Testing Heisenberg-Type Measurement Uncertainty Relations of Three Observables

Heisenberg-type measurement uncertainty relations (MURs) of two quantum observables are essential for contemporary research in quantum foundations and quantum information science. Going beyond, here we report the first experimental study of MUR of three quantum observables. We establish rigorously MURs for triplets of unbiased qubit observables as combined approximation errors lower bounded by an incompatibility measure, inspired by the proposal of Busch et al. [Phys. Rev. A 89, 012129 (2014)]. We develop a convex programming protocol to numerically find the exact value of the incompatibility measure and the optimal measurements. We propose a novel implementation of the optimal joint measurements and present several experimental demonstrations with a single-photon qubit. We stress that our method is universally applicable to the study of many qubit observables. Besides, we theoretically show that MURs for joint measurement can be attained by sequential measurements in two of our explored cases. We anticipate that this work may stimulate broad interests associated with Heisenberg’s uncertainty principle in the case of multiple observables, enriching our understanding of quantum mechanics and inspiring innovative applications in quantum information science.

Figure 1

Experimental optimal joint measurement on triplets of qubit observables. (a) An illustration to approximate measurements $\{M_i\}$ via the implementation of compatible measurement $\{N_i\}$, where the combined approximation errors $\{{\mathrm{\Delta }}_{\rho }(M_i,N_i)\}$ are maximized over all input quantum states ${\rho }_s$ and minimized over all triple jointly measurable observables. (b) A schematics of experimental optimal joint measurement on triplets of qubit observables. We generate a pair of correlated photons with ${\lambda }_{\text{signal}}=1556\mathrm{nm}$ and ${\lambda }_{\text{idler}}=1560\mathrm{nm}$ via the type-0 spontaneous parametric down-conversion process by pumping a periodically poled MgO doped lithium niobate (PPLN) crystal with a laser light at ${\lambda }_p=779\mathrm{nm}$ [58]. The detection of a signal photon heralds the presence of an idler photon. With qubit encoded to the polarization state of the idler photon, we install four polarization-projective detection modules to conduct measurements $\{M_i\}$ and $\{O_k\}$ with variable beam splitter (VBS) to adjust the weight, respectively (see Supplemental Material [59] for details). (c) A realization of VBS with adjustable beam-splitting ratio.

Figure 2

Exact value of incompatibility of 4 triplets of idea qubit observables: (a) ${\mathcal{M}}_o$, (b) ${\mathcal{M}}_{\perp }$, (c) ${\mathcal{M}}_p$, (d) ${\mathcal{M}}_Y$. Red dashed lines are numerical results calculated with protocol 1, blue smooth curves are lower bound of incompatibility in Eq. (3), and red triangles represent experimental results. Error bars stand for 1 standard deviation.

Figure 3

Realization of compatible measurements $\{N_i\}$ by sequential measurements $N_1\to N_2^{\prime }\to N_3^{\prime }$.