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IEEE/ACIS International Conference on Computer and Information Science (ICIS)

A Comparison of Bipartite N-Qubit States to Classify Entangled States under Symmetric Consideration

Alauddin Ahmed, Atiqur Rahman, Mohammed Anwer, Nazmul Haque, Morshed Uddin Chowdhury

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We show the relation between partition and the permutation symmetry for bipartite N-qubits via Young tableaux. The word and weight formalism of Young tableaux provide a simple way of quantum comparison and we use this comparison together with partition to classify bipartite N-qubit states as different entanglement classes each of which has an entanglement index.

DOI
10.1109/icis.2007.11

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We show the relation between partition and the permutation symmetry for bipartite N-qubits via Young tableaux. The word and weight formalism of Young tableaux provide a simple way of quantum comparison and we use this comparison together with partition to classify bipartite N-qubit states as different entanglement classes each of which has an entanglement index.
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.
A general method in constructing a complete set of wave functions for multipartite identical qubits is presented based on the irreducible representations of the permutation group and the nth rank tensors. Particular examples for n =2, 3, and 4 are derived and the entanglement behavior for each state is examined from several criteria. It is found that the states so constructed are all bound entangled states. For the case of even n, all the states are found to have maximum "n-tangle". The symmetry in spin space is found to increase the n-tangle in general. The "n-tangle" for n = 4 is found not always representing 4-way entanglement. It measures the degree of spin-space symmetry instead. A useful relationship in the classification between systems containing different number of qubits is given in terms of the Young's Tableaux based on our analysis.
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.
Using an inductive approach to classify multipartite entangled states under stochastic local operations and classical communication introduced recently by the authors [Phys. Rev. A 74, 052336 (2006)], we give the complete classification of four-qubit entangled pure states. Apart from the expected degenerate classes, we show that there exist eight inequivalent ways to entangle four qubits. In this respect, permutation symmetry is taken into account and states with a structure differing only by parameters inside a continuous set are considered to belong to the same class.