Experimental Test of Residual Error-Disturbance Uncertainty Relations for Mixed Spin-$½$ States

The indeterminacy inherent in quantum measurements is an outstanding character of quantum theory, which manifests itself typically in the uncertainty principle. In the last decade, several universally valid forms of error-disturbance uncertainty relations were derived for completely general quantum measurements for arbitrary states. Subsequently, Branciard established a form that is optimal for spin measurements for some pure states. However, the bound in his inequality is not stringent for mixed states. One of the present authors recently derived a new bound tight in the corresponding mixed state case. Here, a neutron-optical experiment is carried out to investigate this new relation: it is tested whether error and disturbance of quantum measurements disappear or persist in mixing up the measured ensemble. The attainability of the new bound is experimentally observed, falsifying the tightness of Branciard’s bound for mixed spin states.

Figure 1

Illustration of the experimental setup. The neutron polarimeter setup consists of three stages. (i) Preparation (blue region): a monochromatic neutron beam is polarized in the $+z$ direction by passing through a supermirror spin polarizer. In the coil (DC-1) the directions and mixture of the input states are adjusted. (ii) Apparatus $M1$, consisting of a projective $O_A$ measurement (pink region) and a correction operation (light green region): the first measurement is carried out by analyzer 1 together with the coils (DC-$3/4$) followed by a unitary rotation of the output state of the $O_A$ measurement. (iii) Apparatus $M2$, measuring $B$ (dark green region): the second measurement carried out by the coil (DC-4) together with the analyzer 2 is fixed to make a $B$ measurement. Transparent coils are virtual, in practice other dc coils fulfill their tasks.

Figure 2

Influence of the correction procedure on the disturbance. After the projective measurement of $O_A({\theta }_{OA}=5\pi /18)$ plus unitary rotations $U^{\text{corr}}$ [with angle parameters $(\vartheta ,\varphi )$ for the output state of the apparatus $M1$], the measurement of $B={\sigma }_y$ is performed in apparatus $M2$. The angles identify the output states $|\psi (\vartheta ,\varphi )⟩={\mathbf{(}\cos (\vartheta /2),e^{\mathrm{i}\varphi }\sin (\vartheta /2)\mathbf{)}}^T$ and $|-\psi (\vartheta ,\varphi )⟩=|\psi (\pi -\vartheta ,\varphi +\pi )⟩$ of the unitary operation. Blue and red arrow indicate the position of the minimal ($\pi /2$,$3\pi /2$) and maximal ($\pi /2$,$\pi /2$) disturbance.

Figure 3

Error-disturbance uncertainty relation as indicated by inequality Eq. (5) measured with pure states: not only the lower but also upper bounds of the disturbance are found. For a detuning angle of ${\theta }_{OA}=0$ the output observable $O_A$ coincides with $A={\sigma }_z$ at which point $\mathbf{(}\epsilon (A),\eta (B)\mathbf{)}=(0,\sqrt{2})$. For increasing angles ${\theta }_{OA}$ the error increases as well and disturbance spreads between the maximum and minimum values. The extremal points are reached at ${\theta }_{OA}=\pi /2$, at which $O_A$ equals $B$. For angles from $\pi /2$ to $\pi$ the EDUR evolves back. Blue and red arrows indicate the points denoted to the maximal and minimal disturbance in Fig. 2.

Figure 4

Error $\epsilon (A)$ versus disturbance $\eta (B)$ for the standard configuration ($A={\sigma }_z$, $B={\sigma }_y$) with four different mixtures of the state ${\rho }_x(\alpha )=\frac{1}{2}(\mathbb{1}+\alpha {\sigma }_x)$: (a) $\alpha =0.75$, (b) $\alpha =0.5$, (c) $\alpha =0.25$, and (d) $\alpha =0$. The red shaded areas are forbidden according to Eq. (5). The border indicates the theoretical prediction of the lower bound $D_{AB}=1$ which is saturated by our data points. The behavior of the bound $C_{AB}^{\prime }$ is indicated by the colored dashed lines. A change of the mixture parameter $\alpha$ has no effect on the final error-disturbance relation in the standard configuration as initially predicted by the expectation value $C_{AB}^{\prime }$.