arXiv subjects
Publications and source records attributed to Hong-Bin Chen.
We consider a general statistical inference model of finite-rank tensor products. For any interaction structure and any order of tensor products, we identify the limit free energy of the model in terms of a variational formula. Our approach consists of showing first that the limit free energy must be the viscosity solution to a certain Hamilton-Jacobi equation.
It is commonly known that the dephasing in open quantum systems is due to the establishment of bipartite correlations with ambient environments, which are typically difficult to be fully characterized. Recently, a new approach of average over disordered Hamiltonian ensemble is developed and shown to be capable of describing the nonclassicality of incoherent dynamics based on inferring the nonclassical nature of the correlations. Here we further extend the approach of Hamiltonian ensemble in the canonical form to the realm of structural disorder. Under the variable separation of the probability distribution within the Hamiltonian ensemble, the geometrical structure is easily visualized and can be characterized according to the degree of symmetry. We demonstrate four degrees and investigate the effects of different types of symmetry breaking on the incoherent dynamics. We show that these effects are easily understood from the emergences of additional terms in the master equations, leading to rather general master equations and, consequently, going beyond the previous frameworks of pure dephasing or isotropic depolarization.
We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow on polymers pinned at the origin, for energy involving quadratic nearest neighbor interaction and local interaction with random environment. We prove existence and uniqueness of the solution, continuity of the flow, the order-preserving property with respect to the coordinatewise partial order, and the invariance of the asymptotic slope. We establish ordering by noise which means that if two initial conditions have distinct slopes, then the associated solutions eventually get ordered coordinatewise. This, along with the shear-invariance property and existing results on static infinite volume polymer measures, allows to prove that for a fixed asymptotic slope and almost every realization of the environment, the polymer dynamics has a unique invariant distribution given by a unique infinite volume polymer measure, and, moreover, One Force -- One Solution principle holds. We also prove that every polymer measure is concentrated on paths with well-defined asymptotic slopes and give an estimate on deviations from straight lines.
Given posets $\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k$, let the {\em Boolean Ramsey number} $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ be the minimum number $n$ such that no matter how we color the elements in the Boolean lattice $\mathbf{B}_n$ with $k$ colors, there always exists a poset $\mathbf{P}_i$ contained in $\mathbf{B}_n$ whose elements are all colored with $i$. This function was first introduced by Axenovich and Walzer~\cite{AW}. Recently, many results on determining $R(\mathbf{B}_m,\mathbf{B}_n)$ have been published. In this paper, we will study the function $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ for each $\mathbf{P}_i$'s being the $V$-shaped poset. That is, a poset obtained by identifying the minimal elements of two chains. Another major result presented in the paper is to determine the minimal posets $\mathbf{Q}$ contained in $\mathbf{B}_n$, when $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)=n$ is determined, having the Ramsey property described in the previous paragraph. In addition, we define the {\em Boolean rainbow Ramsey number} $RR(\mathbf{P},\mathbf{Q})$ the minimum number $n$ such that when arbitrarily coloring the elements in $\mathbf{B}_n$, there always exists either a monochromatic $\mathbf{P}$ or a rainbow $\mathbf{Q}$ contained in $\mathbf{B}_n$. The upper bound for $RR(\mathbf{P},\mathbf{A}_k)$ was given by Chang, Li, Gerbner, Methuku, Nagy, Patkos, and Vizer for general poset $\mathbf{P}$ and $k$-element antichain $\mathbf{A}_k$. We study the function for $\mathbf{P}$ being the $V$-shaped posets in this paper as well.
We study the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. In general, we bound the limit from above by the unique solution to a certain Hamilton-Jacobi equation. Under additional assumptions on the nonlinearity in the equation which is determined explicitly by the model, we identify the limit with the solution. Two notions of solutions, weak solutions and viscosity solutions, are considered, each of which has its own advantages and requires different treatments. For concreteness, we apply our results to a model with i.i.d. entries and symmetric interactions. In particular, for the first order and even order tensor products, we identify the limit and obtain estimates on convergence rates; for other odd orders, upper bounds are obtained.
We prove that if ${(P_x)}_{x\in \mathscr X}$ is a family of probability measures which satisfy the log-Sobolev inequality and whose pairwise chi-squared divergences are uniformly bounded, and $μ$ is any mixing distribution on $\mathscr X$, then the mixture $\int P_x \, \mathrm{d} μ(x)$ satisfies a log-Sobolev inequality. In various settings of interest, the resulting log-Sobolev constant is dimension-free. In particular, our result implies a conjecture of Zimmermann and Bardet et al. that Gaussian convolutions of measures with bounded support enjoy dimension-free log-Sobolev inequalities.
