We derive a single general Bell inequality which is a necessary and sufficient condition for the correlation function for N particles to be describable in a local and realistic picture, for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables. We also derive a necessary and sufficient condition for an arbitrary N-qubit mixed state to violate this inequality. This condition is a generalization and reformulation of the Horodeccy family condition for two qubits.
First, we present a Bell type inequality for n qubits, assuming that m out of the n qubits are independent. Quantum mechanics violates this inequality by a ratio that increases exponentially with m. Hence an experiment on n qubits violating of this inequality sets a lower bound on the number m of entangled qubits. Next, we propose a definition of maximally entangled states of n qubits. For this purpose we study 5 different criteria. Four of these criteria are found compatible. For any number n of qubits, they determine an orthogonal basis consisting of maximally entangled states generalizing t…
It is shown that even if the linear entropy of mixed two-qubit state is not smaller then 0.457, Bell - CHSH inequalities can be violated. This contradicts the result obtained in the paper of E. Santos [1].
We develop a systematic approach to establish Bell inequalities for qubits based on the Cauchy-Schwarz inequality. We also use the concept of distinct "roots" of Bell function to classify some well-known Bell inequalities for qubits. As applications of the approach, we present three new and tight Bell inequalities for four and three qubits, respectively.
The consistency test and error estimation for the data concerning recently reported violation of Bell inequality for Josephson phase qubits are presented in details. It is pointed out that the deviation of the Bell signal from the classical limit might still be challenged.
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.
There are many different definitions of what a Bell-Kochen-Specker proof with POVMs might be. Here we present and discuss the minimal proof on qubits for three of these definitions and show that they are indeed minimal.
Maximally entangled states should maximally violate the Bell inequality. In this paper, it is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local unitary transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of …
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.
We compare the polynomial invariants for four qubits introduced by Luque and Thibon, PRA {\bf 67}, 042303 (2003), with optimized Bell inequalities and a combination of two qubit concurrences. It is shown for various parameter dependent states from different SLOCC classes that it is possible to measure a genuine 4-qubit entanglement with these polynomials.
A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. Using the qubit portrait method the Bell inequalities for two qubits and two qutrits are discussed in framework of probability representation of quantum mechanics. Semigroup of stochastic matrices is associated with tomographic probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as ansatz to provide a necessary condition f…
It is pointed out that a loophole exists in experimental tests of Bell inequality using atomic qubits, due to possible errors in the rotation angles of the atomic states. A sufficient condition is derived for closing the loophole.
A proof of Bell's theorem without inequalities valid for both inequivalent classes of three-qubit entangled states under local operations assisted by classical communication, namely Greenberger-Horne-Zeilinger (GHZ) and W, is described. This proof leads to a Bell inequality that allows more conclusive tests of Bell's theorem for three-qubit systems. Another Bell inequality involving both tri- and bipartite correlations is introduced which illustrates the different violations of local realism exhibited by the GHZ and W states.
Characterizing trade-offs between simultaneous violations of multiple Bell inequalities in a large network of qubits is computationally demanding. We propose a graph-theoretic approach to efficiently produce Bell monogamy relations in arbitrary arrangements of qubits. All the relations obtained for bipartite Bell inequalities are tight and leverage only a single Bell monogamy relation. This feature is unique to bipartite Bell inequalities, as we show that there is no finite set of such elementary monogamy relations for multipartite inequalities. Nevertheless, many tight monogamy relations for …
Braid theories are applied to quantum computation processes, where to each crossing in the Braid diagram a unitary Yang-Baxter operator R is associated, representing either a Braiding matrix or a universal quantum gate. By operating with Braid operators on the computational basis of n-QUBITS states, orthonormal entangled states are obtained, referred here as general Bell states. The 3-QUBITS Bell states are explicitly developed and the present methods are generalized to any n-QUBITS system. The quantum properties of the general Bell states are analyzed and these properties are related to concu…
In this note, we discuss a closed-form necessary and sufficient condition for any two-qubit state to show hidden nonlocality w.r.t the Bell-CHSH inequality. This is then used to numerically compute the relative volume of states showing hidden Bell-CHSH non-locality , among all two-qubit states with one-sided reduction maximally mixed.
