Xin-Wei Zha, Cun-Bing Huang
In this paper, an unextendible product basis and exact-entanglement bases of three qubit is given, and the properties of entanglement for exact-entanglement bases are also discussed. In addition, the bound entangled mixed state is obtained from the exact-entanglement bases.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Unextendible product bases have been shown to have many important uses in quantum information theory, particularly in the qubit case. However, very little is known about their mathematical structure beyond three qubits. We present several new results about qubit unextendible product bases, including a complete characterization of all four-qubit unextendible product bases, which we show there are exactly 1446 of. We also show that there exist p-qubit UPBs of almost all sizes less than $2^p$.
We investigate the problem of constructing unextendible product bases in the qubit case - that is, when each local dimension equals 2. The cardinality of the smallest unextendible product basis is known in all qubit cases except when the number of parties is a multiple of 4 greater than 4 itself. We construct small unextendible product bases in all of the remaining open cases, and we use graph theory techniques to produce a computer-assisted proof that our constructions are indeed the smallest possible.
It is known that the $n$-qubit system has no unextendible product bases (UPBs) of cardinality $2^n-1$, $2^n-2$ and $2^n-3$. On the other hand the $n$-qubit UPBs of cardinality $2^n-4$ exist for all $n\ge3$. We prove that they do not exist for cardinality $2^n-5$.
We consider the unextendible product bases (UPBs) of fixed cardinality $m$ in quantum systems of $n$ qubits. These UPBs are divided into finitely many equivalence classes with respect to an equivalence relation introduced by N. Johnston. There is a natural partial order `$\leq$' on the set of these equivalence classes for fixed $m$, and we use this partial order to study the topological closure of an equivalence class of UPBs. In the case of four qubits, for $m=8,9,10$, we construct explicitly the Hasse diagram of this partial order.
We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a suffici…
It is known that the states in an unextendible product basis (UPB) cannot be distinguished perfectly when the parties are restricted to local operations and classical communication (LOCC). Previous discussions of such bases have left open the following question: What entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In this paper, I present protocols which use entanglement more efficiently than teleportation to distinguish certain classes of UPB's. The ideas underlying my approach to this problem offer rather general insight into why entanglement is…
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 2x2x2 UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e. all three corresponding 2x4 bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local POVMs and classical commun…
In quantum information theory, it is a fundamental problem to construct multipartite unextendible product bases (UPBs). We show that there exist two families UPBs in Hilbert space $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4$ by merging two different systems of an existing $7$-qubit UPB of size $11$. Moreover, a new family of $7$-qubit positive-partial-transpose (PPT) entangled states of rank $2^7-11$ is constructed. We analytically derive a geometric measure of entanglement of a special PPT entangled states. Also an upper bound a…
The construction of multipartite unextendible product bases (UPBs) is a basic problem in quantum information. We respectively construct two families of $2\times2\times4$ and $2\times2\times2\times4$ 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.
We completely characterize the condition when a tile structure provides an unextendible product basis (UPB), and construct UPBs of different large sizes in $\mathbb{C}^m\otimes\mathbb{C}^n$ for any $n\geq m\geq 3$. This solves an open problem in [S. Halder et al., Phys. Rev. A 99, 062329 (2019)]. As an application, we show that our UPBs of size $(mn-4\lfloor\frac{m-1}{2}\rfloor)$ in $\mathbb{C}^m\otimes\mathbb{C}^n$ can be perfectly distinguished by local operations and classical communications assisted with a $\lceil\frac{m}{2}\rceil\otimes\lceil\frac{m}{2}\rceil$ maximally entangled state.
Any set of states which cannot be perfectly distinguished by local operations and classical communication (LOCC) alone, can always be locally distinguished using quantum teleportation with enough entanglement resource. However, in quantum information theory, entanglement is a very valuable resource, so it leaves the following open question: how to accomplish this task more efficiently than teleportation, that is, design the local discrimination protocol using less entanglement resource. In this paper, we first present two protocols to locally distinguish a set of unextendible product bases (UP…
An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we construct a 7-qubit UPB of size 11. It solves an open problem in [Quantum Information Processing 19:185 (2020)]. Next, we graph-theoretically show that the UPB is locally indistinguishable in the bipartite systems of two qubits and five qubits, respectively. It turns out that the UPB corresponds to a complete graph with 11 vertices constructed by three sorts of nonisomorphic graphs. Taking the graphs as product vectors, we show that they ar…
The construction of multiqubit unextendible product bases
研究提出构造和判定不可扩展正交矩阵（UPBs）不等价的算法，找到了所有大小为8的5-qubit UPBs，并基于此提出构造新不等价UPB的定理。
We generalize the notion of unextendible maximally entangled basis from bipartite systems to multipartite quantum systems. It is proved that there do not exist unextendible maximally entangled bases in three-qubit systems. Moreover,two types of unextendible maximally entangled bases are constructed in tripartite quantum systems and proved to be not mutually unbiased.
