# Gaussian Quantum Marginal Problem

- 163 Downloads
- 12 Citations

## 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.

## Keywords

Gaussian State Unitarily Invariant Norm Marginal Problem Composite Quantum System Main Diagonal Element## Preview

Unable to display preview. Download preview PDF.

## 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 CrossRefADSMathSciNetGoogle Scholar - 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 MathSciNetzbMATHGoogle Scholar - 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 - 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 MathSciNetzbMATHGoogle Scholar - Christandl M. and Winter A. (2004). Squashed entanglement: An additive entanglement measure.
*J. Math. Phys.*45: 829–840 CrossRefADSMathSciNetzbMATHGoogle Scholar - 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 - Nielsen M.A. and Kempe J. (2001). Separable states are more disordered globally than locally.
*Phys. Rev. Lett.*86: 5184–5187 CrossRefADSGoogle Scholar - 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 - Daftuar S. and Hayden P. (2005). Quantum state transformations and the Schubert calculus.
*Ann. Phys.*315: 80–122 CrossRefADSMathSciNetzbMATHGoogle Scholar - Hall W. (2007). Compatibility of subsystem states and convex geometry.
*Phys. Rev. A*75: 032102 CrossRefADSMathSciNetGoogle Scholar - 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 - 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 - Christandl, M.: PhD thesis, (Cambridge, October 2005)Google Scholar
- 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 - Braunstein S.L. and Loock P. (2005). Quantum information with continuous variables.
*Rev. Mod. Phys.*77: 513–577 CrossRefADSGoogle Scholar - 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 - 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 - Sing F.Y. (1976). Some results on matrices with prescribed diagonal elements and singular values.
*Canad. Math. Bull.*19: 89–92 MathSciNetzbMATHGoogle Scholar - Thompson R.C. (1977). Singular values, diagonal elements and convexity.
*SIAM J. Appl. Math.*32: 39–63 CrossRefMathSciNetzbMATHGoogle Scholar - Thompson R.C. (1979). Singular values and diagonal elements of complex symmetric matrices.
*Lin. Alg. Appl.*26: 65–106 CrossRefzbMATHGoogle Scholar - Horn A. (1954). Doubly stochastic matrices and the diagonal of a rotation matrix.
*Amer. J. Math.*76: 620–630 CrossRefMathSciNetzbMATHGoogle Scholar - Mirsky L. (1964). Inequalities and existence theorems in the theory of matrices.
*J. Math. Anal. Appl.*9: 99–118 CrossRefMathSciNetGoogle Scholar - Hyllus P. and Eisert J. (2006). Optimal entanglement witnesses for continuous-variable systems.
*New J. Phys.*8: 51 CrossRefADSGoogle Scholar - Bhatia R. (1997). Matrix Analysis. Springer, Berlin, 254 Google Scholar
- Hiroshima T. (2006). Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs.
*Phys. Rev. A*73: 012330 CrossRefADSGoogle Scholar - Botero A. and Reznik B. (2003). Modewise entanglement of Gaussian states.
*Phys. Rev. A*67: 052311 CrossRefADSGoogle Scholar - 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 - Holevo A.S. and Werner R.F. (2001). Evaluating capacities of bosonic Gaussian channels.
*Phys. Rev. A*63: 032312 CrossRefADSGoogle Scholar - 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 - Adesso G., Serafini A. and Illuminati F. (2004). Extremal entanglement and mixedness in continuous variable systems.
*Phys. Rev. A*70: 022318 CrossRefADSGoogle Scholar - 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 - Holevo A.S. and Werner R.F. (2001). Evaluating capacities of bosonic Gaussian channels.
*Phys. Rev. A*63: 032312 CrossRefADSGoogle Scholar - 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
- Eisert J., Scheel S. and Plenio M.B. (2002). Distilling Gaussian states with Gaussian operations is impossible.
*Phys. Rev. Lett.*89: 137903 CrossRefADSMathSciNetGoogle Scholar - Fiurášek J. (2002). Gaussian transformations and distillation of entangled Gaussian states.
*Phys. Rev. Lett.*89: 137904 CrossRefADSMathSciNetGoogle Scholar - Giedke G. and Cirac J.I. (2002). Characterization of Gaussian operations and distillation of Gaussian states.
*Phys. Rev. A*66: 032316 CrossRefADSGoogle Scholar