Experimental Test of Error-Disturbance Uncertainty Relations by Weak Measurement

We experimentally test the error-disturbance uncertainty relation (EDR) in generalized, strength-variable measurement of a single photon polarization qubit, making use of weak measurement that keeps the initial signal state practically unchanged. We demonstrate that the Heisenberg EDR is violated, yet the Ozawa and Branciard EDRs are valid throughout the range of our measurement strength.

Figure 1

Schematic diagram to test error-disturbance relation using (a) three-state method and (b) weak-probe method. (c) Quantum circuit model of the weak-probe method for single-qubit observables $A=M=Z$ and $B=X$.

Figure 2

(a) Schematic diagram of the experimental setup. Our optical implementation is separated into the state preparation, WP, MA, and post $X$ measurement. In these apparatuses, our signal qubit is the polarization state of a photon, and the path degree of freedom of the same photon is the probe qubit; each apparatus has two output paths (0 or 1) corresponding to the measurement outcomes. For the weak measurement of $X$, additional HWPs (dashed rectangles) are inserted on both sides of the WP. The WP and MA are based on the polarization-division Sagnac interferometer (b) and the corresponding quantum circuit is depicted in (c), where the quantum operations of the PBS, HWPs, and mirrors are indicated (see Supplemental Material [12]). The glass plate compensates the phase difference between the two counterpropagating paths.

Figure 3

Experimental results. (a) The error $\epsilon (Z)$ (blue circles) and disturbance $\eta (X)$ (red squares) as functions of the measurement strength $\cos 2\theta$. Dashed curves are the theoretically calculated error and disturbance for perfect implementation of the quantum circuit presented in Fig. 1. Solid curves are the theoretical values after the nonideal extinction ratio of a PBS is taken into account. (b) Left-hand sides of the EDRs. Blue circles: the Heisenberg EDR in Eq. (1). Red squares: the Ozawa EDR in Eq. (3). Purple triangles: the Branciard EDR in Eq. (5). Green diamonds: the Branciard EDR in Eq. (9). Dashed and solid curves are plotted in the same way as (a). The right-hand side of the EDRs ($C=0.995$) is indicated by the gray line, which is nearly overlapped by the dashed green curve.

Figure 4

Comparison of EDRs’ lower bounds in the error–disturbance plot. Blue (solid) curve: the Heisenberg bound in Eq. (1). Red (short dashed) curve: the Ozawa bound in (3). Purple (long dashed) curve: the Branciard bound in (5). Green (dotted-dashed) curve: the Branciard bound in (9). Black (filled) circles: experimental data shown in Fig. 3. Black (dotted) curve: theoretical prediction for our experiment using imperfect PBSs. The lower-left side of each bound is the forbidden region by the corresponding EDR. Each bound was calculated for $C=1$.