A notion of strength of an unextendible product basis is introduced and a quantitative measure for it is suggested with a view to providing an indirect measure for the bound entanglement of formation of the bound entangled mixed state associated with an unextendible product basis.
We study the local distinguishability of general multi-qubit states and show that local projective measurements and classical communication are as powerful as the most general local measurements and classical communication. Remarkably, this indicates that the local distinguishability of multi-qubit states can be decided efficiently. Another useful consequence is that a set of orthogonal $n$-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension $2^n$. When $n=2$ such a condition is also sufficient. Employing these resu…
We consider a general version of the phenomenon of more nonlocality with less entanglement, within the framework of the unambiguous (i.e., conclusive) quantum state discrimination problem under local quantum operations and classical communication. We show that although the phenomenon was obtained before for two qutrits, it can also be observed for two qubits, while still being at the single-copy level. We establish that the phenomenon is intrinsically connected to the concept of unextendible entangled bases, in the two-qubit case. In the process, we demonstrate a hierarchy of nonlocality among…
We show that there are six inequivalent $4\times4$ unextendible product bases (UPBs) of size eight, when we consider only 4-qubit product vectors. We apply our results to construct Positive-Partial-Transpose entangled states of rank nine. They are at the same 4-qubit, $2\times2\times4$ and $4\times4$ states, and their ranges have product vectors. One of the six UPBs turns out to be orthogonal to an almost genuinely entangled space, in the sense that the latter does not contain $4\times4$ product vector in any bipartition of 4-qubit systems. We also show that the multipartite UPB orthogonal to …
We investigate the relation between unextendible product bases (UPB) and Bell inequalities found recently in [R. Augusiak et al., Phys. Rev. Lett. 107, 070401 (2011)]. We first review the procedure introduced there that associates to any set of mutually orthogonal product vectors in a many-qubit Hilbert space a Bell inequality. We then show that if a set of mutually orthogonal product vectors can be completed to a full basis, then the associated Bell inequality is trivial, in the sense of not being violated by any nonsignalling correlations. This implies that the relevant Bell inequalities tha…
Unextendible stabiliser bases
本文研究不可扩展稳定子基（USB），构造了四量子比特及奇素维度三 qudit 系统的 USB，证明其存在的最小规模，并对比了 USB 与不可扩展乘积基（UPB）的资源理论性质。
We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendibleproduct basis bound entangled states (UPB BE states). The method relies on the fact that an optimalentanglement witness is given by a plane perpendicular to a line between the reference state, entanglementof which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds anapproximation of CSS. In this article, we investigate if this approximation can be good enough toyield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those givenby Ban…
The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size $6$ and six UPBs of size $9$ in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size $9$ in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size $7$ in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a spec…
Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging bound entangled states, quantum nonlocality without entanglement, and Bell inequalities with no quantum violation. 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. [Commun. Math. Phys. 238, 379 (2003)]. As a consequence, we connect such unex…
Unextendible product operator basis
本文将非局域性概念从量子态扩展到算子，提出不可扩展乘积算子基及酉算子基等新概念，构造实例并证明其在LOCC下具有非局域性，且区分它们需要纠缠资源。
Efficient discrimination schemes for unextendible product bases with strong quantum nonlocality
本文提出三种高效区分具有强量子非局域性的不可扩展产物基的方案，利用最大化纠缠态减少资源消耗。
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 $d_A\otimes d_B\otimes d_C$ and $d_A\otim…
We discuss the subject of Unextendible Product Bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e. states of rank 4 in 3 x 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal Unextendible Product Basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general Unextendible Product Bases, nevertheless they can always be locally transformed to the orthogonal form. This fully confirms the surpr…
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…
Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis
