Maximal entanglement increase with single-photon subtraction

Kun Zhang1, Jietai Jing1,2,3,4, Nicolas Treps5, and Mattia Walschaers5

1State Key Laboratory of Precision Spectroscopy, Joint Institute of Advanced Science and Technology, School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
2CAS Center for Excellence in Ultra-intense Laser Science, Shanghai 201800, China
3Department of Physics, Zhejiang University, Hangzhou 310027, China
4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
5Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 place Jussieu, F-75252 Paris, France

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Entanglement is an indispensable quantum resource for quantum information technology. In continuous-variable quantum optics, photon subtraction can increase the entanglement between Gaussian states of light, but for mixed states the extent of this entanglement increase is poorly understood. In this work, we use an entanglement measure based the Rényi-2 entropy to prove that single-photon subtraction increases bipartite entanglement by no more than log 2. This value coincides with the maximal amount of bipartite entanglement that can be achieved with one photon. The upper bound is valid for all Gaussian input states, regardless of the number of modes and the purity.

When developing light-based quantum technologies, Gaussian quantum states -thus called because the measurement statistic of the electromagnetic field follows a Gaussian distribution- are a convenient resource. These states allow us to create large entangled states in a deterministic way. However, to perform interesting quantum protocols that cannot be mimicked with classical resources Gaussian states are insufficient and additional resources are required to make these states non-Gaussian.
Quantum opticians often rely on an operation known as photon subtraction to render quantum states non-Gaussian. In this probabilistic operation, we literally remove one photon from the light in a well-controlled way. In our work, we perform this operation on one subsystem of a large entangled state. Because the operation is probabilistic it is not always successful, but when it is the entanglement between the different subsystems can be increased. The main goal of our work is to provide new quantitative understanding of this entanglement increase.
In our article, we manage to show an intuitive and insightful bound that holds for arbitrary initial Gaussian states, regardless of their purity, mean field, or the number of modes. Our bound can be understood as saying that the increase of entanglement through photon subtraction can never be larger than the maximal amount of entanglement that can be generated by a single photon. This amount of entanglement corresponds to the amount of entanglement that is reached when a photon is sent through a balanced beamsplitter, which creates a Bell state.

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Cited by

[1] Mattia Walschaers, "Non-Gaussian Quantum States and Where to Find Them", PRX Quantum 2 3, 030204 (2021).

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