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Minimum Cardinalities of Multipartite Unextendible Product Bases — arXiv:2609.05657

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.05657
License (as archived): CC0-1.0
Archived: 2026-10-16T14:49:12.940Z
SHA-256 of the archived text: a9f63b402e4957580b8567dc52cd4df3dd06db77abdb7c528bd513c1ea90e80f
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Minimum Cardinalities of Multipartite Unextendible Product Bases — arXiv:2609.05657 — arXiv (en), https://arxiv.org/abs/2609.05657 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_a60e42e693bf (Metraipolis archive dc_a60e42e693bf, sha256 a9f63b402e4957580b8567dc52cd4df3dd06db77abdb7c528bd513c1ea90e80f)

BibTeX

@misc{dca60e42e693bf,
  title = {Minimum Cardinalities of Multipartite Unextendible Product Bases — arXiv:2609.05657},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.05657},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_a60e42e693bf; sha256 a9f63b402e4957580b8567dc52cd4df3dd06db77abdb7c528bd513c1ea90e80f},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.05657. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\mathbb C^{d_1}\otimes\cdots\otimes\mathbb C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as $\lvert \varphi_1\rangle\otimes\cdots\otimes\lvert \varphi_p\rangle$ with $\lvert \varphi_j\rangle\in\mathbb C^{d_j}\setminus\{0\}$. An \emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. For local dimensions $d_1,\ldots,d_p\ge2$, let $f_m(d_1,\ldots,d_p)$ be the minimum cardinality of a UPB and let $f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)$ be the natural lower bound. Alon and Lovász determined exactly when $f_m$ attains the lower bound $f_{LB}$, but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with $p\ge3$, whenever parity prevents the natural lower bound $f_{LB}$ from being attained, the true minimum is exactly $f_{LB}+1$. Equivalently, if the number of even local dimensions is positive and even, and at least one local dimension is greater than two, then $f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1$. The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum-cardinality problem for UPBs in all finite quantum systems.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.CO).
Categorie arXiv: Quantum Physics (quant-ph) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.05657 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:32:29.904Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).