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Communication is essential for advancing science and technology. Quantum communication, in particular, benefits from the use of catalysts. During the communication process, these catalysts enhance performance while remaining unchanged. Although chemical catalysts that undergo deactivation typically perform worse than those that remain unaffected, quantum catalysts, referred to as embezzling catalysts, can surprisingly outperform their non-deactivating counterparts despite experiencing slight alterations. In this work, we employ embezzling quantum catalysts to enhance the transmission of both quantum and classical information. Our results reveal that using embezzling catalysts augments the efficiency of information transmission across noisy quantum channels, ensuring a non-zero catalytic channel capacity. Furthermore, we introduce catalytic superdense coding, demonstrating how embezzling catalysts can enhance the transmission of classical information. Finally, we explore methods to reduce the dimensionality of catalysts, a step toward making quantum catalysis a practical reality., Comment: 16 pages, 9 figures. Comments welcome!

Despite the successful enhancement to the Harrow-Hassidim-Lloyd algorithm by Childs et al., who introduced the Fourier approach leveraging linear combinations of unitary operators, our research has identified non-trivial redundancies within this method. This finding points to a considerable potential for refinement. In this paper, we propose the quantum Krylov-subspace method (QKSM), which is a hybrid classical-quantum algorithm, to mitigate such redundancies. By integrating QKSM as a subroutine, we introduce the quantum Krylov-subspace method based linear solver that not only reduces computational redundancy but also enhances efficiency and accuracy. Extensive numerical experiments, conducted on systems with dimensions up to $2^{10} \times 2^{10}$, have demonstrated a significant reduction in computational resources and have led to more precise approximations.

Classical data with graph structure always exists when dealing with many real-world problems. In parallel, quantum data with graph structure also need to be investigated since they are always produced by common quantum data sources.In this paper, we build a hybrid quantum-classical neural network with deep residual learning (Res-HQCNN) with graph-structured quantum data. Specifically, based on this special graph-structured quantum data, we first find suitable cost functions for Res-HQCNN model to learn semisupervised quantum data with graphs. Then, we present the training algorithm of Res-HQCNN for graph-structured training data in detail. Next, in order to show the learning ability of Res-HQCNN,we perform extensive experiments to show that the using of information about graph structures in quantum data can lead to better learning efficiency compared with the state-of-the-art model. At the same time, we also design comparable experiments to explain that the using of residual block structure can help deeper quantum neural networks learn graph-structured quantum data faster and better., Comment: 30 pages,9 figures

We study the polygamy property for tripartite and multipartite quantum systems. In tripartite system, we build a solution set for polygamy in tripartite system and find a lower bound of the set, which can be a sufficient and necessary condition for any quantum entanglement of assistance $Q$ to be polygamous. In multipartite system, we firstly provide generalized definitions for polygamy in two kind of divisions of $n$-qubit systems, and then build polygamy inequalities with a polygamy power $\beta$, repectively. Moreover, we use right triangle and tetrahedron to explain our polygamy relations according to the new definitions., Comment: 8 pages, 2 figures

Quantum coherence is the key resource in quantum technologies including faster computing, secure communication and advanced sensing. Its quantification and detection are, therefore, paramount within the context of quantum information processing. Having certain priori knowledge on the observables may enhance the efficiency of coherence detection. In this work, we posit that the trace of the observables is a known quantity. Our investigation confirms that this assumption indeed extends the scope of coherence detection capabilities. Utilizing this prior knowledge of the trace of the observables, we establish a series of coherence detection criteria. We investigate the detection capabilities of these coherence criteria from diverse perspectives and ultimately ascertain the existence of four distinct and inequivalent criteria. These findings contribute to the deepening of our understanding of coherence detection methodologies, thereby potentially opening new avenues for advancements in quantum technologies., Comment: 9 pages, 3 figures