A quantum protocol for sharing an arbitrary two-qubit state between N parties is introduced. Any of the members, can retrieve the state, only with collaboration of the other parties. We will show that in terms of resources, i.e. the number of classical bits, the number of Bell pairs shared, and also the type of measurements, our protocol is more efficient. For achieving this, we introduce the basic technique of secure passing of an unknown two qubit state among a sequence of parties, none of which can retrieve the state without authorization of the sender and the other members of the group.
We present a set of Bell inequalities for multiqubit quantum systems. These Bell inequalities are shown to be able to detect multiqubit entanglement better than previous Bell inequalities such as Werner-Wolf-Zukowski- Brukner ones. Computable formulas are presented for calculating the maximal violations of these Bell inequalities for any multiqubit states.
A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin.
In this paper, we define the return operator, the cross product operator and the cross Bell bases. Using the cross Bell bases, we give a protocol of teleportation of arbitrary multipartite qubit entanglement, this scheme is a quite natural generalization of the BBCJPW scheme. We find that this teleportation, in fact, is essentially determined by the teleportation of every single unknown qubit state as in the original scheme of BBCJPW. The calculation in detail is given for the case of tripartite qubit.
A systematic approach is presented to construct non-homogeneous two- and three-qubit Bell-type inequalities. When projector-like terms are subtracted from homogeneous two-qubit CHSH polynomial, non-homogeneous inequalities are attained and the maximal quantum mechanical violation asymptotically equals a constant with the subtracted terms becoming sufficiently large. In the case of three-qubit system, it is found that most significant three-qubit inequalities presented in literature can be recovered in our framework. We aslo discuss the behavior of such inequalities in the loophole-free Bell te…
We investigate the Bell inequalities derived from the graph states with violations detectable even with the presence of noises, which generalizes the idea of error-correcting Bell inequalities [Phys. Rev. Lett. 101, 080501 (2008)]. Firstly we construct a family of valid Bell inequalities tolerating arbitrary $t$-qubit errors involving $3(t+1)$ qubits, e.g., 6 qubits suffice to tolerate single qubit errors. Secondly we construct also a single-error-tolerating Bell inequality with a violation that increases exponentially with the number of qubits. Exhaustive computer search for optimal error-tol…
Bell inequalities for number measurements are derived via the observation that the bits of the number indexing a number state are proper qubits. Violations of these inequalities are obtained from the output state of the nondegenerate optical parametric amplifier.
We study the violations of Bell inequality for thermal states of qubits in a multi-qubit Heisenberg model as a function of temperature and external magnetic fields. Unlike the behaviors of the entanglement the violation can not be obtained by increasing the temperature or the magnetic field. The threshold temperatures of the violation are found be less than that of the entanglement. We also consider a realistic cavity-QED model which is a special case of the mutli-qubit Heisenberg model.
In the case of bipartite two qubits systems, we derive the analytical expression of bound of Bell operator for any given pure state. Our result not only manifest some properties of Bell inequality, for example which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also give the explicit values of maximal violation for any pure state. Finally we point out that for two qubits systems there is no mixed state which can produce maximal violation of Bell inequality.
A protocol is proposed to generate Bell states in two non-directly interacting qubits by means of repeated measurements of the state of a central ancilla connected to both qubits. An optimal measurement rate is found that minimizes the time to stably encode a Bell state in the target qubits, being of advantage in order to reduce detrimental effects from possible interactions with the environment. The quality of the entanglement is assessed in terms of the concurrence and the distance between the qubits state and the target Bell state is quantified by the fidelity.
If we distribute n qubits between two parties, which quantum pure states and distributions of qubits would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell's theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs for any number of qubits, and provide all distinct proofs up to n=7 qubits. Remarkably, there is only one distribution of a state of n=4 qubits, and six distributions, each for a different state of n=6 qubits, which allow these proofs.
