Master Lovas–Andai and equivalent formulas verifying the $$\frac{8}{33}$$ two-qubit Hilbert–Schmidt separability probability and companion rational-valued conjectures

We begin by investigating relationships between two forms of Hilbert–Schmidt two-rebit and two-qubit “separability functions”—those recently advanced by Lovas and Andai (J Phys A Math Theor 50(29):295303, 2017), and those earlier presented by Slater (J Phys A 40(47):14279, 2007). In the Lovas–Andai framework, the independent variable \(\varepsilon \in [0,1]\) is the ratio \(\sigma (V)\) of the singular values of the \(2 \times 2\) matrix \(V=D_2^{1/2} D_1^{-1/2}\) formed from the two \(2 \times 2\) diagonal blocks (\(D_1, D_2\)) of a \(4 \times 4\) density matrix \(D= \left||\rho _{ij}\right||\). In the Slater setting, the independent variable \(\mu \) is the diagonal-entry ratio \(\sqrt{\frac{\rho _{11} \rho _ {44}}{\rho _ {22} \rho _ {33}}}\)—with, of central importance, \(\mu =\varepsilon \) or \(\mu =\frac{1}{\varepsilon }\) when both \(D_1\) and \(D_2\) are themselves diagonal. Lovas and Andai established that their two-rebit “separability function” \(\tilde{\chi }_1 (\varepsilon )\) (\(\approx \varepsilon \)) yields the previously conjectured Hilbert–Schmidt separability probability of \(\frac{29}{64}\). We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we newly find its two-qubit, two-quater[nionic]-bit and “two-octo[nionic]-bit” counterparts, \(\tilde{\chi _2}(\varepsilon ) =\frac{1}{3} \varepsilon ^2 \left( 4-\varepsilon ^2\right) \), \(\tilde{\chi _4}(\varepsilon ) =\frac{1}{35} \varepsilon ^4 \left( 15 \varepsilon ^4-64 \varepsilon ^2+84\right) \) and \(\tilde{\chi _8} (\varepsilon )= \frac{1}{1287}\varepsilon ^8 \left( 1155 \varepsilon ^8-7680 \varepsilon ^6+20160 \varepsilon ^4-25088 \varepsilon ^2+12740\right) \). These immediately lead to predictions of Hilbert–Schmidt separability/PPT-probabilities of \(\frac{8}{33}\), \(\frac{26}{323}\) and \(\frac{44482}{4091349}\), in full agreement with those of the “concise formula” (Slater in J Phys A 46:445302, 2013), and, additionally, of a “specialized induced measure” formula. Then, we find a Lovas–Andai “master formula,” \(\tilde{\chi _d}(\varepsilon )= \frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left( -\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right) }{\Gamma \left( \frac{d}{2}+1\right) ^2}\), encompassing both even and odd values of d. Remarkably, we are able to obtain the \(\tilde{\chi _d}(\varepsilon )\) formulas, \(d=1,2,4\), applicable to full (9-, 15-, 27-) dimensional sets of density matrices, by analyzing (6-, 9, 15-) dimensional sets, with not only diagonal \(D_1\) and \(D_2\), but also an additional pair of nullified entries. Nullification of a further pair still leads to X-matrices, for which a distinctly different, simple Dyson-index phenomenon is noted. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence. A two-qubit separability probability of \(1-\frac{256}{27 \pi ^2}\) is obtained based on the operator monotone function \(\sqrt{x}\), with the use of \(\tilde{\chi _2}(\varepsilon )\).     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit

Implicit–explicit (IMEX) Runge–Kutta (RK) schemes are popular high order time discretization methods for solving stiff kinetic equations. As opposed to the compressible Euler limit (leading order asymptotics of the Boltzmann equation as the Knudsen number \(\varepsilon \) goes to zero), their asymptotic behavior at the Navier–Stokes (NS) level (next order asymptotics) was rarely studied. In this paper, we analyze a class of existing IMEX RK schemes and show that, under suitable initial conditions, they can capture the NS limit without resolving the small parameter \(\varepsilon \), i.e., \(\varepsilon =o(\Delta t)\), \(\Delta t^m=o(\varepsilon )\), where m is the order of the explicit RK part in the IMEX scheme. Extensive numerical tests for BGK and ES-BGK models are performed to verify our theoretical results.     

Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

In this paper, we present unconditionally optimal error estimates of linearized Crank–Nicolson Galerkin finite element methods for a strongly nonlinear parabolic system in \(\mathbb {R}^d\ (d=2,3)\). However, all previous works required certain time-step conditions that were dependent on the spatial mesh size. In order to overcome several entitative difficulties caused by the strong nonlinearity of the system, the proof takes two steps. First, by using a temporal-spatial error splitting argument and a new technique, optimal \(L^2\) error estimates of the numerical schemes can be obtained under the condition \(\tau \ge h\), where \(\tau \) denotes the time-step size and h is the spatial mesh size. Second, we obtain the boundedness of numerical solutions by mathematical induction and inverse inequality when \(\tau \le h\). Then, optimal \(L^2\) and \(H^1\) error estimates are proved in a different way for such case. Numerical results are given to illustrate our theoretical analyses.     

An Advection-Robust Hybrid High-Order Method for the Oseen Problem

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer \(k\geqslant 0\), the discrete velocity unknowns are vector-valued polynomials of total degree \(\leqslant \, k\) on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree \(\leqslant \,k\) on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree \(\leqslant \,(k+1)\), a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity–pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element T of diameter \(h_T\) contributes to the discretization error with an \(\mathcal {O}(h_{T}^{k+1})\)-term in the diffusion-dominated regime, an \(\mathcal {O}(h_{T}^{k+\frac{1}{2}})\)-term in the advection-dominated regime, and scales with intermediate powers of \(h_T\) in between. Numerical results complete the exposition.     

Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation

A linearized Crank–Nicolson Galerkin finite element method with bilinear element for nonlinear Schrödinger equation is studied. By splitting the error into two parts which are called the temporal error and the spatial error, the unconditional superconvergence result is deduced. On one hand, the regularity for a time-discrete system is presented based on the proof of the temporal error. On the other hand, the classical Ritz projection is applied to get the spatial error with order \(O(h^2)\) in \(L^2\)-norm, which plays an important role in getting rid of the restriction of \(\tau \). Then the superclose estimates of order \(O(h^2+\tau ^2)\) in \(H^1\)-norm is arrived at based on the relationship between the Ritz projection and the interpolated operator. At the same time, global superconvergence property is arrived at by the interpolated postprocessing technique. At last, three numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and \(\tau \) is the time step.     

A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

A new weak Galerkin (WG) finite element method is developed and analyzed for solving second order elliptic problems with low regularity solutions in the Sobolev space \(W^{2,p}(\Omega )\) with \(p\in (1,2)\). A WG stabilizer was introduced by Wang and Ye (Math Comput 83:2101–2126, 2014) for a simpler variational formulation, and it has been commonly used since then in the WG literature. In this work, for the purpose of dealing with low regularity solutions, we propose to generalize the stabilizer of Wang and Ye by introducing a positive relaxation index to the mesh size h. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along the interior element edges. When the norm index \(p\in (1,2]\), we strictly derive that the WG error in energy norm has an optimal convergence order \(O(h^{l+1-\frac{1}{p}-\frac{p}{4}})\) by taking the relaxed factor \(\beta =1+\frac{2}{p}-\frac{p}{2}\), and it also has an optimal convergence order \(O(h^{l+2-\frac{2}{p}})\) in \(L^2\) norm when the solution \(u\in W^{l+1,p}\) with \(p\in [1,1+\frac{2}{p}-\frac{p}{2}]\) and \(l\ge 1\). It is recovered for \(p=2\) that with the choice of \(\beta =1\), error estimates in the energy and \(L^2\) norms are optimal for the source term in the sobolev space \(L^2\). Weak variational forms of the WG method give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented without requiring a sufficiently large penalty factor as in the usual discontinuous Galerkin methods. In addition, numerical results illustrate that the proposed WG method with an over-relaxed factor \(\beta (\ge 1)\) converges at optimal algebraic rates for several low regularity elliptic problems.     

Data assimilation and sampling in Banach spaces

This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\), \(j=1,\ldots ,m\), where the \(l_j\) are linear functionals from \(\mathcal{X}^*\). Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\). Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for K the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\), with \(\varepsilon >0\) and V a finite dimensional subspace of \(\mathcal{X}\), which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \). These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.     

