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arXiv:quant-ph/0403060 · Physical Review A

Geometry of three-qubit entanglement

Péter Lévay

Published Jan 27, 2005• 58 citations
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Abstract

A geometrical description of three-qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric $\mathcal{Q}$, a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of $\mathcal{Q}$ with interesting geometric properties. An invariant characterizing the Greenberger-Horne-Zeilinger class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.

DOI
10.1103/physreva.71.012334

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A geometrical description of three-qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric Q\mathcal{Q}Q, a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of Q\mathcal{Q}Q with interesting geometric properties. An invariant characterizing the Greenberger-Horne-Zeilinger class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.
The behavior of quantum states under local unitary transformations (LUTs) and stochastic local quantum operations and classical communication (SLOCC) has proven central to the understanding of entanglement in multipartite quantum systems. In particular, invariants under these operations have provided insight into quantum entanglement in multiple‐qubit states. Relationships between entanglement, mixedness and spin symmetry in multiple‐qubit quantum states can be found by exploiting these properties. For example, concurrence and n‐tangle are naturally expressed in terms of spin‐flip transformations. In the case of specialized entanglement measures and/or special families of states, complementarity relations involving these lengths and state transformations are derivable. Here, the role of geometry in such investigations is explored. Minkowskian geometry is seen to provide elegant representations of entanglement in multiple‐qubit systems, particularly the language of twistors.
The geometry of four-qubit entanglement is investigated. We replace some of the polynomial invariants for four qubits introduced recently by new ones of direct geometrical meaning. It is shown that these invariants describe four points, six lines and four planes in complex projective space CP 3 . For the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables. Moreover, suitable powers of their magnitudes are entanglement monotones that fit nicely into the geometric set of n -qubit ones related to Grassmannians of l -planes found recently. We also show that in terms of these invariants the hyperdeterminant of order 24 in the four-qubit amplitudes takes a more instructive form than the previously published expressions available in the literature. Finally, in order to understand two-, three- and four-qubit entanglement in geometric terms we propose a unified setting based on CP 3 furnished with a fixed quadric.
We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.
We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by Stochastic Local Operations with Classical Communication (SLOCC) invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete et al. are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
We show that the three-qubit entanglement classes, (0) null, (1) separable $A\text{\ensuremath{-}}B\text{\ensuremath{-}}C$, (2a) biseparable $A\text{\ensuremath{-}}BC$, (2b) biseparable $B\text{\ensuremath{-}}CA$, (2c) biseparable $C\text{\ensuremath{-}}AB$, (3) WWW, and (4) Greenberger-Horne-Zeilinger, correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3, and 4 of a Freudenthal triple system defined over the Jordan algebra $\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}$. We also compute the corresponding stochastic local operations and classical communication orbits.