arXiv subjects

Xiande Zhang

Publications and source records attributed to Xiande Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Supersaturation for Eventown via Generator Switching

An eventown family is a family of even-sized subsets of $[n]$ in which every two distinct members have an even-sized intersection. A classical theorem of Berlekamp and Graver shows that the maximum size of such a family is $2^{\lfloor n/2\rfloor}$. The supersaturation problem for eventown asks how many odd-intersection pairs must occur when this extremal bound is exceeded. For a family $\mathcal F$ of even-sized subsets of $[n]$, let $e(\mathcal F)$ denote the number of unordered pairs whose intersection size is odd. O'Neill conjectured that if $|\mathcal F|=2^{\lfloor n/2\rfloor}+s$, then $e(\mathcal F)\ge s\,2^{\lfloor n/2\rfloor-1}$ for \[ 1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}. \] Previously, the conjecture was known for $s=1,2$, and, for $s\le 2^{\lfloor n/8\rfloor}/n$ with $n$ sufficiently large. We prove the conjectured bound for \[ 1\le s\le \frac{2^{\lfloor n/2\rfloor}}{26}, \] extending the known range to a fixed positive proportion of the extremal eventown size. The bound is sharp throughout this range. As further consequences, we derive a lower bound valid for arbitrary excess $s$, which improves the previously known estimate in an additional range. We also establish stability and removal results for families of extremal size satisfying $e(\mathcal F)<2^{\lfloor n/2\rfloor-1}$, showing that such a family is close to an extremal eventown family and can be made eventown by deleting a small number of its members.

math.CO↗

arXiv2026

Robust Quantum Extremal Numbers

Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem $A$, the marginal maximal-mixing defect is defined by \[ D_A=2^{|A|}\operatorname{Tr}(\rho_A^2)-1 =2^{|A|}\left\|\rho_A-\frac{I_A}{2^{|A|}}\right\|_2^2, \] and $Q_{\mathrm{ex},\varepsilon}^{D}(n,k)$ is defined as the maximum number of $k$-body marginals satisfying $D_A\leq\varepsilon$ in an $n$-qubit pure state. This counting problem differs from approximate $k$-uniformity, which requires all $k$-body marginals to obey a common error bound. For pure states on $4m$ qubits, the following local stability inequality is established: \[ \sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1). \] It follows that, whenever $\varepsilon<1/(2m+1)$, the hypergraph of $\varepsilon$-good $2m$-subsets is $K_{2m+1}^{(2m)}$-free. Combined with the known exact eight-qubit construction, this yields the stability plateau \[ Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15. \] For odd systems of $2k+1$ qubits, the exact forbidden hypergraph $H_k$ is used to derive explicit finite-error stability radii. In particular, $Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120$ for $0\leq\varepsilon<1/17$. These results turn exact quantum Tur\'an obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.

quant-ph↗

arXiv2026

A Minimum-Cardinality Genuinely Unextendible Product Basis in Three Qutrits

It has remained an open question whether a genuinely unextendible product basis (GUPB) exists. We resolve this problem by constructing an explicit three-qutrit GUPB of cardinality fourteen in the smallest tripartite Hilbert space in which a GUPB can exist. Together with the nonexistence of three-qutrit GUPBs of cardinality less than fourteen, our construction proves that fourteen is the minimum cardinality. A padding procedure further extends the construction to all tripartite systems whose local dimensions are at least three. As applications, the normalized projector onto the thirteen-dimensional orthogonal complement of the three-qutrit GUPB is positive under partial transposition and bound entangled across every bipartition, while the GUPB exhibits strong quantum nonlocality without entanglement.

quant-ph↗

arXiv2026

Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States

We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies $d\geq 3$. Prior to this work, to the best of our knowledge, $\text{AME}(7,d)$ states were known to exist only when $d$ is a prime power other than $2$, or when $d$ can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no $\text{AME}(7,2)$ state exists, it remains to establish existence for all $d\geq 3$. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to $2$ modulo $4$. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension $d\geq 3$.

