Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.10202

Unextendible stabiliser bases

Markus Frembs

quant-phSubmitted Aug 10, 2026• Announced Aug 12, 2026• 0 citations
arXivHTMLPDF

Abstract

We study incomplete sets of orthogonal stabiliser states that cannot be extended by a further stabiliser state orthogonal to all of its members. Such sets are the stabiliser analogue of unextendible product bases (UPBs), and we thus call them unextendible stabiliser bases (USBs). Leveraging the symplectic geometry underlying the $n$-qudit Pauli group in prime local dimension, we explicitly construct USBs for systems of four qubits and three qudits of odd prime local dimension. Moreover, we show that these are the respective minimal qubit, respectively qudit numbers for which such bases exist, and that USBs exist for all $n\geq4$ qubit and for all $n\geq3$ odd-prime-dimensional qudit systems. Finally, we compare the resource-theoretic aspects of USBs with those of UPBs. We establish that, analogous to the case of UPBs, the orthogonal complement of every unextendible stabiliser set is a stabiliser-free subspace, and its normalised projector is necessarily magic; moreover, it is bound magic in odd prime dimension, yet need not be for qubits. We also show that USB unextendibility alone imposes no uniform quantitative obstruction to discrimination by stabiliser operations.

Comments
4+18 pages, 2 tables; comments welcome

Related papers

Graph-theoretic characterization of unextendible product bases

Fei Shi, Ge Bai, Xiande Zhang +2 more

Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. Here we provide an equivalent characterization of UPBs in graph-theoretic terms. Different from previous graph-theoretic investigations of UPBs, which focused mostly on the orthogonality relations between different product states, our characterization includes a graph-theoretic reformulation of the unextendibility condition. Building on this characterization, we develop a constructive method for building UPBs in low dimensions and shed light on the open question of whether there exist genuinely unextendible product bases (GUPBs), that is, multipartite product bases that are unextendible with respect to every possible bipartition. We derive a lower bound on the size of any candidate GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs and discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension. Finally, we apply our characterization to the problem of distinguishing UPB states under local operations and classical communication, deriving a necessary condition for reliable discrimination in the asymptotic limit.

Physical Review Research•Aug 31, 2023•4 citations

Minimum Cardinalities of Multipartite Unextendible Product Bases

Chenhao Wang

In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as $\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}$ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. 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. Let $f_m(d_1,\ldots,d_p)$ be the minimum cardinality of a UPB in the space, 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 the natural lower bound $f_{LB}$ is unattainable, the true minimum is exactly $f_{LB}+1$. Precisely, if the number of even local dimensions $d_i$ 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 size problem for UPBs in all finite quantum systems.

quant-pharXiv:2609.05657•Sep 8, 2026•0 citations

Unextendible product bases from orthogonality graphs

Shi, Fei, Bai, Ge, Xiande Zhang +2 more

Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. A famous open question is whether there exist genuinely unextendible product bases (GUPBs), namely multipartite product bases that are unextendible with respect to every possible bipartition. Here we shed light on this question by providing a characterization of UPBs and GUPBs in terms of orthogonality graphs. Building on this connection, we develop a method for constructing UPBs in low dimensions, and we derive a lower bound on the size of any GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs. Finally, we discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.

arXiv:2303.02553•Mar 5, 2023•0 citations

Unextendible Product Bases and Bound Entanglement

Charles H. Bennett, David Peter Divincenzo, Tal Mor +3 more

An unextendible product basis (UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled state. We exhibit a tripartite $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e., all three corresponding $2\ifmmode\times\else\texttimes\fi{}4$ bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local positive operator valued measurements and classical communication.

arXiv:quant-ph/9808030•Physical Review Letters•Jun 28, 1999•727 citations

Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system

Guang-Bao Xu, Ying-Hui Yang, Qiao-Yan Wen +2 more

As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.

arXiv:1512.06485•Scientific Reports•Aug 9, 2016•54 citations

Negative result about the construction of genuinely entangled subspaces from unextendible product bases

Maciej Demianowicz

Unextendible product bases (UPBs) provide a versatile tool with various applications across different areas of quantum information theory. Their comprehensive characterization is thus of great importance and has been a subject of vital interest for over two decades now. An open question asks about the existence of UPBs, which are genuinely unextendible, i.e., they are not extendible even with biproduct vectors. In other words, the problem is to verify whether there exist genuinely entangled subspaces (GESs), subspaces composed solely of genuinely multiparty entangled states, complementary to UPBs. We solve this problem in the negative for many sizes of UPBs in different multipartite scenarios. In particular, in the all-important case of equal local dimensions, we show that there are always forbidden cardinalities for such UPBs, including the minimal ones corresponding to GESs of the maximal dimensions.

