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1. Average capacity of quantum entanglement

   https://doi.org/10.1088/1751-8121/acb114  

   Wei, Lu (January 2023, Journal of Physics A: Mathematical and Theoretical)

   Abstract As an alternative to entanglement entropies, the capacity of entanglement becomes a promising candidate to probe and estimate the degree of entanglement of quantum bipartite systems. In this work, we study the statistical behavior of entanglement capacity over major models of random states. In particular, the exact and asymptotic formulas of average capacity have been derived under the Hilbert–Schmidt and Bures-Hall ensembles. The obtained formulas generalize some partial results of average capacity computed recently in the literature. As a key ingredient in deriving the results, we make use of techniques in random matrix theory and our previous results pertaining to the underlying orthogonal polynomials and special functions. Simulations have been performed to numerically verify the derived formulas. 

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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. Bosonic and fermionic Gaussian states from Kähler structures

   https://doi.org/10.21468/SciPostPhysCore.4.3.025  

   Hackl, Lucas; Bianchi, Eugenio (January 2021, SciPost Physics Core)

   We show that bosonic and fermionic Gaussian states (also known as``squeezed coherent states’’) can be uniquely characterized by theirlinear complex structure J J which is a linear map on the classical phase space. This extendsconventional Gaussian methods based on covariance matrices and providesa unified framework to treat bosons and fermions simultaneously. PureGaussian states can be identified with the triple (G,\Omega,J) ( G , Ω , J ) of compatible Kähler structures, consisting of a positive definitemetric G G ,a symplectic form \Omega Ω and a linear complex structure J J with J^2=-\mathbb{1} J 2 = − 1 .Mixed Gaussian states can also be identified with such a triple, butwith J^2\neq -\mathbb{1} J 2 ≠ − 1 .We apply these methods to show how computations involving Gaussianstates can be reduced to algebraic operations of these objects, leadingto many known and some unknown identities. We apply these methods to thestudy of (A) entanglement and complexity, (B) dynamics of stablesystems, (C) dynamics of driven systems. From this, we compile acomprehensive list of mathematical structures and formulas to comparebosonic and fermionic Gaussian states side-by-side. 

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4. On the equivalence between squeezing and entanglement potential for two-mode Gaussian states

   https://doi.org/10.1038/s41598-023-38572-1  

   Li, Bohan; Das, Aritra; Tserkis, Spyros; Narang, Prineha; Lam, Ping Koy; Assad, Syed M. (December 2023, Scientific Reports)

   Abstract The maximum amount of entanglement achievable under passive transformations by continuous-variable states is called the entanglement potential. Recent work has demonstrated that the entanglement potential is upper-bounded by a simple function of the squeezing of formation, and that certain classes of two-mode Gaussian states can indeed saturate this bound, though saturability in the general case remains an open problem. In this study, we introduce a larger class of states that we prove saturates the bound, and we conjecture that all two-mode Gaussian states can be passively transformed into this class, meaning that for all two-mode Gaussian states, entanglement potential is equivalent to squeezing of formation. We provide an explicit algorithm for the passive transformations and perform extensive numerical testing of our claim, which seeks to unite the resource theories of two characteristic quantum properties of continuous-variable systems. 

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5. Non-Abelian symmetry-resolved entanglement entropy

   https://doi.org/10.21468/SciPostPhys.17.5.127  

   Bianchi, Eugenio; Dona, Pietro; Kumar, Rishabh (January 2024, SciPost Physics)

   We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2) symmetry. 

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