This manuscript is a Correction to Becker et al., Trans. Inf. Theory 69, 4128 (2023). Specifically, we add some necessary assumptions inTheorems 2,4, and6therein, which were pointed out to us by Maksim Shirokov, and make some Corrections toFigure 2.

In this article we develop a continuous variable (CV) shadow tomography scheme with wide ranging applications in quantum optics. Our work is motivated by the increasing experimental and technological relevance of CV systems in quantum information, quantum communication, quantum sensing, quantum simulations, quantum computing and error correction. We introduce two experimentally realisable schemes for obtaining classical shadows of CV (possibly non-Gaussian) quantum states using only randomised Gaussian unitaries and easily implementable Gaussian measurements such as homodyne and heterodyne detection. For both schemes, we show thatN=O(poly (1 /ϵ, log (1/δ),Mr+αn, log(m) )) samples of an unknownm-mode state ρ suffice to learn the expected value of anyr-local polynomial in the canonical observables of degree α, both with high probability 1 - δ and accuracy ϵ, as long as the state ρ has moments of ordern> α bounded byMn. By simultaneously truncating states and operators in energy and phase space, we are able to overcome new mathematical challenges that arise due to the infinite-dimensionality of CV systems. We also provide a scheme to learn nonlinear functionals of the state, such as entropies over any small number of modes, by leveraging recent energy-constrained entropic continuity bounds. Finally, we provide numerical evidence of the efficiency of our protocols in the case of CV states of relevance in quantum information theory, including ground states of quadratic Hamiltonians of many-body systems and cat qubit states. We expect our scheme to provide good recovery in learning relevant states of 2D materials and photonic crystals.

We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha $-Rényi and$\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.

The Fashion domain is one of the most profitable domains in most of the e-commerce shops, shoes being one of the top-selling categories within this domain. When shopping for shoes, one of the most important aspects for the buyers is the shoe size. Shoe size charts differ between different brands, geographical regions, genders and age groups. Not providing some of these details, as a buyer or a seller, could lead to a query intent to inventory mismatch and reduced or wrong search results. Furthermore, buying the wrong shoe size is one of the top reasons for product returns, which causes shipping delays and loss in revenue. To address this issue, we propose an approach for shoe size resolution and normalization in search queries and product listings using Knowledge Graphs.