Interconversion of pure Gaussian states requiring non-Gaussian operations

We analyze the conditions under which local operations and classical communication enable entanglement transformations between bipartite pure Gaussian states. A set of necessary and sufficient conditions had been found [G. Giedke et al., Quant. Inf. Comput. 3, 211 (2003)] for the interconversion between such states that is restricted to Gaussian local operations and classical communication. Here, we exploit majorization theory in order to derive more general (sufficient) conditions for the interconversion between bipartite pure Gaussian states that goes beyond Gaussian local operations. While our technique is applicable to an arbitrary number of modes for each party, it allows us to exhibit surprisingly simple examples of $2\times 2$ Gaussian states that necessarily require non-Gaussian local operations to be transformed into each other.

Figure 1

Four possible evolutions of the squeezing parameters $r_1$ and $r_2$ of the two two-mode squeezed vacuum states $|{\Phi }_{r_1}\rangle |{\Phi }_{r_2}\rangle$. Note that since the squeezing vectors are sorted in decreasing order $(r_1\ge r_2$, $r_1^{\prime }\ge r_2^{\prime })$, the two arrows never cross each other.

Figure 2

Transformation from state $|\psi \rangle =|{\Phi }_{r_1}\rangle |{\Phi }_{r_2}\rangle$ to state $|{\psi }^{\prime }\rangle =|{\Phi }_{r_1^{\prime }}\rangle |{\Phi }_{r_2^{\prime }}\rangle$ for fixed values $r_1=1.15$ (9.96 dB), $r_2=0.88$ (7.66 dB), and variable $(r_1^{\prime },r_2^{\prime })$. The two diagonal dashed lines divide the plane in four quadrants corresponding to the four situations depicted in Fig. 1. The final states that are reachable with a LOCC transformation are marked in green (green areas and green points), while the final states that are not reachable because a LOCC transformation exists in the opposite direction are marked in blue (blue areas and blue points). The final states that are incomparable with the initial states (no LOCC exists both ways) are marked as red points. The lower quadrant (light green) corresponds to 1, while the upper quadrant (light blue) corresponds to Fig. 1. Condition (1) implies that a GLOCC exists in the lower quadrant, while a LOCC is ruled out in the upper quadrant since a GLOCC exists in the reverse direction. The right quadrant [Fig. 1] and left quadrant [Fig. 1] correspond to regions where condition (1) is not satisfied both ways, so no GLOCC exists both ways. However, non-Gaussian LOCC are possible and detected by our condition (4) in the dark green region (or nondetected by our condition and represented by green points). Symmetrically, non-Gaussian LOCC are possible in the reverse direction and detected by our condition (4) in the dark blue region (or nondetected by our condition and represented by blue points); this corresponds to cases where no LOCC exists that converts $|\psi \rangle$ into $|{\psi }^{\prime }\rangle$. Our criterion, condition (4), is indicated by a solid green line (direct direction $|\psi \rangle \to |{\psi }^{\prime }\rangle )$ and blue line (reverse direction $|{\psi }^{\prime }\rangle \to |\psi \rangle )$.

Figure 3

A bipartite pure state $|\psi \rangle$ can be transformed into $|{\psi }^{\prime }\rangle$ with a LOCC if and only if the reduced state ${\rho }^{\prime }$ majorizes $\rho$, or equivalently if the vector of eigenvalues ${\lambda }^{\prime }$ of ${\rho }^{\prime }$ can be mapped onto the vector of eigenvalues $\lambda$ of $\rho$ with a column-stochastic matrix $\mathbf{D}$. (a) Case of $1\times 1$ modes, where matrix ${\mathbf{D}}^{\mathcal{A}}$ acts on the vector of eigenvalues of Fock-diagonal states similarly as a quantum-limited amplifier $\mathcal{A}$ of gain $G$ acts on Fock-diagonal states (${\mathbf{D}}^{\mathcal{A}}$ is column-stochastic provided that $G\ge 1)$. (b) Case of $2\times 2$ modes [Fig. 1], where matrix ${\mathbf{D}}^{\mathcal{L}}\otimes {\mathbf{D}}^{\mathcal{A}}$ acts on the vector of eigenvalues of tensor product of Fock-diagonal states similarly as the tensor product of a pure loss channel $\mathcal{L}$ of transmittance $\eta$ with a quantum-limited amplifier $\mathcal{A}$ of gain $G$ acts on tensor products of Fock-diagonal states $({\mathbf{D}}^{\mathcal{L}}\otimes {\mathbf{D}}^{\mathcal{A}}$ is column-stochastic provided that $\eta G\ge 1)$. Note that $\mathcal{L}\otimes \mathcal{A}$ is a Gaussian map, while the corresponding LOCC is non-Gaussian since condition (1) is not satisfied.