arXiv:2202.08356•Physical Review A•Jul 29, 2022•6 citations
We study incomplete sets of orthogonal stabiliser states that cannot be extended by a further stabiliser state orthogonal to all of its members. Such sets are the stabiliser analogue of unextendible product bases (UPBs), and we thus call them unextendible stabiliser bases (USBs). Leveraging the symplectic geometry underlying the nnn-qudit Pauli group in prime local dimension, we explicitly construct USBs for systems of four qubits and three qudits of odd prime local dimension. Moreover, we show that these are the respective minimal qubit, respectively qudit numbers for which such bases exist, and that USBs exist for all n≥4n\geq4n≥4 qubit and for all n≥3n\geq3n≥3 odd-prime-dimensional qudit systems. Finally, we compare the resource-theoretic aspects of USBs with those of UPBs. We establish that, analogous to the case of UPBs, the orthogonal complement of every unextendible stabiliser set is a stabiliser-free subspace, and its normalised projector is necessarily magic; moreover, it is bound magic in odd prime dimension, yet need not be for qubits. We also show that USB unextendibility alone imposes no uniform quantitative obstruction to discrimination by stabiliser operations.
Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. Here we provide an equivalent characterization of UPBs in graph-theoretic terms. Different from previous graph-theoretic investigations of UPBs, which focused mostly on the orthogonality relations between different product states, our characterization includes a graph-theoretic reformulation of the unextendibility condition. Building on this characterization, we develop a constructive method for building UPBs in low dimensions and shed light on the open question of whether there exist genuinely unextendible product bases (GUPBs), that is, multipartite product bases that are unextendible with respect to every possible bipartition. We derive a lower bound on the size of any candidate GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs and discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension. Finally, we apply our characterization to the problem of distinguishing UPB states under local operations and classical communication, deriving a necessary condition for reliable discrimination in the asymptotic limit.
In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as ∣φ1⟩⊗⋯⊗∣φp⟩\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}∣φ1​⟩⊗⋯⊗∣φp​⟩ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. 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. Let fm(d1,…,dp)f_m(d_1,\ldots,d_p)fm​(d1​,…,dp​) be the minimum cardinality of a UPB in the space, and let fLB(d1,…,dp)=1+∑j=1p(dj−1)f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)fLB​(d1​,…,dp​)=1+∑j=1p​(dj​−1) be the natural lower bound. Alon and Lovász determined exactly when fmf_mfm​ attains the lower bound fLBf_{LB}fLB​, but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with p≥3p\ge3p≥3, whenever the natural lower bound fLBf_{LB}fLB​ is unattainable, the true minimum is exactly fLB+1f_{LB}+1fLB​+1. Precisely, if the number of even local dimensions did_idi​ is positive and even, and at least one local dimension is greater than two, then fm(d1,…,dp)=fLB(d1,…,dp)+1f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1fm​(d1​,…,dp​)=fLB​(d1​,…,dp​)+1. The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.
Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. A famous open question is whether there exist genuinely unextendible product bases (GUPBs), namely multipartite product bases that are unextendible with respect to every possible bipartition. Here we shed light on this question by providing a characterization of UPBs and GUPBs in terms of orthogonality graphs. Building on this connection, we develop a method for constructing UPBs in low dimensions, and we derive a lower bound on the size of any GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs. Finally, we discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.
An unextendible product basis (UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled state. We exhibit a tripartite $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e., all three corresponding $2\ifmmode\times\else\texttimes\fi{}4$ bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local positive operator valued measurements and classical communication.
As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.
Unextendible product bases (UPBs) provide a versatile tool with various applications across different areas of quantum information theory. Their comprehensive characterization is thus of great importance and has been a subject of vital interest for over two decades now. An open question asks about the existence of UPBs, which are genuinely unextendible, i.e., they are not extendible even with biproduct vectors. In other words, the problem is to verify whether there exist genuinely entangled subspaces (GESs), subspaces composed solely of genuinely multiparty entangled states, complementary to UPBs. We solve this problem in the negative for many sizes of UPBs in different multipartite scenarios. In particular, in the all-important case of equal local dimensions, we show that there are always forbidden cardinalities for such UPBs, including the minimal ones corresponding to GESs of the maximal dimensions.