Heisenberg's error-disturbance relations: A joint measurement-based experimental test

The original Heisenberg error-disturbance relation was recently shown to be not universally valid and two different approaches to reformulate it were proposed. The first one focuses on how the error and disturbance of two observables $A$ and $B$ depend on a particular quantum state. The second one asks how a joint measurement of $A$ and $B$ affects their eigenstates. Previous experiments focused on the first approach. Here we focus on the second one. First, we propose and implement an extendible method of quantum-walk-based joint measurements of noisy Pauli operators to test the error-disturbance relation for qubits introduced in the work of Busch et al. [Phys. Rev. A 89, 012129 (2014)], where the polarization of the single photon, corresponding to a walker's auxiliary degree of freedom that is commonly known as a coin, undergoes a position- and time-dependent evolution. Then we formulate and experimentally test a universally valid state-dependent relation for three mutually unbiased observables. We therefore establish a method of testing error-disturbance relations.

Figure 1

Experimental setup. (a) Configuration for preparing four input states and the relevant settings of HWP1 and QWP1. (b) Setup for conducting the von Neumann measurement of ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ and the relevant settings of HWP1 and QWP1. Here $|{\sigma }_k+\rangle \langle {\sigma }_k+|$ is the measurement operator corresponding to the outcome ${\sigma }_k=1$, where $k=x$, $y$, and $z$. (c) Optical network for joint measurement of three pairs of Pauli operators: $({\sigma }_x,{\sigma }_y)$, $({\sigma }_x,{\sigma }_z)$, and $({\sigma }_y,{\sigma }_z)$. (d) Optical network for joint measurement of three Pauli operators.

Figure 2

Distance values for joint measurement of two unsharp Pauli observables. Different color bars represent the different input states: $|x+\rangle$, $|y+\rangle$, and $|z+\rangle$. The values of state-independent calibration distances are shown as the blue lines. The error bars are obtained by Monte Carlo simulation (1000 runs).

Figure 3

Normalized probability distributions for joint measurement of three noisy observables $X^{\prime }$, $Y^{\prime }$, and $Z^{\prime }$. Different color bars represent the different input states: $|\psi \rangle$, $|x+\rangle$, $|y+\rangle$, and $|z+\rangle$. The theory values are shown as the star markers and error bars are too small to identify.