Three-term recurrence relations of minimal affinizations of type G2G_2

Minimal affinizations form a class of modules of quantum affine algebras introduced by Chari. We introduce a system of equations satisfied by the $q$-characters of minimal affinizations of type $G_2$ which we call the M-system of type $G_2$. The M-system of type $G_2$ contains all minimal affinizations of type $G_2$ and only contains minimal affinizations. The equations in the M-system of type $G_2$ are three-term recurrence relations. The M-system of type $G_2$ is much simpler than the extended T-system of type $G_2$ obtained by Mukhin and the second author. We also interpret the three-term recurrence relations in the M-system of type $G_2$ as exchange relations in a cluster algebra constructed by Hernandez and Leclerc.Comment: 23 pages. The original name of the paper is: Cluster algebras and minimal affinizations of representations of the quantum group of type $G_2

Maximal Scheduling in Wireless Networks with Priorities

We consider a general class of low complexity distributed scheduling algorithms in wireless networks, maximal scheduling with priorities, where a maximal set of transmitting links in each time slot are selected according to certain pre-specified static priorities. The proposed scheduling scheme is simple, which is easily amendable for distributed implementation in practice, such as using inter-frame space (IFS) parameters under the ubiquitous 802.11 protocols. To obtain throughput guarantees, we first analyze the case of maximal scheduling with a fixed priority vector, and formulate a lower bound on its stability region and scheduling efficiency. We further propose a low complexity priority assignment algorithm, which can stabilize any arrival rate that is in the union of the lower bound regions of all priorities. The stability result is proved using fluid limits, and can be applied to very general stochastic arrival processes. Finally, the performance of the proposed prioritized maximal scheduling scheme is verified by simulation results.Comment: 10 pages, 6 figure

A Necessary and Sufficient Criterion for the Separability of Quantum State

Quantum entanglement has been regarded as one of the key physical resources in quantum information sciences. However, the determination of whether a mixed state is entangled or not is generally a hard issue, even for the bipartite system. In this work we propose an operational necessary and sufficient criterion for the separability of an arbitrary bipartite mixed state, by virtue of the multiplicative Horn's problem. The work follows the work initiated by Horodecki {\it et. al.} and uses the Bloch vector representation introduced to the separability problem by J. De Vicente. In our criterion, a complete and finite set of inequalities to determine the separability of compound system is obtained, which may be viewed as trade-off relations between the quantumness of subsystems. We apply the obtained result to explicit examples, e.g. the separable decomposition of arbitrary dimension Werner state and isotropic state.Comment: 33 pages; published in Scientific Report

Reformulating the Quantum Uncertainty Relation

Uncertainty principle is one of the cornerstones of quantum theory. In the literature, there are two types of uncertainty relations, the operator form concerning the variances of physical observables and the entropy form related to entropic quantities. Both these forms are inequalities involving pairwise observables, and are found to be nontrivial to incorporate multiple observables. In this work we introduce a new form of uncertainty relation which may give out complete trade-off relations for variances of observables in pure and mixed quantum systems. Unlike the prevailing uncertainty relations, which are either quantum state dependent or not directly measurable, our bounds for variances of observables are quantum state independent and immune from the "triviality" problem of having zero expectation values. Furthermore, the new uncertainty relation may provide a geometric explanation for the reason why there are limitations on the simultaneous determination of different observables in $N$-dimensional Hilbert space.Comment: 15 pages, 2 figures; published in Scientific Report

Linear algebraic analogues of the graph isomorphism problem and the Erd\H{o}s-R\'enyi model

A classical difficult isomorphism testing problem is to test isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. It is known that this problem can be reduced to solving the alternating matrix space isometry problem over a finite field in time polynomial in the underlying vector space size. We propose a venue of attack for the latter problem by viewing it as a linear algebraic analogue of the graph isomorphism problem. This viewpoint leads us to explore the possibility of transferring techniques for graph isomorphism to this long-believed bottleneck case of group isomorphism. In 1970's, Babai, Erd\H{o}s, and Selkow presented the first average-case efficient graph isomorphism testing algorithm (SIAM J Computing, 1980). Inspired by that algorithm, we devise an average-case efficient algorithm for the alternating matrix space isometry problem over a key range of parameters, in a random model of alternating matrix spaces in vein of the Erd\H{o}s-R\'enyi model of random graphs. For this, we develop a linear algebraic analogue of the classical individualisation technique, a technique belonging to a set of combinatorial techniques that has been critical for the progress on the worst-case time complexity for graph isomorphism, but was missing in the group isomorphism context. As a consequence of the main algorithm, we establish a weaker linear algebraic analogue of Erd\H{o}s and R\'enyi's classical result that most graphs have the trivial automorphism group. We finally show that Luks' dynamic programming technique for graph isomorphism (STOC 1999) can be adapted to slightly improve the worst-case time complexity of the alternating matrix space isometry problem in a certain range of parameters.Comment: 32 pages, 2 figures. The implication to group enumeration correcte