We experimentally demonstrate a high-efficiency Bell state measurement for time-bin qubits that employs two superconducting nanowire single-photon detectors with short dead-times, allowing projections onto two Bell states, |Psi>- and |Psi+>. Compared to previous implementations for time-bin qubits, this yields an increase in the efficiency of Bell state analysis by a factor of thirty.
We report on the experimental realization of two different Bell inequality tests based on six-qubit linear-type and Y-shape graph states. For each of these states, the Bell inequalities tested are optimal in the sense that they provide the maximum violation among all Bell inequalities with stabilizing observables and possess the maximum resistance to noise.
We introduce a set of Bell inequalities for a three-qubit system. Each inequality within this set is violated by all generalized GHZ states. More entangled a generalized GHZ state is, more will be the violation. This establishes a relation between nonlocality and entanglement for this class of states. Certain inequalities within this set are violated by pure biseparable states. We also provide numerical evidence that at least one of these Bell inequalities is violated by a pure genuinely entangled state. These Bell inequalities can distinguish between separable, biseparable and genuinely entan…
The spectral decomposition is given for the N-qubit Bell operators with two observables per qubit. It is found that the eigenstates (when non-degenerate) are N-qubit GHZ states even for those operators that do not allow the maximal violation of the corresponding inequality. We present two applications of this analysis. In particular, we discuss the existence of pure entangled states that do not violate any Mermin-Klyshko inequality for $N\geq 3$.
By calculating entanglement measures and quantum violation of Bell-type inequality, we reveal the relationship between entanglement measure and the amount of quantum violation for a family of four-qubit entangled states. It has been demonstrated that the Bell-type inequality is completely violated by these four-qubit entangled states. The plot of entanglement measure as a function of the expectation value of Bell operator shows that entanglement measure first decreases and then increases smoothly with increasing quantum violation.
We have studied the analytical dynamics of Bell nonlocality as measured by CHSH inequality and entanglement as measured by concurrence for two noisy qubits that have dipole-dipole interaction. The nonlocal entanglement created by the dipole-dipole interaction is found to be protected from sudden death for certain initial states.
We present a method to derive explicit forms of tight correlation function Bell inequalities for three systems and dichotomic observables, which involve three settings for each observer. We also give sufficient conditions for quantum predictions to satisfy the new inequalities.
We consider quantum systems composed of $N$ qubits, and the family of all Bell's correlation inequalities for two two-valued measurements per site. We show that if a $N$-qubit state $ρ$ violates any of these inequalities, then it is at least bipartite distillable. Indeed there exists a link between the amount of Bell's inequality violation and the degree of distillability. Thus, we strengthen the interpretation of Bell's inequalities as detectors of useful entanglement.
For a class of mixed two -qubit states we show that it is not possible to discriminate between states violating or non - violating Bell - CHSH inequalities, knowing only their entanglement and mixedness. For a large set of possible values of these quantities, we construct pairs of states with the same entanglement and mixedness such that one state is violating but the other is non - violating Bell - CHSH inequality.
