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We provide a complete analysis of mixed three-qubit states composed of a Greenberger-Horne-Zeilinger state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-ta...
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... while a two-qubit state of rank n has always an optimal decomposition of length n, this does not hold for the three-qubit states ρ(p). Remarkably, we have obtained the convex roof of the 3-tangle not only for mixtures as in Eq. (7), but for a considerable part of the Bloch sphere (cf. Fig. 2). We mention that, if any of these density matrices is convexly combined with an arbitrary three-qubit density matrix, our results provide a non-trivial upper bound for the 3-tangle of the resulting ...
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... roofs. -With these findings we can elucidate the affine regions (the convex roofs) of the 3-tangle for any mixture of GHZ and W states. To this end, imagine the two-dimensional space spanned by the GHZ and the W state represented by a Bloch sphere with the GHZ state at its north pole and the W state at the south pole (cf. Fig. 2). All the states |Z(p, ϕ) are located at the unit sphere, with azimuth ϕ and p the distance from the south pole along the north-south line (i.e., p = 0 is the south pole, and p = 1 is the north pole). The states with zero 3-tangle are represented by the simplex S 0 with corners W , Z 0 0 , Z 1 0 , Z 2 0 . Now consider planes P (p) parallel to the ground plane of the simplex (i.e., the plane containing the triangle Z 0 0 , Z 1 0 , Z 2 0 ) and intersection point p with the north-south line. For p 0 ≤ p ≤ p 1 the 3-tangle is constant in a triangle that has its corners at the intersection points of the plane P (p) and the meridians through Z 0 0 , Z 1 0 , and Z 2 0 (see Fig. 2). For p 1 ≤ p ≤ 1 we have another simplex S 1 with affine 3-tangle. The ground plane of this simplex is formed by the last of the "leaves" (at p = p 1 ) described above, and the top corner is the GHZ state. That is, in each plane parallel to the ground plane of this simplex the 3-tangle is constant. The 3-tangle of any point inside S 1 (with distance p ′ from the south pole, i.e., the GHZ weight equals p ′ ) is a convex combination α · g II (p 1 ) + β · 1 of g II (p 1 ) (the 3-tangle in the ground plane of S 1 ) and of 1, the value for the GHZ state. The ...
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... roofs. -With these findings we can elucidate the affine regions (the convex roofs) of the 3-tangle for any mixture of GHZ and W states. To this end, imagine the two-dimensional space spanned by the GHZ and the W state represented by a Bloch sphere with the GHZ state at its north pole and the W state at the south pole (cf. Fig. 2). All the states |Z(p, ϕ) are located at the unit sphere, with azimuth ϕ and p the distance from the south pole along the north-south line (i.e., p = 0 is the south pole, and p = 1 is the north pole). The states with zero 3-tangle are represented by the simplex S 0 with corners W , Z 0 0 , Z 1 0 , Z 2 0 . Now consider planes P (p) parallel to the ground plane of the simplex (i.e., the plane containing the triangle Z 0 0 , Z 1 0 , Z 2 0 ) and intersection point p with the north-south line. For p 0 ≤ p ≤ p 1 the 3-tangle is constant in a triangle that has its corners at the intersection points of the plane P (p) and the meridians through Z 0 0 , Z 1 0 , and Z 2 0 (see Fig. 2). For p 1 ≤ p ≤ 1 we have another simplex S 1 with affine 3-tangle. The ground plane of this simplex is formed by the last of the "leaves" (at p = p 1 ) described above, and the top corner is the GHZ state. That is, in each plane parallel to the ground plane of this simplex the 3-tangle is constant. The 3-tangle of any point inside S 1 (with distance p ′ from the south pole, i.e., the GHZ weight equals p ′ ) is a convex combination α · g II (p 1 ) + β · 1 of g II (p 1 ) (the 3-tangle in the ground plane of S 1 ) and of 1, the value for the GHZ state. The ...
Citations
... Such convex roof extensions are necessary to ensure an entanglement monotone, but since the minimum of a sum of convex functions is not always convex, we need to take the convex hull of Eq. (8) (see also Refs. [63,62]). Finding the optimal ensemble is a non-trivial task as it entails searching over all possible decompositions of the mixed state. ...
