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.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:2609.05657Sep 8, 20260 citations

Constructions of Unextendible Special Entangled Bases

Yanling Wang

Unextendible product basis (UPB), a set of incomplete orthonormal product states whose complementary space has no product state, is very useful for constructing bound entangled states. Naturally, instead of considering the set of product states, Bravyi and Smolin considered the set of maximally entangled states. They introduced the concept of unextendible maximally entangled basis (UMEB), a set of incomplete orthonormal maximally entangled states whose complementary space contains no maximally entangled state [Phys. Rev. A 84, 042,306 (2011)]. An entangled state whose nonzero Schmidt coefficients are all equal to 1/k is called a special entangled state of “type k”. In this paper, we introduce a concept named special unextendible entangled basis of “type k” which generalizes both UPB and UMEB. A special unextendible entangled basis of “type k” (SUEBk) is a set of incomplete orthonormal special entangled states of “type k” whose complementary space has no special entangled state of “type k”. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by k while the other one can be generated by two kinds of the special entangled states of “type k”. This method is very effective when k = pm ≥ 3 where p is a prime number. For these cases, we can obtain sets of SUEBk with continuous integer cardinality when the local dimensions are large.Unextendible product basis (UPB), a set of incomplete orthonormal product states whose complementary space has no product state, is very useful for constructing bound entangled states. Naturally, instead of considering the set of product states, Bravyi and Smolin considered the set of maximally entangled states. They introduced the concept of unextendible maximally entangled basis (UMEB), a set of incomplete orthonormal maximally entangled states whose complementary space contains no maximally entangled state [Phys. Rev. A 84, 042,306 (2011)]. An entangled state whose nonzero Schmidt coefficients are all equal to 1/k is called a special entangled state of “type k”. In this paper, we introduce a concept named special unextendible entangled basis of “type k” which generalizes both UPB and UMEB. A special unextendible entangled basis of “type k” (SUEBk) is a set of incomplete orthonormal special entangled states of “type k” whose complementary space has no special entangled state of “type k”. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by k while the other one can be generated by two kinds of the special entangled states of “type k”. This method is very effective when k = pm ≥ 3 where p is a prime number. For these cases, we can obtain sets of SUEBk with continuous integer cardinality when the local dimensions are large.

Frontiers in PhysicsMay 27, 20220 citations

The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

Jianxin Chen, Nathaniel Johnston

arXiv:1301.1406Communications in Mathematical PhysicsOct 9, 201459 citations

Progress in the study of the (non)existence of genuinely unextendible product bases

Maciej Demianowicz

arXiv:2509.26135Quantum Information ProcessingFeb 13, 20260 citations

\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.