One of the many bizarre features of entanglement is that Alice, by sending a qubit to Bob in a separable state, can generate some entanglement between herself and Bob. This protocol is stripped down to the bare essentials to better elucidate the key properties of the initial resource state that enable this entanglement distribution. The necessary and sufficient conditions under which the correlations of a Bell-diagonal state serve as a useful resource are proven, giving upper and lower bounds on the entanglement that can be distributed when those conditions are met.
We present a set of Bell inequalities that gives rise to a finer classification of the entanglement for tripartite systems. These inequalities distinguish three possible bi-separable entanglements for three-qubit states. The three Bell operators we employed constitute an external sphere of the separable cube.
A set of Bell inequalities classifying the quantum entanglement of four-qubit states is presented. These inequalities involve only two measurement settings per observer and can characterize fully separable, bi-separable and tri-separable quantum states. In addition, a quadratic inequality of the Bell operators for four-qubit systems is derived.
We propose a new type of attack on Ekert protocol in which, rather surprisingly, Eve drives a violation of Bell inequalities in Alice's and Bob's detectors with a mixture of separable states. She does so by sending correlated pulses containing several photons at frequencies where only multiple-photon absorptions are possible in their detectors. Whenever the photons stemming from a same pulse are dispatched in such a way that the number of photons is insufficient to trigger a multiple-photon absorption in either channel, the pulse remains undetected. We show that this simple feature leads to vi…
Chen (quant-ph/0611126) has recently claimed ``exponential violation of local realism by separable states", in the sense that multi-partite separable quantum states are supposed to give rise to correlations and fluctuations that violate a Bell-type inequality that Chen takes to be satisfied by local realism. However, this can not be true since all predictions (including all correlations and fluctuations) that separable quantum states give rise to have a local realistic description and thus satisfy all Bell-type inequalities, and this holds for all number of parties. Since Chen claims otherwise…
By introducing a quantitative `degree of commutativity' in terms of the angle between spin-observables we present two tight quantitative trade-off relations in the case of two qubits: First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to π/2 (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach π/2 the maximal …
We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension $d \geq 3$, a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with spec…
In analogy with Bell's inequality for two-qubit quantum states we propose an inequality criterion for the non-separability of the spin-orbit degrees of freedom of a classical laser beam. A definition of separable and non-separable spin-orbit modes is used in consonance with the one presented in Phys. Rev. Lett. 99, 160401 (2007). As the usual Bell's inequality can be violated for entangled two-qubit quantum states, we show both theoretically and experimentally that the proposed spin-orbit inequality criterion can be violated for non-separable modes. A discussion on the classical-quantum transi…
With a probability of success of $95 \%$ we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be sepa…
We provide an analytical expression for optimal Lewenstein-Sanpera decomposition of Bell decomposable states by using semi-definite programming. Also using the Karush-Kuhn-Tucker optimization method, the minimum relative entropy of entanglement of Bell decomposable states has been evaluated and it is shown that the same separable Bell decomposable state lying at the boundary of convex set of separable Bell decomposable states, optimizes both Lewenstein-Sanpera decomposition and relative entropy of entanglement. {\bf Keywords: Minimum relative entropy of entanglement, Semi-definite programming,…
The statement of E.R. Loubenets, Phys. Rev. A 69, 042102 (2004), that separable states can violate classical probabilistic constraints is based on a misleading definition of classicality, which is much narrower than Bell's concept of local hidden variables. In a Bell type setting the notion of classicality used by Loubenets corresponds to the assumption of perfect correlations if the same observable is measured on both sides. While it is obvious that most separable states do not satisfy this assumption, this does not constitute "non-classical" behaviour in any usual sense of the word.