In recent times, there has been a growing scholarly focus on investigating the intricacies of quantum channel mixing. It has been commonly believed, based on intuition in the literature, that every generalized Pauli channel with dimensionality $d$ could be represented as a convex combination of $(d+1)$ generalized Pauli dephasing channels (see [Phys. Rev. A 103, 022605 (2021)] as a reference). To our surprise, our findings indicate the inaccuracy of this intuitive perspective. This has stimulated our interest in exploring the properties of convex combinations of generalized Pauli channels, beyond the restriction to just $(d+1)$ generalized Pauli dephasing channels. We demonstrate that many previously established properties still hold within this broader context. For instance, any mixture of invertible generalized Pauli channels retains its invertibility. It's worth noting that this property doesn't hold when considering the Weyl channels setting. Additionally, we demonstrate that every Pauli channel (for the case of $d=2$) can be represented as a mixture of $(d+1)$ Pauli dephasing channels, but this generalization doesn't apply to higher dimensions. This highlights a fundamental distinction between qubit and general qudit cases. In contrast to prior understanding, we show that non-invertibility of mixed channels is not a prerequisite for the resulting mapping to constitute a Markovian semigroup., Comment: 9 pages, 2 figures

We study the concurrence for arbitrary N-partite W-class states based on the (N-1)-partite partitions of subsystems by taking account to the structures of W-class states. By using the method of permutation and combination we give analytical formula of concurrence and some elegant relations between the multipartite concurrence and the (N-1)-partite concurrence for arbitrary multipartite W-class states. Applying these relations we present better lower bounds of concurrence for multipartite mixed states. An example is given to demonstrate that our lower bounds can detect more entanglements.

A set of orthogonal multipartite quantum states is said to be distinguishability-based genuinely nonlocal (also genuinely nonlocal, for abbreviation) if the states are locally indistinguishable across any bipartition of the subsystems. This form of multipartite nonlocality, although more naturally arising than the recently popular "strong nonlocality" in the context of local distinguishability, receives much less attention. In this work, we study the distinguishability-based genuine nonlocality of a typical type of genuine multipartite entangled states -- the d-dimensional GHZ states, featuring systems with local dimension not limited to 2. In the three-partite case, we find the existence of small genuinely nonlocal sets consisting of these states: we show that the cardinality can at least scale down to linear in the local dimension d, with the linear factor l = 1. Specifically, the method we use is semidefinite program and the GHZ states to construct these sets are special ones which we call "GHZ-lattices". This result might arguably suggest a significant gap between the strength of strong nonlocality and the distinguishability-based genuine nonlocality. Moreover, we put forward the notion of (s,n)-threshold distinguishability and utilizing a similar method, we successfully construct (2,3)-threshold sets consisting of GHZ states in three-partite systems.

There exist bipartite entangled states whose violations of Clauser-Horne-Shimony-Holt (CHSH) Bell inequality can be observed by a single Alice and arbitrarily many sequential Bobs [Phys. Rev. Lett. 125, 090401 (2020)]. Here we consider its analogues for tripartite systems: a tripartite entangled state is shared among Alice, Bob and multiple Charlies. The first Charlie measures his qubit and then passes his qubit to the next Charlie who measures again with other measurements and so on. The goal is to maximize the number of Charlies that can observe some kind of nonlocality with the single Alice and Bob. It has been shown that at most two Charlies could share genuine nonlocality of the Greenberger-Horne-Zeilinger (GHZ) state via the violation of Svetlichny inequality with Alice and Bob [Quantum Inf. Process. 18, 42 (2019) and Phys. Rev. A 103, 032216 (2021)]. In this work, we show that arbitrarily many Charlies can have standard nonlocality (via violations of Mermin inequality) and some other kind of genuine nonlocality (which is known as genuinely nonsignal nonlocality) with the single Alice and single Bob., Comment: 12 pages, 2figure

Quantum nonlocality without entanglement is a fantastic phenomenon in quantum theory. This kind of quantum nonlocality is based on the task of local discrimination of quantum states. Recently, Bandyopadhyay and Halder [Phys. Rev. A 104, L050201 (2021)] studied the problem: is there any set of orthogonal states which can be locally distinguishable, but under some orthogonality preserving local measurement, each outcome will lead to a locally indistinguishable set. We say that the set with such property has hidden nonlocality. Moreover, if such phenomenon can not arise from discarding subsystems which is termed as local irredundancy, we call it genuine hidden nonlocality. There, they presented several sets of entangled states with genuine hidden nonlocality. However, they doubted the existence of a set without entanglement but with genuine hidden nonlocality. In this paper, we eliminate this doubt by constructing a series of sets without entanglement but whose nonlocality can be genuinely activated. We derive a method to tackle with the local irredundancy problem which is a key tricky for the systems whose local dimensions are composite numbers. As Bandyopadhyay and Halder have been pointed out, sets with genuine hidden nonloclity would lead to some applications on the data hiding., Comment: 13 pages, 6 figures, add one more example