For a smooth vector field in a neighborhood of a critical point with all positive eigenvalues of the linearization, we consider the associated dynamics perturbed by white noise. Using Malliavin calculus tools, we obtain polynomial asymptotics for probabilities of atypically long exit times in the vanishing noise limit.
We consider exit problems for small white noise perturbations of a dynamical system generated by a vector field, and a domain containing a critical point with all positive eigenvalues of linearization. We prove that, in the vanishing noise limit, the probability of exit through a generic set on the boundary is asymptotically polynomial in the noise strength, with exponent depending on the mutual position of the set and the flag of the invariant manifolds associated with the top eigenvalues. Furthermore, we compute the limiting exit distributions conditioned on atypical exit events of polynomially small probability and show that the limits are Radon--Nikodym equivalent to volume measures on certain manifolds that we construct. This situation is in sharp contrast with the large deviation picture where the limiting conditional distributions are point masses.
We investigate contraction of the Wasserstein distances on $\mathbb{R}^d$ under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space---where the heat semigroup corresponds to smoothing the measures by Gaussian convolution---the situation is more subtle. We prove precise asymptotics for the $2$-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is always polynomial, with exponent depending on the moment sequences of the measures. We establish similar results for the $p$-Wasserstein distances for $p \neq 2$ as well as the $χ^2$ divergence, relative entropy, and total variation distance. Together, these results establish the central role of moment matching arguments in the analysis of measures smoothed by Gaussian convolution.
Within the framework of quantum refereed steering games, quantum steerability can be certified without any assumption on the underlying state nor the measurements involved. Such a scheme is termed the measurement-device-independent (MDI) scenario. Here we introduce a measure of steerability in an MDI scenario, i.e., the result merely depends on the observed statistics and the quantum inputs. We prove that such a measure satisfies the convex steering monotone. Moreover, it is robust against not only measurement biases but also losses. We also experimentally estimate the amount of the measure with an entangled photon source. As two by-products, our experimental results provide lower bounds on an entanglement measure of the underlying state and an incompatible measure of the involved measurement. Our research paves a way for exploring one-side device-independent quantum information processing within an MDI framework.
We study the high-dimensional limit of the free energy associated with the inference problem of a rank-one nonsymmetric matrix. The matrix is expressed as the outer product of two vectors, not necessarily independent. The distributions of the two vectors are only assumed to have scaled bounded supports. We bound the difference between the free energy and the solution to a suitable Hamilton-Jacobi equation in terms of two much simpler quantities: concentration rate of this free energy, and the convergence rate of a simpler free energy in a decoupled system. To demonstrate the versatility of this approach, we apply our result to the i.i.d. case and the spherical case. By plugging in estimates of the two simpler quantities, we identify the limits and obtain convergence rates.
A graph is Hamiltonian if it contains a cycle which visits every vertex of the graph exactly once. In this paper, we consider the problem of Hamiltonicity of a graph $G_n$, which will be called the prime difference graph of order $n$, with vertex set $\{1,2,\cdots, n\}$ and edge set $\{uv: |u-v|$ is a prime number$\}$. A recent result, conjectured by Sun and later proved by Chen, asserts that $G_n$ is Hamiltonian for $n\geq 5$. This paper extends their result in three directions. First, we prove that for any two integers $a$ and $b$ with $1\leq a<b\leq n$, there is a Hamilton path in $G_n$ from $a$ to $b$ except some cases of small $n$. This result implies robustness of the Hamiltonicity property of the prime difference graph in a sense that for any edge $e$ in $G_n$ there exists a Hamilton cycle containing $e$. Second, we show that the prime difference graph contains considerably more about the cycle structure than Hamiltonicity; precisely, for any integer $n\geq 7$, the prime difference graph $G_n$ contains any 2-factor of the complete graph of order $n$ as a subgraph. Finally, we find that $G_n$ may contain more edge-disjoint Hamilton cycles. In particular, these Hamilton cycles are generated by two prime differences.
