Automated Construction and Verification of Unextendible Product Bases

• URL: http://arxiv.org/abs/2608.01438v1
• Date: Sun, 02 Aug 2026 18:47:51 GMT
• Title: Automated Construction and Verification of Unextendible Product Bases
• Abstract summary: Unextendible product bases (UPBs) are important structures in quantum information theory.<n>We introduce a SAT-assisted framework based on decompositions of the (N)-dimensional hypercube.<n>We obtain UPBs of several sizes in some tripartite and quadripartite systems.
• Score: 13.46794978663705
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Systematic construction of multiqubit unextendible product bases [4.115763274264731]
  We give a systematic construction of multiqubit unextendible product bases.<n> Row decompositions of small UOMs produce the main intervals.<n>Nonexistence of other UPB sizes follows from counts of fibres and row pairs.
  arXiv  Detail & Related papers  (2026-09-13T20:55:51Z)
• Generalized Numerical Construction of MUBs: A Group Theoretical Investigation [0.0]
  Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory.<n>In non-prime-power dimensions, the existence of maximal sets of MUBs remains an open question.
  arXiv  Detail & Related papers  (2026-04-05T16:11:00Z)
• Graph theoretic quantum contextuality and unextendible Product Bases [0.8324413576411747]
  We establish a graph theoretic connection between contextuality and UPBs.<n>A new class of minimal UPBs in $mathbbC3 otimes mathbbCn$ is constructed.<n>The structural properties of these graphs make them suitable candidates for constructing noncontextuality inequalities.
  arXiv  Detail & Related papers  (2025-10-30T17:21:01Z)
• Light-SQ: Structure-aware Shape Abstraction with Superquadrics for Generated Meshes [60.92139345612904]
  We present Light-SQ, a novel superquadric-based optimization framework.<n>We propose a block-regrow-fill strategy guided by structure-aware volumetric decomposition.<n>Experiments demonstrate that Light-SQ enables efficient, high-fidelity, and editable shape abstraction with superquadrics.
  arXiv  Detail & Related papers  (2025-09-29T16:18:32Z)
• Scheme of quantum communications based on Witting polytope [55.2480439325792]
  Presented paper describes how to use this configuration for a quantum key distribution protocol based on contextuality using some illustrative examples with 40 "quantum cards"<n>In a more general case, two arbitrary quantum systems with four basis states (ququarts) can be used instead.
  arXiv  Detail & Related papers  (2025-03-24T08:26:48Z)
• On The Sample Complexity Bounds In Bilevel Reinforcement Learning [49.19950489963245]
  Bilevel reinforcement learning (BRL) has emerged as a powerful framework for aligning generative models.<n>We present the first sample of $mathcalO(epsilon)$ in continuous state-action complexity.<n>Our analysis also extends complexity improving upon existing bounds of $mathcalO(epsilon)$.
  arXiv  Detail & Related papers  (2025-03-22T04:22:04Z)
• Searching for Efficient Linear Layers over a Continuous Space of Structured Matrices [88.33936714942996]
  We present a unifying framework that enables searching among all linear operators expressible via an Einstein summation. We show that differences in the compute-optimal scaling laws are mostly governed by a small number of variables. We find that Mixture-of-Experts (MoE) learns an MoE in every single linear layer of the model, including the projection in the attention blocks.
  arXiv  Detail & Related papers  (2024-10-03T00:44:50Z)
• Unextendibility, uncompletability, and many-copy indistinguishable ensembles [49.1574468325115]
  We show that the complement of any bipartite pure entangled state is spanned by product states which form a nonorthogonal unextendible product basis (nUPB) of maximum cardinality.<n>We also report a class of multipartite many-copy indistinguishable ensembles for which local indistinguishability property increases with decreasing number of mixed states.
  arXiv  Detail & Related papers  (2023-03-30T16:16:41Z)
• Unextendible product bases from orthogonality graphs [24.740502616119606]
  Unextendible product bases play a key role in the study of quantum entanglement and nonlocality. We show that every minimal GUPB saturating our bound must be associated to regular graphs. We discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.
  arXiv  Detail & Related papers  (2023-03-05T02:33:22Z)
• Negative result about the construction of genuinely entangled subspaces from unextendible product bases [0.0]
  Unextendible product bases (UPBs) provide a versatile tool with various applications across different areas of quantum information theory. An open question asks about the existence of UPBs, which are genuinely unextendible, i.e., they are not extendible even with biproduct vectors. We show that there are always forbidden cardinalities for such UPBs, including the minimal ones corresponding to GESs of the maximal dimensions.
  arXiv  Detail & Related papers  (2022-02-16T22:15:49Z)
• The construction and local distinguishability of multiqubit unextendible product bases [7.238541917115604]
  An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs) We show that the UPB is locally indistinguishable in the bipartite systems of two qubits and five qubits, respectively. Taking the graphs as product vectors, we show that they are in three different orbits up to local unitary equivalence.
  arXiv  Detail & Related papers  (2021-02-23T08:50:19Z)
• Finite-Function-Encoding Quantum States [52.77024349608834]
  We introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions. We investigate some of their structural properties.
  arXiv  Detail & Related papers  (2020-12-01T13:53:23Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.