Quantifying non-Gaussianity of quantum-state correlation

We consider how to quantify non-Gaussianity for the correlation of a bipartite quantum state by using various measures such as relative entropy and geometric distances. We first show that an intuitive approach, i.e., subtracting the correlation of a reference Gaussian state from that of a target non-Gaussian state, fails to yield a non-negative measure with monotonicity under local Gaussian channels. Our finding clearly manifests that quantum-state correlations generally have no Gaussian extremality. We therefore propose a different approach by introducing relevantly averaged states to address correlation. This enables us to define a non-Gaussianity measure based on, e.g., the trace-distance and the fidelity, fulfilling all requirements as a measure of non-Gaussian correlation. For the case of the fidelity-based measure, we also present readily computable lower bounds of non-Gaussian correlation.

Figure 1

Illustration for distinguishing different notions of quantum mutual information. (a) In quantum channel theory, quantum mutual information is the measure of information transfer through a channel, e.g., a loss channel (the case of ${\rho }_{\mathrm{env}}=|\mathrm{vac}\rangle$). It quantifies how much information on the input state ${\rho }_{\mathrm{in}}$ can be retrieved from the output state ${\rho }_{\mathrm{out}}$. More precisely, it is the amount of shareable quantum information between the sender who encodes the information in ${\rho }_{\mathrm{in}}$ at time $t$ and the receiver who decodes the information in ${\rho }_{\mathrm{out}}^{\prime }$ at time $t+\mathrm{\Delta }t$. This is quantified as $S\left({\rho }_{\mathrm{in}}\right)+S\left({\rho }_{\mathrm{out}}^{\prime }\right)-S\left({\rho }_{\mathrm{env}}^{\prime }\right)$ [51]. (b) On the other hand, in this paper, we are interested in the correlation of a bipartite quantum state itself. For example, when two uncorrelated states ${\rho }_1$ and ${\rho }_2$ are mixed at a beam splitter, the output state ${\rho }_{12}^{\prime }$ has the quantum mutual information given by $S\left({\rho }_1^{\prime }\right)+S\left({\rho }_2^{\prime }\right)-S\left({\rho }_{12}^{\prime }\right)$, with the identity $S\left({\rho }_{12}^{\prime }\right)=S\left({\rho }_1\right)+S\left({\rho }_2\right)$. Thus, the Gaussian extremality found in panel (a) has no direct relation to the results in our work.

Figure 2

(a), (b) Contour plots of $\mathrm{\Delta }{\mathcal{I}}_{\alpha }={\mathcal{I}}_{\alpha }\left[{\rho }_{AB}\right]-{\mathcal{I}}_{\alpha }\left[{\sigma }_{AB}\right]$ and $\mathrm{\Delta }{\mathcal{I}}_{\alpha }^{\prime }={\mathcal{I}}_{\alpha }^{\prime }\left({\rho }_{AB}\right)-{\mathcal{I}}_{\alpha }^{\prime }\left({\sigma }_{AB}\right)$ for the entangled coherent state $|\mathrm{\Psi }\rangle$ against the coherent amplitude $\gamma$ and Rényi parameter $\alpha$. Negative regions indicate the breakdown of Gaussian extremality. (c), (d) Contour plots of $\mathrm{\Delta }{\mathcal{I}}_{\alpha }$, $\mathrm{\Delta }{\mathcal{I}}_{\alpha }^{\prime }$ for the entangled coherent state under a symmetric loss channel ${\mathcal{L}}_{\eta }\left[\right|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }\left|\right]$ with $\gamma =1$ as functions of the effective transmittance $\eta$ and the Rényi parameter $\alpha$. (e), (f) Contour plots for the loss dynamics of $\mathrm{\Delta }{\mathcal{I}}_{\mathrm{HS}}={\mathcal{I}}_{\mathrm{HS}}\left({\rho }_{AB}\right)-{\mathcal{I}}_{\mathrm{HS}}\left({\sigma }_{AB}\right)$ and $\mathrm{\Delta }{\mathcal{I}}_{\mathrm{TR}}={\mathcal{I}}_{\mathrm{TR}}\left({\rho }_{AB}\right)-{\mathcal{I}}_{\mathrm{TR}}\left({\sigma }_{AB}\right)$ for the entangled coherent state against the amplitude $\gamma$ and the effective transmittance $\eta$.

Figure 3

Loss dynamics of $\mathrm{\Delta }{\mathcal{I}}_{\mathrm{HS}}={\mathcal{I}}_{\mathrm{HS}}\left({\rho }_{AB}\right)-{\mathcal{I}}_{\mathrm{HS}}\left({\sigma }_{AB}\right)$ for a photon number entangled state $|\psi \rangle ={\sum }_{k=0}^2{c}_k{|k\rangle }_A{|k\rangle }_B$ with $c_0=0.986$, $c_1=0.162$, and $c_2={(1-c_0^2-c_1^2)}^{1/2}$ as a function of the effective transmittance $\eta$.

Figure 4

${\mathcal{J}}_D$ (trace distance, purple dotted curve), ${\mathcal{J}}_{\mathrm{LB}1}$ (black dashed curve) and ${\mathcal{J}}_{\mathrm{LB}2}$ (red solid curve) for the entangled coherent state with amplitude $\gamma =1$ under the symmetric loss channel ${\mathcal{L}}_{\eta }$ plotted against the effective transmittance $\eta$. The difference between the two measures ${\mathcal{J}}_{\mathrm{LB}1}$ and ${\mathcal{J}}_{\mathrm{LB}2}$ is negligible. For comparison, we also plot $\mathrm{\Delta }{\mathcal{I}}_1$ (brown dot-dashed curve) which clearly shows a nonmonotonic behavior.

Figure 5

$\mathrm{\Delta }{E}_F=E_F\left[{\rho }_{AB}\right]-E_F\left[{\sigma }_{AB}\right]$ for the entangled coherent state under a symmetric loss channel versus our non-Gaussianity measure ${\mathcal{J}}_{\mathrm{LB}1}$. We randomly sampled ${10}^5$ states for this parametric plot and observe a positive relation between two measures.

Figure 6

(a) ${\mathcal{J}}_{\mathrm{TR}}$ for CV Werner state in Eq. (13) against the fraction $f$. Black solid and brown dashed curves represent the cases for $r=0.05$ and $r=0.1$, respectively. (b) $\mathrm{\Delta }{E}_N=E_N\left[{\rho }_{AB}\right]-E_N\left[{\sigma }_{AB}\right]$ versus our non-Gaussianity measure ${\mathcal{J}}_{\mathrm{TR}}$ for CV Werner states. We have randomly sampled ${10}^4$ states with $0\le f\le 1$ and $0\le r\le 0.2$ for this parametric plot. (c), (d) entanglement negativity $\mathcal{N}$ for the original CV Werner state (dashed) and an output state from a Gaussian distillation protocol (solid), respectively [(c) $r=0.05$ and (d) $r=0.1$]. Shaded regions represent the degree of distillable entanglement with respect to the fraction $f$.