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1. Wigner non-negative states that verify the Wigner entropy conjecture

   https://doi.org/10.1103/PhysRevA.110.012228  

   Qian, Qipeng; Gagatsos, Christos N (July 2024, Physical Review A)

   We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth by Van Herstraeten and Cerf [Phys. Rev. A 104, 042211 (2021)] and Hertz et al. [J. Phys. A: Math. Theor. 50, 385301 (2017)]. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to 1 + ln π . We prove this conjecture for the qubits formed by Fock states |0⟩ and |1⟩ that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for the conjecture’s validity. Lastly, we elaborate on the states which are in accordance with our condition. 

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2. Liouville partial-differential-equation methods for computing 2D complex multivalued eikonals in attenuating media

   https://doi.org/10.1190/GEO2021-0113.1  

   Leung, Shingyu; Hu, Jiangtao; Qian, Jianliang (March 2022, GEOPHYSICS)

   We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is one essential ingredient for describing wave propagation in attenuating media because this unique quantity summarizes two wave properties into one function: Its real part describes the wave kinematics and its imaginary part captures the effects of phase dispersion and amplitude attenuation. Because some popular ordinary-differential-equation (ODE)-based ray-tracing methods for computing complex-valued eikonals in real space distribute the eikonal function irregularly in real space, we are motivated to develop PDE-based Eulerian methods for computing such complex-valued eikonals in real space on regular meshes. Therefore, we solved novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We call the resulting method the Liouville PDE method for complex-valued multivalued eikonals in attenuating media; moreover, this new method provides a unified framework for Eulerianizing several popular approximate real-space ray-tracing methods for complex-valued eikonals, such as viscoacoustic ray tracing, real viscoelastic ray tracing, and real elastic ray tracing. In addition, we also provide Liouville PDE formulations for computing multivalued ray amplitudes in a weakly viscoacoustic medium. Numerical examples, including a synthetic gas-cloud model, demonstrate that our methods yield highly accurate complex-valued eikonals in the multivalued sense. 

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3. Local navigation-like functions for safe robot navigation in bounded domains with unknown convex obstacles

   https://doi.org/10.1016/j.automatica.2023.111452  

   Farivarnejad, Hamed; Lafmejani, Amir Salimi; Berman, Spring (March 2024, Automatica)

   In this paper, we propose a controller that stabilizes a holonomic robot with single-integrator dynamics to a target position in a bounded domain, while preventing collisions with convex obstacles. We assume that the robot can measure its own position and heading in a global coordinate frame, as well as its relative position vector to the closest point on each obstacle in its sensing range. The robot has no information about the locations and shapes of the obstacles. We define regions around the boundaries of the obstacles and the domain within which the robot can sense these boundaries, and we associate each region with a virtual potential field that we call a local navigation-like function (NLF), which is only a function of the robot’s position and its distance from the corresponding boundary. We also define an NLF for the remaining free space of the domain, and we identify the critical points of the NLFs. Then, we propose a switching control law that drives the robot along the negative gradient of the NLF for the obstacle that is currently closest, or the NLF for the remaining free space if no obstacle is detected. We derive a conservative upper bound on the tunable parameter of the NLFs that guarantees the absence of locally stable equilibrium points, which can trap the robot, if the obstacles’ boundaries satisfy a minimum curvature condition. We also analyze the convergence and collision avoidance properties of the switching control law and, using a Lyapunov argument, prove that the robot safely navigates around the obstacles and converges asymptotically to the target position. We validate our analytical results for domains with different obstacle configurations by implementing the controller in both numerical simulations and physical experiments with a nonholonomic mobile robot. 

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4. Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

   https://doi.org/10.1007/s10440-019-00252-6  

   Gabardo, Jean-Pierre; Han, Deguang (April 2019, Acta Applicandae Mathematicae)

   Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure $$F: \Omega \to B(H)$$ has an integral representation of the form $$F(E) =\sum_{k=1}^{m} \int_{E}\, G_{k}(\omega)\otimes G_{k}(\omega) d\mu(\omega)$$ for some weakly measurable maps $$G_{k} \ (1\leq k\leq m) $$ from a measurable space $$\Omega$$ to a Hilbert space $$\mathcal{H}$$ and some positive measure $$\mu$$ on $$\Omega$$. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic. 

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5. Physical Interpretation of Imaginary Time Delay

   https://doi.org/10.1103/nnk7-xy4v  

   Giovannelli, Isabella L; Anlage, Steven M (July 2025, Physical Review Letters)

   The scattering matrix $S$ linearly relates the vector of incoming waves to outgoing wave excitations, and contains an enormous amount of information about the scattering system and its connections to the scattering channels. Time delay is one way to extract information from $S$ , and the transmission time delay ${\tau }_T$ is a complex (even for Hermitian systems with unitary scattering matrices) measure of how long a wave excitation lingers before being transmitted. The real part of ${\tau }_T$ is a well-studied quantity, but the imaginary part of ${\tau }_T$ has not been systematically examined experimentally, and theoretical predictions for its behavior have not been tested. Here we experimentally test the predictions of Asano [] for the imaginary part of transmission time delay in a nonunitary scattering system. We utilize Gaussian time-domain pulses scattering from a two-port microwave graph supporting a series of well-isolated absorptive modes to show that the carrier frequency of the pulses is changed in the scattering process by an amount in agreement with the imaginary part of the independently determined complex transmission time delay, $\mathrm{Im}\left[{\tau }_T\right]$ , from frequency-domain measurements of the subunitary $S$ matrix. Our results also generalize and extend those of Asano , establishing a means to predict pulse propagation properties of non-Hermitian systems over a broad range of conditions. 

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