Quadrature Coherence Scale of Linear Combinations of Gaussian Functions in Phase Space
Assessing whether a quantum state is nonclassical, meaning that it is not a mixture of coherent states, continues to be a ubiquitous question in quantum optics. Over the years, witnesses, measures, and monotones of such nonclassicality have been designed, with different merits. The recently introduced quadrature coherence scale [1,2,3] has been proven to have several advantages: it is not only an efficient witness but also provides a measure of nonclassicality and can be accessed experimentally using only a linear interferometer, two identical copies of the state and photon number measurements [4]. This protocol has been experimentally tested on the Xanadu cloud computer [5]. The quadrature coherence scale takes a simple form for pure and Gaussian states, which is important since Gaussian states are prominent in continuous-variable quantum information as they are relatively easy to produce experimentally and simple to study on a theoretical level. Nevertheless, non-Gaussian states or operations are essential for performing certain quantum information tasks and in particular, are needed to achieve universal photonic quantum computation. Unfortunately, a general expression for mixed states can be more challenging. In this talk, we introduce a method for computing the quadrature coherence scale of quantum states characterized by a Wigner functions expressible as linear combinations of Gaussian functions [6]. Notable examples within this framework include cat states, GKP states, or states resulting from Gaussian transformations, measurements, or breeding protocols. In particular, we show that the quadrature coherence scale serves as a valuable tool for examining the scalability of nonclassicality in the presence of loss. Our findings lead us to put forth a conjecture suggesting that, under a 50\% loss scenario, all pure states become classical. [1] S. De Bièvre, D. B. Horoshko, G. Patera, and M. I. Kolobov. Measuring nonclassicality of bosonic field quantum states via operator ordering sensitivity, Phys. Rev. Lett., 122:080402, (2019). [2] A. Hertz and S. De Bièvre. Quadrature coherence scale driven fast decoherence of bosonic quantum field states, Phys. Rev. Lett., 124:090402, (2020). [3] A. Hertz and S. De Bièvre. Decoherence and nonclassicality of photon-added/subtracted multimode Gaussian states, Phys. Rev. A 107, 043713, (2023). [4] C. Griffet, M. Arnhem, S. De Bièvre, and N. J. Cerf. Interferometric measurement of the quadrature coherence scale using two replicas of a quantum optical state, Phys. Rev. A 108 023730, (2023). [5] A. Z. Goldberg, G. S. Thekkadath, and K. Heshami. Measuring the quadrature coherence scale on a cloud quantum computer, Phys. Rev. A 107, 042610, (2023). [6] A. Hertz, A. Z. Goldberg, and K. Heshami. In preparation, (2024).