Multipartite entangled states are an essential resource for sensing, quantum error correction, and cryptography. Color centers in solids are one of the leading platforms for quantum networking due to the availability of a nuclear spin memory that can be entangled with the optically active electronic spin through dynamical decoupling sequences. Creating electron-nuclear entangled states in these systems is a difficult task as the always-on hyperfine interactions prohibit complete isolation of the target dynamics from the unwanted spin bath. While this emergent cross-talk can be alleviated by prolonging the entanglement generation, the gate durations quickly exceed coherence times. Here we show how to prepare high-quality GHZ M -like states with minimal cross-talk. We introduce the M -tangling power of an evolution operator, which allows us to verify genuine all-way correlations. Using experimentally measured hyperfine parameters of an NV center spin in diamond coupled to carbon-13 lattice spins, we show how to use sequential or single-shot entangling operations to prepare GHZ M -like states of up to M = 10 qubits within time constraints that saturate bounds on M -way correlations. We study the entanglement of mixed electron-nuclear states and develop a non-unitary M -tangling power which additionally captures correlations arising from all unwanted nuclear spins. We further derive a non-unitary M -tangling power which incorporates the impact of electronic dephasing errors on the M -way correlations. Finally, we inspect the performance of our protocols in the presence of experimentally reported pulse errors, finding that XY decoupling sequences can lead to high-fidelity GHZ state preparation.
... The lines y − wx − b ±1 cross the support vectors, here drawn with a filled shape. The slack variables ζ 1 and ζ 2 penalize the possible misclassification of the two gray triangle points for which the analytic solutions for the CRE of the three-tangle [25,26,35] and the CRE of GME-concurrence [10,23,24] are available. These are the GHZ-symmetric states, the X-states, and the statistical mixture of GHZ and W states, which are defined in Appendix A, which will be used in the following to construct the three-qubit dataset of the SVM algorithm. ...
... GME states are labeled as +1 for both SEP-vs-all and GME-vs-all classifiers. The GME states are labeled according to the three-tangle for generated pure GHZ states [22], defined in Eq. (3), GHZ-symmetric states [35], and statistical mixture of GHZ, W, andW states [25]. In the case of mixed X-state [26], we use the GME-concurrence, Eq. (6). ...
... The analytic calculation of the three-tangle measure for those states has been given in [23,25,35]. ...
Although entanglement is a basic resource for reaching quantum advantage in many computation and information protocols, we lack a universal recipe for detecting it, with analytical results obtained for low-dimensional systems and few special cases of higher-dimensional systems. In this work, we use a machine learning algorithm, the support vector machine with polynomial kernel, to classify separable and entangled states. We apply it to two-qubit and three-qubit systems, and we show that, after training, the support vector machine is able to recognize if a random state is entangled with an accuracy up to $$92\%$$ 92 % for the two-qubit system and up to $$98\%$$ 98 % for the three-qubit system. We also describe why and in what regime the support vector machine algorithm is able to implement the evaluation of an entanglement witness operator applied to many copies of the state, and we describe how we can translate this procedure into a quantum circuit.
... Another good measure for tripartite entanglement is the three-tangle. This is polynomial invariant [39], therefore an optimal decomposition of a mixed density matrix is required, which is a very difficult task in a few exceptional situations [40]. However, there is a crucial defect in the three-tangle as a three-party entanglement measure, i.e., the three-tangle for a tripartite GHZ state is maximal, whereas for the W-state it is zero. ...
Entanglement protection from noisy environments is an essential task in practical quantum information and communication. In this paper, we investigate the entanglement recovery of an amplitude-damped tripartite GHZ state by a weak-measurement reversal procedure. In particular, we emphasize the key importance of the inequivalency of probability amplitudes of the tripartite system under the recovery technique. We explore the maximal and non-maximal tripartite entangled state scenarios under amplitude damping noise in the absence and presence of weak measurement reversal procedures. Importantly, the non-maximal entangled state turns out to be a good choice for the entanglement recovery via weak-measurement reversal procedure.