quant-ph↗

arXiv2026

Automated Construction and Verification of Unextendible Product Bases

Unextendible product bases (UPBs) are important structures in quantum information theory, with applications to completely entangled subspaces, bound entanglement, and local indistinguishability. Since many properties and applications of UPBs are closely related to their cardinalities, one of the central problems in the study of UPBs is to determine whether UPBs of prescribed sizes exist in a given multipartite system. In this paper, we introduce a SAT-assisted framework based on decompositions of the $N$-dimensional hypercube. We define $O_N$-tile decompositions and prove a tile-to-UPB theorem: every $O_N$-tile decomposition induces a UPB through a construction based on tile-wise Fourier product bases and a global stopper state. We then encode the search for such decompositions as a Boolean satisfiability (SAT) problem and use SAT solvers to generate explicit instances. In terms of verification, we also implement a UPB verification algorithm based on local orthogonality graphs and unsaturated subspaces. The algorithm can be used to determine whether an arbitrary finite set of product states forms a UPB. Using this framework, we obtain UPBs of several sizes in some tripartite and quadripartite systems, including sizes $13,14,\ldots,23$ in $\mathbb C^3\otimes\mathbb C^3\otimes\mathbb C^3$. Moreover, the small-dimensional instances obtained here can serve as seed UPBs for recursive constructions, leading to further examples in larger multipartite systems.

quant-ph↗

arXiv2026

Random Reed--Solomon Codes Correcting Permutations, Insertions, and Deletions over Polynomial-Size Alphabets

We study Reed--Solomon codes against adversarial coordinate permutations followed by insertion-deletion (insdel) errors. It was previously shown by Con (2025) that Reed--Solomon codes can attain the exact half-Singleton bound in this setting, but only over exponentially large alphabets. We prove that, by allowing an additive $\epsilon n$ gap from this bound, the alphabet size can be reduced to polynomial. More precisely, for fixed constants $R,\epsilon\in(0,1)$ satisfying $2R+\epsilon<1$ and $k=Rn$, a random Reed--Solomon code of length $n$ and dimension $k$ over an alphabet of size $n^{O_{R,\epsilon}(1)}$ is, with high probability, robust against arbitrary coordinate permutations followed by up to $(1-\epsilon)n-2k+1$ insdel errors. We also prove a complementary alphabet-size lower bound, showing that positive-rate codes, which are robust against linearly many insdel errors in the permutation-insdel setting, require a polynomially superlinear alphabet. Finally, for the explicit two-dimensional Reed--Solomon codes constructed by Con et al. (2024) over alphabet size $O(n^3)$, we give an average $O(n)$-time decoder against arbitrary coordinate permutations followed by $n-3$ insdel errors. Previously, an $O(n)$-time decoder for this code was known only for the deletion setting.

cs.IT↗

arXiv2026

A five-qubit 1-resistant graph state and stabilizer marginal certificates

We study particle-loss resistant entanglement within the framework of stabilizer and graph states. A pure state is $m$-resistant if it remains entangled after the loss of any $m$ particles and becomes fully separable after the loss of any $m+1$ particles. The smallest previously unresolved qubit case was the existence of a five-qubit $1$-resistant pure state, which is resolved here by the five-cycle graph state $\ket{C_5}$. A stabilizer-subgroup method is also developed for verifying $m$-resistance in graph states, using local stabilizers to certify full separability and exact negative partial transpose~(NPT) witnesses to certify entanglement. Applying this to all graph states associated with non-isomorphic graphs on five, six, and seven vertices, we obtain a graph state classification up to local Clifford equivalence, which also classifies stabilizer states up to local Clifford equivalence. Thus, the five-qubit $1$-resistant stabilizer states are exactly the local Clifford class of $C_5$. Six-qubit $2$-resistant stabilizer states exist in three distinct local Clifford classes, whereas no seven-qubit stabilizer state is $m$-resistant for any nonzero admissible $m$. Finally, we prove that the cycle graph states $\ket{C_N}$ with $N\ge 7$ are not $m$-resistant for any $0\le m\le N-2$.

quant-ph↗

arXiv2026

An Explicit Scott-Type Bound for Absolutely Maximally Entangled States with Arbitrary Defect