本文推导了构建展现强量子非局域性的不可扩展可分基（UBB）的规则，解决了Agrawal等人提出的开放问题，为多体系统中真正纠缠子空间的信息论应用提供支持。
We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interestin…
Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs $S$ and $T$ the associated bound entangled states $ρ_S$ and $ρ_T$ can not be converted to each other by LOCC, unless $S$ and $T$ coincide up to local unitaries. More specifically, there exists a finite precision $ε>0$ such that for any LOCC protocol mapping $ρ_S$ into a probabilistic ense…
Genuinely Unextendible Product Bases from Maximum Distance Separable Codes
本文利用经典最大距离可分码，为N≥3的参与者构造出真正不可扩展乘积基，得到真正纠缠子空间与多体束缚纠缠态，建立纠错码与多体纠缠的联系，提供认证真正多体束缚纠缠的代数途径。
We present a generic method to construct a product basis exhibiting Nonlocality Without Entanglement with $n$ parties each holding a system of dimension at least $n-1$. This basis is generated via a quantum circuit made of control-Discrete Fourier Transform gates acting on the computational basis. The simplicity of our quantum circuit allows for an intuitive understanding of this new type of nonlocality. We also show how this circuit can be used to construct Unextendible Product Bases and their associated Bound Entangled States. To our knowledge, this is the first method which, given a general…
Quantifying the unextendibility of entanglement
本文提出一个量化一般两体量子态不可扩展性的框架，定义了不可扩展纠缠度量并证明其在二可扩展量子操作下为纠缠单调，为纠缠或密钥蒸馏的速率与开销提供了高效计算基准。
Genuinely entangled subspaces and strongly nonlocal unextendible biseparable bases in four-partite systems
本文研究四量子比特系统中真正纠缠子空间的构建，证明其强非局域性并展示其在所有双分拆上的可蒸馏性，为量子信息处理提供理论基础。
Mutually unbiased bases (MUBs) provide a standard tool in the verification of quantum states, especially when harnessing a complete set for optimal quantum state tomography. In this work, we investigate the detection of entanglement via inequivalent sets of MUBs, with a particular focus on unextendible MUBs. These are bases for which an additional unbiased basis cannot be constructed and, consequently, are unsuitable for quantum state verification. Here, we show that unextendible MUBs, as well as other inequivalent sets in higher dimensions, can be more effective in the verification of entangl…
Let H[N] denote the tensor product of n finite dimensional Hilbert spaces H(r). A state |phi> of H[N] is separable if |phi> is the tensor product of states in the respective product spaces. An orthogonal unextendible product basis is a finite set B of separable orthonormal states |phi(k)> such that the non-empty space B9perp), the set of vectors orthogonal to B, contains no separable projection. Examples of orthogonal UPB sets were first constructed by Bennett et al [1] and other examples appear, for example, in [2] and [3]. If F denotes the set of convex combinations of the projections |phi(k…
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 unexten…
A Minimum-Cardinality Genuinely Unextendible Product Basis in Three Qutrits
研究解决了真正不可扩展乘积基是否存在的开放问题，构造出三 qutrit 的最小基数 14 的 GUPB，扩展了构造并给出其在量子纠缠与非局域性中的应用。
In quantum mechanics and quantum information, to establish the orthogonal bases is a useful means. The existence of unextendible product bases impels us to study the `entanglement bases' problems. In this paper, the concepts of entanglement bases and exact-entanglement bases are defined, and a theorem about exact-entanglement bases is given. We discuss the general structures of the orthogonal complete bases. Two examples of applications are given. At last, we discuss the problem of transformation of the general structure forms.
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ($d=p^{2}$) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1…
Local indistinguishability and incompleteness of entangled orthogonal bases: Method to generate two-element locally indistinguishable ensembles
本文利用不可扩展纠缠基研究正交态的局域不可区分性，证明所有两量子比特不可扩展纠缠基大小为三且不可被可分离测量完美区分，并提出系统构造二元局域不可区分系综的方法。
Unextendible entanglement of quantum channels
本文利用超信道框架和$k$-可扩展性，定义了在单向LOCC超信道下单调递减的“信道不可扩展纠缠”度量，为资源提纯和零误差通信量提供上界，并提出半正定程序进行计算评估。
Graph theoretic quantum contextuality and unextendible Product Bases
本文通过图论方法建立了量子情境性与不可扩展乘积基（UPB）之间的联系，展示了KCBS向量与Pyramid UPB的等价性，并构造了新的最小UPB类，探讨了其在非情境性不等式中的应用。
We consider the mixed three-qubit bound entangled state defined as the normalized projector on the subspace that is complementary to an Unextendible Product Basis [C. H. Bennett et. al., Phys. Rev. Lett. 82, 5385 (1999)]. Using the fact that no product state lies in the support of that state, we compute its entanglement by providing a basis of its subspace formed by "minimally-entangled" states. The approach is in principle applicable to any entanglement measure; here we provide explicit values for both the geometric measure of entanglement and a generalized concurrence.
