*Introduction.* The state of a bosonic field is classical if it is a statistical mixture of coherent states, or equivalently, if its Glauber-Sudarshan P-function defines a probability on phase space [1]. Otherwise, it is non-classical. Characterizing and measuring such non-classicality remains an important issue in quantum optics and quantum information theory notably. We introduce a new distance-based measure for non-classicality, and show it outperforms existing such measures in several ways.

*Methods.* The quantum states of a bosonic field are characterized by quasi-probability distributions W_{s} which are functions on phase space that depend on an ordering parameter s [2]. The P-function corresponds to s=1 and the Wigner function to s=0. We introduce the ordering sensitivity of the state by

S_{0}=-(d/ds) ln ||W_{s}||^{2}

where ||W_{s}||^{2} is the integral of W_{s}^{2} over phase space. S_{0} evaluates the sensitivity of the state to operator ordering and measures the oscillations in its Wigner function.

*Results.* Using the properties of the quasi-probability distributions W_{s} , we first show that, if the state is classical, then S_{0} is less than 1. This establishes S_{0} as a non-classicality witness. Furthermore, pure states are classical iff S_{0}=1. We then show that S_{0} defines a norm on the space of all density operators and hence induces a distance from any state to the set of all classical states: this distance provides a new measure of non-classicality. We show it is easily computable in terms of field quadratures, captures several intuitive features of non-classicality naturally, and detects in many cases non-classicality more efficiently than previously used indicators [3, 4, 5].

Discussion. Questions arising in quantum information theory drive a continued interest in the exploration of the quantum-classical boundary. There is in this context a need for efficient criteria to determine the strength of the various quantum features of a quantum state. We have concentrated here specifically on the non-classicality question and introduced a new non-classicality criterion that provides an efficient tool for the exploration of the quantum-classical boundary in bosonic systems.

[1] U. M. Titulaer and R. J. Glauber, Phys. Rev. **140**, B676 (1965).

[2] K. Cahill and R. J. Glauber, Phys. Rev., **177**, 5, 1882 (1969).

[3] R. Nair, arXiv:1701.07688 [quant-ph] (2017).

[4] V. V. Dodonov, O. V. Man'ko, V. I. Man'ko, and A. Wünsche, J. Mod. Opt. **47**, 633 (2000).

[5] P. Marian, T. A. Marian, and H. Scutaru, Phys. Rev. Lett. 88, 15, 153601 (2002).