Quasiprobability representation of quantum coherence

We introduce a general method for the construction of quasiprobability representations for arbitrary notions of quantum coherence. Our technique yields a non-negative probability distribution for the decomposition of any classical state. Conversely, quantum phenomena are certified in terms of signed distributions, i.e., quasiprobabilities, and a residual component unaccessible via classical states. Our unifying method combines well-established concepts, such as phase-space distributions in quantum optics, with resources of quantumness relevant for quantum technologies. We apply our approach to analyze various forms of quantum coherence in different physical systems. Moreover, our framework renders it possible to uncover complex quantum correlations between systems, for example, via quasiprobability representations of multipartite entanglement.

Figure 1

Quasiprobability density of a single-photon-added thermal state,$dP\left(\alpha \right)/d\alpha =([1+\overline{n}]{\left|\alpha \right|}^2-\overline{n}\left]\right)e^{-{\left|\alpha \right|}^2/\overline{n}}/\pi {\overline{n}}^3$ for $\overline{n}=1$. The negative contributions (magenta) certify that this state cannot be written as a convex mixture of coherent states.

Figure 2

Schematic decomposition. For classical states $\widehat{\rho }$, i.e., elements of the convex span of $\mathcal{C}$ (green area), the decomposition (4) applies. Nonclassical states can be further separated into two classes [Eq. (5)]: elements in the space spanned by $\mathcal{C}$ (magenta area with $\vec{p}\ngeqq 0$) and elements out of this plane (${\widehat{\rho }}_{\mathrm{res}}\ne 0$). Note that, in general, these classes are not disjoint because we can simultaneously have $\widehat{\rho }\ne 0$ and $\vec{p}\ngeqq 0$.

Figure 3

Decomposition of $\widehat{\rho }$, which is an element of $\mathrm{conv}\mathcal{C}$ (light gray area). The optimal distances (dashed circles) of this state to $\mathcal{C}$ (here the boundary of gray area) results in a subset of points $\mathcal{D}\subset \mathcal{C}$ (green bullets). For an optimal decomposition, we are going to show that $\widehat{\rho }\in \mathrm{conv}\mathcal{D}$ (light green polytope) holds true.

Figure 4

The green volume within the Bloch sphere (right) determines all classical states. The left plot depicts the minimal element of the quasiprobability $\vec{p}$ [cf. Eqs. (22) and (23)] for any cross section of the Bloch sphere, because of symmetry. The states in the lower triangle are decomposed in terms of a non-negative $\vec{p}$, $min\vec{p}\ge 0$. The remaining states are identified as nonclassical, $min\vec{p}<0$ (magenta).

Figure 5

Quasiprobabilities over the Poincaré-sphere phase space. The polar angel is $\vartheta$ and the azimuthal angle is $\phi$. Both define the classical states $|\vartheta ,\phi \rangle \langle \vartheta ,\phi |$ [Eq. (24)]. The radius over the sphere is used to show the quasiprobability $p_{\vartheta ,\phi }$ (bullet points) of the corresponding state; negative and positive values point inside (magenta) and outside (green), respectively. The negativities (magenta bullets) confirm the nonclassicality of the state $|\psi \rangle =\left(\right|s\rangle +|-s\rangle )/\sqrt{2}$.

Figure 6

The top row shows the green regions depicting separable states over $\mathbb{R}$ (left) and $\mathbb{C}$ (right) for the family of states in Eq. (33). The middle row shows, for ${\rho }_x=-{\rho }_y={\rho }_z=3/4$, the quasiprobabilities for $|a\rangle \otimes |b\rangle$; the $x$ and $y$ axes define the states in subsystems $A$ and $B$, respectively. The Hilbert-Schmidt norm of its residual component is depicted on the left for this rebit scenario. The bottom row shows, analogously, the qubit scenario $\parallel {\widehat{\rho }}_{\mathrm{res}}\parallel =0$.