Nonlinear Entanglement Witnesses Research Articles

Using linear and nonlinear entanglement witnesses to generate and detect bound entangled states on an IBM quantum processor

Abstract We investigate bound entanglement in three-qubit mixed states which are diagonal in the Greenberger-Horne-Zeilinger (GHZ) basis. Entanglement in these states is detected using entanglement witnesses and the analysis focuses on states exhibiting positive partial transpose (PPT). We then compare the detection capabilities of optimal linear and nonlinear entanglement witnesses. In theory, both linear and nonlinear witnesses produce non-negative values for separable states and negative values for some entangled GHZ diagonal states with PPT, indicating the presence of entanglement. Our experimental results reveal that in cases where linear entanglement witnesses fail to detect entanglement, nonlinear witnesses are consistently able to identify its presence. Optimal linear and nonlinear witnesses were generated on an IBM quantum computer and their performance was evaluated using two bound entangled states (Kay and Kye states) from the literature, and randomly generated entangled states in the GHZ diagonal form. Additionally, we propose a general quantum circuit for generating a three-qubit GHZ diagonal mixed state using a six-qubit pure state on the IBM quantum processor. We experimentally implemented the circuit to obtain expectation values for three-qubit mixed states and compute the corresponding entanglement witnesses.

Closing loopholes of measurement-device-independent nonlinear entanglement witnesses

Closing loopholes of measurement-device-independent nonlinear entanglement witnesses

Machine-learning-based device-independent certification of quantum networks

Witnessing nonclassical behavior is a crucial ingredient in quantum information processing. For that, one has to optimize the quantum features a given physical setup can give rise to, which is a hard computational task currently tackled with semidefinite programming, a method limited to linear objective functions and that becomes prohibitive as the complexity of the system grows. Here, we propose an alternative strategy, which exploits a feedforward artificial neural network to optimize the correlations compatible with arbitrary quantum networks. A remarkable step forward with respect to existing methods is that it deals with nonlinear optimization constraints and objective functions, being applicable to scenarios featuring independent sources and nonlinear entanglement witnesses. Furthermore, it offers a significant speedup in comparison with other approaches, thus allowing to explore previously inaccessible regimes. We also extend the use of the neural network to the experimental realm, a situation in which the statistics are unavoidably affected by imperfections, retrieving device-independent uncertainty estimates on Bell-like violations obtained with independent sources of entangled photon states. In this way, this work paves the way for the certification of quantum resources in networks of growing size and complexity.

Blindly verifying partially unknown entanglement

SummaryQuantum entanglement has shown distinguished features beyond any classical state. Many methods have been presented to verify unknown entanglement with the complete information about the density matrices by quantum state tomography. In this work, we aim to identify unknown entanglement with only partial information of the state space. The witness consists of a generalized Greenberger-Horne-Zeilinger-like paradox expressed by Pauli observables, and a nonlinear entanglement witness expressed by density matrix elements. First, we verify unknown bipartite entanglement and study the robustness of entanglement witnesses against the white noise. Second, we generalize such verification to partially unknown multipartite entangled states, including the Greenberger-Horne-Zeilinger-type and W-type states. Third, we give a quantum-information application related to the quantum zero-knowledge proof. It further provides a useful method in blindly verifying universal quantum computation resources. These results may be interesting in entanglement theories, quantum communication, and quantum networks.

Non-locality and entanglement in multi-qubit systems from a unified framework

Non-classical probability is the underlying feature of quantum mechanics. The emergence of Bell-CHSH non-locality for bipartite systems and linear entanglement inequalities for two-qubit systems has been shown in Adhikary et al. 2020 [Eur. Phys. J. D 74, 68 (2020)], purely as violations of classical probability rules. In this paper, we improve upon that work by showing that violation of any nonlocality inequality implies violation of classical probability rules, manifested through negative probabilities, without recourse to any underlying theory. Moving on to entanglement, we employ parent pseudoprojections to show how any number of linear and nonlinear entanglement witnesses for multiqubit systems can be obtained as violations of classical probability rules. They include the ones that have been derived earlier by employing different methods. It provides a perspective complementary to the current understanding in terms of the algebraic approaches.

