Communications in Mathematical Physics

, Volume 280, Issue 1, pp 263–280 | Cite as

Gaussian Quantum Marginal Problem

  • Jens Eisert
  • Tomáš Tyc
  • Terry Rudolph
  • Barry C. Sanders


The quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system. This setting, also referred to as the question of the compatibility of local spectra, has several applications in quantum information theory. Here, we introduce the analogue of this statement for Gaussian states for any number of modes, and solve it in generality, for pure and mixed states, both concerning necessary and sufficient conditions. Formally, our result can be viewed as an analogue of the Sing-Thompson Theorem (respectively Horn’s Lemma), characterizing the relationship between main diagonal elements and singular values of a complex matrix: We find necessary and sufficient conditions for vectors (d 1,..., d n ) and (c 1,..., c n ) to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix, respectively. More physically speaking, this result determines what local temperatures or entropies are consistent with a pure or mixed Gaussian state of several modes. We find that this result implies a solution to the problem of sharing of entanglement in pure Gaussian states and allows for estimating the global entropy of non-Gaussian states based on local measurements. Implications to the actual preparation of multi-mode continuous-variable entangled states are discussed. We compare the findings with the marginal problem for qubits, the solution of which for pure states has a strikingly similar and in fact simple form.


Gaussian State Unitarily Invariant Norm Marginal Problem Composite Quantum System Main Diagonal Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Higuchi A., Sudbery A. and Szulc J. (2003). One-qubit reduced states of a pure many-qubit state: polygon inequalities. Phys. Rev. Lett. 90: 107902 CrossRefADSMathSciNetGoogle Scholar
  2. Higuchi, H.: On the one-particle reduced density matrices of a pure three-qutrit quantum state., 2003
  3. Bravyi S. (2004). Compatibility between local and multipartite states. Quant. Inf. Comp. 4: 12–26 MathSciNetzbMATHGoogle Scholar
  4. Han Y.-J., Zhang Y.-S. and Guo G.-C. (2005). Compatibility relations between the reduced and global density matrices Phys. Rev. A 71: 052306 CrossRefGoogle Scholar
  5. Klyachko, A.: Quantum marginal problem and representations of the symmetric group., 2004
  6. Franz M. (2002). Moment polytopes of projective G-varieties and tensor products of symmetric group representations. J. Lie Theory 12: 539–549 MathSciNetzbMATHGoogle Scholar
  7. Christandl M. and Winter A. (2004). Squashed entanglement: An additive entanglement measure. J. Math. Phys. 45: 829–840 CrossRefADSMathSciNetzbMATHGoogle Scholar
  8. Terhal B.M., Koashi M. and Imoto N. (2003). Unconditionally secure key distribution based on two nonorthogonal states. Phys. Rev. Lett. 90: 167904 CrossRefADSGoogle Scholar
  9. Nielsen M.A. and Kempe J. (2001). Separable states are more disordered globally than locally. Phys. Rev. Lett. 86: 5184–5187 CrossRefADSGoogle Scholar
  10. Eisert J., Audenaert K. and Plenio M.B. (2003). Remarks on entanglement measures and non-local state distinguishability. J. Phys. A: Math. Gen. 36: 5605–5615 CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. Daftuar S. and Hayden P. (2005). Quantum state transformations and the Schubert calculus. Ann. Phys. 315: 80–122 CrossRefADSMathSciNetzbMATHGoogle Scholar
  12. Hall W. (2007). Compatibility of subsystem states and convex geometry. Phys. Rev. A 75: 032102 CrossRefADSMathSciNetGoogle Scholar
  13. Liu Y.-K., Christandl M. and Verstraete F. (2007). Quantum computational complexity of the N-Representability Problem: QMA Complete Phys. Rev. Lett. 98: 110503 CrossRefADSGoogle Scholar
  14. Christandl M., Harrow A. and Mitchison G. (2007). On nonzero Kronecker coefficients and their consequences for spectra. Commun. Math. Phys. 270: 575–585 CrossRefADSMathSciNetzbMATHGoogle Scholar
  15. Christandl, M.: PhD thesis, (Cambridge, October 2005)Google Scholar
