Experimental investigation of the stronger uncertainty relations for all incompatible observables

The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However, it does not capture the concept of incompatible observables because it can be trivial even for two incompatible observables. We experimentally demonstrate that the new stronger uncertainty relations proposed by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)] relating to the sum of variances are valid in a state-dependent manner and that the lower bound is guaranteed to be nontrivial when two observables are incompatible on the state of the system being measured. The behavior we find agrees with the predictions of quantum theory and obeys the new uncertainty relations even for the special states which trivialize the Heisenberg-Robertson relation. We realize a direct measurement model and perform an experimental investigation of the strengthened relations.

Figure 1

(a) Representation of a qutrit. Here H and V denote the horizontal and vertical polarizations of the single photons, respectively. The subscripts u and d represent the upper and lower spatial modes of the single photons, respectively. (b) Experimental setup. The herald single photons are created via type I spontaneous parametric down-conversion in a BBO crystal and are injected into the optical network. The polarizing beam splitter (PBS), half-wave plate (HWP, H1), and beam displacer (${\mathrm{BD}}_i$) are used to generate a qutrit state $\left|{\psi }_{\varphi }\right.〉$. The HWPs (H2–H7) and three BDs are used to realize the projective measurements of observables $J_{x\left(y\right)}, J_{x\left(y\right)}^2$, and $C_{\pm }$. To realize that of $D$, H4 and H5 are replaced by the QWPs (Q4 and Q5) at ${45}^{\circ }$.

Figure 2

Experimental results. The solid black line corresponds to the LHS of inequalities (2) and (3), i.e., $\mathrm{\Delta }{J}_x^2+\mathrm{\Delta }{J}_y^2=1$. The black squares represent the sum of the measured uncertainties of $\mathrm{\Delta }{J}_x^2$ and $\mathrm{\Delta }{J}_y^2$ with the 12 states $\left|{\psi }_{\varphi }\right.〉$. The green circles, olive hexagons, magenta diamonds, and purple pentagons represent the experimental results of the RHS of inequality (2) with the optimal state ${\left|{\psi }_{\varphi }^{\perp }\right.〉}_{\text{opt}}$ and three randomly chosen states ${\left|{\psi }_{\varphi }^{\perp }\right.〉}_{\text{1,2,3}}$ for each of the 12 values of $\varphi$. The red dotted line corresponds to the bound of inequality (3) and red stars represent the measured $\langle D\rangle$ for the 12 states ${\left|{\psi }_{\varphi }^{\perp }\right.〉}_{J_x+J_y}$. The blue dot-dashed curves and triangles represent the theoretical predictions and experimental results of the product of the uncertainties and the expectation value of the commutator (Heisenberg-Robertson relation). Error bars indicate the statistical uncertainty.

Figure 3

Experimental setup. The herald single photons are created via type I SPDC in a BBO crystal and are injected into the optical network. The PBS and HWP (H1) are used to generate a qubit state $\left|{\psi }_{\varphi }\right.〉$. The HWP (or QWP) and the following BD are used to realize the projective measurements of observables ${\sigma }_{x\left(y\right)}, {\sigma }_{x\left(y\right)}^2, C_{\pm }$, and $D$.

Figure 4

Experimental results. The solid black line corresponds to the LHS of inequalities (B1) and (B2). The black squares represent the sum of the measured uncertainties of $\mathrm{\Delta }{\sigma }_x^2$ and $\mathrm{\Delta }{\sigma }_y^2$ with the 12 states $\left|{\psi }_{\varphi }\right.〉$. The green circles represent the experimental results of the RHS of inequality (B1) with the optimal and only state $\left|{\psi }_{\varphi }^{\perp }\right.〉$ for each of the 12 values of $\varphi$. The red dotted line corresponds to the bound of inequality (B2) and red stars represent the measured $\langle D\rangle$ for the 12 states $\left|{\psi }_{\varphi }^{\perp }\right.〉$. The blue dot-dashed curves and triangles represent the theoretical predictions and experimental results of the product of the uncertainties and the expectation value of the commutator (Heisenberg-Robertson relation). Error bars indicate the statistical uncertainty.