First, I show explicitly a scheme to {\it faithfully} and {\it deterministically} teleport an arbitrary 2-qubit state from Alice to Bob. In this scheme two same Bell states are sufficient for use. Bob can recover the 2-qubit state by performing at most 4 single-qubit unitary operations conditioned on Alice's 4-bit classical public message corresponding to her two Bell-state measurement outcomes. Then I generalize the 2-qubit teleportation scheme to an aribitrary $N(N\ge 3)$-qubit state teleportation case by using $N$ same Bell states. In the generalized scheme, Alice only needs to identify $N$…
The generation of Bell states of distant objects is of importance for constructing quantum networks. Previous studies have revealed that transient or intermittent Bell states can be generated between remote qubits by exploiting time-continuous measurement on the environments of the systems, e.g., photon counting or homodyne detection. In this paper, we consider a new scheme for achieving long-time sustainable Bell states of two distant qubits mediated by a one-dimension waveguide via continuous photon counting and homodyne detection. In both of cases, different Bell states can be present for d…
Numerous quantum information protocols make use of maximally entangled two-particle states, or Bell states, in which information is stored in the correlations between the two particles rather than their individual properties. Retrieving information stored in this way means distinguishing between different Bell states, yet the well known no-go theorem establishes that projective linear evolution and local measurement (LELM) detection schemes can only reliably distinguish three of the four qubit Bell states. We establish maximum distinguishability of the qutrit Bell states of bosons via projecti…
In this paper, we propose a protocol for complete Bell-state analysis for two superconducting-quantum-interference-device qubits. The Bell-state analysis could be completed by using a sequence of microwave pulses designed by the transition- less tracking algorithm, which is an useful method in the technique of shortcut to adiabaticity. After the whole process, the information for distinguishing four Bell states will be encoded on two auxiliary qubits, while the Bell states keep unchanged. One can read out the information by detecting the auxiliary qubits. Thus the Bell-state analysis is nondes…
An efficient and economical scheme is proposed for the perfect quantum teleportation of n-qubit non-maximally entangled state of generalized Bell-type. A Bell state is used as the quantum channel in the proposed scheme. It is also shown that the controlled teleportation of this n-qubit state can be achieved by using a GHZ state or a GHZ-like state as quantum channel. The proposed schemes are economical because for the perfect and controlled teleportation of n-qubit non-maximally entangled state of generalized Bell-type we only need a Bell state and a tripartite entangled state respectively. It…
We introduce a class of two-level multi-particle Greenberger-Horne-Zeilinger (GHZ) states, and study entanglement swapping between two systems for Bell states and the class of GHZ states in qubit systems, respectively. We give the formulas for the entanglement swapping of Bell states and GHZ states in any number of qubit systems. We further consider entanglement swapping between any number of Bell states and between any number of the introduced GHZ states, and propose a series of entanglement swapping schemes in a detailed way. We illustrate the applications of such schemes in quantum informat…
We introduce a general odd qubit entangled system composed of GHZ and Bell pairs and explicate its usefulness for quantum teleportation, information splitting and superdense coding. After demonstrating the superdense coding protocol on the five qubit system, we prove that '2N +1' classical bits can be sent by sending 'N+1' quantum bits using this channel. It is found that the five-qubit system is also ideal for arbitrary one qubit and two qubit teleportation and quantum information splitting (QIS). For the single qubit QIS, three different protocols are feasible, whereas for the two qubit QIS,…
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.
Entanglement and Bell nonlocality are used to describe quantum inseparabilities. Bell-nonlocal states form a strict subset of entangled states. A natural question arises concerning how much territory Bell nonlocality occupies entanglement for a general two-qubit entangled state. In this work, we investigate the relation between entanglement and Bell nonlocality by using lots of randomly generated two-qubit states, and give out a constraint inequality relation between the two quantum resources. For studying the upper or lower boundary of the inequality relation, we discover maximally (minimally…
We present a family of Bell inequalities involving only two measurement settings of each party for N>2 qubits. Our inequalities include all the standard ones with fewer than N qubits and thus gives a natural generalization. It is shown that all the Greenberger-Horne-Zeilinger states violate the inequalities maximally, with an amount that grows exponentially as 2^{{(N-2)}/2}. The inequalities are also violated by some states that do satisfy all the standard Bell inequalities. Remarkably, our results yield in an efficient and simple way an implementation of nonlocality tests of many qubits favor…
