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arXiv:quant-ph/0211063 · Physical Review Letters

Classifying N -Qubit Entanglement via Bell’s Inequalities

Sixia Yu, Zeng‐Bing Chen, Jian-Wei Pan, Yong-De Zhang

Published Feb 27, 2003• 53 citations
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Abstract

All the states of N qubits can be classified into N-1 entanglement classes from 2-entangled to N-entangled (fully entangled) states. Each class of entangled states is characterized by an entanglement index that depends on the partition of N. The larger the entanglement index of a state, the more entangled or the less separable is the state in the sense that a larger maximal violation of Bell's inequality is attainable for this class of state.

DOI
10.1103/physrevlett.90.080401

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All the states of N qubits can be classified into N-1 entanglement classes from 2-entangled to N-entangled (fully entangled) states. Each class of entangled states is characterized by an entanglement index that depends on the partition of N. The larger the entanglement index of a state, the more entangled or the less separable is the state in the sense that a larger maximal violation of Bell's inequality is attainable for this class of state.
We show that all the N-qubit states can be classified as N entanglement classes each of which has an entanglement index E=N−p=0,1,...,N−1E=N-p=0,1,...,N-1E=N−p=0,1,...,N−1(E=0 corresponds to a fully separate class) where ppp denotes number of groups for a partition of the positive integer N. In other words, for any partition (n1,n2,...,np)(n_1,n_2,...,n_p)(n1​,n2​,...,np​) of N with nj≥1n_j\ge 1nj​≥1 and N=∑j=1pnjN=\sum_{j=1}^{p}n_jN=∑j=1p​nj​, the entanglement index for the corresponding state ρn1⨂ρn2...⨂ρnpρ_{n_1}\bigotimesρ_{n_2}...\bigotimes ρ_{n_p}ρn1​​⨂ρn2​​...⨂ρnp​​ with ρnjρ_{n_j}ρnj​​ denoting a fully entangled state of nj−n_j-nj​−qubits is E(ρn1⨂ρn2...⨂ρnp)=∑j=1p(nj−1)=N−pE(ρ_{n_1}\bigotimesρ_{n_2}...\bigotimes ρ_{n_p})=\sum_{j=1}^{p}(n_j-1)=N-pE(ρn1​​⨂ρn2​​...⨂ρnp​​)=∑j=1p​(nj​−1)=N−p.
We construct a simplex for multipartite qubit states of even number nnn of qubits, which has the same geometry concerning separability, mixedness, kind of entanglement, amount of entanglement, and nonlocality as the bipartite qubit states. We derive the entanglement of the class of states which can be described by only three real parameters with the help of a multipartite measure for all discrete systems. We prove that the bounds on this measure are optimal for the whole class of states and that it reveals that the states possess only nnn-partite entanglement and not, e.g., bipartite entanglement. We then show that this nnn-partite entanglement can be increased by stochastic local operations and classical communication to the purest maximal entangled states. However, pure nnn-partite entanglement cannot be distilled; consequently all entangled states in the simplex are nnn-partite bound entangled. We study also Bell inequalities and find the same geometry as for bipartite qubits. Moreover, we show how the (hidden) nonlocality for all nnn-partite bound entangled states can be revealed.
We study the classification of entanglement in tripartite systems by using Bell-type inequalities and principal basis. By using Bell unctions and the generalized three dimensional Pauli operators, we present a set of Bell inequalities which classifies the entanglement of triqutrit fully separable and bi-separable mixed states. By using the correlation tensors in the principal basis representation of density matrices, we obtain separability criteria for fully separable and bi-separable 2 ⊗ 2 ⊗ 3 quantum mixed states. Detailed example is given to illustrate our criteria in classifying the tripartite entanglement.
Abstract In quantum information science, a major challenge is to look for an efficient means for classifying quantum states. An attractive proposal is to utilize Bell’s inequality as an entanglement witness, for classifying entangled state. The problem is that entanglement is necessary but not sufficient for violating Bell’s inequalities, making these inequalities unreliable in state classification. Furthermore, in general, classifying the separability of states, even for only few qubits, is resource-consuming. Here we look for alternative solutions with the methods of machine learning, by constructing neural networks that are capable of simultaneously encoding convex sets of multiple entanglement witness inequalities. The simulation results indicated that these transformed Bell-type classifiers can perform significantly better than the original Bell’s inequalities in classifying entangled states. We further extended our analysis to classify quantum states into multiple species through machine learning. These results not only provide an interpretation of neural network as quantum state classifier, but also confirm that neural networks can be a valuable tool for quantum information processing.