Absolutely maximally entangled (AME) states and, more generally, $k$-uniform states in $(\C^q)^{\otimes n}$ are central objects in multipartite entanglement theory, with applications to quantum secret sharing, quantum masking, and quantum error correction. In the extremal case $k=\lfloor n/2\rfloor$, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties $n$ exceeds a threshold of order $2q^{2}$ (with a parity dependence on $n$), where $q$ is the local dimension. Recently, Ning et al.\ studied \emph{defective} AME states (i.e., $k=\lfloor n/2\rfloor-l$ with $l>0$), gave explicit Scott-type bounds for defects $l=1,2$ and conjectured a general $(2l+2)q^{2}+o(q^{2})$ behavior. In this paper, we solve this conjecture and establish a fully explicit Scott-type upper bound for AME states with arbitrary defect $l\ge 0$, yielding Scott's bound for $l=0$ and Ning et al.'s bounds for $l=1,2$ as special cases. Equivalently, this gives nonexistence bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof uses a truncated MacWilliams linear-programming system and an explicit infeasibility certificate. As a direct application, we derive explicit asymptotic upper bounds on $k/n$ for fixed local dimension $q$, improving the implicit upper bounds given by Ning et al.

quant-ph↗

arXiv2026

On the Smallest Eigenvalues and Quantum Chromatic Numbers of Hamming Graphs and Generalizations

The smallest eigenvalues of (distance-j) Hamming graphs with distance parameter j at least half the length were completely determined by Brouwer et al. (2018). In the present work, we address the complementary regime, namely distances j strictly less than half the length, and derive asymptotic lower bounds on the smallest eigenvalue of binary Hamming graphs. For certain natural generalizations, specifically Cayley graphs defined over quaternary vector spaces, we asymptotically determine the smallest eigenvalue as well. As an application, we obtain lower bounds on the quantum chromatic number of these graphs. In particular, for the aforementioned Cayley graphs over quaternary vectors, our lower bounds for the quantum chromatic number coincide with known upper bounds.

math.CO↗

arXiv2026

A Hypergraph Container Method for Spread SAT: Approximation and Speedup

We develop a hypergraph container method for the Boolean Satisfiability Problem (SAT) via the newly developed container results [Campos and Samotij (2026)]. This provides an explicit connection between the extent of spread of clauses and the efficiency of container-based algorithms. Informally, the more evenly the clauses are distributed, the stronger the shrinking effect of the containers, which leads to faster algorithms for SAT. To quantify the extent of spread, we use a weighted point of view, in which a clause of size $s$ receives weight $p^s$ for some $0<p\le 1$.In this way, we introduce the notion of $(\lambda,p)_k$-structure for SAT formulas, where $\lambda$ is the spread parameter and $k$ is the maximum size of clauses. By the almost-independence property of containers, we prove that for formulas with $(\lambda,p)_k$-structures, one can distinguish between ``unsatisfiable formulas'' and ``formulas satisfying at least a $(1-\delta)$-fraction of clauses'' in sub-exponential time. This shows that sufficiently spread formulas are not worst-case instances for Gap-ETH. Moreover, we show that the speedup is directly controlled by the spread parameter $\lambda$, yielding faster exact algorithms for SAT formulas containing a $(\lambda,p)_k$-structure. This result extends previous work [Zamir (STOC 2023)] to the non-uniform case.

