arXiv subjects

Zhiyu Tian

Publications and source records attributed to Zhiyu Tian.

At least 19 recordsLinked to original sources

arXiv2024

Local-global principle and integral Tate conjecture for certain varieties

We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a separably rationally connected variety defined over a global function field. We prove that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on all geometrically rational surfaces defined over a global function field, and to the Hasse principle for rational points on del Pezzo surfaces of degree four defined over a global function field of odd characteristic. Along the way, we also prove some results about the space of one-cycles on a smooth projective variety that is separably rationally connected in codimension one, which leads to the equality of the coniveau filtration and the strong coniveau filtration on degree $3$ homology of such varieties.

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arXiv2024

Experimental demonstration of 4-state reference-frame-independent quantum key distribution over 200km

Reference frame independent quantum key distribution (RFI-QKD) has gained widespread attention due to the unique advantage for practical application, as it circumvents the need for active reference frame alignment within the system. However, in comparison to the standard BB84 protocol, the original 6-state RFI protocol requires a greater number of quantum states to be operated by Alice and Bob, which is an aspect that merits optimization. In this work, we propose a 4-state RFI protocol and illustrate that Alice and Bob each require only four quantum states to perform channel estimation that remains independent of reference frame deviation, which can proficiently reduce the system complexity. Furthermore, through numerical simulations taking the finite-size key effect into consideration, we show that 4-state RFI protocol can achieve a secure key rate and transmission distance on par with the original 6-state RFI protocol. Finally, a experiment over 200 km is inplemented to conducted the feasibility of our scheme. We believe that our protocol can streamline the implementation of RFI-QKD and thereby contribute to the practical advancement of RFI-QKD.

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arXiv2024

Experimental demonstration of improved reference-frame-independent quantum key distribution over 175km

Reference-frame-independent (RFI) quantum key distribution (QKD) presents promising advantages, especially for mobile-platform-based implementations, as it eliminates the need for active reference frame calibration. While RFI-QKD has been explored in various studies, limitations in key rate and distance persist due to finite data collection. In this study, we experimentally demonstrate an improved RFI-QKD protocol proposed by Zhu \textit{et al.} [Opt. Lett. 47, 4219 (2022)], featuring a statistical quantity for bounding information leaked to Eve that exhibits more insensitivity to statistical fluctuations and more robustness to variations in the reference frame. Taking into account finite-size considerations and potential general attacks, RFI-QKD is implemented over a distance of 175 \si{\kilo\meter} in this work. We believe that our study extends the communication distance achievable by RFI-QKD, thereby constituting a notable advancement for its practical application.

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arXiv2023

Space of one cycles and coniveau filtrations

We prove a structural result about the space of one cycles of a separably rationally connected variety or a separably rationally connected fibration over a curve, either as a topological group or as an h-sheaf. This has the following consequences: a proof that the strong coniveau filtration agrees with the coniveau filtration on degree 3 homology, and a result on the integral Tate conjecture for homologically trivial one cycles.

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arXiv2023

Stable maps of curves and algebraic equivalence of 1-cycles

We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern $A_1(X)$, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If $K/k$ is an algebraic extension, then the kernel of $A_1(X_k)\to A_1(X_K)$ is at most ${\mathbb Z}/2{\mathbb Z}$. If $k$ is finite, then the image equals the subgroup of Galois invariant cycles. This paper replaces Sections~2--3 of 2211.15915.v.1 and Sections~2--3 of 2211.15911.v.1. The other Sections are retained in the revised versions of these papers.

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arXiv2022

Experimental test of Tsirelson's bound with a single photonic qubit

For many protocols, quantum strategies have advantages compared with their classical counter-partners, and these advantages have attracted many interests and applications. One of the famous examples is the Clauser-Horne-Shimony-Holt (CHSH) game, which recasts Bell's theorem~\cite{2} into the framework of a game. In the CHSH game, two space-like separated players, Alice and Bob are each assigned a classical bit $a$ and $b$ respectively. Then they return bits $x$ and $y$ according to some pre-agreed strategies. They will win the game when $x\oplus y= a\cdot b$. In the game, if the players use the classical strategies, the optimal success probability $w(\text{CHSH})=0.75$.However, if they add some quantum resources, the success probability will increase and up to maximal value $cos^2(π/8)$, which is know as the Tsirelson's bound. Moreover, Popescu and Rohrlich noted that the perfect success probability $1$ can also be achieved in a more general theory without violating the no-signaling assumption

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arXiv2022

Shannon Entropy and Diffusion Coeffcient in Parity-Time Symmetric Quantum Walks

Non-Hermitian topological edge states have many intriguing properties, but have so far mainly been discussed in terms of bulk-boundary correspondence. Here we propose to use a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient is found to show unique features with the topological phase transitions driven by paritytime( PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite artificial boundaries are not constructed by inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated to the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented here may open up a new avenue for studying the topological state in Non-Hermitian quantum walk systems.

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arXiv2020

Zero cycles on rationally connected varieties over Laurent fields

We study zero cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined over such fields. In general, the degree map is an isomorphism if rationally connected varieties defined over algebraically closed fields of characteristic zero satisfy the integral Hodge/Tate conjecture for one cycles, or if the Tate conjecture is true for divisor classes on surfaces defined over finite fields. To prove these results, we introduce techniques from the minimal model program to study the homology of certain complexes defined by Kato/Bloch-Ogus.

