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arXiv:1501.06639 · Progress of Theoretical and Experimental Physics

Entanglement entropy and Schmidt number as measures of delocalization of α clusters in one-dimensional nuclear systems

Yoshiko Kanada-En’yo

Published Apr 1, 2015• 9 citations
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Abstract

We calculate the von Neumann entanglement entropy and the Schmidt number of one-dimensional (1D) cluster states and show that these are useful measures to estimate entanglement caused by delocalization of clusters. We analyze system size dependence of these entanglement measures in the linear-chain |$n\alpha$| states given by Tohsaki–Horiuchi–Schuck–Röpke wave functions for 1D cluster gas states. We show that the Schmidt number is an almost equivalent measure to the von Neumann entanglement entropy when the delocalization of clusters occurs in the entire system but it shows different behaviors in a partially delocalized state containing localized clusters and delocalized ones. Thus the Rényi-2 entanglement entropy, which relates to the Schmidt number, is found to be almost equivalent to the von Neumann entanglement entropy for the full delocalized cluster system but it is less sensitive to the partially delocalized cluster system than the von Neumann entanglement entropy. We also propose a new entanglement measure which is a generalized form of the Schmidt number. The sensitivity of these measures of entanglement to the delocalization of clusters in low-density regions is discussed.

DOI
10.1093/ptep/ptv050

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We calculate the von Neumann entanglement entropy and the Schmidt number of one-dimensional (1D) cluster states and show that these are useful measures to estimate entanglement caused by delocalization of clusters. We analyze system size dependence of these entanglement measures in the linear-chain |nαn\alphanα| states given by Tohsaki–Horiuchi–Schuck–Röpke wave functions for 1D cluster gas states. We show that the Schmidt number is an almost equivalent measure to the von Neumann entanglement entropy when the delocalization of clusters occurs in the entire system but it shows different behaviors in a partially delocalized state containing localized clusters and delocalized ones. Thus the Rényi-2 entanglement entropy, which relates to the Schmidt number, is found to be almost equivalent to the von Neumann entanglement entropy for the full delocalized cluster system but it is less sensitive to the partially delocalized cluster system than the von Neumann entanglement entropy. We also propose a new entanglement measure which is a generalized form of the Schmidt number. The sensitivity of these measures of entanglement to the delocalization of clusters in low-density regions is discussed.
I investigate entanglement entropy of one-dimensional (1D) cluster states to discuss the delocalization of clusters in linear-chain $3\ensuremath{\alpha}$- and $4\ensuremath{\alpha}$-cluster states. In analysis of entanglement entropy of 1D Tohsaki-Horiuchi-Schuck-R\"opke (THSR) and Brink-Bloch cluster wave functions, I show clear differences in the entanglement entropy between localized cluster wave functions and delocalized cluster wave functions. To clarify spatial regions where the entanglement entropy is generated by the delocalization of clusters, I analyze the spatial distribution of entanglement entropy. In the linear-chain $3\ensuremath{\alpha}$-cluster state, the delocalization occurs dominantly in a low-density tail region while it is relatively suppressed in an inner region because of the Pauli blocking effect between clusters. In the linear-chain $4\ensuremath{\alpha}$ state having a larger system size than the linear-chain $3\ensuremath{\alpha}$ state, the delocalization occurs in the whole system. The entanglement entropy is found to be a measure of the delocalization of clusters in the 1D cluster systems.
We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles. This implies definite constraints on the mathematical form of the state vector associated with the whole system. We then analyze separately the cases of fermion and boson systems, and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neumann entropy of the one-particle reduced density operators can supply us with a consistent criterion for detecting entanglement. In particular, the consideration of the von Neumann entropy is particularly useful in deciding whether the correlations of the considered states are simply due to the indistinguishability of the particles involved or are a genuine manifestation of the entanglement. The treatment leads to a full clarification of the subtle aspects of entanglement of two identical constituents which have been a source of embarrassment and of serious misunderstandings in the recent literature.
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.
We derive exact relations between the Rényi entanglement entropies and the particle-number fluctuations of (connected and disjoint) spatial regions in systems of N non-interacting fermions in arbitrary dimension. We prove that the asymptotic large- N behavior of the entanglement entropies is proportional to the variance of the particle number. We also consider 1D Fermi gases with a localized impurity, where all particle cumulants contribute to the asymptotic large- N behavior of the entanglement entropies. The particle cumulant expansion turns out to be convergent for all integer-order Rényi entropies (except for the von Neumann entropy) and the first few cumulants provide already a good approximation. Since the particle cumulants are accessible to experiments, these relations may provide a measure of entanglement in these systems.
Abstract We explore a variational characterization of the bipartite entanglement entropy such as von Neumann entropy for pure states in terms of general projective measurements of an arbitrary system. By properly choosing the measurement basis, we show that a general estimation of the bipartite entanglement measure can be explicitly written. The number of projective measurement operators of the subsystem necessary for calculating an upper bound of the bipartite entanglement entropy is less than or equal to the dimension of the smaller subsystem. This number is much smaller than that of the full tomography of the density matrix. As an application, our method is useful to estimate the entanglement entropy of each energy eigenstate of a given Hamiltonian.
This study introduces a quantum information perspective to analyze the internal structure of atomic nuclei, focusing on the quantum entanglement between ααα clusters in the 0+^++ state of 8^88Be. A wave function based on angular momentum coupling is developed to transform the two-cluster wave function from the conventional center of mass and relative coordinate basis (HRc.m.⊗Hr\mathcal{H}_{R_{c.m.}} \otimes \mathcal{H}_rHRc.m.​​⊗Hr​) into the individual particle basis (H1⊗H2\mathcal{H}_1 \otimes \mathcal{H}_2H1​⊗H2​), which is essential for a precise quantification of entanglement. Within this method, the von Neumann entropy is employed to quantify the entanglement arising from the mixing of angular momentum channels. Additionally, we introduce the concept of spatially resolved entropy, which measures entanglement as a function of the radial separation between clusters. Our analysis reveals that the entanglement is strongly correlated with the spatial configuration at the femtometer scale. As the inter cluster separation vanishes, the system approaches a separable SSS-wave state, indicating that entanglement is dynamically generated during the spatial separation of the clusters. This research provides a new tool for investigating nonlocal quantum correlations in nuclear structure, complementing existing descriptions.