Persistent Tensors and Multiqudit Entanglement Transformation

Year: 2024

Authors: Gharahi M., Lysikov V.

Autors Affiliation: INO CNR, QSTAR, Largo Enrico Fermi 2, I-50125 Florence, Italy; Ruhr Univ Bochum, Ruhr Univ Bochum, D-44801 Bochum, Germany.

Abstract: We construct a lower bound of the tensor rank for a new class of tensors, which we call persistent tensors. We present three specific families of persistent tensors, of which the lower bound is tight. We show that there is a chain of degenerations between these three families of minimal -rank persistent tensors that can be used to study the entanglement transformation between them. In addition, we show that these three families of persistent tensors are indeed different generalizations of multiqubit W states within multiqudit systems and are geometrically in the orbit closure of multiqudit GHZ states. Consequently, we show that one can obtain every one of the generalizations of W state from a multiqudit GHZ state via asymptotic Stochastic Local Operations and Classical Communication (SLOCC) with rate one. Finally, we extend the obtained lower bound of the tensor rank to direct sums with persistent summands and to even more general combinations of tensors, which we call block pyramidal tensors. As a result, we show that the tensor rank is multiplicative under the Kronecker and tensor products of minimal -rank persistent tensors with the GHZ tensor.

Journal/Review: QUANTUM

Volume: 8      Pages from: 1238-1  to: 1238-19

More Information: We would like to thank Matthias Christandl for helpful discussions and anonymous reviewers for their comments. Part of the work was done while M. G. was at the University of Camerino and V. L. was at the University of Copenhagen. M. G. acknowledges the hospitality of the QMATH at the Univer- sity of Copenhagen where this work was carried out. M. G. 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. V. L. acknowledges financial support from VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the European Union (ERC Grant Agreements 818761 and 101040907) . Views and opinions expressed are however those of the author (s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
KeyWords: Cubic Matrices; Rank; Orbits
DOI: 10.22331/q-2024-01-31-1238

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