arXiv subjects
Publications and source records attributed to Naihuan Jing.
The quantum discord of bipartite systems is one of the best-known measures of non-classical correlations and an important quantum resource. In the recent work appeared in [Phys. Rev. Lett 2020, 124:110401], the quantum discord has been generalized to multipartite systems. In this paper, we give analytic solutions of the quantum discord for tripartite states with fourteen parameters.
We study the monogamy and polygamy inequalities of unified entanglement in multipartite quantum systems. We first derive the monogamy inequality of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition, and then obtain the monogamy inequalities of the $α$th ($0\leqα\leq\frac{r}{2}, r\geq\sqrt{2}$) power of entanglement of formation for tripartite states and their generalizations in multi-qubit quantum states. We also generalize the polygamy inequalities of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition. Moreover, we investigate polygamy inequalities of the $β$th ($β\geq \max\{1, s\}, 0\leq s\leq s_0, 0\leq s_0\leq\sqrt{2}$) power of the entanglement of formation for $2\otimes2\otimes2$ and $n$-qubit quantum systems. Finally, using detailed examples, we show that the results are tighter than previous studies.
We study the genuine multipartite entanglement in tripartite quantum systems. By using the Schmidt decomposition and local unitary transformation, we convert the general states to simpler forms and consider certain matrices from correlation tensors in the Bloch representation of the simplified density matrices. Using these special matrices, we obtain new criteria for genuine multipartite entanglement. Detail examples show that our criteria are able to detect more tripartite entangled and genuine tripartite entangled states than some existing criteria.
In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $μ$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $μ$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. When $μ=\mathrm{id}$ and $\mathfrak g$ in affine type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$. Second, we give a simple characterization of the affine quantum Serre relations on restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$-modules in terms of "normal order products". Third, we prove that the category of restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$. Last, we study the classical limit of $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the $\hbar$-deformation of all nullity $2$ extended affine Lie algebras.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg's uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as `uncertainty relation', plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several `near optimal' bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an \emph{uncertainty interval}, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.
We study the monogamy and polygamy inequalities of quantum correlations in arbitrary dimensional multipartite quantum systems. We first derive the monogamy inequality of the $α$th ($0\leqα\leq\frac{r}{2}, r\geq2$) power of concurrence for any $2\otimes2\otimes2^{n-2}$ tripartite states and generalize it to the $n$-qubit quantum states. In addition to concurrence, we show that the monogamy relations are satisfied by other quantum correlation measures such as entanglement of formation. Moreover, the polygamy inequality of the $β$th ($β\leq0$) power of concurrence and the $β$th ($β\geq s, 0\leq s\leq1$) power of the negativity are presented for $2\otimes2\otimes2^{n-2}$. We then obtain the polygamy inequalities of quantum correlations for multipartite states. Finally, our results are shown to be tighter than previous studies using detailed examples.
In this paper, we give an RTT presentation of the twisted quantum affine algebra of type $A_{2n-1}^{(2)}$ and show that it is isomorphic to the Drinfeld new realization via the Gauss decomposition of the L-operators. This provides the first such presentation for twisted quantum affine algebras with nontrivial central element.
We derive strong variance-based uncertainty relations for arbitrary two and more unitary operators by re-examining the mathematical foundation of the uncertainty relation. This is achieved by strengthening the celebrated Cauchy-Schwarz inequality using a method of brackets and convex functions. The unitary uncertainty relations outperform several strong unitary uncertainty relations, notably better than some recent best lower bounds such as [Phys. Rev. Lett. 120, 230402 (2018)] and [Phys. Rev. A. 100, 022116 (2019)].
An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup.
In this paper we study the integral form of the lattice vertex algebra $V_L$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $\mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.
We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Ram's combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Luduc-Ram formula.
An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the Kostka-Foulkes polynomials is shown. We also use our algorithm to give a formula of $K_{λμ}(t)$ for $μ$ being hook-shaped.
This is a continuation of a previous study initiated by one of us on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right $H$-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra $H$ and give a construction of deformations of vertex algebras with a right $H$-comodule nonlocal vertex algebra structure and a compatible $H$-module nonlocal vertex algebra structure. We also give a construction of $ϕ$-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.
We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the $R$-matrix and Drinfeld presentations of the Yangians.
We study bosonic tau functions in relation with the charged free bosonic fields. It is proved that up to a constant the only tau function in the Fock space M is the vacuum vector, and some tau functions were given in the completion of M using Schur functions. We also give a new proof of Borchardt's identity and obtain several q-series identities by using the boson-boson correspondence.
In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type $A_{2n-1}, D_{n+1}, A_{2n}$ and $D_4$ based on the newly found Moody-Rao-Yokonuma-like presentation.
Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also obtain a level-one module of the quantum toroidal algebra for type C as a special case.
In this paper, we study $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules. As an application, we also construct (equivariant) $ϕ$-coordinated quasi modules for lattice vertex algebras by using Lepowsky's work on twisted vertex operators.