math.CO↗

arXiv2025

The Hilton-Milner type results of $(k, \ell)$-sum-free sets in $\mathbb F_p^n$

For a prime $p \equiv 2 \pmod 3$, it is well known that the largest sum-free subsets of $\mathbb{F}_p^n$ have size $\frac{p+1}{3} p^{n-1}$, and the extremal sets must be a cuboid of the form $\{\frac{p+1}{3}, \frac{p+1}{3}+1, \ldots, \frac{2p-1}{3}\} \times \mathbb{F}_p^{n-1}$ up to isomorphism. Recently, Reiner and Zotova proved a Hilton--Milner type stability result showing that for large $p$, any sum-free set not contained in the extremal cuboid has size at most $\frac{p-2}{3} p^{n-1}$, and all possible structures attaining this bound were classified. In this paper, we develop a general Hilton--Milner theory for $(k,\ell)$-sum-free sets in $\mathbb{F}_p^n$ for $k > \ell \ge 1$. We determine the maximum size of such sets for all $p \equiv μ\pmod{k+\ell}$ with $2 \le μ\le k+\ell-1$, and show that the extremal configurations are precisely $\lceil (μ-1)/2 \rceil$ non-isomorphic cuboids. Beyond the extremal regime, we prove sharp Hilton--Milner type stability results showing that, for all sufficiently large $p$, a $(k,\ell)$-sum-free set not contained in any of these extremal cuboids is uniformly bounded away from the maximum by a gap of order $p^{n-1}$, and we determine the full structure of all sets achieving this second-best bound in several broad parameter ranges. In particular, when $2 \le μ\le k+\ell-3$ (which is tight), only two structural types occur for all $k+\ell \ge 5$; and when $μ= 2$ or $3$, we obtain a complete classification for all $k > \ell \ge 1$. Our arguments combine additive combinatorics and Fourier-analytic methods, and make use of recent progress toward the long-standing $3k-4$ conjecture, highlighting new connections between inverse additive number theory and extremal problems over finite vector spaces.

math.CO↗

arXiv2025

Quantum Chromatic Number of Subgraphs of Orthogonality Graphs and the Distance-2 Hamming Graph

The determination of the quantum chromatic number of graphs has attracted considerable attention recently. However, there are few families of graphs whose quantum chromatic numbers are determined. A notable exception is the family of orthogonality graphs, whose quantum chromatic numbers are fully determined. In this paper, we extend these results by determining the exact quantum chromatic number of several subgraphs of the orthogonality graphs. Using the technique of combinatorial designs, we also determine the quantum chromatic number of the distance-2 Hamming graph, whose edges consist of binary vectors of Hamming distance 2, for infinitely many length.

math.CO↗

arXiv2025

Quantitative Frameproof Codes and Hypergraphs

Frameproof codes are a class of secure codes introduced by Boneh and Shaw in the context of digital fingerprinting, and have been widely studied from a combinatorial point of view. In this paper, we study a quantitative extension of frameproof codes and hypergraphs, referred to as {\it quantitative frameproof codes and hypergraphs}. We give asymptotically optimal bounds on the maximum sizes of these structures and determine their exact sizes for a broad range of parameters. In particular, we introduce a generalized version of the Erdős matching number in our proof and derive relevant estimates for it.

math.CO↗

arXiv2025

Exploring quantum weight enumerators from the $n$-qubit parallelized SWAP test

Quantum weight enumerators are fundamental tools for analyzing quantum error-correcting codes and multipartite entanglement, offering insights into the existence of quantum error-correcting codes and $k$-uniform states. In this work, we establish a connection between quantum weight enumerators and the $n$-qubit parallelized SWAP test. We demonstrate that each shadow enumerator corresponds to a probability derived from this test, providing a physical interpretation for the shadow enumerators. Leveraging the non-negativity of these probabilities, we present an elegant proof for the shadow inequalities. Additionally, we show that the Shor-Laflamme weight enumerators and the Rains unitary enumerators can be calculated using the $n$-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine the $k$-uniformity of pure states, and evaluate multipartite entanglement measures. Our results indicate that quantum weight enumerators can be efficiently estimated on quantum computers, opening a path to calculate and verify the distances of quantum error-correcting codes.

quant-ph↗

arXiv2025

New constructions of multipartite entanglement resistant to particle loss

An entangled state is called m-resistant if it remains entangled after losing an arbitrary subset of mparticles but becomes fully separable after losing any number of particles larger than m. Quinta et al. [Phys. Rev. A (2019)] conjectured that for any N-particle systems, there always exists an m-resistant pure state. In this paper, we give two general constructions of m-resistant pure states. One is from the mixtures of Dicke states, which provides strong (N - k)-resistant pure N-qubit states with k = 4 or 5. The other is from classical error correcting codes, which provides new m-resistant qudit states for certain m < N/2.