Visual cryptography encrypts the secret image into $n$ shares (transparency) so that only stacking a qualified number of shares can recover the secret image by the human visual system while no information can be revealed without a large enough number of shares. This paper investigates the $(k,n)$-threshold Visual Secret Sharing (VSS) model, where one can decrypt the original image by stacking at least $k$ shares and get nothing with less than $k$ shares. There are two main approaches in the literature: codebook-based schemes and random-grid-based schemes; the former is the case of this paper. In general, given any positive integers $k$ and $n$, it is not easy to design a valid scheme for the $(k,n)$-threshold VSS model. In this paper, we propose a simple strategy to construct an efficient scheme for the $(k,n)$-threshold VSS model for any positive integers $2\leq k\leq n$. The crucial idea is to establish a seemingly unrelated connection between the $(k,n)$-threshold VSS scheme and a mathematical structure -- the generalized Pascal's triangle. This paper improves and extends previous results in four aspects: Our construction offers a unified viewpoint and covers several known results; The resulting scheme has a progressive-viewing property that means the more shares being stacked together the clearer the secret image would be revealed. The proposed scheme can be constructed explicitly and efficiently based on the generalized Pascal's triangle without a computer. Performance of the proposed scheme is comparable with known results.
Exceptional points, resulting from non-Hermitian degeneracies, have the potential to enhance the capabilities of quantum sensing. Thus, finding exceptional points in different quantum systems is vital for developing such future sensing devices. Taking advantage of the enhanced light-matter interactions in a confined volume on a metal nanoparticle surface, here we theoretically demonstrate the existence of exceptional points in a system consisting of quantum emitters coupled to a metal nanoparticle of subwavelength scale. By using an analytical quantum electrodynamics approach, exceptional points are manifested as a result of a strong coupling effect and observable in a drastic splitting of originally coalescent eigenenergies. Furthermore, we show that exceptional points can also occur when a number of quantum emitters is collectively coupled to the dipole mode of localized surface plasmons. Such a quantum collective effect not only relaxes the strong-coupling requirement for an individual emitter, but also results in a more stable generation of the exceptional points. Furthermore, we point out that the exceptional points can be explicitly revealed in the power spectra. A generalized signal-to-noise ratio, accounting for both the frequency splitting in the power spectrum and the system's dissipation, shows clearly that a collection of quantum emitters coupled to a nanoparticle provides a better performance of detecting exceptional points, compared to that of a single quantum emitter.
Motivated by the paper of Axenovich and Walzer [2], we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring on elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also give some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.
One of the central problems in quantum theory is to characterize, detect, and quantify quantumness in terms of classical strategies. Dephasing processes, caused by non-dissipative information exchange between quantum systems and environments, provides a natural platform for this purpose, as they control the quantum-to-classical transition. Recently, it has been shown that dephasing dynamics itself can exhibit (non)classical traits, depending on the nature of the system-environment correlations and the related (im)possibility to simulate these dynamics with Hamiltonian ensembles---the classical strategy. Here we establish the framework of detecting and quantifying the nonclassicality for pure dephasing dynamics. The uniqueness of the canonical representation of Hamiltonian ensembles is shown, and a constructive method to determine the latter is presented. We illustrate our method for qubit, qutrit, and qubit-pair pure dephasing and describe how to implement our approach with quantum process tomography experiments. Our work is readily applicable to present-day quantum experiments.
The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture is that every connected graph, except a single edge, is antimagic. Some graphs are known to be antimagic, but little has been known about sparse graphs, not even trees. This paper studies a weak version called $k$-shifted-antimagic labelings which allow the consecutive numbers starting from $k+1$, instead of starting from 1, where $k$ can be any integer. This paper establishes connections among various concepts proposed in the literature of antimagic labelings and extends previous results in three aspects: $\bullet$ Some classes of graphs, including trees and graphs whose vertices are of odd degrees, which have not been verified to be antimagic are shown to be $k$-shifted-antimagic for sufficiently large $k$. $\bullet$ Some graphs are proved $k$-shifted-antimagic for all $k$, while some are proved not for some particular $k$. $\bullet$ Disconnected graphs are also considered.
The fact that nonlocality implies steering enables one to certify steerability by using a Bell inequality violation. Such a certification is device-independent (DI), i.e., one makes no assumption neither on the underlying state nor on the measurements. However, not all steerable states can violate a Bell inequality. Here, we systematically construct a collection of witnesses for steerable resources, defined by assemblages, in a measurement-device-independent (MDI) scenario. The inputs driving the measurement are replaced by a set of tomographically complete quantum states, and neither the detectors nor the underlying state is characterized. We show that all steerable assemblages can be detected by properly chosen witnesses. Furthermore, we introduce the first measure of steerability in an MDI scenario and show that such a measure is a standard one, i.e., a steering monotone, by proving that it is equivalent to the steering robustness.