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The moduli space of three qutrit states

Briand, E, Jean-Gabriel Luque, Jean‐Yves Thibon, Verstraete, F

Published Jun 18, 2003• 38 citations
OpenAlex

Abstract

We consider pure entangled three-partite qutrit states and classify all possible states according to the (semi-stable) orbits generated by local filtering operations (SLOCC). We give a complete description of the orbit space both in its affine and projective versions. This leads to a very symmetric normal form for pure states of three qutrits, parameterized by three complex numbers.

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We consider pure entangled three-partite qutrit states and classify all possible states according to the (semi-stable) orbits generated by local filtering operations (SLOCC). We give a complete description of the orbit space both in its affine and projective versions. This leads to a very symmetric normal form for pure states of three qutrits, parameterized by three complex numbers.
We study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behavior of pure entangled three-partite qutrit states and their normal forms under local filtering operations (SLOCC). We describe the orbit space of the SLOCC group SL(3,C)×3 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers. The parameters of the possible normal forms of a given state are roots of an algebraic equation, which is proved to be solvable by radicals. The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes.
Classification and quantification of bipartite qubit, tripartite qubit, and bipartite qutrit entanglement are studied in detail based on the entanglement measure of multipartite pure states. The results show that different ways of pure state entanglement can be characterized by extremal values of the corresponding entanglement measure under the stochastic local operation and classical communication(SLOCC). Detailed analysis shows that there are three different kinds of entanglement under the SLOCC in both tripartite qubit and bipartite qutrit pure state systems. Comparison of the measure with the recently proposed entropy-product measure is made and briefly discussed.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems. PACS: 03.67.-a; 03.67.Mn; 03.65.Ud; 03.65.Bz Quantum entanglement is an important resource to be implemented in quantum information processing and quantum communication protocols. [1] The interests in studying multi-dimensional entangled states come from both the foundations of quantum mechanics and the development of new protocols in quantum communication. For example, it was shown that maximally entangled states of two quantum systems in a high-dimensional Hilbert space, qudits, violate local realism stronger than qubits, and entangled qudits are less affected by noise than entangled qubits. [2,3] In quantum cryptography [4], the use of entangled qutrits [4,5] or qudits [6,7] instead of qubits is more secure against eavesdropping attacks. Furthermore, the protocols for quantum teleportation or for quantum cryptography, work best with maximally entangled states. All these facts motivate the development of techniques to generate entangled states among quantum systems in higher dimensional Hilbert space with a good quality of entanglement.
Abstract In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety X of separable three-qutrit states within the projective Hilbert space P (  ) = P 26 . Given a quantum pure state ∣ φ 〉 ∈ P (  ) we define the X φ -hypersuface by cutting X with a hyperplane H φ defined by the linear form 〈 φ ∣ (the X φ -hypersurface of X is X ∩ H φ ⊂ X ). We prove that when ∣ φ 〉 ranges over the stochastic local operation and classical communication entanglement classes, the ‘worst’ possible singular X φ -hypersuface with isolated singularities, has a unique singular point of type D 4 .