Lagrangian representation for fermionic linear optics

Details Lagrangian representation for fermionic linear optics Lagrangian representation for fermionic linear optics

2004-04-30

arxiv.org/abs/quant-ph/0404180v2

Quantum Inf. and Comp., Vol. 5, No. 3 (2005) pp.216-238

Physics

Quantum Physics

12 pages, Revtex4, significantly improved version

Scientific paper

Notions of a Gaussian state and a Gaussian linear map are generalized to the case of anticommuting (Grassmann) variables. Conditions under which a Gaussian map is trace preserving and (or) completely positive are formulated. For any Gaussian map an explicit formula relating correlation matrices of input and output states is presented. This formalism allows to develop the Lagrangian representation for fermionic linear optics (FLO). It covers both unitary operations and the single-mode projectors associated with FLO measurements. Using the Lagrangian representation we reduce a classical simulation of FLO to a computation of Gaussian integrals over Grassmann variables. Explicit formulas describing evolution of a quantum state under FLO operations are put forward.

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