Certification of Non-Gaussian States with Operational Measurements

We derive a theoretical framework for the experimental certification of non-Gaussian features of quantum states using double homodyne detection. We rank experimental non-Gaussian states according to the recently defined stellar hierarchy and we propose practical Wigner negativity witnesses. We simulate various use-cases ranging from fidelity estimation to witnessing Wigner negativity. Moreover, we extend results on the robustness of the stellar hierarchy of non-Gaussian states. Our results illustrate the usefulness of double homodyne detection as a practical measurement scheme for retrieving information about continuous-variable quantum states, and show that certification of high-order non-Gaussian features can be carried out experimentally with current technology.

Popular Summary

Quantum technologies are predicted to offer dramatic advantages over existing technologies for computing, communication, and cryptography. However, there is still a long way to go before this theorized revolution becomes a reality. Indeed, quantum information is fragile and can easily be lost when the physical system that carries it interacts with its environment. The development of quantum technologies is therefore difficult and greatly depends on the existence of efficient tools to assess the relevant properties of quantum systems. In particular, the so-called non-Gaussian properties are essential features for any quantum computational advantage using quantum states of light, but they have remained challenging to detect and characterize in complex quantum systems. In this work, we resolve this challenge and provide efficient methods for the experimental investigation of non-Gaussian properties of quantum states.

By studying the robustness of non-Gaussian properties of quantum states, we deduce how to compute robust benchmarks for these properties from simple experimental measurements of the states. Importantly, our results show that the certification of non-Gaussian features can actually be carried out efficiently experimentally with current technology and is not prohibited by an unreachable amount of required samples. We demonstrate the applicability of our methods with numerical and experimental data for both single-mode and multimode systems.

Our findings pave the way for the development of light-based quantum technologies, with direct applications in existing quantum-optics experiments. Our work also motivates future studies of fundamental problems such as the intriguing link between two key quantum properties: non-Gaussianity and entanglement.

Figure 1

Phase-space measurements. (a) Homodyne detection of the rotated quadrature ${\hat{x}}_{\varphi }$. (b) Double homodyne detection with unbalancing parameter $\zeta$. The dashed red lines represent beam splitters and LO stands for local oscillator, i.e., a strong coherent state.

Figure 2

A pictorial depiction of the stellar hierarchy. The right column gives examples of pure states of each stellar rank, which corresponds to the minimal number of photon additions ${\hat{a}}^{†}$ necessary to obtain the state from the vacuum. All states of finite stellar rank are robust, i.e., they have only states of equal or higher stellar rank in their close vicinity.

Figure 3

Achievable fidelities with target Fock states $|1⟩$, $|2⟩$, $|3⟩$, and $|4⟩$, using states of finite stellar rank. In each fidelity region, the number indicates the minimal stellar rank of states achieving these fidelities, i.e., it corresponds to the minimal number of photon additions and/or subtractions needed to build these states, together with Gaussian operations.

Figure 4

Estimation of fidelity with target Fock states at 95% confidence, using simulated samples from double homodyne detection of various states. When the lowest point of an error bar is higher than a threshold value, the property corresponding to this threshold value is certified with more than 95% confidence. The precision of the estimate varies depending on the number of samples used. We display the stellar rank 1 (respectively, 2) threshold in red (respectively, purple) and the Wigner negativity witness threshold (see Sec. 4) in cyan. Note that the rank-$1$ threshold varies depending on whether the target state is the Fock state $|1⟩$ or $|2⟩$, as can be checked with Fig. 3. (a) Lossy two-photon Fock states $(1-\eta {)}^2|0⟩⟨0|+2\eta (1-\eta )|1⟩⟨1|+{\eta }^2|2⟩⟨2|$, where $\eta$ is an efficiency parameter, and target Fock state $|2⟩$. For $\eta =0.9$, $0.8$, and $0.6$, the true fidelities to $|2⟩$ are $0.81$, $0.64$, and $0.36$, respectively. The Wigner-negativity threshold is obtained numerically [40]. (b) Photon-subtracted squeezed vacuum states with 3 dB of squeezing and target Fock state $|1⟩$, for several values of the preparation purity, i.e., the purity of the squeezed vacuum state before the photon subtraction. (c) A photon subtracted two-mode EPR state with experimental covariance matrix taken from Ref. [41] and target Fock state $|1⟩$. Simulated photon subtraction is done on the first mode and double homodyne measurement on the second mode of the EPR state.

Figure 5

Negativity witnesses for a photon-subtracted squeezed vacuum state with 3 dB of squeezing and $0.95$ preparation purity. Using simulated samples from double homodyne detection, the witnesses in Eq. (12) are estimated for a thousand values of the displacement amplitude $\alpha$, which are interpolated for clarity, with a fixed precision $ϵ=0.1$ and a fixed number of samples $N=5.5\times {10}^5$. The curve is shifted upwards by the value of $ϵ$, and the red points witness the negativity of the state with at least $98\mathrm{\%}$ confidence.

Figure 6

Efficiency of the various estimation methods for a target Fock state $|2⟩$. The curves represent the number of samples necessary to obtain a given precision on the fidelity estimate with 95% confidence. The top curve is obtained using the protocol from Ref. [32], the middle one using the optimized protocol with analytical bounds, and the bottom one using the improved bounds from the CLT (for which a Fock state $|2⟩$ is sampled).

Figure 7

Achievable fidelities with target core states $\cos \varphi |0⟩+e^{i\chi }\sin \varphi |1⟩$ using Gaussian states, for all $\varphi ,\chi \in [0,2\pi ]$, as a function of $\varphi$.

Figure 8

Achievable fidelities with the two-mode target Fock state $|1⟩\otimes |1⟩$. Note that the first value $0.342$ is greater than the square of the maximum achievable fidelity with a single-photon Fock state: this means using a state of the form $\hat{S}({\xi }_1)\hat{D}({\alpha }_1)|0⟩\otimes \hat{S}({\xi }_2)\hat{D}({\alpha }_2)|0⟩$ is not an optimal way to approximate the state $|1⟩\otimes |1⟩$ with a two-mode state of stellar rank $0$. Similarly, the second value $0.532$ is greater than the maximum achievable fidelity with a single-photon Fock state: this means using a state of the form $|1⟩\otimes \hat{S}(\xi )\hat{D}(\alpha )|0⟩$ is not an optimal way to approximate the state $|1⟩\otimes |1⟩$ with a two-mode state of stellar rank $1$. In both cases, the multimode operation $\hat{U}$ plays a crucial role in the approximation.