Quantum key distribution(QKD) might be the most famous application of quantum information theory. The idea of QKD is not difficult to understand but in practical implementations, many problems are needed to be solved, for example, the noise of the channels. Previous works usually discuss the estimate of the channels and employ error-correcting procedures, whose feasibility and efficiency depend on the strength of the noise, or assist with entanglement distillation procedures, which often result in a large consumption of states while not all states can be distilled. This paper aims to study QKD over noisy channels including Pauli noises, amplitude damping noises, phase damping noises, collective noises as well as mixtures of them, in any strength without distillations. We provide a method, called the testing state method, to implement QKD protocols without errors over arbitrarily strength noisy channels. The method can be viewed as an error-correcting procedure, and can also be employed for other tasks., Comment: 11 pages

Quantum multipartite entangled states play significant roles in quantum information processing. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant mixed orthogonal arrays (IrMOAs) and thus provide positive answers to two open problems. The first is the extension of the method for constructing homogeneous systems from orthogonal arrays (OAs) to heterogeneous multipartite systems with different individual levels. The second is the existence of $k$-uniform states in heterogeneous quantum systems. We present explicit constructions of two and three-uniform states for arbitrary heterogeneous multipartite systems with coprime individual levels, and characterize the entangled states in heterogeneous systems consisting of subsystems with nonprime power dimensions as well. Moreover, we obtain infinite classes of $k$-uniform states for heterogeneous multipartite systems for any $k\geq2$. The non-existence of a class of IrMOAs is also proved.

Inspired by the success of classical neural networks, there has been tremendous effort to develop classical effective neural networks into quantum concept. In this paper, a novel hybrid quantum-classical neural network with deep residual learning (Res-HQCNN) is proposed. We firstly analysis how to connect residual block structure with a quantum neural network, and give the corresponding training algorithm. At the same time, the advantages and disadvantages of transforming deep residual learning into quantum concept are provided. As a result, the model can be trained in an end-to-end fashion, analogue to the backpropagation in classical neural networks. To explore the effectiveness of Res-HQCNN , we perform extensive experiments for quantum data with or without noisy on classical computer. The experimental results show the Res-HQCNN performs better to learn an unknown unitary transformation and has stronger robustness for noisy data, when compared to state of the arts. Moreover, the possible methods of combining residual learning with quantum neural networks are also discussed., Comment: 37 pages, 13 figures

We study the norms of the Bloch vectors for arbitrary $n$-partite quantum states. A tight upper bound of the norms is derived for $n$-partite systems with different individual dimensions. These upper bounds are used to deal with the separability problems. Necessary conditions are presented for $\mathbf m$-separable states in $n$-partite quantum systems. Based on the upper bounds, classification of multipartite entanglement is illustrated with detailed examples., Comment: 14 pages

We investigate monogamy relations and upper bounds for generalized $W$-class states related to the R\'{e}nyi-$\alpha$ entropy. First, we present an analytical formula on R\'{e}nyi-$\alpha$ entanglement (R$\alpha$E) and R\'{e}nyi-$\alpha$ entanglement of assistance (REoA) of a reduced density matrix for a generalized $W$-class states. According to the analytical formula, we show monogamy and polygamy relations for generalized $W$-class states in terms of R$\alpha$E and REoA. Then we give the upper bounds for generalized $W$-class states in terms of R$\alpha$E. Next, we provide tighter monogamy relations for generalized $W$-class states in terms of concurrence and convex-roof extended negativity and obtain the monogamy relations for R$\alpha$E by the analytical expression between R$\alpha$E and concurrence. Finally, we apply our results into quantum games and present a new bound of the nonclassicality of quantum games restricting to generalized $W$-class states., Comment: 17 pages, 3 figures

