Abstract
AbstractWe introduce the bosonic and fermionic ensembles of density matrices and study their entanglement. In the fermionic case, we show that random bipartite fermionic density matrices have non-positive partial transposition; hence, they are typically entangled. The similar analysis in the bosonic case is more delicate, due to a large positive outlier eigenvalue. We compute the asymptotic ratio between the size of the environment and the size of the system Hilbert space for which random bipartite bosonic density matrices fail the PPT criterion, being thus entangled. We also relate moment computations for tensor-symmetric random matrices to evaluations of the circuit counting and interlace graph polynomials for directed graphs.
Funder
Agence Nationale de la Recherche
Australian Research Council
Publisher
Springer Science and Business Media LLC
Subject
Electrical and Electronic Engineering,Modeling and Simulation,Signal Processing,Theoretical Computer Science,Statistical and Nonlinear Physics,Electronic, Optical and Magnetic Materials
Reference29 articles.
1. Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial of a graph. J. Comb. Theory Ser. B 92(2), 199–233 (2004)
2. Aubrun, G., Nechita, I.: Realigning random states. J. Math. Phys. 53(10), 102210 (2012)
3. Aubrun, G.: Partial transposition of random states and non-centered semicircular distributions. Random Matrices: Theory Appl. 1(02), 1250001 (2012)
4. Biamonte, J., Bergholm, V.: Tensor networks in a nutshell. (2017). arXiv preprint arXiv:1708.00006
5. Brijder, R., Hoogeboom, H.J.: Nullity invariance for pivot and the interlace polynomial. Linear Algebra Appl. 435(2), 277–288 (2011)