Non-Gaussian quantum states of a multimode light field


Advanced quantum technologies require scalable and controllable quantum resources1,2. Gaussian states of multimode light, such as squeezed states and cluster states, are scalable quantum systems3,4,5, which can be generated on demand. However, non-Gaussian features are indispensable in many quantum protocols, especially to reach a quantum computational advantage6. Embodying non-Gaussianity in a multimode quantum state remains a challenge as non-Gaussian operations generally cannot maintain coherence among multiple modes. Here, we generate non-Gaussian quantum states of a multimode light field by removing a single photon in a mode-selective manner from a Gaussian state7. To highlight the potential for continuous-variable quantum technologies, we first demonstrated the capability to generate negativity of the Wigner function in a controlled mode. Subsequently, we explored the interplay between non-Gaussianity and quantum entanglement and verify a theoretical prediction8 about the propagation of non-Gaussianity along the nodes of photon-subtracted cluster states. Our results demonstrate large-scale non-Gaussianity with great flexibility along with an ensured compatibility with quantum information protocols. This range of features makes our approach ideal to explore the physics of non-Gaussian entanglement9,10 and to develop quantum protocols, which range across quantum computing11,12, entanglement distillation13 and quantum simulations14.

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Fig. 1: Mode-selective photon subtraction from a multimode quantum state.
Fig. 2: Wigner function reconstructed from experimental data.
Fig. 3: Effect of photon subtraction on cluster states.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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We thank Valentina Parigi for fruitful discussions. This work is supported by the French National Research Agency project COMB and the European Union Grant QCUMbER (no. 665148). N.T. is a member of the Institut Universitaire de France. Y.-S.R. acknowledges support from the European Commission through Marie Skłodowska-Curie actions (no. 708201) and the National Research Foundation of Korea funded by the Ministry of Education (NRF-2018R1A6A3A03012129) and the Ministry of Science and ICT (NRF-2019R1C1C1005196). M.W. acknowledges funding through research fellowship WA 3969/2-1 from the German Research Foundation.

Author information

Y.-S.R. and A.D. conducted the experiments with help from C.J. and T.M. Y.-S.R. and M.W. analysed the data. M.W. developed the theoretical model. Y.-S.R., M.W. and N.T. wrote the manuscript with input from all the authors. C.F. and N.T. supervised the project. All the authors contributed to scientific discussions.

Correspondence to Young-Sik Ra.

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The authors declare no competing interests.

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Peer Review Information Nature Physics thanks Marco Bellini and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Wigner function reconstructed with optical loss correction.

Optical loss only by the homodyne detection (\(12.5 \%\)) has been corrected. Comparing with Fig. 2, non-Gaussian Wigner functions show reduced \({W}_{0}\). Errors noted in parentheses are one standard deviations calculated by bootstrapping. Source data

Extended Data Fig. 2 Purities of the Wigner functions in Fig. 2(a).

For comparison, the purity of a Wigner function by the ideal photon subtraction is provided in square brackets, which agrees well with the experimental result. Low purity in a photon-subtracted mode is attributed to a non-ideal input state19. No optical loss is corrected in the calculation. Errors noted in parentheses are one standard deviations calculated by bootstrapping.

Extended Data Fig. 3 Effect of mode mismatch between photon subtraction and measurement.

When a single photon is subtracted in \({{\rm{HG}}}_{0}-i{{\rm{HG}}}_{1}\), a Wigner function (without optical loss correction) is obtained in a measurement mode having (a) full match (\({{\rm{HG}}}_{0}-i{{\rm{HG}}}_{1}\)), (b) partial match (\(i{{\rm{HG}}}_{1}\)), and (c) no match (\({{\rm{HG}}}_{0}+i{{\rm{HG}}}_{1}\)). Errors noted in parentheses are one standard deviations calculated by bootstrapping. Source data

Extended Data Fig. 4 Experimental covariance matrix.

(a) is for \(x\) quadratures, seen from above, and (b) is for \(p\) quadratures, seen from below. Mode indexes are \({{\rm{HG}}}_{0}\), \(i{{\rm{HG}}}_{1}\), \({{\rm{HG}}}_{2}\), and \(i{{\rm{HG}}}_{3}\), where \(i\) is added for the odd-index HG modes to have \(p\)-squeezed vacua in all modes. For clarity, the vacuum noise (corresponding to the identity matrix) is subtracted from the covariance matrix. In the covariance matrix, variances of (\(x\), \(p\)) quadratures are (\(2.8{\rm{dB}}\), \(-1.8{\rm{dB}}\)) in mode 0, (\(2.1{\rm{dB}}\), \(-1.6{\rm{dB}}\)) in mode 1, (\(1.6{\rm{dB}}\), \(-1.0{\rm{dB}}\)) in mode 2, and (\(1.4{\rm{dB}}\), \(-0.7{\rm{dB}}\)) in mode 3.

Source data

Source Data Fig. 2

Data of W\({}_{0}\) and F shown in Fig. 2.

Source Data Fig. 3

Data used to plot Fig. 3.

Source Data Extended Data Fig. 1

Data of W\({}_{0}\) and F shown in Extended Data Fig. 1.

Source Data Extended Data Fig. 3

Data of W\({}_{0}\) shown in Extended Data Fig. 3.

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Ra, Y., Dufour, A., Walschaers, M. et al. Non-Gaussian quantum states of a multimode light field. Nat. Phys. (2019).

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