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arXiv2020

Moduli space of quasi-polarized K3 surfaces of degree 6 and 8

In this paper, we study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and we have natural GIT constructions for the corresponding moduli spaces and we show that the K3 surfaces with at worst ADE singularities are GIT stable. We give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. We compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga's compactifications from arithmetic perspective. One of the main ingredients is the study of the projective models of K3 surfaces in terms of Noether-Lefschetz divisors.

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arXiv2019

Separable rational connectedness and weak approximation in positive characteristic

In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the uniruling index for a smooth Fano variety of Picard number one to be separably rationally connected. As an application, we prove some weak approximation results for Fano complete intersections in positive charactersitic. For example, we show that weak approximation holds at place of strong potentially good reduction for a Fano complete intersection in $\mathbb{P}^n$ of type $(d_1, \ldots, d_c)$ in characteristic $p$ such that $n>d_1+\ldots +d_c, p>d_1, \ldots, d_c.$

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arXiv2018

Weak approximation for Fano complete intersections in positive characteristic

For a smooth curve $B$ over an algebraically closed field $k$, for every $B$-flat complete intersection $X_B$ in $B\times_{\text{Spec}\ k} \mathbb{P}^n_k$ of type $(d_1,\dots,d_c)$, if the Fano index is $\geq 2$ and if $\text{char}(k)>\max(d_1,\dots,d_c)$, we prove weak approximation of $\widehat{\mathcal{O}}_{B,b}$-points of $X_B$ by $k(B)$-points at all places of (strong) potentially good reduction, including all places of good reduction. The key step is the proof that such complete intersections are \emph{separably uniruled by lines}, and even \emph{separably rationally connected}, whenever smooth. We prove that the inequality is close to sharp. We prove a similar theorem for Fano manifolds of Picard number $1$ and Fano index $1$.

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arXiv2018

2-cycles sur les hypersurfaces cubiques de dimension 5

On étudie les cycles algébriques de codimension 3 sur les hypersurfaces cubiques lisses de dimension 5. Pour une telle hypersurface, on démontre d'une part que son groupe de Griffiths des cycles de codimension 3 est trivial et d'autre part que l'application d'Abel-Jacobi induit un isomorphisme entre son groupe de Chow des cycles de codimension 3 algébriquement equivalents à zéro et sa jacobienne intermédiaire. ---------- We study 2-cycles of a smooth cubic hypersurface of dimension 5. We show that the Griffiths group of 2-cycles is trivial and the Abel-Jacobi map induces an isomorphism between the Chow group of algebraically trivial 2-cycles and the intermediate Jacobian.

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arXiv2018

Motivic HyperKähler Resolution Conjecture : I. Generalized Kummer varieties

Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-Göttsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an algebra object in the category of Chow motives. Inspired by Ruan, one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture. We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura, in two situations related to an abelian surface $A$ and a positive integer $n$. Case (A) concerns Hilbert schemes of points of $A$ : the Chow motive of $A^{[n]}$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $[A^{n}/\mathfrak{S}_{n}]$. Case (B) for generalized Kummer varieties : the Chow motive of the generalized Kummer variety $K_n(A)$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $[A_{0}^{n+1}/\mathfrak {S}_{n+1}]$, where $A_{0}^{n+1}$ is the kernel abelian variety of the summation map $A^{n+1}\to A$. As a byproduct, we prove the original Cohomological HyperKähler Resolution Conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow-Künneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch-Beilinson-Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville.

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arXiv2018

Motivic multiplicative McKay correspondence for surfaces

We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the minimal resolution is isomorphic to the complex orbifold Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the orbifold surface. This confirms the two-dimensional Motivic Crepant Resolution Conjecture.

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arXiv2016

Irregular varieites with geometric genus one, theta divisors, and fake tori

We study the Albanese image of a compact Kähler manifold whose geometric genus is one. We prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact Kähler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.

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arXiv2015

Weak Approximation for Cubic Hypersurfaces and Degree 4 del Pezzo Surfaces

In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 11 elements. (2) For del Pezzo surfaces of degree 4 defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 13 elements. (3) Weak approximation holds for cubic hypersurfaces of dimension at least 10 defined over a global function field of characteristic not equal to 2, 3, 5 or a purely imaginary number field.

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arXiv2015

Finiteness of fundamental groups

We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of klt singularities. As an application, we verify that the local fundamental group of a three dimensional klt singularity and the fundamental group of the smooth locus of a three dimensional Fano variety with canonical singularities are always finite.

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arXiv2015

Hasse principle for three classes of varieties over global function fields

We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cubic hypersurfaces of dimension at least $4$ in characteristic at least $7$, and smooth complete intersections of two quadrics of dimension at least $3$ in odd characteristics. In Appendix A we explain how to modify a previous argument of the author to prove weak approximation for cubic hypersurfaces defined over function fields of curves over algebraically closed fields of characteristic at least $7$. In Appendix B we prove some corollaries of Kollár's results on the fundamental group of separably rationally connected varieties.

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