We investigate the entanglement measures of tripartite W-State and GHZ-state in noninertial frame through the coordinate transformation between Minkowski and Rindler. First it is shown that all three qubits undergo in a uniform acceleration $a$ of W-State, we find that the one-tangle, two-tangle, and $\pi$-tangle decrease when the acceleration parameter $r$ increases, and the two-tangle cannot arrive to infinity of the acceleration. Next we show that the one qubit goes in a uniform acceleration $a_{1}$ and the other two undergo in a uniform acceleration $a$ of GHZ-state, we find that the two-tangle is equal to zero and $N_{B_I (A_I C_I)} = N_{C_I (A_I B_I)}\neq N_{A_I (B_I C_I)}$, but one-tangle and $\pi$-tangle never reduce to zero for any acceleration., Comment: 9 pages, 9 figures

We investigate monogamy relations related to the R\'{e}nyi-$\alpha$ entanglement and polygamy relations related to the R\'{e}nyi-$\alpha$ entanglement of assistance. We present new entanglement monogamy relations satisfied by the $\mu$-th power of R\'{e}nyi-$\alpha$ entanglement with $\alpha\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]$ for $\mu\geqslant2$, and polygamy relations satisfied by the $\mu$-th power of R\'{e}nyi-$\alpha$ entanglement of assistance with $\alpha\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]$ for $0\leq\mu\leq1$. These relations are shown to be tighter than the existing ones., Comment: 8 pages, 2 figures. Some typoes are corrected. arXiv admin note: substantial text overlap with arXiv:1703.02858 by other authors

We study the distinguishability of a particular type of maximally entangled states -- the "lattice states" using a new approach of semidefinite program. With this, we successfully construct all sets of four ququad-ququad orthogonal maximally entangled states that are locally indistinguishable and find some curious sets of six states having interesting property of distinguishability. Also, some of the problems arose from \cite{CosentinoR14} about the PPT-distinguishability of "lattice" maximally entangled states can be answered., Comment: It's rewritten. We deleted the original section II about PPT-distinguishability of three ququad-ququad MESs. Moreover, we have joined new section V which discuss PPT-distinguishability of lattice MESs for cases $t=3$ and $t=4$ . As a result, the sequence of the theorems in our article has been changed. And we revised the title of our article

In this paper we introduce the notion of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence associated with Hecke algebras is representation stable., Comment: 10 pages,no figures

In recent papers, the theory of representations of finite groups has been proposed to analyzing the violation of Bell inequalities. In this paper, we apply this method to more complicated cases. For two partite system, Alice and Bob each make one of $d$ possible measurements, each measurement has $n$ outcomes. The Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements $d$, but the quantum bound is dependent both on $n$ and $d$. Even so, when $d$ is large enough, the quantum bound is only dependent on $d$. The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. Moreover, all the corresponding nonlocal games are analyzed., Comment: The name of our paper is changed. 1 figures is added. And some typos are fixed. 15 pages, 4 figures

We study the problem of distinguishing maximally entangled quantum states by using local operations and classical communication (LOCC). A question of fundamental interest is whether any three maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are distinguishable by LOCC. In this paper, we restrict ourselves to consider the generalized Bell states. And we prove that any three generalized Bell states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are locally distinguishable., Comment: 11 pages, 1 figure, accepted by Quantum Information Processing

We studied the construction problem of the unextendible product basis (UPB). We mainly give a method to construct a UPB of a quantum system through the UPBs of its subsystem. Using this method and the UPBs which are known for us, we construct different kinds of UPBs in general bipartite quantum system. Then we use these UPBs to construct a family of UPBs in multipartite quantum system. The UPBs can be used to construct the bound entangled states with different ranks., Comment: 6 pages, 1 figures

In this paper, we give a BLM realization of the positive part of the quantum group of $U_v(gl_n)$ with respect to RTT relations., Comment: 14 pages, No figures

In \cite{fl}, the authors get a new presentation of two-parameter quantum algebra $U_{v,t}(\mathfrak{g})$. Their presentation can cover all Kac-Moody cases. In this paper, we construct a suitable Hopf pairing such that $U_{v,t}(sl_{n})$ can be realized as Drinfeld double of certain Hopf subalgebras with respect to the Hopf pairing. Using Hopf pairing, we construct a $R$-matrix for $U_{v,t}(sl_{n})$ which will be used to give the Schur-Weyl dual between $U_{v,t}(sl_{n})$ and Hecke algebra $H_{k}(v,t)$. Furthermore, using the Fusion procedure we construct the primitive orthogonal idempotents of $H_{k}(v,t)$. As a corollary, we give the explicit construction of irreducible $U_{v,t}(sl_{n})$-representations of $V^{\otimes k}$., Comment: 13 pages

