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arXiv:quant-ph/0006107

Group-theoretical Structure of the Entangled States of N Identical Particles

Suranjana Rai, Jagdish Rai

Published Jun 23, 2000• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0006107

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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.
We explore the connection between quantum entanglement and the exchange symmetry of the states of N identical particles. Each particle has n-levels. The N particles span the nN 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 that are found to have a measure of entanglement which is related to the representation of the permutation group. The maximally entangled states are generated from the linear combinations of fully correlated but unentangled states. The states of lower entanglement are generated from the manifold of partially correlated states. The degree of exchange symmetry is found to be related to a group theoretical measure of entanglement. Email: Jagdish_Rai@hotmail.com, jrai@iitk.ac.in Email: srai56@hotmail.com
We analyze entanglement classes for permutation-symmetric states for nnn qudits (i.e., ddd-level systems), with respect to local unitary operations (LU equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the Jordan form of such one-particle operators is a SLOCC invariant. We find, as representatives of those classes, a discrete set of entangled states that generalize the Greenberger-Horne-Zeilinger and WWW states for the many-particle qudit case. In the latter case, we introduce the excitation state as a natural generalization of the WWW state for d>2d>2d>2.
With the advent of general relativity and quantum mechanics, physics has in many cases shifted the emphasis from dynamics to symmetry principles associated with different types of mathematical groups. We speak of the group structure of crystals, the group invariants of the equations of motion, the group classification of elementary particles, and also of the group homeomorphisms of fractals. The objective of this article is to show how the group theoretical structure of entanglement can be used to classify atomic and molecular structure. The article begins by establishing a working definition of entanglement and then using it to derive the Pauli-exclusion principle by showing that Fermi-Dirac statistics can be considered as an entangled state invariant under the action of the SL(n, C)With the advent of general relativity and quantum mechanics, physics has in many cases shifted the emphasis from dynamics to symmetry principles associated with different types of mathematical groups. We speak of the group structure of crystals, the group invariants of the equations of motion, the group classification of elementary particles, and also of the group homeomorphisms of fractals. The objective of this article is to show how the group theoretical structure of entanglement can be used to classify atomic and molecular structure. The article begins by establishing a working definition of entanglement and then using it to derive the Pauli-exclusion principle by showing that Fermi-Dirac statistics can be considered as an entangled state invariant under the action of the SL(n,C)SL(n, C)SL(n,C) group. Based on this, a group classification scheme of entangled states is proposed to distinguish atomic and molecular structures associated with chemical bonds. group. Based on this, a group classification scheme of entangled states is proposed to distinguish atomic and molecular structures associated with chemical bonds.
Permutation symmetries of multipartite quantum states are defined only when the constituent subsystems are of equal dimensions. In this work we extend this notion of permutation symmetry to heterogeneous systems, that is, systems composed of subsystems having unequal dimensions. Given a tensor product space of kkk subsystems (of arbitrary dimensions) and a permutation operation σσσ over kkk symbols, these states are such that they have identical decompositions (up to an overall phase) in the given tensor product space and the tensor product space obtained by the permuting the subsystems by σσσ. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation σσσ. Eigenvectors of this matrix have the required symmetry. We then examine entanglement of states in the igenspaces of these matrices. It is found that all nonsymmetric eigenspaces of such matrices are completely entangled subspaces, with states being equally entangled in both the given tensor product space and the permuted tensor product space.
Permutation symmetries of multipartite quantum states are defined only when the constituent subsystems are of equal dimensions. In this work we extend this notion of permutation symmetry to heterogeneous systems, that is, systems composed of subsystems having unequal dimensions. Given a tensor product space of kkk subsystems (of arbitrary dimensions) and a permutation operation σ\sigmaσ over kkk symbols, these states are such that they have identical decompositions (up to an overall phase) in the given tensor product space and the tensor product space obtained by the permuting the subsystems by σ\sigmaσ. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation σ\sigmaσ. Eigenvectors of this matrix have the required symmetry. We then examine entanglement of states in the igenspaces of these matrices. It is found that all nonsymmetric eigenspaces of such matrices are completely entangled subspaces, with states being equally entangled in both the given tensor product space and the permuted tensor product space.
We consider entanglement in a system with a fixed number of identical particles. Since any operation should be symmetrized over all the identical particles and there is the precondition that the spatial wave functions overlap, the meaning of identical-particle entanglement is fundamentally different from that of distinguishable particles. The identical-particle counterpart of the Schmidt basis is shown to be the single-particle basis in which the one-particle reduced density matrix is diagonal. But it does not play a special role in the issue of entanglement, which depends on the single-particle basis chosen. The nonfactorization due to (anti)symmetrization is naturally excluded by using the (anti)symmetrized basis or, equivalently, the particle number representation. The natural degrees of freedom in quantifying the identical-particle entanglement in a chosen single-particle basis are occupation numbers of different single-particle basis states. The entanglement between effectively distinguishable spins is shown to be a special case of the occupation-number entanglement.