Experimental investigation of multi-observable uncertainty relations

The uncertainty relation is a distinguishing feature of quantum theory, characterizing the incompatibility of noncommuting observables in the preparation of quantum states. Recently, many uncertainty relations were proposed with improved lower bounds and were deemed capable of incorporating multiple observables. Here we report an experimental verification of seven uncertainty relations of this type with single-photon measurements. The results, while confirming these uncertainty relations, show as well the relative stringency of various uncertainty lower bounds.

Figure 1

Experimental setup. In the state preparation stage, a photon pair is produced via type-II spontaneous parametric down-conversion (SPDC) in a BBO crystal. Half-wave plate (HWP) and 1-mm BBO in each path are used to compensate the birefringent walk-off effect in the 2-mm BBO. Polarizing beam splitter (PBS), quarter wave plate (QWP, Q1), H1, and Q2 are used to generate qubit states. In the projective measurement stage, Q3, H2, and PBS are used for measuring the expectation values of the observables.

Figure 2

Experimental results of uncertainty relations for three observables. (a) Experimental results for $A={\sigma }_x$, $B={\sigma }_y$, and $C={\sigma }_z$, with a family of states $\left|\psi \right(\theta ,0)\rangle =cos(\theta /2\left)\right|0\rangle +sin(\theta /2)|1\rangle$. (b) Experimental results for ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$, with states $|\psi (\pi /3,\varphi )\rangle =\sqrt{3}/2|0\rangle +e^{i\varphi }/2|1\rangle$. In (a) and (b), the solid black line corresponds to the LHS of inequalities (4), (5), and (6), i.e., the sum of the variances (SV) ${\left(\mathrm{\Delta }{\sigma }_x\right)}^2+{\left(\mathrm{\Delta }{\sigma }_y\right)}^2+{\left(\mathrm{\Delta }{\sigma }_z\right)}^2$. The black circles represent the measured SV. The dot-dashed green, dashed purple, and dotted red curves, in turn, represent the theoretical values of T1, T2, and T3, where T1, T2, and T3 denote the RHS of relations (4), (5), and (6), respectively. The green stars, purple squares, and red triangles, in turn, represent the experimental values of T1, T2, and T3. Error bars represent $\pm 1$ standard deviation.

Figure 3

Experimental results of uncertainty relations for $N$ observables ($N=3$). (a) Experimental results for $A={\sigma }_x$, $B={\sigma }_y$, and $C={\sigma }_z$, with a family of states $\left|\psi \right(\theta ,0)\rangle =cos(\theta /2\left)\right|0\rangle +sin(\theta /2)|1\rangle$. (b) Experimental results for ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$, with states $|\psi (\pi /3,\varphi )\rangle =\sqrt{3}/2|0\rangle +e^{i\varphi }/2|1\rangle$. In (a) and (b), the solid black line corresponds to the LHS of inequalities (7), (8), (9), and (10), i.e., the sum of the variances (SV) ${\left(\mathrm{\Delta }{\sigma }_x\right)}^2+{\left(\mathrm{\Delta }{\sigma }_y\right)}^2+{\left(\mathrm{\Delta }{\sigma }_z\right)}^2$. The black circles represent the measured SV. The dashed magenta, dot-dashed blue, dotted gray, and dashed cyan curves, in turn, represent the theoretical values of M1, M2, M3, and M4, where M1, M2, M3, and M4 denote the RHS of relations (7), (8), (9), and (10), respectively. The magenta up-triangles, blue down-triangles, gray stars, and cyan squares, in turn, represent the experimental values of M1, M2, M3, and M4.