Deterministic Gaussian conversion protocols for non-Gaussian single-mode resources

In the context of quantum technologies over continuous variables, Gaussian states and operations are typically regarded as freely available, as they are relatively easily accessible experimentally. In contrast, the generation of non-Gaussian states, as well as the implementation of non-Gaussian operations, pose significant challenges. This divide has motivated the introduction of resource theories of non-Gaussianity. As for any resource theory, it is of practical relevance to identify free conversion protocols between resources, namely, Gaussian conversion protocols between non-Gaussian states. Via systematic numerical investigations, we address the approximate conversion between experimentally relevant single-mode non-Gaussian states via arbitrary deterministic one-to-one mode Gaussian maps. First we show that cat and binomial states are approximately equivalent for finite energy, while this equivalence was previously known only in the infinite-energy limit. Then we consider the generation of cat states from photon-added and photon-subtracted squeezed states, improving over known schemes by introducing additional squeezing operations. The numerical tools that we develop also allow one to devise conversions of trisqueezed into cubic-phase states beyond previously reported performances. Finally, we identify various other conversions which instead are not viable.

Figure 1

Wigner plots of the types of states and codes used in this paper. The color bar indicates the Wigner positivity (blue) and negativity (red), of the Wigner function $W(q,p)$ in normalized units, see Eq. (19). (a) Binomial code with rotation symmetry $N=2$, truncation $K=3$, and logical encoding $\mu =0$. (b) Cat code with displacement $\alpha =2, \mu =0, N=2$, and $r=\varphi =0$. (c) Even-parity cat state with displacement $\alpha =2$. (d) Photon-subtracted squeezed state with $L=-2$ photons and squeezing $\xi =5\mathrm{dB}$. (e) Photon-added squeezed state with $L=+3$ photons and squeezing $\xi =3\mathrm{dB}$. (f) Cubic-phase state with cubicity $c=0.551$ and squeezing $-5\mathrm{dB}$. (g) Trisqueezed state (three-photon squeezed state), with triplicity $t=0.1$. (h) GKP code with squeezing $\mathrm{\Delta }=14\mathrm{dB}$, and $\mu =0$.

Figure 2

Fidelity between cat and binomial codes as function of the amplitude $\alpha$ of the cat code for different truncations $K$ of the binomial code. $N$ gives the rotation symmetry of the investigated code words and is always the same for both. The dotted vertical lines give the value of ${\alpha }_{\text{iso}}$ for which both code words have the same energy, whereas the solid vertical lines show the value of $\alpha$ corresponding to the maximum fidelity.

Figure 3

Values of the amplitude ${\alpha }_{\text{max}}$ for the respective cat code at maximum fidelity (top panel) as well as maximum fidelity (bottom panel) as a function of the truncation $K$ of the binomial code for various rotation symmetries $N$. The values next to the markers in the first panel are the values of maximal fidelity, while they are the values of ${\alpha }_{\text{max}}$ in the second panel.

Figure 4

Wigner plots of binomial codes and cat codes, both with rotation symmetry $N=2$, and logical encoding $\mu =0$. The truncation $K$ for the binomial code and the displacement $\alpha$ for the cat code are indicated by the labels. This phase-space representation makes the approximate equivalence between binomial and cat codes become apparent.

Figure 5

Fidelities of conversion from PASS to cat states as a function of the squeezing $\xi$ of the PASS state. Please note that the lowest value of the displacement ($\alpha =1.5$) was chosen such that the cat state no longer looks like a continuous rotationally symmetric state—it has well-separated coherence peaks with fringes in-between. The dashed lines are the fidelity between the target cat state and a PASS state with the indicated $L$ and $\xi$. The solid lines are the fidelities after the application of the optimal Gaussian CPTP map that consists of squeezing or antisqueezing. As can be seen, the fidelity with an additional squeezing operation is higher than without the application of a protocol.

Figure 6

Conversion between a PASS state with squeezing $\xi$ and $L$ photon subtractions and an even-parity cat state with $\alpha =2$. The first panel shows the fidelities between the converted PASS state and the cat state, using a Gaussian CPTP, extracted from the central panel in the bottom row of Fig. 5. The Gaussian map is effectively only squeezing with strength $\mathrm{\Xi }$. In the second panel we plot the optimal squeezing $\mathrm{\Xi }$ of the map as a function of the input squeezing $\xi$ of the PASS state. The vertical lines show the squeezing for which the PASS state and the cat state have maximal fidelity without application of the map (they correspond to the points where the solid and dashed curves coincide in the central panel in the bottom row of Fig. 5). As can be seen, this corresponds to when the Gaussian map does not add any squeezing itself.

Figure 7

Fidelity for the conversion from trisqueezed to cubic-phase state as a function of the cubicity of the target state. Different markers correspond to different squeezing values $\xi$ of the cubic-phase state, when using an optimal Gaussian CPTP map (solid lines), and no map (dashed lines). The triplicity is such that the Wigner negativity is the same for the input and target state (see values in Table 1).

Figure 8

Fidelity for the conversion from a PASS state with $L=2$ added photons to a cat code with rotation symmetry $N=2$, as a function of squeezing $\xi$ of the PASS state. The logical encoding of the cat code is $\mu =1$, and the displacement $\alpha$ is indicated by the legend. For low squeezing $\xi$, the states initially look very similar and have a high initial fidelity, but closer inspection in Fig. 9 shows that they have different qualitative features.

Figure 9

Comparison of rotationally symmetric codes. (a) PASS state with $L=+2$ photons and zero squeezing, which is the Fock state $|2\rangle$. (b) Cat code with $\alpha =1$, logical encoding $\mu =1$, and rotation symmetry $N=2$. As the squeezing of the PASS increases, the protocol can still yield a high fidelity by “undoing” the squeezing. As the magnitude of $\alpha$ increases significantly, the protocol can also yield a relatively high fidelity, but the cat code breaks the continuous rotational symmetry, displaying a clear fourfold discrete rotational symmetry in phase space. This means that the features are not captured by the protocol, and the conversion is in this sense a no-go. This is illustrated in panel (c), where the amplitude of the Wigner function is plotted along a circle of fixed radius, as indicated by the dashed line in panel (a). The oscillations in the cat code are due to the discrete rotational symmetry, while the photon state is completely rotationally symmetric.

Figure 10

Conversion of a PASS state with $L=-2$ photons and squeezing 3 dB to a cat code with displacement $\alpha =1$ and $\mu =0$. The state (left) obtained after applying the conversion protocol to the input state has increased fidelity from roughly 0.67 to 0.98 but misses the Wigner negativity in the top-left and bottom-right corner regions of the target state (right).

Figure 11

Conversion of a cubic-phase state with cubicity 0.05 and squeezing $-5\mathrm{dB}$ to a GKP state with squeezing noise 5 dB. The state (left) obtained after applying the conversion protocol to the input state has increased fidelity from roughly 0.55 to 0.93, but as can be seen, the qualitative features of the target (right) are captured very poorly.