Universal Subspaces for Local Unitary Groups of Fermionic Systems

Abstract

Let $\mathcal{V}={\wedge }^{N}V$ be the $N$-fermion Hilbert space with $M$-dimensional single particle space $V$ and $2N\le M$. We refer to the unitary group $G$ of $V$ as the local unitary (LU) group. We fix an orthonormal (o.n.) basis $|v_1⟩,...,|v_M⟩$ of $V$. Then the Slater determinants $e_{i_1,...,i_N}:= \ket{v_{i_1}\we v_{i_2}\we...\we v_{i_N}}$ with $i_1<...<i_N$ form an o.n. basis of $\cV$. Let $\cS\subseteq\cV$ be the subspace spanned by all $e_{i_1,...,i_N}$ such that the set $\{i_1,...,i_N\}$ contains no pair $\{2k-1,2k\}$, $k$ an integer. We say that the $\ketψ\in\cS$ are single occupancy states (with respect to the basis $|v_1⟩,...,|v_M⟩$). We prove that for N=3 the subspace $\cS$ is universal, i.e., each $G$-orbit in $\cV$ meets $\cS$, and that this is false for N>3. If $M$ is even, the well known BCS states are not LU-equivalent to any single occupancy state. Our main result is that for N=3 and $M$ even there is a universal subspace $\cW\subseteq\cS$ spanned by $M(M-1)(M-5)/6$ states $e_{i_1,...,i_N}$. Moreover the number $M(M-1)(M-5)/6$ is minimal.

25 pages, 2 figures. Abstract has been rewritten

References in corpus (4)

• N-representability is QMA-complete
• The Pauli principle revisited
• Connecting N-representability to Weyl's problem: The one particle density matrix for N = 3 and R = 6
• Bipartite entanglement, spherical actions and geometry of local unitary orbits

Cited by in corpus (4)

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• Quantum marginals from pure doubly excited states
• Optimal Slater-determinant approximation of fermionic wave functions
• Geometric entanglement in the Laughlin wave function