In this paper, we give an geometric description of the Schur-Weyl duality for two-parameter quantum algebras $U_{v, t}(gl_n)$, where $U_{v, t}(gl_n)$ is the deformation of $U_v(I, \cdot)$, the classic Shur-Weyl duality $(U_{r, s}(gl_n), V^{\otimes d}, H_d(r, s))$ can be seen as a corollary of the Shur-Weyl duality $(U_{v, t}(gl_n), V^{\otimes d}, H_d(v, t))$ by using the galois descend approach. we also establish the Shur-Weyl duality between the algebras $\widetilde{U_{v, t}(gl_N)^m}$, $\widehat{U_{v, t}(gl_N)^m}$ and Heck algebra $H_k(v, t)$., Comment: 26 papes, No figures. This is a draft of the first paper of our series papers on geometric Schur Duality of two parameter quantum groups. arXiv admin note: substantial text overlap with arXiv:1408.6740 by other authors

In this paper, we introduce and study the quantum deformations of the cluster superalgebra. Then we prove the quantum version of the Laurent phenomenon for the super-case., Comment: 15 pages, no figures. Modify some topes

In this paper, we present some generalized monogamy inequalities based on negativity and convex-roof extended negativity (CREN). These monogamy relations are satisfied by the negativity of $N$-qubit quantum systems $ABC_1\cdots C_{N-2}$, under the partitions $AB|C_1\cdots C_{N-2}$ and $ABC_1|C_2\cdots C_{N-2}$. Furthermore, the $W$-class states are used to test these generalized monogamy inequalities., Comment: 10 pages, two figures. The paper has been rewriteen and one auther is added. We add 2 examples to show the two conclusions in the reference [29] is not correct. And we modify our previous conclusions. These make our paper more complete

We study the unextendible maximally entangled bases (UMEB) in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a partial Hadamard matrix can not extend to a complete Hadamard matrix in $\mathbb{C}^{d}$. As a corollary, any $(d-1)\times d$ partial Hadamard matrix can extend to a complete Hadamard matrix. Then we obtain that for any $d$ there is an UMEB except $d=p\ \text{or}\ 2p$, where $p\equiv 3\mod 4$ and $p$ is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in $\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times5 \times7$., Comment: 6 pages, no figures

In this paper, we study the concurrence of arbitrary dimensional tripartite quantum systems. An explicit operational lower bound of concurrence is obtained in terms of the concurrence of sub-states. A given example show that our lower bound may improve the well known existing lower bounds of concurrence. The significance of our result is to get a lower bound when we study the concurrence of arbitrary dimensional multipartite quantum systems., Comment: 1 figures, Quantum Information Processing 2016

In this paper, we mainly study the local indistinguishability of multipartite product states. Firstly, we follow the method of Z.-C. Zhang \emph{et al}[Phys. Rev. A 93, 012314(2016)] to give another more concise set of $2n-1$ orthogonal product states in $\mathbb{C}^m\otimes\mathbb{C}^n(4\leq m\leq n)$ which can not be distinguished by local operations and classical communication(LOCC). Then we use the 3 dimension cubes to present some product states which give us an intuitive view how to construct locally indistinguishable product states in tripartite quantum system. At last, we give an explicit construction of locally indistinguishable orthogonal product states for general multipartite system., Comment: 5 pages, 6 figures

V.S.Guba had proved that the R.Thompson group $T$ satisfies polynomial isoperimetric inequality and $\Phi_T(n)\preceq n^7$, where $\Phi_T$ is the Dehn function of group $T$. In this paper, we show that $\Phi_T(n)\preceq n^5$., Comment: 9 pages, 8 figures. We correct some typos and some gramma problems

