Entanglement transformations of pure Gaussian states

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In particular, this holds for the 'eigenstates' of the position operator, which are elements of the dual space of Schwartz space.

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Care must be taken when extending state transformation results to infinite dimensional systems. E.g., it is no longer true, that all states with equal Schmidt coefficients can be converted into each other by local unitaries, as the counterexample ψ = Σ _n c_n |n〉 |n〉, ψ′ = Σ _n c_n |n + 1〉 |n + 1〉 demonstrates. Nevertheless, the majorisation criterion can be used in this infinite dimensional setting when "ρ can be converted into ρ′ is now defined to mean that there is a sequence of local transformations T_k that for each k map ρ to ρ_k′ such that ρ_k′ approaches ρ′ in trace norm; i.e., while it may not be possible to transform the states exactly, the target state can be approximated to arbitrary accuracy and with probability 1 via LOCC. In the finite dimensional case this definition of convertibility is equivalent to the usual one. Then the equivalence of the majorisation condition and convertibility can be formally derived for all norm-approachable states, e.g., by investigating sequences of the finite-dimensional pure states ρ _k and ρ _k ′ obtained by projecting ρ, ρ′ onto the subspaces corresponding to the k largest Schmidt coefficients. Considering the finite case for all k ∈ N, one immediately arrives at the majorisation criterion in the infinite-dimensional case.

Note that transformations that are both symplectic and orthogonal reflect all those unitary Gaussian operations that can be implemented by means of passive optical elements (beam splitters and phase shifts).