A new non-classicality measure for the quantum states of a bosonic field

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... [ view full abstract ]

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

S₀=-(d/ds) ln ||W_s||²

where ||W_s||² is the  integral of W_s² over phase space. S₀ 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 S₀ is less than 1. This establishes S₀ as a non-classicality witness.  Furthermore, pure states are classical iff  S₀=1. We then show that S₀ 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.

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[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).