The present methods for obtaining the optimal Lewenestein- Sanpera decomposition of a mixed state are difficult to handle analytically. We provide a simple analytical expression for the optimal Lewenstein-Sanpera decomposition by using semidefinite programming. Specially, we obtain the optimal Lewenstein-Sanpera decomposition for some examples such as: Bell decomposable state, Iso-concurrence state, generic two qubit state in Wootters's basis, $2\otimes 3$ Bell decomposable state, $d\otimes d$ Werner and isotropic states, a one parameter $3\otimes 3$ state and finally multi partite isotropic s…
One of the fundamental concepts of quantum information theory is that of entanglement purification; that is, the transformation of a partially entangled state into a smaller-dimensional, more completely entangled state. Of particular interest are protocols for entanglement purification (EPPs) that alternate purely local operations with one- or two-way classical communication. In the present work, we consider a more general, but simpler, class of transformations, called separable superoperators. Since every EPP is a separable superoperator, bounds on separable superoperators apply as well to EP…
In this study, using the concept of relative entropy as a distance measure of cor- relations we investigate the important issue of evaluating quantum correlations such as entanglement, dissonance and classical correlations for 2n-dimensional Bell-diagonal states. We provide an analytical technique, which describes how we find the closest classical states(CCS) and the closest separable states(CSS) for these states. Then analytical results are obtained for quantum discord of 2n-dimensional Bell-diagonal states. As illustration, some special cases are examined. Finally, we investigate the additiv…
In contrast to the wide-spread opinion that any separable quantum state satisfies every classical probabilistic constraint, we present a simple example where a separable quantum state does not satisfy the original Bell inequality although the latter inequality, in its perfect correlation form, is valid for all joint classical measurements. In a very general setting, we discuss inequalities for joint experiments upon a bipartite quantum system in a separable state. We derive quantum analogues of the original Bell inequality and specify the conditions sufficient for a separable state to satisfy …
It was shown that two distant particles can be entangled by sending a third particle never entangled with the other two [T. S. Cubitt et al., Phys. Rev. Lett. 91, 037902 (2003)]. In this paper, we investigate a class of three-qubit separable states to distribute entanglement by the same way, and calculate the maximal amount of entanglement which two particles of separable states in the class can have after applying the way.
It is known from Bell's theorem that quantum predictions for some entangled states cannot be mimicked using local hidden variable (LHV) models. From a computer science perspective, LHV models may be interpreted as classical computers operating on a potentially infinite number of correlated bits originating from a common source. As such, Bell inequality violations achieved through entangled states are able to characterise the quantum advantage of certain tasks, so long as the task itself imposes no restriction on the availability of correlated bits. However, if the number of shared bits is limi…
We show that no entanglement is necessary to distribute entanglement; that is, two distant particles can be entangled by sending a third particle that is never entangled with the other two. Similarly, two particles can become entangled by continuous interaction with a highly mixed mediating particle that never itself becomes entangled. We also consider analogous properties of completely positive maps, in which the composition of two separable maps can create entanglement.
We strengthen the bound on the correlations of two spin-1/2 particles (qubits) in separable (non-entangled) states for locally orthogonal spin directions by much tighter bounds than the well-known Bell inequality. This provides a sharper criterion for the experimental distinction between entangled and separable states, and even one which is a necessary and sufficient condition for separability. However, these improved bounds do not apply to local hidden-variable theories, and hence they provide a criterion to test the correlations allowed by local hidden-variable theories against those allowed…
We show how nonclassical correlations in local bipartite states can act as a resource for quantum information processing. Considering the task of quantum random access codes (RAC) through separable Bell-diagonal states, we demonstrate the advantage of superunsteerability over classical protocols assisted with two-bits of shared randomness. We propose a measure of superunsteerability, which quantifies nonclassicality beyond quantum steering, and obtain its analytical expression for Bell-diagonal states in the context of the two- and three-setting steering scenarios that are directly related to …
We generate the non-separable state of polarization and orbital angular momentum (OAM) using a laser beam. The generated state undergoes a cyclic polarization evolution which introduces a Pancharatnam geometric phase to the polarization state and in turn a relative phase in the non-separable state. We experimentally study the violation of Bell - CHSH inequality for different Pancharatnam phases introduced by various cyclic polarization evolutions with linear and circular states as measurement bases. While measuring in linear bases, the Bell-CHSH parameter oscillates with Pancharatnam phase. On…
Quantum nonlocal correlations of anomalous weak values
Faculty of Engineering and Institute of Nanotechnology and Advanced Materials, Bar-Ilan University（巴伊兰大学工程学院与纳米技术和先进材料研究所）；Institute for Quantum Studies, Chapman University（查普曼大学量子研究所）；Iyar, The Israeli Institute for Advanced Research（以色列高级研究院）
We propose a simple geometrical approach for finding the robustness of entanglement for Bell decomposable states of 2 otimes 2 quantum systems. It is shown that the robustness of entanglement is equal to the concurrence. We also present an analytical expression for two separable states that wipe out all entanglement of these states. Finally the random robustness of these states is also obtained.
