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1. Quantum Query Complexity of Entropy Estimation

   https://doi.org/10.1109/TIT.2018.2883306  

   Li, Tongyang; Wu, Xiaodi (April 2019, IEEE Transactions on Information Theory)

   Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing. In this paper, we give the first quantum algorithms for estimating α-Rényi entropies (Shannon entropy being 1-Rényi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for α-Rényi entropy estimation for all α ≥ 0, including tight bounds for the Shannon entropy, the Hartley entropy (α = 0), and the collision entropy (α = 2). We also provide quantum upper bounds for estimating min-entropy (α = +∞) as well as the Kullback-Leibler divergence. We complement our results with quantum lower bounds on α- Rényi entropy estimation for all α ≥ 0. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH) [1], however, with many new technical ingredients: (1) we improve the error dependence of the BHH framework by a fine-tuned error analysis together with Montanaro’s approach to estimating the expected output of quantum subroutines [2] for α = 0, 1; (2) we develop a procedure, similar to cooling schedules in simulated annealing, for general α ≥ 0; (3) in the cases of integer α ≥ 2 and α = +∞, we reduce the entropy estimation problem to the α-distinctness and the ⌈log n⌉-distinctness problems, respectively. 

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2. Euclidean and complex geometries from real-time computations of gravitational Rényi entropies

   https://doi.org/10.1007/JHEP02(2025)136  

   Held, Jesse; Liu, Xiaoyi; Marolf, Donald; Wang, Zhencheng (February 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> Gravitational Rényi computations have traditionally been described in the language of Euclidean path integrals. In the semiclassical limit, such calculations are governed by Euclidean (or, more generally, complex) saddle-point geometries. We emphasize here that, at least in simple contexts, the Euclidean approach suggests an alternative formulation in terms of the bulk quantum wavefunction. Since this alternate formulation can be directly applied to the real-time quantum theory, it is insensitive to subtleties involved in defining the Euclidean path integral. In particular, it can be consistent with many different choices of integration contour. Despite the fact that self-adjoint operators in the associated real-time quantum theory have real eigenvalues, we note that the bulk wavefunction encodes the Euclidean (or complex) Rényi geometries that would arise in any Euclidean path integral. As a result, for any given quantum state, the appropriate real-time path integral yields both Rényi entropies and associated complex saddle-point geometries that agree with Euclidean methods. After brief explanations of these general points, we use JT gravity to illustrate the associated real-time computations in detail. 

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3. On the α-q-Mutual Information and the α-q-Capacities

   https://doi.org/10.3390/e23060702  

   Ilić, Velimir M.; Djordjević, Ivan B. (June 2021, Entropy)

   null (Ed.)

   The measures of information transfer which correspond to non-additive entropies have intensively been studied in previous decades. The majority of the work includes the ones belonging to the Sharma–Mittal entropy class, such as the Rényi, the Tsallis, the Landsberg–Vedral and the Gaussian entropies. All of the considerations follow the same approach, mimicking some of the various and mutually equivalent definitions of Shannon information measures, and the information transfer is quantified by an appropriately defined measure of mutual information, while the maximal information transfer is considered as a generalized channel capacity. However, all of the previous approaches fail to satisfy at least one of the ineluctable properties which a measure of (maximal) information transfer should satisfy, leading to counterintuitive conclusions and predicting nonphysical behavior even in the case of very simple communication channels. This paper fills the gap by proposing two parameter measures named the α-q-mutual information and the α-q-capacity. In addition to standard Shannon approaches, special cases of these measures include the α-mutual information and the α-capacity, which are well established in the information theory literature as measures of additive Rényi information transfer, while the cases of the Tsallis, the Landsberg–Vedral and the Gaussian entropies can also be accessed by special choices of the parameters α and q. It is shown that, unlike the previous definition, the α-q-mutual information and the α-q-capacity satisfy the set of properties, which are stated as axioms, by which they reduce to zero in the case of totally destructive channels and to the (maximal) input Sharma–Mittal entropy in the case of perfect transmission, which is consistent with the maximum likelihood detection error. In addition, they are non-negative and less than or equal to the input and the output Sharma–Mittal entropies, in general. Thus, unlike the previous approaches, the proposed (maximal) information transfer measures do not manifest nonphysical behaviors such as sub-capacitance or super-capacitance, which could qualify them as appropriate measures of the Sharma–Mittal information transfer. 