Decoupled,Unconditionally Stable,Higher Order Discretizations for MHD Flow Simulation

We propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep yet maintains unconditional stability with respect to the time step size, is optimally accurate in space, and behaves like second order in time in practice. The proposed method chooses a parameter \(\theta \in [0,1]\), dependent on the viscosity \(\nu \) and magnetic diffusivity \(\nu _m\), so that the explicit treatment of certain viscous terms does not cause instabilities, and gives temporal accuracy \(O(\Delta t^2 + (1-\theta )|\nu -\nu _m|\Delta t)\). In practice, \(\nu \) and \(\nu _m\) are small, and so the method behaves like second order. When \(\theta =1\), the method reduces to a linearized BDF2 method, but it has been proven by Li and Trenchea that such a method is stable only in the uncommon case of \(\frac{1}{2}< \frac{\nu }{\nu _m} < 2\). For the proposed method, stability and convergence are rigorously proven for appropriately chosen \(\theta \), and several numerical tests are provided that confirm the theory and show the method provides excellent accuracy in cases where usual BDF2 is unstable.     

Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with $${\textit{EQ}}_1^{rot}$$ Nonconforming Finite Element

Nonlinear parabolic equation is studied with a linearized Galerkin finite element method. First of all, a time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, a rigorous analysis for the regularity of the time-discrete system is presented based on the proof of the temporal error skillfully. On the other hand, the spatial error is derived \(\tau \)-independently with the above achievements. Then, the superclose result of order \(O(h^2+\tau ^2)\) in broken \(H^1\)-norm is deduced without any restriction of \(\tau \). The two typical characters of the \({\textit{EQ}}_1^{rot}\) nonconforming FE (see Lemma 1 below) play an important role in the procedure of proof. At last, numerical results are provided in the last section to confirm the theoretical analysis. Here, h is the subdivision parameter, and \(\tau \), the time step.     

Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary Conditions

A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The \(L1\) discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The convergence order is \(\mathcal{O }(\tau ^{3-\alpha }+h^4)\) in the maximum norm, where \(\tau \) is the temporal grid size and \(h\) is the spatial grid size, respectively. In addition, a Crank–Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of \(\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)\) is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank–Nicolson scheme are presented to show the effectiveness of the compact scheme.     

Monogamy and polygamy for multi-qubit W-class states using convex-roof extended negativity of assistance and Rényi-$$\alpha $$ entropy

We present some new analytical polygamy inequalities satisfied by the x-th power of convex-roof extended negativity of assistance with \(x\ge 2\) and \(x\le 0\) for multi-qubit generalized W-class states. Using Rényi-\(\alpha \) entropy (R\(\alpha \)E) with \(\alpha \in [(\sqrt{7}-1)/2, (\sqrt{13}-1)/2]\), we prove new monogamy and polygamy relations. We further show that the monogamy inequality also holds for the \(\mu \)th power of Rényi-\(\alpha \) entanglement. Moreover, we study two examples in multipartite higher-dimensional system for those new inequalities.     

Recursive Computation of Logarithmic Derivatives,Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and Applications

Spheroidal harmonics and modified Bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics (\(P_n^m(x)\), \(Q_n^m(x)\), \(n\ge m\ge 0\), \(x>1\)), the oblate spheroidal harmonics (\(P_n^m(ix)\), \(Q_n^m(ix)\), \(n\ge m\ge 0\), \(x>0\)), and the modified Bessel functions (\(I_n(x)\), \(K_n(x)\), \(n\ge 0\), \(x>0\)) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.     

Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives

In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally \(\ell ^1\) contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, \( (\Delta t)^{\alpha } = O(\Delta x)\), where \(\alpha \in (0,1]\) is the fractional exponent in the derivative. When \(\alpha \) is small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally \(\ell ^1\) contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.     

Bimodal behavior of post-measured entropy and one-way quantum deficit for two-qubit X states

A method for calculating the one-way quantum deficit is developed. It involves a careful study of post-measured entropy shapes. We discovered that in some regions of X-state space the post-measured entropy \(\tilde{S}\) as a function of measurement angle \(\theta \in [0,\pi /2]\) exhibits a bimodal behavior inside the open interval \((0,\pi /2)\), i.e., it has two interior extrema: one minimum and one maximum. Furthermore, cases are found when the interior minimum of such a bimodal function \(\tilde{S}(\theta )\) is less than that one at the endpoint \(\theta =0\) or \(\pi /2\). This leads to the formation of a boundary between the phases of one-way quantum deficit via finite jumps of optimal measured angle from the endpoint to the interior minimum. Phase diagram is built up for a two-parameter family of X states. The subregions with variable optimal measured angle are around 1\(\%\) of the total region, with their relative linear sizes achieving \(17.5\%\), and the fidelity between the states of those subregions can be reduced to \(F=0.968\). In addition, a correction to the one-way deficit due to the interior minimum can achieve \(2.3\%\). Such conditions are favorable to detect the subregions with variable optimal measured angle of one-way quantum deficit in an experiment.     

