Quantum Models à la Gabor for the Space-Time Metric

As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space into Hilbertian operators. The values are space-time variables, and the values are their conjugate frequency-wave vector variables. The procedure is first applied to the variables and produces essentially canonically conjugate self-adjoint operators. It is next applied to the metric field gμν(x) of general relativity and yields regularized semi-classical phase space portraits gˇμν(x). The latter give rise to modified tensor energy density. Examples are given with the uniformly accelerated reference system and the Schwarzschild metric. Interesting probabilistic aspects are discussed.     

Unconditional Authentication Based on Physical Layer Offered Chain Key in Wireless Communication

Authentication is a critical issue in wireless communication due to the impersonation and substitution attacks from the vulnerable air interface launched by the malicious node. There are currently two kinds of authentication research in wireless communication. One is based on cryptography and relies on computational complexity, the other is based on physical layer fingerprint and can not protect data integrity well. Both of these approaches will become insecure when facing attackers with infinite computing power. In this paper, we develop a wireless unconditional authentication framework based on one-time keys generated from wireless channel. The proposed unconditional authentication framework provides a new perspective to resist infinite computing power attackers. We study the performance of the unconditional authentication framework in this paper. First, a physical layer offered chain key (PHYLOCK) structure is proposed, which can provide one-time keys for unconditional authentication. The physical layer offered chain keys are generated by XORing the physical layer updated keys extracted from the current channel state information (CSI) and the previous chain keys. The security of PHYLOCK is analyzed from the perspective of information theory. Then, the boundary of the deception probability is conducted. It is shown that unconditional authentication can achieve a probability of deception 2−12Hk, where Hk is the entropy of the one-time key used for one message. Finally, the conditions for unconditional authentication are listed. Our analysis shows that the length of the key and the authentication code need to be twice the length of the message and the encoding rules of the authentication code need to satisfy the restrictions we listed.     

Enhancing entanglement of entangled coherent states by single-mode coherent superposition of photon subtraction and addition

We introduce a new entangled quantum state generated by applying single-mode coherent superposition of photon subtraction and addition (a^† cos θ + a sin θ)^m to the entangled coherent state |Ψ_±(α,0)〉$|{\Psi }_{\pm }\left(\alpha ,0\right)〉$, and then investigate the entanglement properties affected by coherent superposition operation. It is shown that this operation can be applied to enhance the entanglement of the state |Ψ₊(α,0)〉$|{\Psi }_{+}\left(\alpha ,0\right)〉$. In addition, the effects of the coherent operation is better to improve the entanglement than that of the creation operation (a^†m) for |Ψ₊(α,0)〉$|{\Psi }_{+}\left(\alpha ,0\right)〉$ in a small-amplitude regime and for |Ψ₋(α,0)〉$|{\Psi }_{-}\left(\alpha ,0\right)〉$ in any regime.     

Charged Shear-Free Fluids and Complexity in First Integrals

The equation yxx=f(x)y2+g(x)y3 is the charged generalization of the Emden-Fowler equation that is crucial in the study of spherically symmetric shear-free spacetimes. This version arises from the Einstein–Maxwell system for a charged shear-free matter distribution. We integrate this equation and find a new first integral. For this solution to exist, two integral equations arise as integrability conditions. The integrability conditions can be transformed to nonlinear differential equations, which give explicit forms for f(x) and g(x) in terms of elementary and special functions. The explicit forms and arise as repeated roots of a fourth order polynomial. This is a new solution to the Einstein-Maxwell equations. Our result complements earlier work in neutral and charged matter showing that the complexity of a charged self-gravitating fluid is connected to the existence of a first integral.     

A Semi-Deterministic Random Walk with Resetting

We consider a discrete-time random walk (xt) which, at random times, is reset to the starting position and performs a deterministic motion between them. We show that the quantity determines if the system is averse, neutral or inclined towards resetting. It also classifies the stationary distribution. Double barrier probabilities, first passage times and the distribution of the escape time from intervals are determined.     

First Integrals of Shear-Free Fluids and Complexity

A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of yxx=f(x)y2, find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function f(x). We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions.     

On the Achievable Rate Region of the K-Receiver Broadcast Channels via Exhaustive Message Splitting

This paper focuses on K-receiver discrete-time memoryless broadcast channels (DM-BCs) with private messages, where the transmitter wishes to convey K private messages to K receivers. A general inner bound on the capacity region is proposed based on an exhaustive message splitting and a K-level modified Marton’s coding. The key idea is to split every message into submessages each corresponding to a set of users who are assigned to recover them, and then send these submessages via codewords chosen from a K-level structure codebooks. To guarantee the joint typicality among all transmitted codewords, a sufficient condition on the subcodebooks’ sizes is derived through a newly establishing hierarchical covering lemma, which extends the 2-level multivariate covering lemma to the K-level case with more intricate dependences. As the number of auxiliary random variables and rate conditions both increase exponentially with K, the standard Fourier–Motzkin elimination procedure becomes infeasible when K is large. To tackle this problem, we obtain a closed form of achievable rate region with a special observation of disjoint unions of sets that constitute the power set of {1,⋯,K}. The proposed achievable rate region allows arbitrary input probability mass functions and improves over previously known achievable (closed form) rate regions for K-receiver (K≥3) BCs.     

Robust Quantum Search with Uncertain Number of Target States

The quantum search algorithm is one of the milestones of quantum algorithms. Compared with classical algorithms, it shows quadratic speed-up when searching marked states in an unsorted database. However, the success rates of quantum search algorithms are sensitive to the number of marked states. In this paper, we study the relation between the success rate and the number of iterations in a quantum search algorithm of given λ=M/N, where M is the number of marked state and N is the number of items in the dataset. We develop a robust quantum search algorithm based on Grover–Long algorithm with some uncertainty in the number of marked states. The proposed algorithm has the same query complexity ON as the Grover’s algorithm, and shows high tolerance of the uncertainty in the ratio M/N. In particular, for a database with an uncertainty in the ratio M±MN, our algorithm will find the target states with a success rate no less than 96%.     

The Solvability of the Discrete Boundary Value Problem on the Half-Line

This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form with αx(0)+βa(0)Δx(0)=0,x(∞)=d, where d,α,β∈R, α2+β2>0. To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of (α,β)-complex fuzzy sets and then define -complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (α,β)-complex fuzzy subgroup and define (α,β)-complex fuzzy normal subgroups of given group. We extend this ideology to define (α,β)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (α,β)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (α,β)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (α,β)-complex fuzzy subgroup of the classical quotient group and show that the set of all (α,β)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of -complex fuzzy subgroups and investigate the (α,β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.