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4. Subaru/Hyper Suprime-Cam z-Broadband Excess Selection of Extreme Emission Line Galaxies at z < 1

   https://doi.org/10.3847/1538-4357/ac5117  

   Rosenwasser, Benjamin E.; Taylor, Anthony J.; Barger, Amy J.; Cowie, Lennox L.; Hu, Esther M.; Jones, Logan H.; Songaila, Antoinette (March 2022, The Astrophysical Journal)

   Abstract We present a search for extreme emission line galaxies (EELGs) atz< 1 in the COSMOS and North Ecliptic Pole (NEP) fields with imaging from Subaru/Hyper Suprime-Cam (HSC) and a combination of new and existing spectroscopy. We select EELGs on the basis of substantial excess flux in thezbroad band, which is sensitive to Hαat 0.3 ≲z≲ 0.42 and [Oiii]λ5007 at 0.7 ≲z≲ 0.86. We identify 10,470 galaxies withzexcesses in the COSMOS data set and 91,385 in the NEP field. We cross-reference the COSMOS EELG sample with the zCOSMOS and DEIMOS 10k spectral catalogs, finding 1395 spectroscopic matches. We made an additional 71 (46 unique) spectroscopic measurements withY< 23 using the HYDRA multiobject spectrograph on the WIYN 3.5 m telescope, and 204 spectroscopic measurements from the DEIMOS spectrograph on the Keck II telescope, providing a total of 1441/10,470 spectroscopic redshifts for the EELG sample in COSMOS (∼14%). We confirm that 1418 (∼98%) are Hαor [Oiii]λ5007 emitters in the above stated redshift ranges. We also identify 240 redshifted Hαand [Oiii]λ5007 emitters in the NEP using spectra taken with WIYN/HYDRA and Keck/DEIMOS. Using broadband-selection techniques in theg−r−icolor space, we distinguish between Hαand [Oiii]λ5007 emitters with 98.6% accuracy. We test our EELG selection by constructing Hαand [Oiii]λ5007 luminosity functions and comparing to recent literature results. We conclude that broadband magnitudes from HSC, the Vera C. Rubin Observatory, and other deep optical multiband surveys can be used to select EELGs in a straightforward manner. 

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5. Large N algebras and generalized entropy

   https://doi.org/10.1007/JHEP04(2023)009  

   Chandrasekaran, Venkatesa; Penington, Geoff; Witten, Edward (April 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We construct a Type II_∞von Neumann algebra that describes the largeNphysics of single-trace operators in AdS/CFT in the microcanonical ensemble, where there is no need to include perturbative 1/Ncorrections. Using only the extrapolate dictionary, we show that the entropy of semiclassical states on this algebra is holographically dual to the generalized entropy of the black hole bifurcation surface. From a boundary perspective, this constitutes a derivation of a special case of the QES prescription without any use of Euclidean gravity or replicas; from a purely bulk perspective, it is a derivation of the quantum-corrected Bekenstein-Hawking formula as the entropy of an explicit algebra in theG →0 limit of Lorentzian effective field theory quantum gravity. In a limit where a black hole is first allowed to equilibrate and then is later potentially re-excited, we show that the generalized second law is a direct consequence of the monotonicity of the entropy of algebras under trace-preserving inclusions. Finally, by considering excitations that are separated by more than a scrambling time we construct a “free product” von Neumann algebra that describes the semiclassical physics of long wormholes supported by shocks. We compute Rényi entropies for this algebra and show that they are equal to a sum over saddles associated to quantum extremal surfaces in the wormhole. Surprisingly, however, the saddles associated to “bulge” quantum extremal surfaces contribute with a negative sign. 

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