Genuinely entangled subspaces beyond strongly nonlocal unextendible biseparable bases
本文提出一个充分条件，用于构造高维真正纠缠子空间，发现最大已知的GES，并证明所有状态在任意双分区下1可蒸馏，同时证明UBB在LOCC下不可区分。
Assuming the validity of a conjecture in quant-ph/9910026 and quant-ph/9910022 we show that the distillable entanglement for two bipartite states, each of which individually has zero distillable entanglement, can be nonzero. We show that this also implies that the distillable entanglement is not a convex function. Our example consists of the tensor product of a bound entangled state based on an unextendible product basis with a Werner state which lies in the class of conjectured undistillable states.
In this paper, an optimal scheme of four-level quantum teleportation and swapping of quantum entanglement is given. We construct a complete orthogonal basis of the bipartite ququadrit systems. Using this basis, the four-level quantum teleportation and swapping can be achieved according to the standard steps. In addition, associate the above bases with the unextendible product bases and the exact entanglement bases, we prove that in the $2\times 2\times 2$ systems or $3\times 3$ systems the collective translations of multipartite quantum entanglement can be realized. PACC numbers: 03.67.Mn, 03.…
Automated Construction and Verification of Unextendible Product Bases
该研究提出基于超立方体分解与SAT的框架，构造并验证了多体系统中多种大小的不可扩展乘积基，其小维实例可用于递归构造更大系统的相关基。
Useful variants and perturbations of completely entangled subspaces and spans of unextendible product bases
本文研究完全纠缠子空间与不可扩展乘积基的变体和扰动，通过线性张成构造新空间，并给出具有无穷多个纯乘积态的示例，发展了相关变分与扰动理论。
Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$
针对多体系统中无纠缠强非定域正交集是否存在的问题，本文利用$N$维超立方体分解，为所有奇数$N\geq 3$构造强非定域正交集和不可扩展乘积基，并应用于量子秘密共享等场景。
Construction of three-qubit positive-partial-transpose entangled states of rank four
本文研究了三量子比特系统中秩为四的正部分转置纠缠态，通过洛伦兹不变量将此类态分为两种类型，并提供了利用不可扩展产基构造正部分转置纠缠态的方法，同时分析了其在局部操作和经典通信下的等价关系。
We investigate the teleportation of an entangled two-qubit state using three-qubit GHZ and W channels. The effects of white noise on the average teleportation fidelity and amount of entanglement transmitted are also studied.
In this Brief Report we show the relation between three-qubit entanglement invariants and two-qubit concurrence with the help of projective measurements. How to use these invariants to represent the entanglement property of three-qubit pure states is also investigated.
It is shown that the entanglement-structure of 3- and 4-qubit states can be characterized by optimized operators of the Mermin-Klyshko type. It is possible to discriminate between pure 2-qubit entanglements and higher entanglements. A comparison with a global entanglement measure and the i-concurrence is made.
We study the entanglement creation between two flux qubits interacting with electromagnetic field modes. No direct interaction between the qubits exists. Entanglement is reached using entanglement swapping method by an interference measurement performed on photons. We discuss the influence of off-resonance and multi-photon initial states on the qubit-qubit entanglement. The presented scheme is able to drive an initially separable state of two qubits into an highly entangled state suitable for quantum information processing.
Entanglement of four qubit pure states defined by the $η$-trigonometric functions is studied. We analyze the behavior of two recently proposed symmetric entanglement monotones on the chosen $n=4$ qubit states .
The concurrence of two alternate qubits in a four-qubit Heisenberg XX chain is investigated when a uniform magnetic field B is included. It is found that there is no thermal entanglement between alternate qubits if B is close to zero. Magnetic field can induce entanglement in a certain range both for the antiferromagnetic and ferromagnetic cases. Near zero temperature, the entanglement undergoes two sudden changes with increasing value of the magnetic field B. This is due to the changes in the ground state. This novel property may be used as quantum entanglement switch. The anisotropy in the s…
Single-photon triggered quantum entanglement between two qubits or at least 2000 identical qubits
该研究探究单光子光场对量子纠缠的调控作用，分析了激发态权重、初始相干性的影响，发现单光子可在至少2000个全同量子比特的系统中触发任意两比特间纠缠，且纠缠程度与比特数无关。
A generic algorithm is developed to reduce the problem of obtaining linear and nonlinear entanglement witnesses of a given quantum system, to convex optimization problem. This approach is completely general and can be applied for the entanglement detection of any N-partite quantum system. For this purpose, a map from convex space of separable density matrices to a convex region called feasible region is defined, where by using exact convex optimization method, the linear entanglement witnesses can be obtained from polygonal shape feasible regions, while for curved shape feasible regions, envel…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。