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1. Cutoff phenomenon and entropic uncertainty for random quantum circuits

   https://doi.org/10.1088/2516-1075/acf2d3  

   Oh, Sangchul; Kais, Sabre (September 2023, Electronic Structure)

   Abstract How fast a state of a system converges to a stationary state is one of the fundamental questions in science. Some Markov chains and random walks on finite groups are known to exhibit the non-asymptotic convergence to a stationary distribution, called the cutoff phenomenon. Here, we examine how quickly a random quantum circuit could transform a quantum state to a Haar-measure random quantum state. We find that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform (QFT). Thus the entropic uncertainty of random quantum states has balanced Shannon entropies for the computational basis and the QFT basis. By calculating the Shannon entropy for random quantum states and the Wasserstein distances for the eigenvalues of random quantum circuits, we show that the cutoff phenomenon occurs for the random quantum circuit. It is also demonstrated that the Dyson-Brownian motion for the eigenvalues of a random unitary matrix as a continuous random walk exhibits the cutoff phenomenon. The results here imply that random quantum states could be generated with shallow random circuits. 

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2. 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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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. Thermalization of randomly coupled SYK models

   https://doi.org/10.1088/1742-5468/ac416b  

   Sohal, Ramanjit; Nie, Laimei; Sun, Xiao-Qi; Fradkin, Eduardo (January 2022, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract We investigate the thermalization of Sachdev–Ye–Kitaev (SYK) models coupled via random interactions following quenches from the perspective of entanglement. Previous studies have shown that when a system of two SYK models coupled by random two-body terms is quenched from the thermofield double state with sufficiently low effective temperature, the Rényi entropies do not saturate to the expected thermal values in the large- N limit. Using numerical large- N methods, we first show that the Rényi entropies in a pair SYK models coupled by two-body terms can thermalize, if quenched from a state with sufficiently high effective temperature, and hence exhibit state-dependent thermalization. In contrast, SYK models coupled by single-body terms appear to always thermalize. We provide evidence that the subthermal behavior in the former system is likely a large- N artifact by repeating the quench for finite N and finding that the saturation value of the Rényi entropy extrapolates to the expected thermal value in the N → ∞ limit. Finally, as a finer grained measure of thermalization, we compute the late-time spectral form factor of the reduced density matrix after the quench. While a single SYK dot exhibits perfect agreement with random matrix theory, both the quadratically and quartically coupled SYK models exhibit slight deviations. 

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5. Rényi mutual information in quantum field theory, tensor networks, and gravity

   https://doi.org/10.1007/JHEP06(2024)195  

   Kudler-Flam, Jonah; Nie, Laimei; Vijay, Akash (June 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We explore a large class of correlation measures called theα−zRényi mutual informations (RMIs). Unlike the commonly used notion of RMI involving linear combinations of Rényi entropies, theα−zRMIs are positive semi-definite and monotonically decreasing under local quantum operations, making them sensible measures of total (quantum and classical) correlations. This follows from their descendance from Rényi relative entropies. In addition to upper bounding connected correlation functions between subsystems, we prove the much stronger statement that for certain values ofαandz, theα−zRMIs also lower bound certain connected correlation functions. We develop an easily implementable replica trick which enables us to compute theα−zRMIs in a variety of many-body systems including conformal field theories, free fermions, random tensor networks, and holography. 

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