## Mattia Walschaers

*Laboratoire Kastler Brossel*

Mattia Walschaers is a post-doctoral researcher at the Laboratoire Kastler Brossel in Paris. After doing a PhD on benchmarks for quantum transport in complex systems, his research is now centred on continuous-variable quantum optics. As a theorist in an experimental group, he develops a theoretical framework for studying multimode non-Gaussian states, in particular those that can be experimentally generated through photon-subtraction. On the level of methods, he merges tools from quantum statistics mechanics with more standard aspects of quantum optics.

Quantum entanglement, one of the key resources for quantum information processing, can be deterministically generated in a scalable manner in continuous variable (CV) systems. However such CV entangled states typically display... [ view full abstract ]

Quantum entanglement, one of the key resources for quantum information processing, can be deterministically generated in a scalable manner in continuous variable (CV) systems. However such CV entangled states typically display Gaussian statistics, which limits their use for quantum computing. It is experimentally feasible to overcome this problem by the mode-selective subtraction of photons from multimode Gaussian states, thus rendering them non-Gaussian.

In multimode setups, however, the theoretical properties of the resulting non-Gaussian states are still surrounded by open questions. In the present contribution we use techniques from quantum statistical mechanics to obtain the multimode *Wigner function* for photon-subtracted states. This directly allows us to uncover a general condition for the negativity of the Wigner function, and to explore the state's entanglement properties.

As a key result, we use correlation functions to derive the general Wigner function of a non-displaced photon-subtracted state:

W_{-}(β)=[(β,V^{-1}A_{g}V^{-1}β) - tr(V^{-1}A_{g})+2]W_{G}(β)/2,

where W_{G}(β) is the Wigner function of the initial Gaussian state from which the photon was subtracted, and V is its covariance matrix. All non-Gaussian features are introduced by the matrix A_{g}, that depends on the mode g in which the photon is subtracted, and which can be expressed analytically. From the expression for W_{-}(β) we directly deduce an elegant condition for the negativity of the Wigner function. Moreover, for pure states, we show that coherent subtraction of a photon can enhance entanglement between modes. We prove that this entanglement can persist in *all* mode bases, contrary to the Gaussian case.

These methods also allow us to understand more details about specific classes of states. In particular, we will treat the case of photon subtraction from CV graph states. Such graph states can be implemented by generating gaussian entanglement between the optical modes of a multimode squeezed vacuum. These states display Gaussian statistics upon measurement of the field quadratures. Moreover, they form the backbone of measurement-based quantum computation, and can be produced in present-day experiments. Photon subtraction can introduce non-Gaussian features in these states, hence making them appropriate for applications in quantum information processing.

In this contribution, we take a first step in exploring the physical properties of such experimentally achievable photon-subtracted CV graph states. By exploring the spread of non-Gaussian features through the graph upon subtraction of a photon in one of its vertices, we pave the way for future experimental developments.

Quantum information processing and computing , Fundamental science for quantum technologies , Quantum optics and non-classical light sources