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1. Predicting quantum channels over general product distributions

   Chen, Sitan; de_Dios_Pont, Jaume; Hsieh, Jun-Ting; Huang, Hsin-Yuan; Lange, Jane; Li, Jerry (September 2024, https://doi.org/10.48550/arXiv.2409.03684)

   We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an n-qubit channel E and an observable O, we aim to learn the mapping ρ↦Tr(OE[ρ]) to within a small error for most ρ sampled from a distribution D. Previously, Huang, Chen, and Preskill proved a surprising result that even if E is arbitrary, this task can be solved in time roughly nO(log(1/ϵ)), where ϵ is the target prediction error. However, their guarantee applied only to input distributions D invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states ρ. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution D, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information. 

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2. Recoverability for Holevo's Just-as-Good Fidelity

   https://doi.org/10.1109/ISIT.2018.8437346  

   Wilde, Mark M. (June 2018, Proceedings of the 2018 International Symposium on Information Theory)

   Holevo's just-as-good fidelity is a similarity measure for quantum states that has found several applications. One of its critical properties is that it obeys a data processing inequality: the measure does not decrease under the action of a quantum channel on the underlying states. In this paper, I prove a refinement of this data processing inequality that includes an additional term related to recoverability. That is, if the increase in the measure is small after the action of a partial trace, then one of the states can be nearly recovered by the Petz recovery channel, while the other state is perfectly recovered by the same channel. The refinement is given in terms of the trace distance of one of the states to its recovered version and also depends on the minimum eigenvalue of the other state. As such, the refinement is universal, in the sense that the recovery channel depends only on one of the states, and it is explicit, given by the Petz recovery channel. The appendix contains a generalization of the aforementioned result to arbitrary quantum channels. 

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3. Optimized Quantum F-Divergences

   https://doi.org/10.1109/ISIT.2018.8437925  

   Wilde, Mark M. (June 2018, Proceedings of the 2018 IEEE International Symposium on Information Theory)

   null (Ed.)

   The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz--Renyi relative entropies. In this contribution, I introduce the optimized quantum f-divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Interestingly, the sandwiched Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz--Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. 

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4. From Blackwell Dominance in Large Samples to Rényi Divergences and Back Again

   https://doi.org/10.3982/ECTA17548  

   Mu, Xiaosheng; Pomatto, Luciano; Strack, Philipp; Tamuz, Omer (January 2021, Econometrica)

   null (Ed.)

   We study repeated independent Blackwell experiments; standard examples include drawing multiple samples from a population, or performing a measurement in different locations. In the baseline setting of a binary state of nature, we compare experiments in terms of their informativeness in large samples. Addressing a question due to Blackwell (1951), we show that generically an experiment is more informative than another in large samples if and only if it has higher Rényi divergences. We apply our analysis to the problem of measuring the degree of dissimilarity between distributions by means of divergences. A useful property of Rényi divergences is their additivity with respect to product distributions. Our characterization of Blackwell dominance in large samples implies that every additive divergence that satisfies the data processing inequality is an integral of Rényi divergences. 

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5. Channel activation of CHSH nonlocality

   https://doi.org/10.1088/1367-2630/ab7bef  

   Zhang, Yujie; Bravo, Rodrigo Araiza; Lorenz, Virginia O.; Chitambar, Eric (April 2020, New Journal of Physics)

   Abstract Quantum channels that break CHSH nonlocality on all input states are known as CHSH-breaking channels. In quantum networks, such channels are useless for distributing correlations that can violate the CHSH Inequality. Motivated by previous work on activation of nonlocality in quantum states, here we demonstrate an analogous activation of CHSH-breaking channels. That is, we show that certain pairs of CHSH-breaking channels are no longer CHSH-breaking when used in combination. We find that this type of activation can emerge in both uni-directional and bi-directional communication scenarios. 

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