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Extendibility of bosonic Gaussian states

Lami, Ludovico; Khatri, Sumeet; Adesso, Gerardo; M. Wilde, Mark


Ludovico Lami

Sumeet Khatri

Mark M. Wilde


Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is k-extendible if and only if it has a Gaussian k-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide k-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a k-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and R'enyi relative entropies. These bounds, which can be seen as "Gaussian de Finetti theorems," exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian k-extendible states, which has no analogue in the finite-dimensional setting.


Lami, L., Khatri, S., Adesso, G., & M. Wilde, M. (2019). Extendibility of bosonic Gaussian states. Physical Review Letters, 123(5),

Journal Article Type Article
Acceptance Date Jul 2, 2019
Online Publication Date Jul 31, 2019
Publication Date Aug 2, 2019
Deposit Date Sep 13, 2019
Publicly Available Date Sep 13, 2019
Journal Physical Review Letters
Print ISSN 0031-9007
Electronic ISSN 1079-7114
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 123
Issue 5
Article Number 050501
Keywords Quantum Physics; Other Condensed Matter; Mathematical Physics
Public URL
Publisher URL
Additional Information Extendibility of Bosonic Gaussian States, Ludovico Lami, Sumeet Khatri, Gerardo Adesso, and Mark M. Wilde, Phys. Rev. Lett. 123, 050501


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