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1. Page curves and typical entanglement in linear optics

   https://doi.org/10.22331/q-2023-05-23-1017  

   Iosue, Joseph T.; Ehrenberg, Adam; Hangleiter, Dominik; Deshpande, Abhinav; Gorshkov, Alexey V. (May 2023, Quantum)

   Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity. 

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2. Bosonic pair production and squeezing for optical phase measurements in long-lived dipoles coupled to a cavity

   https://doi.org/10.48550/arXiv.2204.13090  

   Sundar, B.; Barberena, D.; Piñeiro Orioli, A.; Chu, A.; Thompson, J.K.; Rey, A.M.; Lewis-Swan, R.J. (October 2022, ArXivorg)

   We propose to simulate bosonic pair creation using large arrays of long-lived dipoles with multilevel internal structure coupled to an undriven optical cavity. Entanglement between the atoms, generated by the exchange of virtual photons through a common cavity mode, grows exponentially fast and is described by two-mode squeezing (TMS) of effective bosonic quadratures. The mapping between an effective bosonic model and the natural spin description of the dipoles allows us to realize the analog of optical homodyne measurements via straightforward global rotations and population measurements of the electronic states, and we propose to exploit this for quantum-enhanced sensing of an optical phase (common and differential between two ensembles). We discuss a specific implementation based on Sr atoms and show that our sensing protocol is robust to sources of decoherence intrinsic to cavity platforms. Our proposal can open unique opportunities for the observation of continuous variable entanglement in atomic systems and associated applications in next-generation optical atomic clocks. 

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3. Safeguarding Oscillators and Qudits with Distributed Two-Mode Squeezing

   https://doi.org/10.22331/q-2024-09-19-1478  

   Brady, Anthony J; Wu, Jing; Zhuang, Quntao (September 2024, Quantum)

   Recent advancements in multi-mode Gottesman-Kitaev-Preskill (GKP) codes have shown great promise in enhancing the protection of both discrete and analog quantum information. This broadened range of protection brings opportunities beyond quantum computing to benefit quantum sensing by safeguarding squeezing — the essential resource in many quantum metrology protocols. However, the potential for quantum sensing to benefit quantum error correction has been less explored. In this work, we provide a unique example where techniques from quantum sensing can be applied to improve multi-mode GKP codes. Inspired by distributed quantum sensing, we propose the distributed two-mode squeezing (dtms) GKP codes that offer benefits in error correction with minimal active encoding operations. Indeed, the proposed codes rely on a $single$(active) two-mode squeezing element and an array of beamsplitters that effectively distributes continuous-variable correlations to many GKP ancillae, similar to continuous-variable distributed quantum sensing. Despite this simple construction, the code distance achievable with dtms-GKP qubit codes is comparable to previous results obtained through brute-force numerical search \cite{lin2023closest}. Moreover, these codes enable analog noise suppression beyond that of the best-known two-mode codes \cite{noh2020o2o} without requiring an additional squeezer. We also provide a simple two-stage decoder for the proposed codes, which appears near-optimal for the case of two modes and permits analytical evaluation. 

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4. Generation of time-domain-multiplexed two-dimensional cluster state

   https://doi.org/10.1126/science.aay2645  

   Asavanant, Warit; Shiozawa, Yu; Yokoyama, Shota; Charoensombutamon, Baramee; Emura, Hiroki; Alexander, Rafael N.; Takeda, Shuntaro; Yoshikawa, Jun-ichi; Menicucci, Nicolas C.; Yonezawa, Hidehiro; et al (October 2019, Science)

   Entanglement is the key resource for measurement-based quantum computing. It is stored in quantum states known as cluster states, which are prepared offline and enable quantum computing by means of purely local measurements. Universal quantum computing requires cluster states that are both large and possess (at least) a two-dimensional topology. Continuous-variable cluster states—based on bosonic modes rather than qubits—have previously been generated on a scale exceeding one million modes, but only in one dimension. Here, we report generation of a large-scale two-dimensional continuous-variable cluster state. Its structure consists of a 5- by 1240-site square lattice that was tailored to our highly scalable time-multiplexed experimental platform. It is compatible with Bosonic error-correcting codes that, with higher squeezing, enable fault-tolerant quantum computation. 

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5. Average Rényi entanglement entropy in Gaussian boson sampling

   https://doi.org/10.1103/PhysRevResearch.7.023125  

   Youm, Jason; Iosue, Joseph T; Ehrenberg, Adam; Wang, Yu-Xin; Gorshkov, Alexey V (May 2025, Physical review research)

   Recently, many experiments have been conducted with the goal of demonstrating a quantum advantage over classical computation. One popular framework for these experiments is Gaussian boson sampling, where quadratic photonic input states are interfered via a linear optical unitary and subsequently measured in the Fock basis. In this paper, we study the modal entanglement of the output states in this framework just before the measurement stage. Specifically, we compute Page curves as measured by various Rényi- $\alpha$entropies, where the Page curve describes the entanglement between two partitioned groups of output modes averaged over all linear optical unitaries. We derive these formulas for $\alpha =1$(i.e., the von Neumann entropy) and, more generally, for all positive integer $\alpha$, in the asymptotic limit of infinite number of modes and for input states that are composed of single-mode-squeezed-vacuum state with equal squeezing strength. We then analyze the limiting behaviors when the squeezing is small and large. Having determined the averages, we then explicitly calculate the Rényi- $\alpha$variance for integers $\alpha >1$and are able to show that these entropies are weakly typical. Published by the American Physical Society2025 

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