Narrow Results

Entanglement is an important resource for quantum technologies that allow us to go beyond the classical world. Quantum systems can be entangled in various ways. Also, the degree of entanglement between the subsystems of a quantum system can be calculated in different ways. In our recent paper [Prog. Theor. Exp. Phys. (2022)], we directly measure the three-tangle of a pure three-qubit quantum state. In this paper, we extend this method to pure N-qubit quantum states (N is any odd number greater than 1) by considering polynomial invariant of degree 4 as a measure of entanglement. In this algorithm, we use quantum gates and output probabilities to calculate polynomial invariant of degree 4. The importance of this method lies in the possibility of experimental implementation of quantum calculations in the language of quantum gates.

A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems.

Entanglement and occupation probabilities along the yrast line in the interacting Bose–Einstein condensates with contact interaction in two dimension is studied for the angular momentum quantum number L ≤ n with the help of a simple improved Mathematica package. Results of the entanglement along the yrast line up to L = 6 and occupation probabilities for the yrast states up to L = 5 are calculated, which show that the system undergoes a quantum phase transition for all the yrast states, of which the total number of bosons serves as the control parameter. The critical point is near n = L + 1.

Entanglement measures quantify the amount of quantum entanglement that is contained in quantum states. Typically, different entanglement measures do not have to be partially ordered. The presence of a definite partial order between two entanglement measures for all quantum states, however, allows for meaningful conceptualization of sensitivity to entanglement, which will be greater for the entanglement measure that produces the larger numerical values. Here, we have investigated the partial order between the normalized versions of four entanglement measures based on Schmidt decomposition of bipartite pure quantum states, namely, concurrence, tangle, entanglement robustness and Schmidt number. We have shown that among those four measures, the concurrence and the Schmidt number have the highest and the lowest sensitivity to quantum entanglement, respectively. Further, we have demonstrated how these measures could be used to track the dynamics of quantum entanglement in a simple quantum toy model composed of two qutrits. Lastly, we have employed state-dependent entanglement statistics to compute measurable correlations between the outcomes of quantum observables in agreement with the uncertainty principle. The presented results could be helpful in quantum applications that require monitoring of the available quantum resources for sharp identification of temporal points of maximal entanglement or system separability.

The simple entanglement of N-body N-particle pure states is extended to the more general M-body or M-body N-particle states where N ≠ M. Some new features of the M-body N-particle pure states are discussed. An application of the measure to quantify quantum correlations in a Bose–Einstien condensate model is demonstrated.

The Schmidt number is of crucial importance in characterizing the bipartite pure states. We explore and propose here a generalization of Schmidt number for states in multipartite systems. It is shown to be entanglement monotonic and valid for both pure and mixed states. In addition, the corresponding generalization of multipartite Schmidt coefficients is introduced. Our approach is applicable for systems with arbitrary number of parties and for arbitrary dimensions.

In this paper, we develop a mathematical framework for describing entanglement quantitatively and qualitatively for multipartite qudit states in terms of rows or columns of coefficient matrices. More specifically, we propose an entanglement measure and separability criteria based on rows or columns of coefficient matrices. This entanglement measure has an explicit mathematical expression by means of exterior products of all pairs of rows or columns in coefficient matrices. It is introduced via our result that the $I$-concurrence coincides with the entanglement measure based on two-by-two minors of coefficient matrices. Depending on our entanglement measure, we obtain the separability criteria and maximal entanglement criteria in terms of rows or columns of coefficient matrices. Our conclusions show that just like every two-by-two minor in a coefficient matrix of a multipartite pure state, every pair of rows or columns can also exhibit its entanglement properties, and thus can be viewed as its smallest entanglement contribution unit too. The great merit of our entanglement measure and separability criteria is two-fold. First, they are very practical and convenient for computation compared to other methods. Second, they have clear geometric interpretations.

We geometrically examine the entanglement in symmetric multiquibt states using the spin coherent states properties. We employ the Majorana representation to examine how coherent (polarized) and unpolarized states can be represented in the Bloch sphere and subsequently to quantify entanglement in multipartite systems. Entanglement can be viewed as a distance separate two points in Bloch sphere and how this notion can be extended for multipartite states in W class. This provides us with an entanglement measure for $W$-states analogue to the notion of Concurrence.

Our study employs a connected correlation matrix to quantify quantum entanglement. The matrix encompasses all necessary measures for assessing the degree of entanglement between particles. We begin with a three-qubit state and involve obtaining a mixed state by performing partial tracing over one qubit. Our goal is to exclude the non-connected sector by focusing on the connected correlation. This suggests that the connected correlation is deemed crucial for capturing relevant entanglement degrees. The study classifies mixed states and observes that separable states exhibit the lowest correlation within each class. We demonstrate that the entanglement measure monotonically increases concerning the correlation measure. This implies that connected correlation serves as an effective measure of quantum entanglement. Finally, our proposal suggests that interpreting quantum entanglement from a local perspective is possible. The observable is described as a vector with locality but violates freedom of choice.

We present a set of entanglement measures R_m for n-qubit quantum states. Since the measure R_m vanishes if and only if a state is separable into m subsystems, this measure R_m indicates whether the state can be decomposed into m subsystems. This R_m for m = n is equivalent to Meyer–Wallach measure while R_m for m = 2 is effectively equivalent to the one recently introduced by Love et al. In order to show the utility of our measures for characterising globally entangled states, the measures R_m are evaluated explicitly for the GHZ, W and Dicke states.