Classifying entanglement by algebraic geometry

Year: 2024

Authors: Gharahi., M.

Autors Affiliation: INO CNR, QSTAR, Largo Enr Fermi 2, I-50125 Florence, Italy.

Abstract: Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by k-secant varieties of the Segre variety and l-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.

Journal/Review: INTERNATIONAL JOURNAL OF QUANTUM INFORMATION

Volume: 22 (3)      Pages from:   to:

More Information: The work was done while the author was at the University of Camerino. The author also acknowledges financial support from the CNR-INO grant agreement n. 4125 and the European Commission through the H2020 QuantERA ERA-NET Co-fund in Quantum Technologies project MENTA.
KeyWords: Classification; entanglement; multipartite; multiqubit; multiqudit; SLOCC; asymptotic SLOCC; tensors; tensor rank; tensor border rank; persistent tensors; W states; GHZ states
DOI: 10.1142/S0219749923500478

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