All phase-space linear bosonic channels are approximately Gaussian dilatable

Abstract

We compare two sets of multimode quantum channels acting on a finite collection of harmonic oscillators: (a) the set of linear bosonic channels, whose action is described as a linear transformation at the phase space level; and (b) Gaussian dilatable channels, that admit a Stinespring dilation involving a Gaussian unitary. Our main result is that the set (a) coincides with the closure of (b) with respect to the strong operator topology. We also present an example of a channel in (a) which is not in (b), implying that taking the closure is in general necessary. This provides a complete resolution to the conjecture posed in Sabapathy and Winter (2017 Phys. Rev. A 95 062309). Our proof technique is constructive, and yields an explicit procedure to approximate a given linear bosonic channel by means of Gaussian dilations. It turns out that all linear bosonic channels can be approximated by a Gaussian dilation using an ancilla with the same number of modes as the system. We also provide an alternative dilation where the unitary is fixed in the approximating procedure. Our results apply to a wide range of physically relevant channels, including all Gaussian channels such as amplifiers, attenuators, phase conjugators, and also non-Gaussian channels such as additive noise channels and photon-added Gaussian channels. The method also provides a clear demarcation of the role of Gaussian and non-Gaussian resources in the context of linear bosonic channels. Finally, we also obtain independent proofs of classical results such as the quantum Bochner theorem, and develop some tools to deal with convergence of sequences of quantum channels on continuous variable systems that may be of independent interest.

Characterizing the performance of continuous-variable Gaussian quantum gates

“…which implies that the sequence { D η, α √ 1−η } η∈[0,1) converges to D α in the strong sense [LSW18,Lemma 8].…”

“…A connection between the notion of strong convergence and the notion of uniform convergence over energy-bounded states was established in [Shi18a]. Later, these different topologies of convergence were studied in the context of linear bosonic channels and Gaussian dilatable channels in [LSW18]. Furthermore, topologies of convergence in the context of teleportation simulation of physically relevant phaseinsensitive bosonic Gaussian channels have been investigated in [Wil18].…”

Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

“…Let us consider for instance the case λ = 0. Since χ σ,0 achieves it maximum modulus at 0 (see [96,Proposition 14] and [97,Lemma 10]), if it is a Gaussian it must be centred, i.e. [44,Eq.…”

Refined diamond norm bounds on the emergence of objectivity of observables

“…( 21) corresponds to a pure loss channel of parameter 1 N [43]. Varying the environment state one obtains instead a general attenuator [44][45][46][47][48][49]. The reduced map Λ j : D(A) → D(B j ) is given by Λ j = Tr B\B j • Λ and has the same form for all j, as shown in Appendix B.1.…”