Unextendible Product Bases Research Articles

Useful variants and perturbations of completely entangled subspaces and spans of unextendible product bases

Finite-dimensional entanglement for pure states has been used extensively in quantum information theory. Depending on the tensor product structure, even a set of separable states can show non-intuitive characters. Two situations are well studied in the literature, namely, the unextendible product basis (UPB) by Bennett et al.,4 and completely entangled subspaces explicitly given by Parthasarathy.22 More recently, Boyer et al.,6 Boyer and Mor7 and Liss et al.21 studied spaces which have only finitely many pure product states. We carry this further and consider the problem of perturbing different spaces, such as the orthogonal complement of an UPB and also Parthasarathy’s completely entangled spaces, by taking linear spans with specified product vectors. To this end, we develop methods and theory of variations and perturbations of the linear spans of certain UPBs, their orthogonal complements, and also Parthasarathy’s completely entangled subspaces. Finally, we give examples of perturbations with infinitely many pure product states.

Strong quantum nonlocality and unextendibility without entanglement in N-partite systems with odd N

A set of orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. Although such a phenomenon has been shown to any three-, four-, and five-partite systems, the existence of strongly nonlocal orthogonal product sets in multipartite systems remains unknown. In this paper, by using a general decomposition of the N-dimensional hypercubes, we present strongly nonlocal orthogonal product sets in N-partite systems for all odd N≥3. Based on this decomposition, we give explicit constructions of unextendible product bases in N-partite systems for odd N≥3. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.

The construction and weakly local indistinguishability of multiqubit unextendible product bases

The construction and weakly local indistinguishability of multiqubit unextendible product bases

Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis

Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis

Graph-theoretic characterization of unextendible product bases

Graph-theoretic characterization of unextendible product bases

Extension of the unextendible product bases of 5-qubit under coarsening the system

Extension of the unextendible product bases of 5-qubit under coarsening the system

Unextendible product bases from tile structures in bipartite systems

Unextendible product basis (UPB) is one of the most fundamental concepts in the field of quantum information theory, and its constructions attract much attention. Recently, Shi et al (2020 Phy. Rev. A 101 062329) constructed UPBs via a tile structure approach. They showed that a tile structure in an m × n grid with certain properties leads to explicit constructions of UPBs in . The size of the UPB is if the tile structure contains exactly s tiles. In this paper, we deeply analyze the extent of the tile structure approach, by showing that a desired tile structure in an m × m grid contains at most tiles, and thus the smallest size of a UPB in constructed by this approach is . Such tile structures and UPBs are referred to as extremal tile structures and extremal UPBs. We provide algorithms both for constructing possible candidates of extremal tile structures and checking their validity. For applications, we connect the extremal UPBs to local irreducibility and bound entangled states with large ranks.

Local approximation for perfect discrimination of quantum states

Quantum state discrimination involves identifying a given state out of a set of possible states. When the states are mutually orthogonal, perfect state discrimination is alway possible using a global measurement. In the case of multipartite systems when the parties are constrained to use multiple rounds of local operations and classical communications (LOCC), perfect state discrimination is often impossible even with the use of \emph{asymptotic LOCC}, wherein an error is allowed but must vanish in the limit of an infinite number of rounds. Utilizing our recent results on asymptotic LOCC, we derive a lower bound on the error probability for LOCC discrimination of any given set of mutually orthogonal pure states. We use this lower bound to prove necessary conditions for perfect state discrimination by asymptotic LOCC. We then illustrate by example the power of these necessary conditions in significantly simplifying the determination of whether perfect discrimination of a given set of states can be accomplished arbitrarily closely using LOCC. The latter examples include a proof that perfect discrimination by asymptotic LOCC is impossible for any \emph{minimal} unextendible product basis (UPB), where minimal means that for the given multipartite system, no UPB with a smaller number of states can exist. We also give a simple proof that what has been called \emph{strong nonlocality without entanglement} is considerably stronger than had previously been demonstrated.

Constructing unextendible product bases from multiqubit ones

The construction of multipartite unextendible product bases (UPBs) is a basic problem in quantum information. We respectively construct two families of 2 × 2 × 4 and 2 × 2 × 2 × 4 UPBs of size eight by using the existing four-qubit and five-qubit UPBs. As an application, we construct novel families of multipartite positive-partial-transpose entangled states, as well as their entanglement properties in terms of the geometric measure of entanglement.

Small set of orthogonal product states with nonlocality

A set of orthogonal states in multipartite systems is called to be locally stable if to preserve the orthogonality of the states, only trivial local measurement can be performed for each partite. Locally stable sets of states are always locally indistinguishable yielding a form of nonlocality which is different from the Bell-type nonlocality. In this work, we study the locally stable set of product states with small size. First, we give a lower bound on the size of locally stable sets of product states. It is well known that unextendible product basis (UPB) is locally indistinguishable. But we find that some of them are not locally stable. On the other hand, there exists some small subsets of minimum size UPB that are also locally stable which implies the nonlocality of such UPBs are stronger than other form.

Unextendible product operator basis

Quantum nonlocality is associated with the local indistinguishability of orthogonal states. Unextendible product basis (UPB), a widely used tool in quantum information, exhibits nonlocality, which is the powerful resource for quantum information processing. In this work, we extend the definitions of nonlocality and genuine nonlocality from states to operators. We also extend UPB to the notions of unextendible product operator basis, unextendible product unitary operator basis (UPUOB), and strongly UPUOB. We construct their examples and show the nonlocality of some strongly UPUOBs under local operations and classical communications. We study the phenomenon of these operators acting on quantum states. As an application, we distinguish the two-dimensional strongly UPUOB, which only consumes three ebits of entanglement. Our results imply that such UPUOBs exhibit nonlocality as UPBs, and the distinguishability of them requires entanglement resources.