We discuss here the best disentanglement processes of states of two two-level systems which belong to (i) the universal set, (ii) the set in which the states of one party lie on a single great circle of the Bloch sphere, and (iii) the set in which the states of one party commute with each other, by teleporting the states of one party (on which the disentangling machine is acting) through three particular type of separable channels, each of which is a mixture of Bell states. In the general scenario, by teleporting one party's state of an arbitrary entangled state of two two-level parties throug…
We study the relation between violation of Bell inequalities and distillability properties of quantum states. Recently, Dür has shown that there are some multiparticle bound entangled states, non-separable and non-distillable, that violate a Bell inequality. We prove that for all the states violating this inequality there exist at least one splitting of the parties into two groups such that some pure-state entanglement can be distilled, obtaining a connection between Bell inequalities and bipartite distillable entanglement.
It is shown that the outcomes of measurements on systems in separable mixed states can be partitioned, via subsequent measurements on a disentangled extraneous system, into subensembles that display the statistics of entangled states. This motivates the introduction of the concept of "counterfactual" entanglement, which can be associated with all separable mixed states including those that are factorable. This type of entanglement gives rise to a new kind of postselection-induced Bell inequality violation. The significance of counterfactual entanglement, and its physical implications, are asse…
We investigate the internal logic of a quantum computer with two qubits, in the two particular cases of non-entanglement (separable states) and maximal entanglement (Bell's states). To this aim, we consider an internal (reversible) measurement which preserves the probabilities by mirroring the states. We then obtain logical judgements for both cases of separable and Bell's states.
An analogous model system for high-dimensional quantum entanglement is proposed, based on the angular and radial degrees of freedom of the improved Laguerre Gaussian mode. Experimentally, we observed strong violations of the Bell-CGLMP inequality for maximally non-separable states of dimension 2 through 10. The results for violations in classical non-separable state are in very good agreement with quantum instance, which illustrates that our scheme can be a useful platform to simulate high-dimensional non-local entanglement. Additionally, we found that the Bell measurements provide sufficient …
All the states of N qubits can be classified into N-1 entanglement classes from 2-entangled to N-entangled (fully entangled) states. Each class of entangled states is characterized by an entanglement index that depends on the partition of N. The larger the entanglement index of an state, the more entangled or the less separable is the state in the sense that a larger maximal violation of Bell's inequality is attainable for this class of state.
In general the calculation of robustness of entanglement for the mixed entangled quantum states is rather difficult to handle analytically. Using the the convex semi-definite programming method, the robustness of entanglement of some mixed entangled quantum states such as: $2\otimes 2$ Bell decomposable (BD) states, a generic two qubit state in Wootters basis, iso-concurrence decomposable states, $2\otimes 3$ Bell decomposable states, $d\otimes d$ Werner and isotropic states, a one parameter $3\otimes 3$ state and finally multi partite isotropic state, is calculated exactly, where thus obtaine…
In this paper we classify the four-qubit states that commute with $U\otimes{U}\otimes{V}\otimes{V}$, where $U$ and $V$ are arbitrary members of the Pauli group. We characterize the set of separable states for this class, in terms of a finite number of entanglement witnesses. Equivalently, we characterize the two-qubit, Bell-diagonal-preserving, completely positive maps that are separable. These separable completely positive maps correspond to protocols that can be implemented with stochastic local operations assisted by classical communication (SLOCC). This allows us to derive a complete set o…
We investigate the physical mechanism behind the counterintuitive phenomenon, the distribution of continuous-variable entanglement between two distant modes by sending a third separable auxiliary mode between them. For this purpose, we propose a new more simple and more efficient protocol resulting in distributed entanglement with more than an order of the magnitude higher logarithmic negativity than in the previously proposed protocol. This new protocol shows that the distributed entanglement originates from the entanglement of one mode and the auxiliary mode used for distribution which is fi…
Doubts regarding the completeness of quantum mechanics as raised by Einstein, Podolsky and Rosen(EPR) have predominantly been resolved by resorting to a measurement of correlations between entangled photons which clearly demonstrate violation of Bell's inequality. This article is an attempt to reconcile incompatibility of hidden variable theories with reality by demonstrating experimentally a violation of Bell's inequality in locally correlated systems whose two degrees of freedom, the spin and orbital angular momentum, are maximally correlated. To this end we propose and demonstrate a linear,…
Entanglement boosting: Low-volume logical Bell pair preparation for distributed fault-tolerant quantum computation
Nanofiber Quantum Technologies, Inc. (NanoQT)（纳米纤维量子技术有限公司）；Clarendon Laboratory, University of Oxford（牛津大学克拉伦登实验室）；Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo（东京大学大学院信息理工学研究系计算机科学专业）
Resource-adaptive distributed fault tolerance with very noisy Bell pairs
University of Bremen（不来梅大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；University of Oxford（牛津大学）；German Research Center for Artificial Intelligence GmbH（德国人工智能研究中心有限公司）
In a general setting, we introduce a new bipartite state property sufficient for the validity of the perfect correlation form of the original Bell inequality for any three bounded quantum observables. A bipartite quantum state with this property does not necessarily exhibit perfect correlations. The class of bipartite states specified by this property includes both separable and nonseparable states. We prove analytically that, for any dimension d>2, every Werner state, separable or nonseparable, belongs to this class.