On the Distribution of Photon Counts with Censoring in Two-Photon Laser Scanning Microscopy

We address the problem of counting emitted photons in two-photon laser scanning microscopy. Following a laser pulse, photons are emitted after exponentially distributed waiting times. Modeling the counting process is of interest because photon detectors have a dead period after a photon is detected that leads to an underestimate of the count of emitted photons. We describe a model which has a Poisson \((\alpha )\) number N of photons emitted, and a dead period \(\Delta \) that is standardized by the fluorescence time constant \(\tau (\delta = \Delta /\tau )\), and an observed count D. The estimate of \(\alpha \) determines the intensity of a single pixel in an image. We first derive the distribution of D and study its properties. We then use it to estimate \(\alpha \) and \(\delta \) simultaneously by maximum likelihood. We show that our results improve the signal-to-noise ratio, hence the quality of actual images.     

Indistinguishability of pure orthogonal product states by LOCC

We construct two sets of incomplete and extendible quantum pure orthogonal product states (POPS) in general bipartite high-dimensional quantum systems, which are all indistinguishable by local operations and classical communication. The first set of POPS is composed of two parts which are \(\mathcal {C}^m\otimes \mathcal {C}^{n_1}\) with \(5\le m\le n_1\) and \(\mathcal {C}^m\otimes \mathcal {C}^{n_2}\) with \(5\le m \le n_2\), where \(n_1\) is odd and \(n_2\) is even. The second one is in \(\mathcal {C}^m\otimes \mathcal {C}^n\) \((m, n\ge 4)\). Some subsets of these two sets can be extended into complete sets that local indistinguishability can be decided by noncommutativity which quantifies the quantumness of a quantum ensemble. Our study shows quantum nonlocality without entanglement.     

Numerical Method for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Equation with the Temperature-Jump Boundary Condition

This article proposes a new nanoscale heat transfer model based on the Caputo type fractional dual-phase-lagging (DPL) heat conduction equation with the temperature-jump boundary condition. The model is proved to be well-posed. A finite difference scheme based on the L1 approximation for the Caputo derivative is then presented for solving the fractional DPL model. Unconditional stability and convergence of the scheme are proved by using the discrete energy method. Three numerical examples are given to verify the accuracy of the scheme. Results show the convergence order to be \(O(\tau ^{2-\alpha }+h^2)\) , which coincides with the theoretical analysis. A simple nanoscale semiconductor silicon device is illustrated to show the applicability of the model. It is seen from the numerical result that when \(\alpha =1\), the fractional DPL reduces to the conventional DPL and the obtained peak temperature is almost identical to those obtained in the literature. However, when \(\alpha <1\), the model predicts a higher peak temperature level than that when \(\alpha =1\). In particular, when \(\alpha = 0.7\) and 0.9, an oscillatory temperature at the beginning is observed. This indicates that the fractional DPL model can be an excellent candidate for analyzing the temperature instability appearing in electronic nano-semiconductor devices.     

Extremal properties of conditional entropy and quantum discord for XXZ,symmetric quantum states

For the XXZ subclass of symmetric two-qubit X states, we study the behavior of quantum conditional entropy \(S_{cond}\) as a function of measurement angle \(\theta \in [0,\pi /2]\). Numerical calculations show that the function \(S_{cond}(\theta )\) for X states can have at most one local extremum in the open interval from zero to \(\pi /2\) (unimodality property). If the extremum is a minimum, the quantum discord displays region with variable (state-dependent) optimal measurement angle \(\theta ^*\). Such \(\theta \)-regions (phases, fractions) are very tiny in the space of X-state parameters. We also discover the cases when the conditional entropy has a local maximum inside the interval \((0,\pi /2)\). It is remarkable that the maxima exist in surprisingly wide regions, and the boundaries for such regions are defined by the same bifurcation conditions as for those with a minimum.