We study separability problem using general symmetric informationally complete measurements and propose separability criteria in $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ and $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\cdots\otimes\mathbb{C}^{d_{n}}$. Our criteria just require less local measurements and provide experimental implementation in detecting entanglement of unknown quantum states., Comment: 10 pages, no figures. arXiv admin note: text overlap with arXiv:1406.7820, arXiv:1407.0314 by other authors

We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of $\mathbb{C}^d\otimes\mathbb{C}^d$, where $d$ is odd, Zhang \emph{et al} have constructed $d^2$ orthogonal product basis quantum states which are locally indistinguishable in [Phys. Rev. A. {\bf 90}, 022313(2014)]. We find a subset contains with $6d-9$ orthogonal product states which are still locally indistinguishable. Then we generalize our method to arbitrary bipartite quantum system $\mathbb{C}^m\otimes\mathbb{C}^n$. We present a small set with only $3(m+n)-9$ orthogonal product states and prove these states are LOCC indistinguishable. Even though these $3(m+n)-9$ product states are LOCC indistinguishable, they can be distinguished by separable measurements. This shows that separable operations are strictly stronger than the local operations and classical communication., Comment: 5 pages

In this paper, we study the one-way local operations and classical communication (LOCC) problem. In $\mathbb{C}^d\otimes\mathbb{C}^d$ with $d\geq4$, we construct a set of $3\lceil\sqrt{d}\rceil-1$ one-way LOCC indistinguishable maximally entangled states which are generalized Bell states. Moreover, we show that there are four maximally entangled states which cannot be perfectly distinguished by one-way LOCC measurements for any dimension $d\geq 4$., Comment: 10 pages.Very recently, we became aware of related work \cite{Zhang2} in which the same $\lceil\frac{d}{2}\rceil+2$ states in $\mathbb{C}^d\otimes\mathbb{C}^d$ is proved to be one-way LOCC indistinguishable. arXiv admin note: text overlap with arXiv:1310.4220 by other author

We study the concurrence of arbitrary dimensional multipartite quantum systems. An explicit analytical lower bound of concurrence for four-partite mixed states is obtained in terms of the concurrences of tripartite mixed states. Detailed examples are given to show that our lower bounds improve the existing lower bounds of concurrence. The approach is generalized to five-partite quantum systems., Comment: 9 pages,2 figures, correct some errors

We give a explicit construction of $d$ locally indistinguishable orthogonal maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d$ for any $d\geq 4$. This gives an answer to the conjecture proposed by S. Bandyopadhyay in 2009. Thus it reflects the nonlocality of the fundamental feature of quantum mechanics., Comment: The paper is rewrited, especally the introduction. 5 pages, no figures

We solved the unextendible maximally entangled basis (UMEB) problem in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d'}(d\neq d')$,the results turn out to be that there always exist a UMEB.In addition,there might be two sets of UMEB with different numbers.The main difficult is to prove the unextendibility of the set of states.We give an explicit construction of UMEB by considering the Schmidt number of the complementary space of the states we construct., Comment: 4 pages,Physical Review A, June, 2014

In this paper, we give a method for the local unitary equivalent problem which is more efficient than that was proposed by Bin Liu $et \ al$ \cite{bliu}., Comment: Accepted by International Journal of Theoretical Physics

In this paper, we give a quantum cluster algebra structure on the deformed Grothendieck ring of $\CC_{n}$, where $\CC_{n}$ is a full subcategory of finite dimensional representations of $U_q(\widehat{sl_{2}})$ defined in section II., Comment: 11pages, no figures

In this paper, we prove one case of the conjecture given by Hernandez and Leclerc\cite{HL0}. Specifically, we give a cluster algebra structure on the Grothendieck ring of a full subcategory of the finite dimensional representations of a simply-laced quantum affine algebra $U_q(\widehat{\g})$. In the procedure, we also give a specific description of compatible subsets of type $E_{6}$. As a conclusion, for every exchange relation of cluster algebra there exists a exact sequence of the full subcategory corresponding to it., Comment: 58 pages, 2 figures. Correct some typoes and add a new example. arXiv admin note: text overlap with arXiv:0903.1452 by other authors