Representations of Rota-Baxter algebras and regular singular decompositions

There is a Rota-Baxter algebra structure on the field $A=\mathbf{k}((t))$ with $P$ being the projection map $A=\mathbf{k}[[t]]\oplus t^{-1}\mathbf{k}[t^{-1}]$ onto $\mathbf{k}[[ t]]$. We study the representation theory and regular-singular decompositions of any finite dimensional $A$-vector space. The main result shows that the category of finite dimensional representations is semisimple and consists of exactly three isomorphism classes of irreducible representations which are all one-dimensional. As a consequence, the number of $GL_A(V)$-orbits in the set of all regular-singular decompositions of an $n$-dimensional $A$-vector space $V$ is $(n+2)(n+1)/2$. We also use the result to compute the generalized class number, i.e., the number of the $GL_n(A)$-isomorphism classes of finitely generated $\mathbf{k}[[t]]$-submodules of $A^n$

Equivalence theorem of uncertainty relations

We present an equivalence theorem to unify the two classes of uncertainty relations, i.e., the variance-based ones and the entropic forms, which shows that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators. That means an uncertainty relation in the language of entropy may be mapped onto a variance-based one, and vice versa. Employing the equivalence theorem, alternative formulations of entropic uncertainty relations stronger than existing ones in the literature are obtained for qubit system, and variance based uncertainty relations for spin systems are reached from the corresponding entropic uncertainty relations.Comment: 18 pages, 1 figure; published in J. Phys. A: Math. Theo

Separable Decompositions of Bipartite Mixed States

We present a practical scheme for the decomposition of a bipartite mixed state into a sum of direct products of local density matrices, using the technique developed in Li and Qiao (Sci. Rep. 8: 1442, 2018). In the scheme, the correlation matrix which characterizes the bipartite entanglement is first decomposed into two matrices composed of the Bloch vectors of local states. Then we show that the symmetries of Bloch vectors are consistent with that of the correlation matrix, and the magnitudes of the local Bloch vectors are lower bounded by the correlation matrix. Concrete examples for the separable decompositions of bipartite mixed states are presented for illustration.Comment: 22 pages; published in Quantum Inf. Proces

Greedy Maximal Scheduling in Wireless Networks

In this paper we consider greedy scheduling algorithms in wireless networks, i.e., the schedules are computed by adding links greedily based on some priority vector. Two special cases are considered: 1) Longest Queue First (LQF) scheduling, where the priorities are computed using queue lengths, and 2) Static Priority (SP) scheduling, where the priorities are pre-assigned. We first propose a closed-form lower bound stability region for LQF scheduling, and discuss the tightness result in some scenarios. We then propose an lower bound stability region for SP scheduling with multiple priority vectors, as well as a heuristic priority assignment algorithm, which is related to the well-known Expectation-Maximization (EM) algorithm. The performance gain of the proposed heuristic algorithm is finally confirmed by simulations.Comment: 6 pages, 3 figures. A shorter version will appear in IEEE Globecom 201

Representations of Polynomial Rota-Baxter Algebras

A Rota--Baxter operator is an algebraic abstraction of integration, which is the typical example of a weight zero Rota-Baxter operator. We show that studying the modules over the polynomial Rota--Baxter algebra $(k[x],P)$ is equivalent to studying the modules over the Jordan plane, and we generalize the direct decomposability results for the $(k[x],P)$-modules in [Iy] from algebraically closed fields of characteristic zero to fields of characteristic zero. Furthermore, we provide a classification of Rota--Baxter modules up to isomorphism based on indecomposable $k[x]$-modules