Communications in Mathematical Physics

, Volume 333, Issue 2, pp 541–563 | Cite as

Universal Subspaces for Local Unitary Groups of Fermionic Systems

  • Lin ChenEmail author
  • Jianxin Chen
  • Dragomir Ž. Đoković
  • Bei Zeng


Let \({\mathcal{V}=\wedge^{N} V}\) be the N-fermion Hilbert space with M-dimensional single particle space V and 2N ≤ 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 \({e_{i_1,\cdots,i_N}:= |{v_{i_1}\wedge v_{i_2} \wedge\cdots\wedge v_{i_N}}\rangle}\) with i 1 < ... < i N form an o.n. basis of \({\mathcal{V}}\) . Let \({\mathcal{S}\subseteq\mathcal{V}}\) be the subspace spanned by all \({e_{i_1,\cdots,i_N}}\) such that the set {i 1,...,i N } contains no pair {2k−1,2k}, k an integer. We say that the \({|{\psi}\rangle \in\mathcal{S}}\) are single occupancy states (with respect to the basis |v 1⟩,...,|v M ⟩). We prove that for N = 3 the subspace \({\mathcal{S}}\) is universal, i.e., each G-orbit in \({\mathcal{V}}\) meets \({\mathcal{S}}\) , 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 \({\mathcal{W}\subseteq\mathcal{S}}\) spanned by M(M−1)(M−5)/6 states \({e_{i_1,\ldots,i_N}}\) . Moreover, the number M(M−1)(M−5)/6 is minimal.


Unitary Group Fermionic System Complex Hilbert Space Slater Determinant Entanglement Property 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Lin Chen
    • 1
    • 2
    • 3
    • 5
    Email author
  • Jianxin Chen
    • 2
    • 4
  • Dragomir Ž. Đoković
    • 1
    • 2
  • Bei Zeng
    • 2
    • 4
  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  3. 3.Center for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  4. 4.Department of Mathematics and StatisticsUniversity of GuelphGuelphCanada
  5. 5.Singapore University of Technology and DesignSingaporeSingapore

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