# Internship: Steering non-Gaussian quantum states

**Contact:** Mattia Walschaers – mattia.walschaers@lkb.upmc.fr

Light offers a vast potential to develop modern quantum technologies due to its intrinsic resilience to decoherence effects that tend to scramble quantum information in matter-based setups. One avenue for employing light to process quantum information focuses on the continuous variable regime, where the observables of interest are the quadratures of the electric field. These continuous variables have proven their worth as a platform for creating huge entangled states (entangling up to one million optical modes).

Additionally, this entanglement can be created in a deterministic fashion, and the resulting states are easily manipulated with standard techniques in optics [1,2] To reach a quantum advantage, and perform a task that cannot be efficiently simulated with a classical device, we require more than just entanglement. The additional ingredient is non-Gaussian statistics in the outcomes of the quadrature measurements. More specifically, we must create quantum states with a negative Wigner functions. At LKB, we have recently developed a mode-tunable photon subtractor as a device for creating such states [3,4]. As such, we now have the possibility to produce large entangled states and to render them non-Gaussian. This opens up a whole new realm of research, where a vast amount of question on the interplay between entanglement and non-Gaussian effects remain unanswered [5,6]. Within this internship, we will explore some of these questions. In particular, we will investigate the interplay between non-Gaussian aspects of quantum states and quantum steering [7-9]. The latter is a special feature of certain quantum correlations, and is in some sense “stronger” than quantum entanglement. In general, when two subsystems, A (Alice) and B (Bob) are correlated, a measurement on A improves the precision of predictions for a measurement on B. In classical statistical theories, there is a limit on the amount of information that can be extracted in this way. However, in quantum physics, these limits can be overcome, and in some cases, we can find quantum correlations that allow us to make predictions about system B are more precise than possible with any classical correlation. Such quantum correlations are said to be steerable, and A is said to be able to steer B. In continuous variable quantum optics, a profound example of a steerable quantum correlation can be found in Einstein-Podolsky-Rosen states.

The internship offers two possible directions of research to probe how quantum steering and nonGuassian effects are intertwined:

● On the one hand, we explore how Gaussian quantum states with steerable quantum correlations react to non-Gaussian operations such as photon subtraction. This work will directly build on recent result [10].

● On the other hand, we explore how quantum states with manifestly non-Gaussian entanglement can be steered in a systematic way (see Figure). Brute force numerical methods are available to check whether a state is steerable or not, but our goal is rather to acquire an analytical understanding that allows us to learn something about the properties of these non-Gaussian states.

Finally, we will explore the possible connection between non-Gaussian entanglement and transfer of negativity of the Wigner function. The above figure shows the scenario where all the entanglement between Alice and Bob is non-Gaussian. However, we might consider modifying the beam splitter after the single-photon detector, such that Alice and Bob are working in a different basis. An understanding of the signatures of non-Gaussian entanglement in such different mode is still missing. Gaining insight in this open question may provide a route to experimentally observe non-Gaussian entanglement.

**Subsequent PhD possibilities:** The subject of non-Gaussian states in quantum optics is vast, and interested students who choose this project have the possibility to apply for a PhD position in the group. Even though the internship is mainly theoretical, a potential PhD can also involve experimental work in the multimode quantum optics group.

**Specificities related to the COVID 19 epidemic:** This is a theoretical project that can be carried out by remotely if necessary. In this case, the intern will be integrated in the group and meet with his/her supervisor through video-calls.

**Practicalities:** The starting and ending dates of the internship are flexible, but the internship is expected to last between two and six months (depending on the intern’s prior knowledge of quantum optics).

[1] J. Roslund, R. M. de Araujo, S. Jiang, C. Fabre, N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs”, Nat. Photon. 8, 109 (2014).

[2] Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, N. Treps, “Multimode entanglement in reconfigurable graph states using optical frequency combs”, Nat. Commun. 8, 15645 (2017).

[3] Y.-S. Ra, C. Jacquard, A. Dufour, C. Fabre, N. Treps, “Tomography of a Mode-Tunable Coherent Single-Photon Subtractor” Phys. Rev. X 7, 031012 (2017).

[4] Y.-S. Ra, A Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, N. Treps, “Non Gaussian quantum states of a multimode light field”, Nat. Phys. 16, 144–147 (2020)

[5] M. Walschaers, C. Fabre, V. Parigi, and N. Treps, “Entanglement and Wigner Function Negativity of Multimode Non-Gaussian States”, Phys. Rev. Lett. 119, 183601 (2017).

[6] M. Walschaers, S. Sarkar, V. Parigi, and N. Treps “Tailoring Non-Gaussian Continuous Variable Graph States” Phys. Rev. Lett. 121, 220501 (2018).

[7] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the EinsteinPodolsky-Rosen paradox”, Phys. Rev. A 80, 032112 (2009).

[8] I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian Quantum Steering”, Phys. Rev. Lett. 114, 060403 (2015).

[9] X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of Monogamy Relations for Einstein-Podolsky-Rosen Steering in Gaussian Cluster States”, Phys. Rev. Lett. 118, 230501 (2017).

[10] M. Walschaers and N. Treps, “Remote Generation of Wigner Negativity through Einstein Podolsky-Rosen Steering”, Phys. Rev. Lett. 124, 150501 (2020).