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arXiv:1606.05537 · Journal of Physics A Mathematical and Theoretical

Three-qutrit entanglement and simple singularities

Frédéric Holweck, Hamza Jaffali

Published Oct 26, 2016• 11 citations
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Abstract

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 .

DOI
10.1088/1751-8113/49/46/465301

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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 .
An intriguing correspondence between four-qubit systems and simple singularity of type D 4 is established. We first consider the algebraic variety X of separable states within the projective Hilbert space . Then, cutting X with a specific hyperplane H , we prove that the X -hypersurface, defined from the section X ∩ H ⊂ X , has an isolated singularity of type D 4 ; it is also shown that this is the ‘worst-possible’ isolated singularity that one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version, by proving that the equation of the dual variety of X , which is nothing but the Cayley hyperdeterminant of type 2 × 2 × 2 × 2, can be expressed in terms of the stochastic local operation and classical communication-invariant polynomials as the discriminant of the miniversal deformation of the D 4 -singularity. As a consequence of this correspondence one obtains a finer-grained classification of entanglement classes of four-qubit systems.
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.
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).
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.