Abstract
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.
Similar content being viewed by others
References
Nielsen M., Chuang I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)
Amico L., Fazio R., Osterloh A., Vedral V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008)
Acin A., Andrianov A., Costa L., Jane E., Latorre J.I., Tarrach R.: Generalized Schmidt decomposition and classification of three-quantum-bit States. Phys. Rev. Lett. 85, 1560–1563 (2000)
Acin A., Andrianov A., Jané E., Tarrach R.: Three-qubit pure-state canonical forms. J. Phys. A Math. Gen. 34, 6725 (2001)
An J., Đoković D.Ž.: Universal subspaces for compact Lie groups. J. Reine Angew. Math. 647, 151–173 (2010)
Carteret H.A., Higuchi A., Sudbery A.: Multipartite generalisation of the Schmidt decomposition. J. Math. Phys. 41, 7932–7939 (2000)
Schliemann J., Loss D., MacDonald A.H.: Double-occupancy errors, adiabaticity, and entanglement of spin qubits in quantum dots. Phys. Rev. B 63, 085311 (2001)
Schliemann J., Cirac J., Kus M., Lewenstein M., Loss D.: Quantum correlations in two-fermion systems. Phys. Rev. A 64, 022303 (2001)
Li Y.S., Zeng B., Liu X.S., Long G.L.: Entanglement in a two-identical-particle system. Phys. Rev. A 64, 054302 (2001)
Paskauskas P., You L.: Quantum correlations in two-boson wave functions. Phys. Rev. A 64, 042310 (2001)
Eckert K., Schliemann J., Bruss D., Lewenstein M.: Quantum correlations in systems of indistinguishable particles. Ann. Phys. 229, 88–127 (2002)
Wootters W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)
Bourbaki N.: Chapitres 1 à à 3, Hermann, Paris (1970)
Hasset B.: Introduction to Algebraic Geometry. Cambridge University Press, Cambridge (2007)
Weng Z.-Y.: Superconducting ground state of a doped Mott insulator. New J. Phys. 13, 103039 (2011)
Liu Y.-K., Christandl M., Verstraete F.: Quantum computational complexity of the N-representability problem: QMA complete. Phys. Rev. Lett. 98, 110503 (2007)
Ocko S.A., Chen X., Zeng B., Yoshida B., Ji Z., Ruskai M.B., Chuang I.L.: Quantum codes give counterexamples to the unique pre-image conjecture of the N-representability problem. Phys. Rev. Lett. 106, 110501 (2011)
Coleman A.J.: Structure of fermion density matrices. Rev. Mod. Phys. 35, 668–686 (1963)
Borland R.E., Dennis K.: The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six. J. Phys. B Atom. Mol. Phys. 5, 7–15 (1972)
Klyachko, A.: Quantum marginal problem and N-representability. A talk at 12 Central European workshop on Quantum Optics, Bilkent University, Turkey. J. Phys. Conf. Ser. 36, 72–86 (2006)
Ruskai M.B.: Connecting N-representability to Weyl’s problem: the one-particle density matrix for N = 3 and R = 6. J. Phys. A Math. Theor. 40, F961–F967 (2007)
Klyachko, A.: The Pauli exclusion principle and beyond (2009). quant-ph/0904.2009
Horn R., Johnson C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Huckleberry A., Kus M., Sawicki A.: Bipartite entanglement, spherical actions and geometry of local unitary orbits. J. Math. Phys. 54, 022202 (2013)
Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-1—a computer algebra system for polynomial computations (2010). http://www.singular.uni-kl.de
Popov V.L., Vinberg E.B. (1994) Invariant theory. In: Parshin, A.N., Shafarevich, I.R. (eds.) Algebraic Groups IV. Encycl. Math. Sci., vol. 55, Springer, Berlin
Bardeen J., Cooper L.N., Schrieffer J.R.: Microscopic theory of superconductivity. Phys. Rev. 106, 162–164 (1957)
Yang C.N.: Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors. Rev. Mod. Phys. 34, 694704 (1962)
Altunbulak Murat, Klyachko Alexander: The Pauli principle revisited. Commun. Math. Phys. 282, 287–322 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Winter
Rights and permissions
About this article
Cite this article
Chen, L., Chen, J., Đoković, D.Ž. et al. Universal Subspaces for Local Unitary Groups of Fermionic Systems. Commun. Math. Phys. 333, 541–563 (2015). https://doi.org/10.1007/s00220-014-2187-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2187-6