Unextendible and uncompletable product bases in every bipartition

Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging from quantum nonlocality to quantum cryptography. A generalized concept called uncompletable product basis also attracts much attention. In this paper, we find some unextendible product bases that are uncompletable product bases in every bipartition, which answers a 19 year-old open question proposed by DiVincenzo et al (2003 Commun. Math. Phys. 238 379–410). As a consequence, we connect such unextendible product bases to local hiding of information, positive-partial-transpose entangled states and genuinely entangled states. Furthermore, we give a sufficient condition for the existence of an unextendible product basis that is still unextendible in every bipartition, and the existence of such a UPB is another open question proposed by Demianowic et al (2018 Phys. Rev. A 98 012313). Our results advance the understanding of the geometry of unextendible product bases.

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

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.

Constructions of Unextendible Special Entangled Bases

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.

Local approximation of multipartite quantum measurements

We provide a necessary condition that a quantum measurement can be implemented by the class of protocols known as Local Operations and Classical Communication, or LOCC, including when an error is allowed but must vanish in the limit of an infinite number of rounds, a case referred to as asymptotic LOCC. Our condition unifies, extends, and provides an intuitive, geometric justification for previous results on asymptotic LOCC. We use our condition to answer a variety of long-standing, unsolved problems, including for distinguishability of certain sets of states by LOCC. These include various classes of unextendible product bases, for which we prove they cannot be distinguished by LOCC even when infinite resources are available and asymptotically vanishing error is allowed.

Strongly nonlocal unextendible product bases do exist

A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving local measurement can be performed to this set. In general, it is difficult to show that such an orthogonality-preserving local measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully show the existence of unextendible product bases (UPBs) that are locally irreducible in every bipartition in d⊗d⊗d for any d≥3, and 3⊗3⊗3 achieves the minimum dimension for the existence of such UPBs. These UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Our result solves an open question given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. It also sheds new light on the connections between UPBs and strong quantum nonlocality.

Strong quantum nonlocality for unextendible product bases in heterogeneous systems

A set of multipartite orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. It is known that unextendible product bases (UPBs) can show the phenomenon of quantum nonlocality without entanglement. Thus it is interesting to investigate the strong quantum nonlocality for UPBs. Most of the UPBs with the minimum size cannot demonstrate strong quantum nonlocality. In this paper, we construct a series of UPBs with different large sizes in d A ⊗ d B ⊗ d C and d A ⊗ d B ⊗ d C ⊗ d D for d A , d B , d C , d D ⩾ 3, and we also show that these UPBs have strong quantum nonlocality, which answers an open question given by Halder et al (2019 Phys. Rev. Lett. 122 040403) and Yuan et al (2020 Phys. Rev. A 102 042228) for any possible three and four-partite systems. Furthermore, we propose an entanglement-assisted protocol to locally discriminate the UPB in 3 ⊗ 3 ⊗ 4, and it consumes less entanglement resource than the teleportation-based protocol. Our results build the connection between strong quantum nonlocality and UPBs.

New results on unextendible product bases

Unextendible product bases (UPB) is an important concept and has extensive applications in various fields of quantum information theory. The construction of UPBs is closely related to combinatorics. Alon and Lovász (2001) first used a series of graph theoretic tools to characterize the necessary and sufficient conditions when the size of a UPB can attain the trivial lower bound. Later Feng (2006) brought the 1-factorization of graphs into the study of UPBs. In this paper, we further apply some graph theoretic tools to analyze the minimum size of a UPB under certain parameters and obtain a series of new results. Moreover, we have an almost complete characterization of all the potential sizes of UPBs in $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^k$.

Positively factorizable maps

We initiate a study of linear maps on Mn(C) that have the property that they factor through a tracial von Neumann algebra (A,τ) via operators Z∈Mn(A) whose entries consist of positive elements from the von-Neumann algebra. These maps often arise in the context of non-local games especially in the synchronous case. We establish a connection with the convex sets in Rn containing self-dual cones and the existence of these maps. The Choi matrix of a map of this kind which factor through an abelian von-Neumann algebra turns out to be a completely positive (CP) matrix. We fully characterize positively factorizable maps whose Choi rank is 2. We also provide some applications of this analysis in finding doubly nonnegative matrices which are not CPSD. A special class of these examples are found from the concept of Unextendible Product Bases in quantum information theory.

State space structure of tripartite quantum systems

State space structure of tripartite quantum systems is analyzed. In particular, it has been shown that the set of states separable across all the three bipartitions [say $\mathcal{B}^{int}(ABC)$] is a strict subset of the set of states having positive partial transposition (PPT) across the three bipartite cuts [say $\mathcal{P}^{int}(ABC)$] for all the tripartite Hilbert spaces $\mathbb{C}_A^{d_1}\otimes\mathbb{C}_B^{d_2}\otimes\mathbb{C}_C^{d_3}$ with $\min\{d_1,d_2,d_3\}\ge2$. The claim is proved by constructing state belonging to the set $\mathcal{P}^{int}(ABC)$ but not belonging to $\mathcal{B}^{int}(ABC)$. For $(\mathbb{C}^{d})^{\otimes3}$ with $d\ge3$, the construction follows from specific type of multipartite unextendible product bases. However, such a construction is not possible for $(\mathbb{C}^{2})^{\otimes3}$ since for any $n$ the bipartite system $\mathbb{C}^2\otimes\mathbb{C}^n$ cannot have any unextendible product bases [Phys. Rev. Lett. 82, 5385 (1999)]. For the $3$-qubit system we, therefore, come up with a different construction.