Nonlinear entanglement witnesses for four qubits in mutually unbiased bases

Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartite entanglement of mixed states. Nonlinear entanglement witnesses have the advantage of a wider detection range in the entangled region. We construct genuine entanglement witnesses for four qubits density matrices in the mutually unbiased basis. First, we find the convex feasible region with positive partial transpose states. Then, to reveal the entangled regions, we present some appropriate linear entanglement witnesses and, we find the envelope of this family of linear witnesses as a nonlinear witness. Finally, we study thermal entanglement and we show for some Hamiltonians the witnesses can detect the entanglement at all temperatures.

Closing the detection loophole in nonlinear entanglement witnesses

Detectors in the laboratory are often unlike their ideal theoretical cousins. They have nonideal efficiencies, which may then lead to nontrivial implications. We show how it is possible to predict correct answers about whether a shared quantum state is entangled in spite of finite detector efficiencies, when the tool for entanglement detection is a nonlinear entanglement witness. We first consider the detection loophole for shared quantum states with nonpositive partial transpose. We subsequently find nonlinear witness operators for bound entangled states with positive partial transpose, and show how the detection loophole can be closed also in such instances.

Experimental measurement of a nonlinear entanglement witness by hyperentangling two-qubit states

We demonstrate that non-linear entanglement witnesses can be made particularly useful for entanglement detection in hyper-entangled or multilevel states. We test this idea experimentally on the platform of linear optics using a hyper-entangled state of two photons. Instead of several simultaneous copies of two-photon entangled states, one can directly measure the witness on single copy of a hyper-entangled state. Our results indicate that hyper-entanglement can be used for quick entanglement detection and it provides a practical testbed for experiments with non-linear entanglement witnesses.

Exploring Tripartite Quantum Correlations: Entanglement Witness and Quantum Discord

In this study, we explore the tripartite quantum correlations by employing the quantum relative entropy as a distance measure. First, we evaluate the explicit expression for nonlinear entanglement witness (EW) of tripartite systems in the four dimensional space that lends itself to a straightforward algorithm for finding closest separable state (CSS) to the generic state. Then using nonlinear EW with specific feasible regions (FRs), quantum discord is derived analytically for the three-qubit and tripartite systems in the four dimensional space. Furthermore, we explicitly figure out the additivity relation of quantum correlations in tripartite systems.

Experimental measurement of collective nonlinear entanglement witness for two qubits

We present a proof-of-principle experiment demonstrating measurement of the collectibility, a nonlinear entanglement witness proposed by Rudnicki et al. [Phys. Rev. Lett. 107, 150502 (2011)]. This entanglement witness works for both mixed and pure two-qubit states. In the later case it can be used to measure entanglement in terms of the negativity. We measured the collectibility for three distinct classes of photonic polarization-encoded two-qubit states, i.e., maximally entangled, separable and maximally mixed states. We demonstrate that the measurement procedure is feasible and robust against typical experimental shortcomings such as imperfect two-photon indistinguishability.

Nonlinear entanglement witnesses based on continuous-variable local orthogonal observables for bipartite systems

In this paper, a nonlinear entanglement witness criterion based on continuous-variable local orthogonal observables for bipartite states is established, which is strictly stronger than the the linear entanglement witnesses criterion introduced by Zhang et al. (Phys. Rev. Lett. 111:190501, 2013). This criterion is particularly applied to two-mode Gaussian states yielding a criterion in terms of the covariance matrix. Comparison with CCNR criterion is discussed.

Generation of Orbital Angular Momentum Bell States and Their Verification via Accessible Nonlinear Witnesses

The controlled generation of entangled states and their subsequent detection are integral aspects of quantum information science. In this Letter, we implement a simple and precise technique that produces any of the four Bell states in the orbital angular momentum degree of freedom. We then use these states to perform the first experimental demonstration of an accessible nonlinear entanglement witness. Such a witness determines entanglement by using the same measurements as required for a linear witness but can detect, in this case, twice as many states as a single linear witness can. We anticipate that our method of state preparation and nonlinear witnesses will have further uses in areas of quantum science, such as superdense coding and quantum key distribution.