... The definitions of the Greenberger-Horne-Zeilinger (GHZ) state and the W state can be found in [19]. The analytic solution for the mixture is F (s) = (5s 2 − 4s + 8)/9, provided by Lohmayer et al. [20]. For this state, the dimension of the exterior supremum for X in (2) is reduced to 8 (compared to 64) due to the state's symmetries. ...
Entanglement is known to decay even in isolated systems, an effect attributed to spontaneous emission. This fragility of entanglement can be exacerbated by entanglement sudden death (ESD), where entanglement drops to zero abruptly within a finite time. It is natural to assume that multipartite entanglement is more vulnerable to ESD, as it involves more parties experiencing spontaneous emission. In this work, we challenge this assumption and present a contrasting observation, which provides compelling evidence that multipartite entanglement demonstrates increased robustness against ESD from spontaneous emission.
... To probe the entangling properties of the coin operator in (24) further, we analyze its Entangling power [14], [15], [24], [25,26], [27], which describes the capacity of the coin operator to produce an entangled state from the initial tensor-product states. For concreteness, consider the initial state, ...
... • The generalized version of entangling power is through Concurrence matrix [15,24] or n−tangle operators for higher qubits and higher dimensions [25][26][27]. It would be interesting to see if the generalizations can be used as an order parameter to determine entanglement evolutions in real systems. ...
We analyze the effect of a simple coin operator, built out of Bell pairs, in a 2d Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the Entangling Power of the coin operator. Secondly, we compute the Generalized Relative Rényi Entropy between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the Entangling Power and Generalized Relative Rényi Entropy behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the 2d DQRW into two 1d massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.
... Second, when the parameter g is outside the region of 0.297 ≤ g ≤ 0.612, the criterion in Eq. (7) is satisfied. This parameter region is very close to other wellknown regions using two other entanglement measures [39,40]. On the one hand, the three-tangle τ vanishes for 0 ≤ g ≤ g τ ≈ 0.627, where τ measures residual (three-partite) entanglement that cannot be expressed as two-body entanglement [41]. ...
In recent years, analysis methods for quantum states based on randomized measurements have been investigated extensively. Still, in the experimental implementations these methods were typically used for characterizing strongly entangled states and not to analyze the different families of multiparticle or weakly entangled states. In this work, we experimentally prepare various entangled states with path-polarization hyper-entangled photon pairs, and study their entanglement properties using the full toolbox of randomized measurements. First, we successfully characterize the correlations of a series of GHZ-W mixed states using the second moments of the random outcomes, and demonstrate the advantages of this method by comparing it with the well-known three-tangle and squared concurrence. Second, we generate bound entangled chessboard states of two three-dimensional systems and verify their weak entanglement with a criterion derived from moments of randomized measurements.
... In this space, both criteria were shown to be much more effective in certifying entanglement than criteria based only on the full sector length S 3 . In particular, Eq. (64) can detect multipartite entanglement for mixtures of GHZ states and W states, even if the three-tangle [109] and the bipartite entanglement in the reduced subsystems vanish simultaneously [211]. ...
Randomised measurements provide a way of determining physical quantities without the need for a shared reference frame nor calibration of measurement devices. Therefore, they naturally emerge in situations such as benchmarking of quantum properties in the context of quantum communication and computation where it is difficult to keep local reference frames aligned. In this review, we present the advancements made in utilising such measurements in various quantum information problems focusing on quantum entanglement and Bell inequalities. We describe how to detect and characterise various forms of entanglement, including genuine multipartite entanglement and bound entanglement. Bell inequalities are discussed to be typically violated even with randomised measurements, especially for a growing number of particles and settings. Additionally, we provide an overview of estimating other relevant nonlinear functions of a quantum state or performing shadow tomography from randomised measurements. Throughout the review, we complement the description of theoretical ideas by explaining key experiments.