  16. Eisert J. and Plenio M.B. (2003). Introduction to the basics of entanglement theory in continuous-variable systems. Int. J. Quant. Inf. 1: 479–506 CrossRefzbMATHGoogle Scholar
  17. Braunstein S.L. and Loock P. (2005). Quantum information with continuous variables. Rev. Mod. Phys. 77: 513–577 CrossRefADSGoogle Scholar
  18. Adesso G., Serafini A. and Illuminati F. (2007). Optical state engineering, quantum communication, and robustness of entanglement promiscuity in three-mode Gaussian states. New J. Phys. 9: 60 CrossRefADSGoogle Scholar
  19. Adesso G., Serafini A. and Illuminati F. (2006). Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence. Phys. Rev. A 73: 032345 CrossRefADSGoogle Scholar
  20. Sing F.Y. (1976). Some results on matrices with prescribed diagonal elements and singular values. Canad. Math. Bull. 19: 89–92 MathSciNetzbMATHGoogle Scholar
  21. Thompson R.C. (1977). Singular values, diagonal elements and convexity. SIAM J. Appl. Math. 32: 39–63 CrossRefMathSciNetzbMATHGoogle Scholar
  22. Thompson R.C. (1979). Singular values and diagonal elements of complex symmetric matrices. Lin. Alg. Appl. 26: 65–106 CrossRefzbMATHGoogle Scholar
  23. Horn A. (1954). Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math. 76: 620–630 CrossRefMathSciNetzbMATHGoogle Scholar
  24. Mirsky L. (1964). Inequalities and existence theorems in the theory of matrices. J. Math. Anal. Appl. 9: 99–118 CrossRefMathSciNetGoogle Scholar
  25. Hyllus P. and Eisert J. (2006). Optimal entanglement witnesses for continuous-variable systems. New J. Phys. 8: 51 CrossRefADSGoogle Scholar
  26. Bhatia R. (1997). Matrix Analysis. Springer, Berlin, 254 Google Scholar
  27. Hiroshima T. (2006). Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs. Phys. Rev. A 73: 012330 CrossRefADSGoogle Scholar
  28. Botero A. and Reznik B. (2003). Modewise entanglement of Gaussian states. Phys. Rev. A 67: 052311 CrossRefADSGoogle Scholar
  29. Giedke G., Eisert J., Cirac J.I. and Plenio M.B. (2003). Entanglement transformations of pure Gaussian states. Quant. Inf. Comp. 3: 211–223 MathSciNetzbMATHGoogle Scholar
  30. Holevo A.S. and Werner R.F. (2001). Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63: 032312 CrossRefADSGoogle Scholar
  31. Arvind Dutta B., Mukunda N. and Simon R. (1995). The real symplectic groups in quantum mechanics and optics. Pramana 45(6): 471–497 CrossRefADSGoogle Scholar
  32. Adesso G., Serafini A. and Illuminati F. (2004). Extremal entanglement and mixedness in continuous variable systems. Phys. Rev. A 70: 022318 CrossRefADSGoogle Scholar
  33. Reck M., Zeilinger A., Bernstein H.J. and Bertani P. (1994). Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73: 58–61 CrossRefADSGoogle Scholar
  34. Holevo A.S. and Werner R.F. (2001). Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63: 032312 CrossRefADSGoogle Scholar
  35. Eisert J. and Wolf M.M. (2007). Gaussian quantum channels. In: Cerf, N.J., Leuchs, G. and Polzik, E.J. (eds) Quantum Information with Continuous Variables of Atoms and Light, pp 23–42. Imperial College Press, London Google Scholar
  36. Eisert J., Scheel S. and Plenio M.B. (2002). Distilling Gaussian states with Gaussian operations is impossible. Phys. Rev. Lett. 89: 137903 CrossRefADSMathSciNetGoogle Scholar
  37. Fiurášek J. (2002). Gaussian transformations and distillation of entangled Gaussian states. Phys. Rev. Lett. 89: 137904 CrossRefADSMathSciNetGoogle Scholar
  38. Giedke G. and Cirac J.I. (2002). Characterization of Gaussian operations and distillation of Gaussian states. Phys. Rev. A 66: 032316 CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Jens Eisert
    • 1
    • 2
    • 3
  • Tomáš Tyc
    • 4
  • Terry Rudolph
    • 2
  • Barry C. Sanders
    • 5
  1. 1.Institute for Mathematical SciencesImperial College LondonLondonUK
  2. 2.Blackett LaboratoryImperial College LondonLondonUK
  3. 3.Physics DepartmentUniversity of PotsdamPotsdamGermany
  4. 4.Institute of Theoretical PhysicsMasaryk UniversityBrnoCzech Republic
  5. 5.Institute for Quantum Information ScienceUniversity of CalgaryAlbertaCanada

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