Operational quasiprobabilities for qudits

We propose an operational quasiprobability function for qudits, enabling a comparison between quantum and hidden-variable theories. We show that the quasiprobability function becomes positive semidefinite if consecutive measurement results are described by a hidden-variable model with locality and noninvasive measurability assumed. Otherwise, it is negative valued. The negativity depends on the observables to be measured as well as a given state, as the quasiprobability function is operationally defined. We also propose a marginal quasiprobability function and show that it plays the role of an entanglement witness for two qudits. In addition, we discuss an optical experiment of a polarization qubit to demonstrate its nonclassicality in terms of the quasiprobability function.

Figure 1

Schematic representation of measurement setups. When two observables $A_1$ and $A_2$ are selectively and consecutively measured, four measurement setups can be implemented as shown in (a): (i) no measurement, (ii) the measurement of $A_1$, (iii) the measurement of $A_2$, and (iv) the consecutive measurement of $A_1$ and $A_2$. When there are $K$ possible observables, $2^K$ measurement setups can be implemented. As an example, the consecutive measurement of $A_1$ and $A_3$ is displayed in (b).

Figure 2

Degree of nonclassicality $\mathcal{N}$ of pure qubit states with mutually unbiased measurements. The Bloch vector $\vec{\rho }=({\rho }_1,{\rho }_2,{\rho }_3)$ of a pure qubit state corresponds to a point on the Bloch sphere. In (a), where two complementary observables are employed, the nonclassicality is displayed as a function of the Bloch vector, i.e., $\mathcal{N}({\rho }_1,{\rho }_2,{\rho }_3)$. In (b), where three complementary observables are employed instead, the nonclassicality is displayed in a similar way. In both (a) and (b), the maximal nonclassicality is given by ${\mathcal{N}}_{\max }=(\sqrt{2}-1)/4\approx 0.103$.

Figure 3

Degree of nonclassicality $\mathcal{N}$ of pure qubit states with mutually biased measurements. In (a), where two mutually biased measurements are employed, the nonclassicality is displayed as a function of the Bloch vector. In this case the maximal nonclassicality is given by ${\mathcal{N}}_{\max }=0.125$, which is higher than the maximal value attainable by mutually unbiased measurements, $(\sqrt{2}-1)/4\approx 0.103$, as shown in (b).

Figure 4

An optical experiment of a polarization qubit. A red (light gray) square represents a polarizing beam splitter (PBS) that transmits a horizontally polarized photon (H) and reflects a vertically polarized one (V). A blue (dark gray) square denotes a PBS that transmits a diagonally polarized photon (D) and reflects an antidiagonally polarized one (A). A black half circle represents a photon detector, and its position is denoted by $D_{a_1,a_2}$. The commensurate quasiprobability function can be obtained by Fourier transformation of the expectations of the measurement setups shown in (a)–(d).