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Ping Xu

Publications and source records attributed to Ping Xu.

At least 91 records · Page 5Linked to original sources
arXiv2016

Grover Walks on a Line with Absorbing Boundaries

In this paper, we study Grover walks on a line with one and two absorbing boundaries. In particular, we present some results for the absorbing probabilities both in a semi-finite and finite line. Analytical expressions for these absorbing probabilities are presented by using the combinatorial approach. These results are perfectly matched with numerical simulations. We show that the behavior of Grover walks on a line with absorbing boundaries is strikingly different from that of classical walks and that of Hadamard walks.

quant-ph↗
arXiv2015

From Atiyah Classes to Homotopy Leibniz Algebras

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ into a Lie algebra object in $D^+(X)$, the bounded below derived category of coherent sheaves on $X$. Furthermore Kapranov proved that, for a Kähler manifold $X$, the Dolbeault resolution $Ω^{\bullet-1}(T_X^{1,0})$ of $T_X[-1]$ is an $L_\infty$ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we define the Atiyah class $α_E$ of an $A$-module $E$ (relative to $L$) as the obstruction to the existence of an $A$-compatible $L$-connection on $E$. We prove that the Atiyah classes $α_{L/A}$ and $α_E$ respectively make $L/A[-1]$ and $E[-1]$ into a Lie algebra and a Lie algebra module in the bounded below derived category $D^+(\mathcal{A})$, where $\mathcal{A}$ is the abelian category of left $\mathcal{U}(A)$-modules and $\mathcal{U}(A)$ is the universal enveloping algebra of $A$. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of $L/A$ and $E$, and inducing the aforesaid Lie structures in $D^+(\mathcal{A})$.

math.DG↗
arXiv2015

Periodic Cyclic Homology and Equivariant Gerbes

This paper is our first step in establishing a de Rham model for equivariant twisted $K$-theory using machinery from noncommutative geometry. Let $G$ be a compact Lie group, $M$ a compact manifold on which $G$ acts smoothly. For any $α\in H^3_G (M, {\mathbb Z})$ we introduce a notion of localized equivariant twisted cohomology $H^\bullet ({\barΩ}^\bullet (M, G, L)_g, d^α_{G^g})$, indexed by $g\in G$. We prove that there exists a natural family of chain maps, indexed by $g\in G$, inducing a family of morphisms from the equivariant periodic cyclic homology $HP^G_\bullet ( C^\infty (M, α) )$, where $C^\infty (M, α)$ is a certain smooth algebra constructed from an equivariant bundle gerbe defined by $α\in H^3_G (M,{\mathbb Z} )$, to $H^\bullet ( {\barΩ}^\bullet (M, G, L)_g, d^α_{G^g})$. We formulate a conjecture of Atiyah-Hirzebruch type theorem for equivariant twisted $K$-theory.

math.KT↗
arXiv2015

The Atiyah class of a dg-vector bundle

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary bracket, and the Atiyah cocycle corresponding to a torsion-free affine connection as binary bracket.

math.DG↗
arXiv2015

Rozansky-Witten-type invariants from symplectic Lie pairs

We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams.

math.DG↗
arXiv2014

A Hopf algebra associated to a Lie pair

The quotient $L/A[-1]$ of a pair $A\hookrightarrow L$ of Lie algebroids is a Lie algebra object in the derived category $D^b(\mathscr{A})$ of the category $\mathscr{A}$ of left $\mathcal{U}(A)$-modules, the Atiyah class $α_{L/A}$ being its Lie bracket. In this note, we describe the universal enveloping algebra of the Lie algebra object $L/A[-1]$ and we prove that it is a Hopf algebra object in $D^b(\mathscr{A})$.

math.AG↗
arXiv2014

Implementation of a Measurement-Device-Independent Entanglement Witness

Entanglement, the essential resource in quantum information processing, should be witnessed in many tasks such as quantum computing and quantum communication. The conventional entanglement witness method, relying on an idealized implementation of measurements, could wrongly conclude a separable state to be entangled due to imperfect detections. Inspired by the idea of a time-shift attack, we construct an attack on the conventional entanglement witness process and demonstrate that a separable state can be falsely identified to be entangled. To close such detection loopholes, based on a recently proposed measurement-device-independent entanglement witness method, we design and experimentally demonstrate a measurement-device-independent entanglement witness for a variety of two-qubit states. By the new scheme, we show that an entanglement witness can be realized without detection loopholes.

quant-ph↗
arXiv2013

Weak Lie 2-bialgebra

We introduce the notion of weak Lie 2-bialgebra. Roughly, a weak Lie 2-bialgebra is a pair of compatible 2-term $L_\infty$-algebra structures on a vector space and its dual. The compatibility condition is described in terms of the big bracket. We prove that (strict) Lie 2-bialgebras are in one-one correspondence with crossed modules of Lie bialgebras.

