Assisted concentration of Gaussian resources

In spite of their outstanding experimental relevance, Gaussian operations in continuous-variable quantum systems are subjected to fundamental limitations, as it is known that general resources cannot be distilled within the Gaussian paradigm. We show that these limitations can be overcome by considering a collaborative setting where one party increases the amount of local resource with the aid of another party, whose operations are assumed to be Gaussian but are otherwise unrestricted; the two parties can only communicate classically. We show that in single-shot scenarios, unlike in the well-known case of entanglement theory, two-way classical communication does not lead to any improvement over one-way classical communication from the aiding party to the aided party. We then provide a concise general expression for the Gaussian resource of assistance, i.e., the maximum amount of resource that can be obtained when the aiding party holds a purification of the aided party's state, as measured by a general monotone. To demonstrate its usefulness, we apply our result to two important kinds of resources, squeezing and entanglement, and find some simple analytic solutions. In the case of entanglement theory, we are able to find general upper bounds on the regularized Gaussian entanglement of assistance, and to establish additivity for tensor powers of thermal states. This allows us to draw a quantitative and enlightening comparison with the performance of assisted entanglement distillation in the non-Gaussian setting. On the technical side, we develop some variational expressions to handle functions of symplectic eigenvalues that may be of independent interest. Our results suggest further potential for Gaussian operations to play a major role in practical quantum information processing protocols.

Figure 1

A pictorial representation of the one-way collaborative Gaussian concentration paradigm. Alice and Bob's initial Gaussian state has QCM $V_{AB}$. Bob applies a Gaussian measurement with seed ${\gamma }_B$ to his share of the system and then communicates the outcome to Alice, who then obtains a Gaussian state with QCM $V_{A^{\prime }}^{\prime }=(V_{AB}+{\gamma }_B)/(V_B+{\gamma }_B)$.

Figure 2

A pictorial representation of a two-way collaborative Gaussian concentration scheme. Now Alice and Bob are allowed to manipulate their initial Gaussian state with QCM $V_{AB}$ using arbitrary Gaussian operations on Bob's side (filled boxes), free Gaussian operations on Alice's (patterned boxes), and two-way classical communication.

Figure 3

Equivalence between the entanglement potential of assistance and the localizable entanglement: if one thinks of the measurement as performed before the beam splitter, then the amount of entanglement between $A$ and $C$ at the output is the entanglement potential of assistance. If instead one sees the measurement as performed after the beam splitter, then the maximum entanglement between $A$ and $C$ is the localizable entanglement of ${|\varphi \rangle }_{ABC}:=U_{\mathrm{BS}}^{AC}{|\psi \rangle }_{AB}{|0\rangle }_C$.