It is possible to reformulate the three state method (see details here) in terms of mixed states for determination of of error and disturbance by measuring the mean values of and in three different states. We have and , for error and disturbance, respectively. For each projector combination of and an intensity output is acquired and the expectation value is calculated by combination of four intensities. We label these intensities as where take values . The expectation value of and for any state are obtained by and . To determine the error these intensities have to be measured for the state , the reflected state and the pure state . The same applies to the measurement of disturbance where the input states are , and the pure state . The prefactors and are obtained separately in the state preparation adjustment process. If is a general mixed qubit state then the polarization of the state is given by . This relation allows to prepare and check the initial state’s direction and the degree of mixtures. For a general mixed state given by , the degree of polarization is given by . In order to prepare the mixed states, required for the determination of error and disturbance , this has to be varied. This is achieved by applying a random noisy magnetic field in addition to the static one in DC1. That is, neutrons with different arriving times at the coil DC1 experience different magnetic field strengths. This is equivalent to apply different unitary operators , describing the noisy -rotation about the -axes, at each time: this is written in a form of . Detected count rates of the successive measurements carried out by apparatuses M1 and M2 are depicted blow (left). The successive measurements of and have four outcomes, denoted as , which are recorded for four input states, i.e. , and . The initial states are given by with five different mixtures . The observable is set as and the polar angle of the observable is tuned as . The detuned observable are adjusted within the -plane.
In the middle we have the squared disturbance vs. squared error . For detuning angle of the output observable coincides with with . For increasing angles the error increases, while disturbance either minimizes when corrected or maximizes when anti-corrected.
On the right side error-disturbance uncertainty relation measured with pure states is depicted: not only the lower but also upper bounds of the disturbance (at fixed error) are found. For a detuning angle of the output observable coincides with at which point . For increasing angles the error increases as well and disturbance spreads between the maximum and minimum values. The extremal points are reached at at which equals .