Algebraic geometry tools for the study of entanglement: an application to spin squeezed states

Year: 2012

Authors: Bernardi A., Carusotto I.

Autors Affiliation: GALAAD, INRIA Méditerranée, 2004 route del Lucioles, BP 93, F-06902 Sophia Antipolis Cedex, France; INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, via Sommarive 14, I-38123 Povo, Trento, Italy

Abstract: A short review of algebraic geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics. Examples of application to assemblies of indistinguishable two-level bosonic atoms are discussed using modern formulations of the classical Sylvester algorithm for the decomposition of homogeneous polynomials in two variables. In particular, the symmetric rank and symmetric border rank of spin squeezed states are calculated as well as their Schrodinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different.


Volume: 45 (10)      Pages from: 105304  to: 105304

More Information: AB was partially supported by Project Galaad of INRIA Sophia Antipolis Mediterranee (France) and Marie Curie Intra-European Fellowships for Career Development (FP7-PEOPLE-2009-IEF): \’DECONSTRUCT\’. IC is grateful to C Miniatura and P Vignolo for the kind hospitality at INLN and acknowledges financial support from the ERC via the QGBE grant. Stimulating discussions with P Hyllus are warmly acknowledged.
KeyWords: algebraic geometry; quantum mechanics; matrix multiplication; symmetric tensors; complexity
DOI: 10.1088/1751-8113/45/10/105304

Citations: 13
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