math-ph↗
arXiv2013

Poisson 2-groups

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a "universal lifting theorem" for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2-algebra on the other hand.

math.DG↗
arXiv2013

Quantum teleportation and entanglement distribution over 100-kilometre free-space channels

A long standing goal for quantum communication is to transfer a quantum state over arbitrary distances. Free-space quantum communication provides a promising solution towards this challenging goal. Here, through a 97-km free space channel, we demonstrate long distance quantum teleportation over a 35-53 dB loss one-link channel, and entanglement distribution over a 66-85 dB high-loss two-link channel. We achieve an average fidelity of {80.4(9)}% for teleporting six distinct initial states and observe the violation of the Clauser-Horne-Shimony-Holt inequality after distributing entanglement. Besides being of fundamental interest, our result represents a significant step towards a global quantum network. Moreover, the high-frequency and high-accuracy acquiring, pointing and tracking technique developed in our experiment provides an essential tool for future satellite-based quantum communication.

quant-ph↗
arXiv2012

Exponential map and $L_\infty$ algebra associated to a Lie pair

In this note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair $(L,A)$ of algebroids. In particular, we prove that the quotient $L/A$ of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid $A$, which we call Kapranov module.

math.QA↗
arXiv2012

Glanon groupoids

We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that there is a bijection between Glanon Lie algebroids and source-simply connected and source-connected Glanon groupoids. As a consequence, we recover various integration theorem and obtain the integration theorem for holomorphic Poisson groupoids.

math.DG↗
arXiv2012

Experimental measurement-based quantum computing beyond the cluster-state model

The paradigm of measurement-based quantum computation opens new experimental avenues to realize a quantum computer and deepens our understanding of quantum physics. Measurement-based quantum computation starts from a highly entangled universal resource state. For years, clusters states have been the only known universal resources. Surprisingly, a novel framework namely quantum computation in correlation space has opened new routes to implement measurement-based quantum computation based on quantum states possessing entanglement properties different from cluster states. Here we report an experimental demonstration of every building block of such a model. With a four-qubit and a six-qubit state as distinct from cluster states, we have realized a universal set of single-qubit rotations, two-qubit entangling gates and further Deutsch's algorithm. Besides being of fundamental interest, our experiment proves in-principle the feasibility of universal measurement-based quantum computation without using cluster states, which represents a new approach towards the realization of a quantum computer.

quant-ph↗
arXiv2012

Realizing bending waveguides with anisotropic epsilon-near-zero metamaterials

We study metamaterials with an anisotropic effective permittivity tensor in which one component is near zero. We find that such an anisotropic metamaterial can be used to control wave propagation and construct almost perfect bending waveguides with a high transmission rate (>95%). This interesting effect originates in the power flow redistribution by the surface waves on the input and output interfaces, which smoothly matches with the propagating modes inside the metamaterial waveguide. We also find that waves in such anisotropic epsilon-near-zero materials can be reflected by small-sized perfect magnetic conductor defects. Numerical calculations have been performed to confirm the above effects.

physics.optics↗
arXiv2012

Experimental demonstration of topological error correction

Topological error correction--a novel method to actively correct errors based on cluster states with topological properties--has the highest order of tolerable error rates known to date (10^{-2}). Moreover, the scheme requires only nearest-neighbour interaction, particularly suitable for most physical systems. Here we report the first experimental demonstration of topological error correction with an 8-qubit optical cluster state. In the experiment, it is shown that a correlation can be protected against a single error on any single qubit. In addition, when all qubits are simultaneously subjected to errors with equal probability, the effective error rate is significantly reduced, clearly verifying the advantage of topological error correction. The quantum gate with the error rate below the threshold is within the current experimental technology. We believe topological error correction should be a critical ingredient for the future large-scale quantum computation.

quant-ph↗
arXiv2011

Compact source of narrow-band counterpropagating polarization-entangled photon pairs using a single dual-periodically poled crystal

We propose a scheme for the generation of counterpropagating polarization-entangled photon pairs from a dual-periodically poled crystal. Compared with the usual forward-wave type source, this source, in the backward-wave way, has a much narrower bandwidth. With a 2-cm-long bulk crystal, the bandwidths of the example sources are estimated to be 3.6 GHz, and the spectral brightnesses are more than 100 pairs/(s GHz mW). Two concurrent quasi-phase-matched spontaneous parametric down-conversion processes in a single crystal enable our source to be compact and stable. This scheme does not rely on any state projection and applies to both degenerate and non-degenerate cases, facilitating applications of the entangled photons.

quant-ph↗
arXiv2011

Observation of eight-photon entanglement

Using ultra-bright sources of pure-state entangled photons from parametric down conversion, an eight-photon interferometer and post-selection detection, we demonstrate the ability to experimentally manipulate eight individual photons and report the creation of an eight-photon Schrödinger cat state with an observed fidelity of $0.708 \pm 0.016$.

quant-ph↗