quant-ph↗

arXiv2025

Improved Bounds for Codes over Trees

Codes over trees were introduced recently to bridge graph theory and coding theory with diverse applications in computer science and beyond. A central challenge lies in determining the maximum number of labelled trees over $n$ nodes with pairwise distance at least $d$, denoted by $A(n,d)$, where the distance between any two labelled trees is the minimum number of edit edge operations in order to transform one tree to another. By various tools from graph theory and algebra, we show that when $n$ is large, $A(n,d)=O((Cn)^{n-d})$ for any $d\leq n-2$, and $A(n,d)=Ω((cn)^{n-d})$ for any $d$ linear with $n$, where constants $c\in(0,1)$ and $C\in [1/2,1)$ depending on $d$. Previously, only $A(n,d)=O(n^{n-d-1})$ for fixed $d$ and $A(n,d)=Ω(n^{n-2d})$ for $d\leq n/2$ were known, while the upper bound is improved for any $d$ and the lower bound is improved for $d\geq 2\sqrt{n}$. Further, for any fixed integer $k$, we prove the existence of codes of size $Ω(n^k)$ when $n-d=o(n)$, and give explicit constructions of codes which show $A(n,n-4)=Ω(n^2)$ and $A(n,n-13)=Ω(n^3)$.

math.CO↗

arXiv2025

Counting cliques with prescribed intersection sizes

We study the generalized Turán problem regarding cliques with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subset [0,r-1]$ be a fixed integer set with $|L|\notin \{1,r\}$ and $\ell_1<\dots<\ell_s$, and let $Ψ_r(n,L)$ denote the maximum number of $r$-cliques in an $n$-vertex graph whose $r$-cliques are $L$-intersecting as a family of $r$-subsets. Helliar and Liu recently initiated the systematic study of the function $Ψ_r(n,L)$ and showed that $Ψ_r(n,L)\le \left(1-\frac{1}{3r}\right) \prod_{\ell\in L}\frac{n-\ell}{r-\ell}$ for large $n$, improving the trivial bound from the Deza--Erdős--Frankl theorem by a factor of $1-\frac{1}{3r}$. In this article, we improve their result by showing that as $n$ goes to infinity $Ψ_r(n,L)=Θ_{r,L}(n^{|L|})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and fully determining the corresponding exact values of $Ψ_r(n,L)$ for sufficiently large $n$ in this case. Moreover, when $L=[t,r-1]$, for the generalized Turán extension of the Erdős--Ko--Rado theorem given by Helliar and Liu, we show a Hilton--Milner-type stability result.

math.CO↗

arXiv2025

Linear Programming Bounds on $k$-Uniform States

The existence of $k$-uniform states has been a widely studied problem due to their applications in several quantum information tasks and their close relation to combinatorial objects like Latin squares and orthogonal arrays. With the machinery of quantum enumerators and linear programming, we establish several improved non-existence results and bounds on $k$-uniform states. 1. First, for any fixed $l\geq 1$ and $q\geq 2$, we show that there exists a constant $c$ such that $(\left\lfloor{n/2}\right\rfloor-l)$-uniform states in $(\mathbb{C}^q)^{\otimes n}$ do not exist when $n\geq cq^2+o(q^2)$. The constant $c$ equals $4$ when $l=1$ and $6$ when $l=2$, which generalizes Scott's bound (2004) for $l=0$. 2. Second, when $n$ is sufficiently large, we show that there exists a constant $θ<1/2$ for each $q \le 9$, such that $k$-uniform states in $(\mathbb{C}^q)^{\otimes n}$ exist only when $k\leq θn$. In particular, this provides the first bound (to the best of our knowledge) of $k$ for $4\leq q\leq 9$ and confirms a conjecture posed by Shi et al. (2023) when $q=5$ in a stronger form. 3. Finally, we improve the shadow bounds given by Shi et al. (2023) by a constant for $q = 3,4,5$ and small $n$. When $q=4$, our results can update some bounds listed in the code tables maintained by Grassl (2007--2024).

quant-ph↗