The most popular, bipartite methods approach to the analysis of entanglement included in genuine entangled quantum states, describing n-partite, n > 2 quantum systems are based on the notion of entropy of the corresponding reduced density matrices or/and the use of Negativity measure. However the equivalence of these two different measures is not clear. In particular, the equivalence of these notions of being maximally entangled in the sense of amount of entanglement included is not known. In the case of special, maximally entangled, with respect to the Negativity measure, pure states, describing 5 and 6 qubit systems, the equivalence to the entropy measure based definitions was shown by the straightforward calculations based on Schmidt’s decomposition method.
The most popular, bipartite methods approach to the analysis of entanglement included in genuine entangled quantum states, describing n-partite, n > 2 quantum systems are based on the notion of entropy of the corresponding reduced density matrices or/and the use of Negativity measure. However the equivalence of these two different measures is not clear. In particular, the equivalence of these notions of being maximally entangled in the sense of amount of entanglement included is not known. In the case of special, maximally entangled, with respect to the Negativity measure, pure states, describing 5 and 6 qubit systems, the equivalence to the entropy measure based definitions was shown by the straightforward calculations based on Schmidt’s decomposition method.
The thesis includes the original results of our articles [30, 37, 40, 42, 51, 53, 75]. A method is developed to compute analytically entanglement measures of three-qubit pure states. Owing to it closed-form expressions are presented for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space and the Hilbert space of three-qubits consists of three different entangled regions. The states that lie on joint surfaces separating different entangled regions, designated as shared states, have particularly interesting features and are dual quantum channels for the perfect teleportation and superdense coding [42]. A powerful method is developed to compute analytically multipartite entanglement measures. The method uses the duality concept and creates a bijection between highly entangled quantum states and their nearest separable states. The bijection gives explicitly the geometric entanglement measure of arbitrary generalized W states of n qubits [30, 75]. Additionaly, the behavior of the geometric entanglement measure of many-qubit W states is analyzed and an interpolating formula is derived [51]. Generalized Schmidt decomposition of pure three-qubit states has four positive and one complex coefficients. In contrast to the bipartite case, they are not arbitrary and the largest Schmidt coefficient restricts severely other coefficients. It is derived a non-strict inequality between three-qubit Schmidt coefficients, where the largest coefficient defines the least upper bound for the three nondiagonal coefficients or, equivalently, the three nondiagonal coefficients together define the greatest lower bound for the largest coefficient [53].
A quantum state's entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.
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.
It is known that ${\ensuremath{\rho}}^{AB}$ as a bipartite reduced state of the three-qubit Greenberger-Horne-Zeilinger (GHZ) state is separable, but part A and part B indeed ``share tripartite entanglement'' in the GHZ state. Namely, whether a state can ``share'' more entanglement is dependent on the global system it lives in. Here We explore such kind of entanglement in any n-partite system with arbitrary dimensions, $n\ensuremath{\geqslant}3$, and call it partitewise entanglement, which includes pairwise entanglement proposed by Dong et al. [Phys. Rev. A 110, 032420 (2024)] as a special case. We propose three classes of the partitewise entanglement measures, which are based on the genuine entanglement measure, the minimal bipartition, and the minimal distance from the partitewise separable states, respectively. The former two methods are far ranging since all of them are defined by the reduced function. [Note: If a positive function h satisfies $h({\ensuremath{\rho}}^{A})=E(|\ensuremath{\psi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\psi}{|}^{AB})$ for some bipartite entanglement measure E,h is called the reduced function (of E).] Consequently, we establish the framework of the resource theory of the partitewise entanglement. In addition, we investigate the partitewise entanglement extensibility and give a measure of such extensibility, from which we find that the maximal partitewise entanglement extension is its purification. Lastly, the relation between this extensibility and the partitewise entanglement is discussed.
Bipartite mixed-state quantum entanglement (QE) and its measures play a crucial role in both theoretical research and practical quantum applications. Its internal structure is far more complex and less well understood compared with bipartite pure-state QE. Some existing measures involve inherently intractable global optimizations, while others are only applicable to highly limited-dimensional quantum systems. Here based on the inherent feature that bipartite QE systems nonseparable necessarily implies that local reduced density matrix differs from its \textquotedblleft native\textquotedblright density matrix, we propose a more physical and intuitive measure termed Reduction-induced Minimal Partial Entropy Difference to quantify arbitrary bipartite mixed-state QE. Partial Von Neumann Entropy is only a pure-state special case of this method. This measure offers intrinsic structural insights into bipartite QE characterization, thereby establishing itself as a valuable complementary measure. Its intuitive and clear physical picture, combined with wide applicability and relatively low computational complexity, facilitates exploring its potential quantum information applications, hence its conceptual framework and line of thought deserve to be further developed to describe and quantify multipartite QE in the future.
The entanglement of a pure bipartite state is uniquely measured by the von-Neumann entropy of its reduced density matrices. Though it cannot specify all the non-local characteristics of pure entangled states. It was proven that for every possible value of entanglement of a bipartite system, there exists an infinite number of equally entangled pure states, not comparable(satisfies Nielsen’s criteria) to each other. In this work, we investigate other correlation measures of pure bipartite states that are able to differentiate the quantum correlations of the states with entropy of entanglement. In Schmidt rank 3, we consider the whole set of states having same entanglement and compare how minutely such states can be distinguished by other correlation measures. Then for different values of entanglement we compare the sets of states belonging to the same entanglement and also investigate the graphs of different correlation measures. We extend our search to Schmidt rank 4 and 5 also.