Experimental detection of quantum entanglement

In this review, we introduce some methods for detecting or measuring entanglement. Several nonlinear entanglement witnesses are presented. We derive a series of Bell inequalities whose maximally violations for any multipartite qubit states can be calculated by using our formulas. Both the nonlinear entanglement witnesses and the Bell inequalities can be operated experimentally. Thus they supply an effective way for detecting entanglement. We also introduce some experimental methods to measure the entanglement of formation, and the lower bound of the convex-roof extension of negativity.

Determining lower bounds on a measure of multipartite entanglement from few local observables

We introduce a method to lower bound an entropy-based measure of genuine multipartite entanglement via nonlinear entanglement witnesses. We show that some of these bounds are tight and explicitly work out their connection to a framework of nonlinear witnesses that were published recently. Furthermore we provide a detailed analysis of these lower bounds in the context of other possible bounds and measures. In exemplary cases we show that only a few local measurements are necessary to determine these lower bounds.

Entanglement detection and lower bound of the convex-roof extension of the negativity

We present a set of inequalities based on mean values of quantum mechanical observables similar to nonlinear entanglement witnesses for bipartite quantum systems. These inequalities give rise to sufficient and necessary conditions for separability of all bipartite pure states and even some mixed states. In terms of these mean values of quantum mechanical observables, a measurable lower bound of the convex-roof extension of the negativity is derived.

Relativistic entanglement in single-particle quantum states using non-linear entanglement witnesses

In this study, the spin-momentum correlation of one massive spin-1/2 and spin-1 particle states, which are made based on projection of a relativistic spin operator into timelike direction is investigated. It is shown that by using Non-Linear entanglement witnesses (NLEWs), the effect of Lorentz transformation would decrease both the amount and the region of entanglement.

Nonlinear and linear entanglement witnesses for bipartite systems via exact convex optimization

Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5.

General algorithm for manipulating nonlinear and linear entanglement witnesses by using exact convex optimization

We consider the problem of obtaining the entanglement witnesses for a given quantum system. We present a method to convert this problem to a standard convex optimization problem by defining a feasible region. Then we develop a generic two-step algorithm for this problem which can be applied to the entanglement detection of $N$-partite quantum system. This permits an organized construction of nonlinear and linear entanglement witnesses for a given quantum system. To demonstrate the capability of this approach, we apply our results to several three-qubit systems which in some cases lead to necessary and sufficient conditions for separability.

Detecting some three-qubit MUB diagonal entangled states via nonlinear optimal entanglement witnesses

The three qubits mutually unbiased bases (MUB) diagonal density matrices with maximally entanglement in Greenberger-Horne-Zeilinger (GHZ) basis are studied. These are a natural generalization of Bell-state diagonal density matrices. The linearity of positive partial transpose (PPT) conditions allows one to specify completely PPT states or feasible region (FR) which form a polygon, where the projection of the feasible region to some two dimensional planes has lead to better visualization. To reveal the PPT entangled regions of these density matrices, we manipulate some appropriate optimal non-decomposable linear entanglement witnesses (EWs) as the envelope of family of linear optimal non-decomposable EWs. These nonlinear EWs are nonlinear functional of MUB diagonal states, so that they are nonnegative valued over all separable, but they are negative valued over some PPT entangled MUB diagonal states. Even though, these nonlinear EWs can not separate completely, the PPT entanglement region from separable one, but however in special cases they lead to necessary and sufficient condition for separability. To support the evidence, we study three categories for special choices of parameters in density matrices, and using the nonlinear EWs we can distinguish the region of PPT entangled states and separable states, completely. At the end some numerical simulations are provided to show the practical applicability of these nonlinear EWs in detecting some PPT entangled MUB diagonal states.

Investigating a class of2⊗2⊗dbound entangled density matrices via linear and nonlinear entanglement witnesses constructed by exact convex optimization

Here we consider a class of $2\otimes2\otimes d$ chessboard density matrices starting with three-qubit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically. {\bf Keywords: chessboard density matrices, optimal non-linear entanglement witnesses, convex optimization} {\bf PACs Index: 03.65.Ud}