... A non-linear entanglement witness operator has been constructed to identify all three types of three-qubit states [149]. Using entanglement measures, biseparability in mixed three-qubit systems has also been analyzed in detail in [150,151]. The necessary and sufficient condition for the detection of permutationally invariant threequbit biseparable states has been studied in [152]. ...
The detection and classification of entanglement properties in a two-qubit and a multi-qubit system is a topic of great interest. This topic has been extensively studied, and as a result, we discovered various approaches for detecting and classifying multi-qubit, in particular three-qubit entangled states. The emphasis of this work is on a formalism of methods for the detection and classification of bipartite as well as multipartite quantum systems. We have used the method of structural physical approximation of partially transposed matrix (SPA-PT) for the detection of entangled states in arbitrary dimensional bipartite quantum systems. Also, we have proposed criteria for the classification of all possible stochastic local operations and classical communication (SLOCC) inequivalent classes of a pure and mixed three-qubit state using the SPA-PT map. To quantify entanglement, we have defined a new measure of entanglement based on the method of SPA-PT, which we named as "structured negativity". We have shown that this measure can be used to quantify entanglement for negative partial transposed entangled states (NPTES). Since the methods for detection, classification and quantification of entanglement, defined in this thesis are based on SPA-PT, they may be realized in an experiment.
... Such convex roof extensions are necessary to ensure an entanglement monotone, but since the minimum of a sum of convex functions is not always convex, we need to take the convex hull of Eq. (8) (see also Refs. [59,60]). Finding the optimal ensemble is a non-trivial task as it entails searching over all possible decompositions of the mixed state. ...
Multipartite entangled states are an essential resource for sensing, quantum error correction, and cryptography. Color centers in solids are one of the leading platforms for quantum networking due to the availability of a nuclear spin memory that can be entangled with the optically active electronic spin through dynamical decoupling sequences. Creating electron-nuclear entangled states in these systems is a difficult task as the always-on hyperfine interactions prohibit complete isolation of the target dynamics from the unwanted spin bath. While this emergent cross-talk can be alleviated by prolonging the entanglement generation, the gate durations quickly exceed coherence times. Here we show how to prepare high-quality GHZ$_M$-like states of up to $M=10$ qubits, with minimal cross-talk. We introduce the $M$-way entangling power of an evolution operator, which allows us to verify genuine all-way correlations. We show how to use sequential or single-shot entangling operations to prepare GHZ$_M$-like states within the coherence times that saturate bounds on $M$-way correlations. Finally, we study the entanglement of mixed electron-nuclear states and develop a non-unitary $M$-way entangling power which additionally captures correlations arising from all spectator nuclear spins.
... This is in general difficult to calculate, and numerical solutions are usually very costly. Therefore, we use particular types of mixed states for which the analytic solutions for the CRE of the three-tangle [26,27,34], and the CRE of GME-concurrence [9,24,25] are available. These are the GHZ-symmetric states, the X-states, and the statistical mixture of GHZ and W states which are defined in Appendix A, which will be used in the following to construct the three-qubit dataset of the SVM algorithm. ...
... GME states are labeled as +1 for both SEP-vs-all and GME-vs-all classifiers. The GME states are labeled according to the three-tangle for generated pure GHZ states [23], defined in Eq. (3), GHZ-symmetric states [34], and statistical mixture of GHZ, W, andW states [26]. In the case of mixed Xstate [27] we use the GME-concurrence, Eq. (6). ...
... The analytic calculation of the three-tangle measure for those states has been given in [24,26,34]. ...
Although entanglement is a basic resource for reaching quantum advantange in many computation and information protocols, we lack a universal recipe for detecting it, with analytical results obtained for low dimensional systems and few special cases of higher dimensional systems. In this work, we use a machine learning algorithm, the support vector machine with polynomial kernel, to classify separable and entangled states. We apply it to two-qubit and three-qubit systems, and we show that, after training, the support vector machine is able to recognize if a random state is entangled with an accuracy up to 92% for the two-qubit system and up to 98% for the three-qubit system. We also describe why and in what regime the support vector machine algorithm is able to implement the evaluation of an entanglement witness operator applied to many copies of the state, and we describe how we can translate this procedure into a quantum circuit.