This paper studies the Kazhdan-Lusztig coefficients $\mu(u,w)$ of the Kazhdan-Lusztig polynomials $P_{u,w}$ for the lowest cell ${c_{0}}$ of an affine Weyl group of type $\widetilde{G_{2}}$ and gives an estimation $\mu(u,w)\leqslant 3$ for $u,w\in c_{0}$., Comment: 96 pages

In this paper, we will construct the injective resolution of any $\cR$-2-module, define the right derived 2-functor, and give some related properties of the derived 2-functor in ($\cR$-2-Mod)., Comment: 29 pages, 57 figures. This paper is the fifth paper of the series of our works on higher dimensional homology algebra. In our coming papers, we shall define $\cE$xt 2-functor and spectral sequence in an abelian 2-category, try to give the relation between $\cE$xt 2-functor and the extension of 2-modules

In this paper, we will construct the projective resolution of any $\cR$-2-module, define the derived 2-functor and give some related properties of the derived 2-functor., Comment: 17pages, 32 figures, This is the fourth paper of the series of our works on higher dimensional homological algebra. In the coming papers, we shall define the right derived 2-functor in the 2-categories (2-SGp) and ($\cR$-2-Mod), and give some relations of left derived 2-functors and right derived 2-functors

In this paper, we will define the derived 2-functor by projective resolution of any symmetric 2-group, and give some related properties of the derived 2-functor., Comment: 30 pages, 50 figures. This is the third paper of the series of our works on higher dimensional homological algebra. In the coming papers, we define the right derived 2-functor in the 2-categories (2-SGp) and ($\cR$-2-Mod). In this version, we correct some errors in last version, add more results, such as 2-chain homotopy, its related results, etc

In this paper, we will prove that the 2-category (2-SGp) of symmetric 2-groups and 2-category ($\cR$-2-Mod) of $\cR$-2-modules(\cite{5}) have enough projective objects, respectively., Comment: 10 pages, 4 figures. This is the second paper of the series works on higher dimensional homology algebra. The first paper is "2-Modules and the Representation of 2-Rings\cite{4}". In the coming papers, we shall give the definition of injective object in the 2-category ($\cR$-2-Mod), prove that this 2-category has enough injective objects and develop the (co)homology theory of it

In this paper, we develop 2-dimensional algebraic theory which closely follows the classical theory of modules. The main results are giving definitions of 2-module and the representation of 2-ring. Moreover, for a 2-ring $\cR$, we prove that its modules form a 2-Abelian category., Comment: 78 pages, 99 figures

Analogue to commutants in the theory of associative algebras, one can construct a new subalgebra of vertex algebra known as a vertex algebra commutant. In this paper, for the adjoint representation $V$ of Lie algebra $sl(2,\C)$, we describe a commutant of $\beta\gamma$- System $S(V)$ by giving its generators, moreover, we get a new Howe pair of vertex algebras., Comment: 15 pages, No figures. This is our first paper of series on this topic

Let $L_{D_8}(1, 0)$ and $L_{E_8}(1, 0)$ be the simple vertex operator algebras associated to untwisted affine Lie algebra $\widehat{{\mathbf g}}_{D_{8}}$ and $\widehat{{\mathbf g}}_{E_8}$ with level 1 respectively. In the 1980s by I. Frenkel, Lepowsky and Meurman as one of the many important preliminary steps toward their construction of the moonshine module vertex operator algebra, they use roots lattice showing that $L_{D_8}(1, 0)$ can embed into $L_{E_8}(1, 0)$ as a vertex operator subalgebra(\cite{5, 6, 8}). Their construct is a base of vertex operator theory. But the embedding they gave using the fact $L_{\mathbf g}(1,0)$ is isomorphic to its root lattice vertex operator algebra $V_L$. In this paper, we give an explicitly construction of the embedding and show that as an $L_{D_8}(1, 0)$-module, $L_{E_8}(1, 0)$ is isomorphic to the extension of $L_{D_8}(1, 0)$ by its simple module $L_{D_8}(1, \overline{\omega}_8)$. It may be convenient to be used for conformal field theory., Comment: The abstrat and section 1 are modified. We give more informations about the embedding

In this paper, we study the following problem $$ \{{ll} \Delta_{H^n} u-u+u^p=0 & in H^n u>0& in H^n u(x)\to 0 &\rho(x)\to\infty}. $$ where $1