1301.3421 (Lin Chen et al.)

Universal Subspaces for Local Unitary Groups of Fermionic Systems    [PDF]

Lin Chen, Jianxin Chen, Dragomir Z. Djokovic, Bei Zeng

For the representation $\rho$ of the local unitary (LU) group $G$ on the $N$-fermion Hilbert space $\mathcal{V}=\wedge^N V$ with $M$-dimensional single particle space $V$, a universal subspace $\mathcal{W}\subseteq\mathcal{V}$ is a subspace that meets all $G$-orbits in $\mathcal{V}$. We focus on the case that $\mathcal{W}$ is spanned by Slater determinants $\ket{v_1}\wedge\ket{v_2}\we...\wedge\ket{v_N}$ built from orthonormal local basis ${\ket{v}}$. Hence our discussion is a fermionic analogue of the universal subspace spanned by local basis product states studied in quantum information theory. While the N=2 case has a straightforward solution based on the existence of Slater decomposition, which is an analogue of the Schmidt decomposition, for $N>2$ case the problem is nontrivial due to the lack of a similar decomposition. We introduce a configuration for the Slater determinants called single occupancy, where the single particle states ${\ket{v}}$ are paired and none of such pairs occurs entirely in $\ket{v_1}\wedge\ket{v_2}\we...\wedge\ket{v_N}$. We show that, for N=3 the subspace $\mathcal{S}$ spanned by all the single occupancy states is universal. For $M$ even, we also obtain universal subspaces of $\mathcal{S}$ whose dimensions are minimal. For the smallest non-trivial case of $N=3,M=6$ where $\dim\wedge^3 V=20$, a direct application of our results gives universal subspaces with dimension 5. We further show that it is not the case for $N>3$, i.e. not all fermionic states are LU equivalent to a single occupancy state. For $N$ even, BCS states provide explicit examples for states that are not LU equivalent to any single occupancy state.

View original: http://arxiv.org/abs/1301.3421