Abstract
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.
Similar content being viewed by others
References
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
Higuchi, H.: On the one-particle reduced density matrices of a pure three-qutrit quantum state. http://arxiv.org/list/quant-ph/0309186, 2003
Bravyi S. (2004). Compatibility between local and multipartite states. Quant. Inf. Comp. 4: 12–26
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
Klyachko, A.: Quantum marginal problem and representations of the symmetric group. http://arxiv.org/list/quant-ph/0409113, 2004
Franz M. (2002). Moment polytopes of projective G-varieties and tensor products of symmetric group representations. J. Lie Theory 12: 539–549
Christandl M. and Winter A. (2004). Squashed entanglement: An additive entanglement measure. J. Math. Phys. 45: 829–840
Terhal B.M., Koashi M. and Imoto N. (2003). Unconditionally secure key distribution based on two nonorthogonal states. Phys. Rev. Lett. 90: 167904
Nielsen M.A. and Kempe J. (2001). Separable states are more disordered globally than locally. Phys. Rev. Lett. 86: 5184–5187
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
Daftuar S. and Hayden P. (2005). Quantum state transformations and the Schubert calculus. Ann. Phys. 315: 80–122
Hall W. (2007). Compatibility of subsystem states and convex geometry. Phys. Rev. A 75: 032102
Liu Y.-K., Christandl M. and Verstraete F. (2007). Quantum computational complexity of the N-Representability Problem: QMA Complete Phys. Rev. Lett. 98: 110503
Christandl M., Harrow A. and Mitchison G. (2007). On nonzero Kronecker coefficients and their consequences for spectra. Commun. Math. Phys. 270: 575–585
Christandl, M.: PhD thesis, (Cambridge, October 2005)
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
Braunstein S.L. and Loock P. (2005). Quantum information with continuous variables. Rev. Mod. Phys. 77: 513–577
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
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
Sing F.Y. (1976). Some results on matrices with prescribed diagonal elements and singular values. Canad. Math. Bull. 19: 89–92
Thompson R.C. (1977). Singular values, diagonal elements and convexity. SIAM J. Appl. Math. 32: 39–63
Thompson R.C. (1979). Singular values and diagonal elements of complex symmetric matrices. Lin. Alg. Appl. 26: 65–106
Horn A. (1954). Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math. 76: 620–630
Mirsky L. (1964). Inequalities and existence theorems in the theory of matrices. J. Math. Anal. Appl. 9: 99–118
Hyllus P. and Eisert J. (2006). Optimal entanglement witnesses for continuous-variable systems. New J. Phys. 8: 51
Bhatia R. (1997). Matrix Analysis. Springer, Berlin, 254
Hiroshima T. (2006). Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs. Phys. Rev. A 73: 012330
Botero A. and Reznik B. (2003). Modewise entanglement of Gaussian states. Phys. Rev. A 67: 052311
Giedke G., Eisert J., Cirac J.I. and Plenio M.B. (2003). Entanglement transformations of pure Gaussian states. Quant. Inf. Comp. 3: 211–223
Holevo A.S. and Werner R.F. (2001). Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63: 032312
Arvind Dutta B., Mukunda N. and Simon R. (1995). The real symplectic groups in quantum mechanics and optics. Pramana 45(6): 471–497
Adesso G., Serafini A. and Illuminati F. (2004). Extremal entanglement and mixedness in continuous variable systems. Phys. Rev. A 70: 022318
Reck M., Zeilinger A., Bernstein H.J. and Bertani P. (1994). Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73: 58–61
Holevo A.S. and Werner R.F. (2001). Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63: 032312
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
Eisert J., Scheel S. and Plenio M.B. (2002). Distilling Gaussian states with Gaussian operations is impossible. Phys. Rev. Lett. 89: 137903
Fiurášek J. (2002). Gaussian transformations and distillation of entangled Gaussian states. Phys. Rev. Lett. 89: 137904
Giedke G. and Cirac J.I. (2002). Characterization of Gaussian operations and distillation of Gaussian states. Phys. Rev. A 66: 032316
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.B. Ruskai
Rights and permissions
About this article
Cite this article
Eisert, J., Tyc, T., Rudolph, T. et al. Gaussian Quantum Marginal Problem. Commun. Math. Phys. 280, 263–280 (2008). https://doi.org/10.1007/s00220-008-0442-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0442-4