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1. Exploring the possibility of a complex-valued non-Gaussianity measure for quantum states of light

   https://doi.org/10.1063/5.0219011  

   Pizzimenti, Andrew J; Dhara, Prajit; Van_Herstraeten, Zacharie; Cheng, Sijie; Gagatsos, Christos N (September 2024, APL Quantum)

   We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements. 

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2. 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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3. Cryptography in the Common Haar State Model: Feasibility Results and Separations

   Ananth, Prabhanjan; Gulati, Aditya; Lin, Yao-Ting (November 2024, Springer Nature)

   Common random string model is a popular model in classi- cal cryptography. We study a quantum analogue of this model called the common Haar state (CHS) model. In this model, every party participating in the cryptographic system receives many copies of one or more i.i.d Haar random states. We study feasibility and limitations of cryptographic primitives in this model and its variants: – We present a construction of pseudorandom function-like states with security against computationally unbounded adversaries, as long as the adversaries only receive (a priori) bounded number of copies. By suitably instantiating the CHS model, we obtain a new approach to construct pseudorandom function-like states in the plain model. – We present separations between pseudorandom function-like states (with super-logarithmic length) and quantum cryptographic primitives, such as interactive key agreement and bit commitment, with classical communication. To show these separations, we prove new results on the indistinguishability of identical versus independent Haar states against LOCC (local operations, classical communication) adversaries. 

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4. Entanglement dynamics in U(1) symmetric hybrid quantum automaton circuits

   https://doi.org/10.22331/q-2023-12-06-1200  

   Han, Yiqiu; Chen, Xiao (December 2023, Quantum)

   We study the entanglement dynamics of quantum automaton (QA) circuits in the presence of U(1) symmetry. We find that the second Rényi entropy grows diffusively with a logarithmic correction as $\sqrt{t\ln t}$, saturating the bound established by Huang \cite{Huang_2020}. Thanks to the special feature of QA circuits, we understand the entanglement dynamics in terms of a classical bit string model. Specifically, we argue that the diffusive dynamics stems from the rare slow modes containing extensively long domains of spin 0s or 1s. Additionally, we investigate the entanglement dynamics of monitored QA circuits by introducing a composite measurement that preserves both the U(1) symmetry and properties of QA circuits. We find that as the measurement rate increases, there is a transition from a volume-law phase where the second Rényi entropy persists the diffusive growth (up to a logarithmic correction) to a critical phase where it grows logarithmically in time. This interesting phenomenon distinguishes QA circuits from non-automaton circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an area-law phase transition exists, and any non-zero rate of projective measurements in the volume-law phase leads to a ballistic growth of the Rényi entropy. 

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5. Quantum Curved Tetrahedron, Quantum Group Intertwiner Space, and Coherent States

   https://doi.org/10.1088/1361-6382/adb533  

   Hsiao, Chen-Hung; Pan, Qiaoyin (February 2025, Classical and Quantum Gravity)

   Abstract In this paper, we construct the phase space of a constantly curved tetrahedron with fixed triangle areas in terms of a pair of Darboux coordinates called the length and twist coordinates, which are in analogy to the Fenchel-Nielsen coordinates for flat connections, and their quantization. The curvature is identified to the value of the cosmological constant, either positive or negative. The physical Hilbert space is given by the $$\mathcal{U}_q(\mathfrak{su}(2))$$ intertwiner space. We show that the quantum trace of quantum monodromies, defining the quantum length operators, form a fusion algebra and describe their representation theory. We also construct the coherent states in the physical Hilbert space labeled by the length and twist coordinates. These coherent states describe quantum curved tetrahedra and peak at points of the tetrahedron phase space. This work is closely related to 3+1 dimensional Loop Quantum Gravity with a non-vanishing cosmological constant. The coherent states constructed herein serve as good candidates for the application to the spinfoam model with a cosmological constant. 

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