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arXiv:quant-ph/0211139

N-qubit Entanglement Index and Classification

Ying Wu

Published Nov 21, 2002• 1 citation
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Abstract

We show that all the N-qubit states can be classified as N entanglement classes each of which has an entanglement index $E=N-p=0,1,...,N-1$(E=0 corresponds to a fully separate class) where $p$ denotes number of groups for a partition of the positive integer N. In other words, for any partition $(n_1,n_2,...,n_p)$ of N with $n_j\ge 1$ and $N=\sum_{j=1}^{p}n_j$, the entanglement index for the corresponding state $ρ_{n_1}\bigotimesρ_{n_2}...\bigotimes ρ_{n_p}$ with $ρ_{n_j}$ denoting a fully entangled state of $n_j-$qubits is $E(ρ_{n_1}\bigotimesρ_{n_2}...\bigotimes ρ_{n_p})=\sum_{j=1}^{p}(n_j-1)=N-p$.

DOI
10.48550/arxiv.quant-ph/0211139

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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.
Entanglement Classification in the Graph States: The generalization to nnn-Qubits States using the Entanglement Matrix
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We provide a classification of entangled states that uses new discrete entanglement invariants. The invariants are defined by algebraic properties of linear maps associated with the states. We prove a theorem on a correspondence between the invariants and sets of equivalent classes of entangled states. The new method works for an arbitrary finite number of finite-dimensional state subspaces. As an application of the method, we considered a large selection of cases of three subspaces of various dimensions. We also obtain an entanglement classification of four qubits, where we find 27 fundamental sets of classes.
We provide a group-theoretical classification of the entangled states of N identical particles. The connection between quantum entanglement and the exchange symmetry of the states of N identical particles is made explicit using the duality between the permutation group and the simple unitary group. Each particle has n-levels and spans the n-dimensional Hilbert space. We shall call the general state of the particle as a qunit. The direct product of the N qunit space is given a decomposition in terms of states with definite permutation symmetry. The nature of fundamental entanglement of a state can be related to the classes of partitions of the integer N. The maximally entangled states are generated from linear combinations of the less entangled states of the direct product space. We also discuss the nature of maximal entanglement and its measures.
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