Decoherence is one of the main obstacles in long-distance quantum communication. Recently, the theoretical work of Fröwis and W. Dür (Phys. Rev. Lett. \textbf{106}, 110402 (2011)) and the experiment of Lu \emph{et al.} (Nat. Photon. \textbf{8}, 364 (2014)) both showed that the logic qubits entanglement say the concatenated Greenberger-Horne-Zeilinger (C-GHZ) state is more robust under decoherence. In this paper, we describe a protocol for Bell-state analysis for this logic qubits entanglement. This protocol can also be extended to the multipartite C-GHZ state analysis. Also, we discuss its app…
We present a Bell-type inequality for four-qubit systems. Using the inequality we investigate quantum nonlocality of a generic family of states $\left|G_{abcd}\right >$ [Phys. Rev. A 65, 052112 (2002)] and several canonical four-qubit entangled states. It has been demonstrated that the inequality is maximally violated by so called "four-qubit maximal entangled state $\left|G_{m}\right >$" and it is also violated by the four-qubit W state and a special family of states $\left|G_{ab00}\right >$. Moreover, a useful entanglement-nonlocality relationship for the family of states $\left|G_{ab00}\rig…
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the com…
We study the problem of closing the detection loophole in three-qubit Bell tests, the experimentally most relevant case beyond the usual bipartite scenario, and show that the minimal detection efficiencies required can be considerably lowered compared to the two-qubit case. The lowest reported detection efficiency thresholds for two and three qubits so far are $\sim66.7\%$ and $60\%$, respectively. Using the three-qubit W state and a 3-setting Bell inequality, we beat these thresholds and with an 8-setting Bell inequality we reach $50.13\%$. We also investigate generic three-qubit states which…
Generation and read-out of many-body Bell correlations with a probe qubit
Qilimanjaro Quantum Tech（奇力马扎罗量子技术）；ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学技术研究院）；Faculty of Physics, University of Warsaw（华沙大学物理学院）
Sufficient conditions for (the non-violation of) the Bell-CHSH inequalities in a mixed state of a two-qubit system are: 1) The linear entropy of the state is not smaller than 0.5, 2) The sum of the conditional linear entropies is non-negative, 3) The von Neumann entropy is not smaller than 0.833, 4) The sum of the conditional von Neumann entropies is not smaller than 0.280.
We observe violation of a Bell inequality between the quantum states of two remote Yb ions separated by a distance of about one meter with the detection loophole closed. The heralded entanglement of two ions is established via interference and joint detection of two emitted photons, whose polarization is entangled with each ion. The entanglement of remote qubits is also characterized by full quantum state tomography.
In this study we investigate the spin squeezing in superposition of a Biaxial state [1,2] and Bell state. Numerical and analytical solutions for the length of mean spin, mean spin direction and spin squeezing are given. It is shown that both the mean spin direction and spin squeezing parameter are determined by the coefficients of superposition and the relative phase.
We find a single parameter family of genuinely entangled three qubit pure states, called the maximally Bell inequality violating states (MBV), which exhibit maximum Bell inequality violation by the reduced bipartite system for a fixed amount of genuine tripartite entanglement quantified by the so called tangle measure. This in turn implies that there holds a complementary relation between the Bell inequality violation by the reduced bipartite systems and the tangle present in the three qubit states, not necessarily pure. The MBV states also exhibit maximum Bell inequality violation by the redu…
We propose a new scheme in which perfect transmission of 1-qubit information is achieved with high success using purposefully delayed sharing of non-maximally entangled 2-qubit resource and repeated generalized Bell-state measurements (GBSM). Alice possesses initially all qubits and she makes repeated GBSM on the pair of qubits, consisting of (1) the qubit of information state and (2) one of the two entangled resource qubits (taken alternately) until transmission with perfect fidelity is indicated. Alice then sends to Bob, the qubit not used in the last GBSM and also the result of this GBSM an…
We develop a scheme for generating a universal qubit cluster state using probabilistic Bell measurements without the need for feed-forward or long-time quantum memories. Borrowing ideas from percolation theory we numerically show that using unambiguous Bell measurements that succeed with 75% success probability, one could build a cluster state with an underlying pyrochlore geometry such that the probability of having a spanning cluster in a chosen direction approaches unity in the limit of an infinite lattice size. The initial resources required for the generation of a universal state in our p…
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.