We demonstrate entanglement generation of two neutral atoms trapped inside an optical cavity. Entanglement is created from initially separable two-atom states through carving with weak photon pulses reflected from the cavity. A polarization rotation of the photons heralds the entanglement. We show the successful implementation of two different protocols and the generation of all four Bell states with a maximum fidelity of (90+-2)%. The protocol works for any distance between cavity-coupled atoms, and no individual addressing is required. Our result constitutes an important step towards applica…
We study the entanglement evolution of the set of Bell diagonal states for a two-qubit system coupled to two independent vacuum noise sources. This set can be represented geometrically as the set of points inside a tetrahedron in a three-dimensional Euclidean space and contains the maximally entangled states for bipartite systems. We show that the set of entangled Bell diagonal states can be divided into two bounded subsets in this representation: states that evolve into separable states in a finite time and states that lose their entanglement asymptotically. Additionally, we find that the fin…
We derive tight quadratic inequalities for all kinds of hybrid separable-inseparable $n$-particle density operators on an arbitrary dimensional space. This methodology enables us to truly derive a tight quadratic inequality as tests for full $n$-partite entanglement in various Bell-type correlation experiments on the systems that may not be identified as a collection of qubits, e.g., those involving photons measured by incomplete detectors. It is also proved that when the two measured observables are assumed to precisely anti-commute, a stronger quadratic inequality can be used as a witness of…
Three distant labs A, B and C, having no prior entanglement can establish a shared GHZ state, when one of them say A sends two particles to B and C for their local actions. The mediating particles remain separable from each other and from the particles of A, B and C. We prove that in this way, GHZ states are shared with a probability $\frac{1}{7}$. We also show how separable particles can be mediated to establish arbitrary $d-$ dimensional Bell states between distant labs. Our method is constructive and allows generaization of GHZ sharing between any number of parties and in any dimension. The…
We study information-disturbance trade-off in generalized entanglement swapping protocols wherein starting from Bell pairs $\left(1,2\right)$ and $\left(3,4\right)$, one performs an arbitrary joint measurement on $\left(2,3\right)$, so that $\left(1,4\right)$ now becomes correlated. We obtain trade-off inequalities between information gain in correlations of $\left(1,4\right)$ and residual information in correlations of $\left(1,2\right)$ and $\left(3,4\right)$ respectively and argue that information contained in correlations (information) is conserved if each inequality is an equality. We sho…
A new pure quantum state, isotropic in spin variables, is defined in an extended spin phase space beyond standard quantum mechanics. It allows to represent the entangled singlet state in separable form. The statistical correlations between Alice and Bob measurements become self correlations between hidden spin values for each particle, together with perfect anti correlation between spin values on the pair. Alice determines through measurement on her particle the value of spin in some direction. Spin in another direction is inferred from Bob measurement on the companion. Bell's inequalities are…
We investigate noise effects on the performance of entanglement distribution by separable state. We consider a realistic situation in which the mediating particle between two distant nodes of the network goes through a noisy channel. For a large class of noise models we show that the average value of distributed entanglement between two parties is equal to entanglement between particular bipartite partitions of target qubits and exchange qubit in intermediate steps of the protocol. This result is valid for distributing two qubit/qudit and three qubit entangled states. In explicit examples of t…
Two kinds of Bell-states diagonal (BSD) entanglement witnesses (EW) are constructed by using the algebra of Dirac $γ$ matrices in the space-time of arbitrary dimension $d$, where the first kind can detect some BSD relativistic and non-relativistic $m$-partite multispinor bound entangled states in Hilbert space of dimension $2^{m\lfloor d/2\rfloor}$, including the bipartite Bell-type and iso-concurrence type states in the four-dimensional space-time ($d=4$). By using the connection between Hilbert-Schmidt measure and the optimal EWs associated with states, it is shown that as far as the spin qu…
