematic investigation of the spoiling knowledge proof technique to obtain lower bounds on smooth entropy and to show new connections to Renyi entropy of order /spl alpha/>1.
Extraction of uniform randomness from (noisy) non-uniform sources is an important primitive in many security applications, e.g. (pseudo-)random number generators, privacy-preserving biometrics, and key storage based on Physical Unclonable Functions. Generic extraction methods exist, using universal hash functions. There is a trade-off between the length of the extracted bit string and the uniformity of the string. In the literature there are proven lower bounds on this length as a function of the desired uniformity. The best known bound involves a quantity known as smooth min-entropy. Unfortunately, there exist at least three definitions of smooth entropy. In this paper we compare three of these definitions, and we derive improved lower bounds on the extractable randomness.\nWe also investigate the use of almost universal hash functions, which are slightly worse at extracting randomness than universal hash functions, but are preferable in practice because they require far less resources in devices. We show that using them has negligible effect on the extractable randomness.\nKeywords: Leftover Hash Lemma; Universal hash function; Randomness extraction
Random circuit sampling experiments, the leading demonstrations of quantum computational advantage, are tested with the linear cross-entropy benchmark (XEB), which scores outcomes by their ideal probabilities. We study what a high XEB score certifies about an untrusted quantum device. In the quantum query model, we show that the honest sampler's ideal score acts as a Tsirelson bound for efficient devices and certified randomness. First, exceeding the ideal score by a constant requires
Ω(N1/3) queries for the Haar and Fourier oracle ensembles with
N outcomes, and this bound is tight; this proves an earlier conjecture for Haar-random states and extends it to Fourier sampling. Second, for Fourier sampling and for Haar-random states accessed through the canonical state-preparation oracle, the
n-bit output of a device that makes polynomially many queries and scores within
o(1) of the ideal score has
n−O(logn) bits of smooth min-entropy, even against an adversary who is entangled with the device and later learns the oracle; this is nearly optimal. As a corollary, sampling the collision distribution, which is proportional to the square of the ideal distribution, has query complexity
Θ(N1/3), and a coherent version of the same algorithm approximately prepares quantum Hadamard products of states with uniform amplitude magnitudes. Both lower bounds rest on progress measures that bound the excess over the ideal score and increase only slightly with each query. For Fourier sampling the measure is the norm of the coherent amplitudes for removing pairs in a purification of the random function, combining the polynomial method with the compressed oracle technique; for Haar-random states it is the square root of the average level in an expansion in Dirichlet orthogonal polynomials, controlled by an exact identity for the effect of one observation.
We find a tight characterization of the strong converse exponent for randomness extraction against quantum side information. In contrast to previous tight bounds, we employ a composable error criterion given by the fidelity (or purified distance) to a uniform distribution in product with the marginal state. Our characterization is formulated in terms of the club-sandwiched conditional entropy, recently introduced by Rubboli, Goodarzi, and Tomamichel. It extends the corresponding result for classical side information, whose strong converse exponent was first established by Berta and Yao and later recovered by Li, Li, and Yu within this entropic framework. This provides the first precise operational interpretation of this family of conditional entropies in the quantum setting.
Abstract. We revisit the classical problem: given a memoryless source having a certain amount of Shannon Entropy, how many random bits can be extracted? This question appears in works studying random number generators built from physical entropy sources. Some authors use a heuristic estimate obtained from the Asymptotic Equipartition Property, which yields roughly n extractable bits, where n is the total Shannon entropy amount. However the best known pre-cise form gives only n − O(√log(1/)n), where is the distance of the extracted bits from uniform. In this paper we show a matching n−Ω(√log(1/)n) upper bound. Therefore, the loss of Θ(√log(1/)n) bits is necessary. As we show, this theoretical bound is of practical rele-vance. Namely, applying the imprecise AEP heuristic to a mobile phone accelerometer one might overestimate extractable entropy even by 100%, no matter what the extractor is. Thus, the “AEP extracting heuristic” should not be used without taking the precise error into account.