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arXiv:quant-ph/0210117 · Physical Review A

Characterizing the entanglement of symmetric many-particle spin- 1 2 systems

John K. Stockton, J. M. Geremia, Andrew C. Doherty, Hideo Mabuchi

Published Feb 28, 2003• 225 citations
arXivHTMLPDF

Abstract

Analyzing the properties of entanglement in many-particle spin-1/2 systems is generally difficult because the system's Hilbert space grows exponentially with the number of constituent particles, N. Fortunately, it is still possible to investigate a many-particle entanglement when the state of the system possesses sufficient symmetry. In this paper, we present a practical method for efficiently computing various bipartite entanglement measures for states in the symmetric subspace and perform these calculations for $N\ensuremath{\sim}{10}^{3}.$ By considering all possible bipartite splits, we construct a picture of the multiscale entanglement in large symmetric systems. In particular, we characterize dynamically generated spin-squeezed states by comparing them to known reference states (e.g., Greenberger-Horne-Zeilinger and Dicke states), and families of states with near-maximal bipartite entropy. We quantify the trade-off between the degree of entanglement and its robustness to particle loss, emphasizing that substantial entanglement need not be fragile.

DOI
10.1103/physreva.67.022112

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Analyzing the properties of entanglement in many-particle spin-1/2 systems is generally difficult because the system's Hilbert space grows exponentially with the number of constituent particles, N. Fortunately, it is still possible to investigate a many-particle entanglement when the state of the system possesses sufficient symmetry. In this paper, we present a practical method for efficiently computing various bipartite entanglement measures for states in the symmetric subspace and perform these calculations for N\ensuremath∼103.N\ensuremath{\sim}{10}^{3}.N\ensuremath∼103. By considering all possible bipartite splits, we construct a picture of the multiscale entanglement in large symmetric systems. In particular, we characterize dynamically generated spin-squeezed states by comparing them to known reference states (e.g., Greenberger-Horne-Zeilinger and Dicke states), and families of states with near-maximal bipartite entropy. We quantify the trade-off between the degree of entanglement and its robustness to particle loss, emphasizing that substantial entanglement need not be fragile.
Quantifying mixed-state entanglement in many-body systems has been a formidable task. In this work, we quantify the entanglement of states in unresolvable spin ensembles, which are inherently mixed. By exploiting their permutationally invariant properties, we show that the bipartite entanglement of a wide range of unresolvable ensemble states can be calculated exactly. Our formalism is versatile; it can be used to evaluate the entanglement in an ensemble with an arbitrary number of particles, effective angular momentum, and bipartition. We apply our method to explore the characteristics of entanglement in different physically motivated scenarios, including states with definite magnetization and metrologically useful superpositions such as Greenberger-Horne-Zeilinger (GHZ) states and spin-squeezed states. Our method can help understand the role of entanglement in spin-ensemble-based quantum technologies.
We construct an entanglement witness for many-qubit systems, based on symmetric two-body correlations with two measurement settings. This witness is able to detect the entanglement of some Dicke states for any number of particles and such detection exhibits some robustness against white noise and thermal noise under the Lipkin-Meshkov-Glick Hamiltonian. In addition, it detects the entanglement of spin-squeezed states, with a detection strength that approaches the maximal value for sufficiently large numbers of particles. As spin-squeezed states can be experimentally generated, the properties of the witness with respect to these states may be amenable to experimental investigation. We show that while the witness is unable to detect Greenberger-Horne-Zeilinger (GHZ) states, it is instead able to detect superpositions of Dicke states with GHZ states.
One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.
We present a method to detect bipartite entanglement based on number-phase-like uncertainty relations in split spin ensembles. First, we derive an uncertainty relation that plays the role of a number-phase uncertainty for spin systems. It is important that the relation is given with well-defined and easily measurable quantities, and that it does not need assuming infinite dimensional systems. Based on this uncertainty relation, we show how to detect bipartite entanglement in an unpolarized Dicke state of many spin-1/2 particles. The particles are split into two subensembles, then collective angular momentum measurements are carried out locally on the two parts. First, we present a bipartite Einstein-Podolsky-Rosen (EPR) steering criterion. Then, we present an entanglement condition that can detect bipartite entanglement in such systems. We demonstrate the utility of the criteria by applying them to a recent experiment given in K. Lange et al. [Science 360, 416 (2018)] realizing a Dicke state in a Bose-Einstein condensate of cold atoms, in which the two subensembles were spatially separated from each other. Our methods also work well if split spin-squeezed states are considered. We show in a comprehensive way how to handle experimental imperfections, such as the nonzero particle number variance including the partition noise, and the fact that, while ideally BECs occupy a single spatial mode, in practice the population of other spatial modes cannot be fully suppressed.
Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.
Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500 atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.