Measure of the non-Gaussian character of a quantum state

We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a reference Gaussian state. We analyze in detail the properties of the proposed measure and exploit it to evaluate the non-Gaussianity of some relevant single-mode and multimode quantum states. The evolution of non-Gaussianity is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows one to define a corresponding measure for the non-Gaussian character of a quantum operation.

Figure 1

(Top) Non-Gaussianity of single mode Fock states (gray line) $|p⟩$ and of multimode Fock states ${|p⟩}^{\otimes n}$ (black line) as a function of $p$. Non-Gaussianity for multimode states has been maximized over the number of copies, $n$. (Bottom) Non-Gaussianity, as a function of the parameter $\varphi$, for the two-mode superpositions $|\Phi ⟩$ (dashed gray line), $|\Psi ⟩$ (solid gray line), and for the single-mode superposition of coherent states, $|{\psi }_S⟩$, for $\alpha =0.5$ (solid black line) and $\alpha =5$ (dashed black line).

Figure 2

Distribution of non-Gaussianity $\delta \left[{\varrho }_d\right]$ as evaluated for ${10}^5$ random quantum states, for three different value of the maximum number of photons, $d$. Top: $d=2$ (left), $d=10$ (right). Bottom: $d=20$ (left). (Bottom right) Mean values and variances of the non-Gaussianities evaluated for ${10}^5$ random quantum states, as a function of the maximum number of photons, $d$.

Figure 3

(Color online) (Top) Non-Gaussianity of Fock states $|p⟩$ undergoing Gaussification by the loss mechanism due to the interaction with a bath of oscillators at zero temperature. We show ${\delta }_{\eta p}$ as a function of $1-\eta$ for different values of $p$: from bottom to top, $p=1,10,100,1000$. (Bottom) Non-Gaussianity of ${\varrho }_{IPS}$ as a function of $T$ for $r=0.5$ and for different values of $ϵ=0.2,0.4,0.6,0.8$ (from bottom to top). ${\delta }_{IPS}$ results to be a monotonous increasing function of $T$, while $ϵ$ only slightly changes the non-Gaussian character of the state.

Figure 4

Non-Gaussianity of a Schrödinger-cat-like state as a function of the superposition parameter $\varphi$, with either $C$ obtained by numerical minimization (solid line) or with $C=\text{Tr}\left[a\varrho \right]$ (dotted line). (Left) $\alpha =0.5$. (Right) $\alpha =5$.