Like a silver thread, quantum entanglement [1] runs through the foundations and breakthrough applications of quantum information theory. It cannot arise from local operations and classical communication (LOCC) and therefore represents a more intimate relationship among physical systems than we may encounter in the classical world. The `nonlocal' character of entanglement manifests itself through a number of counterintuitive phenomena encompassing Einstein-Podolsky-Rosen paradox [2,3], steering [4], Bell nonlocality [5] or negativity of entropy [6,7]. Furthermore, it extends our abilities to pr…
Quantum correlations resulting in violations of Bell inequalities have generated a lot of interest in quantum information science and fundamental physics. In this paper, we address some questions that become relevant in Bell-type tests involving systems with local dimension greater than 2. For CHSH-Bell tests within 2-dimensional subspaces of such high-dimensional systems, it has been suggested that experimental violation of Tsirelson's bound indicates that more than 2-dimensional entanglement was present. We explain that the overstepping of Tsirelson's bound is due to violation of fair sampli…
Following the linear programming prescription of Ref. \cite{PRA72}, the $d\otimes d$ Bell diagonal entanglement witnesses are provided. By using Jamiolkowski isomorphism, it is shown that the corresponding positive maps are the generalized qudit Choi maps. Also by manipulating particular $d\otimes d$ Bell diagonal separable states and constructing corresponding bound entangled states, it is shown that thus obtained $d\otimes d$ BDEW's (consequently qudit Choi maps) are non-decomposable in certain range of their parameters.
Classifying states as entangled or separable is a highly challenging task, while it is also one of the foundations of quantum information processing theory. This task is higly nontrivial even for relatively simple cases, such as two-qutrit Bell-diagonal states, i.e., mixture of nine mutually orthogonal maximally entangled states. In this article we apply the Gilbert algorithm to revise previously obtained results for this class. In particular we use ``cartography of entanglement'' to argue that most states left in [Hiesmayr, B. C. {\em Scientific Reports} {\bf 11}, 19739 (2021)] as unknown to …
We derive Bohm's trajectories from Bell's beables for arbitrary bipartite systems composed by dissipative noninteracting harmonic oscillators at finite temperature. As an application of our result, we calculate the Bohmian trajectories of particles described by a generalized Werner state, comparing the trajectories when the sate is either separable or entangled. We show that qualitative differences appear in the trajectories for entangled states as compared with those for separable states.
The distribution of entanglement between macroscopically separated parties represents a crucial protocol for future quantum information networks. Surprisingly, it has been theoretically shown that two distant systems can be entangled by sending a third mediating system that is not entangled with either of them. Such a possibility seems to contradict the intuition that to distribute entanglement, the transmitted system always needs to be entangled with the sender. Here, we experimentally distribute entanglement by exchanging a subsystem and successfully prove that this subsystem is not entangle…
The state of a quantum system, consisting of two distinct subsystems, is called separable if it can be prepared by two distant experimenters who receive instructions from a common source, via classical communication channels. A necessary condition is derived and is shown to be more sensitive than Bell's inequality for detecting quantum inseparability. Moreover, collective tests of Bell's inequality (namely, tests that involve several composite systems simultaneously) may sometimes lead to a violation of Bell's inequality, even if the latter is satisfied when each composite system is tested sep…
We report a nonlocal interferometer capable of detecting entanglement and identifying Bell states statistically. This is possible due to the interferometer's unique correlation dependence on the anti-diagonal elements of the density matrix, which have distinct bounds for separable states and unique values for the four Bell states. The interferometer consists of two spatially separated balanced Mach-Zehnder or Sagnac interferometers that share a polarization entangled source. Correlations between these interferometers exhibit non-local interference, while single photon interference is suppresse…
Distribution of quantum correlations among remote users is a key procedure underlying many quantum information technologies. Einstein-Podolsky-Rosen steering, which is one kind of such correlations stronger than entanglement, has been identified as a resource for secure quantum networks. We show that this resource can be established between two and even more distant parties by transmission of a system being separable from all the parties. For the case with two parties, we design a protocol allowing to distribute one-way Gaussian steering between them via a separable carrier; the obtained steer…
