FIG. 4. Probabilities \( P ( m _ { 1 } , m _ { 2 } | \psi ) \) for the outcomes of the sequential measurement of \( m _ { 1 } \) (PM-polarization) and \( m _ { 2 } \) (HV-polarization) on an input state polarized at \( 6 7 . 5 ^ { \circ } \), halfway between P and V .

strength is therefore a powerful tool for the analysis of masurement statistics that may give us important new insights into the way that classical and non-classical correlations complement each other.

FIG. 5. Conditional averages \( A _ { \mathrm { o p t . } } ( m _ { 1 } , m _ { 2 } ) \) as a function of measurement strength \( \theta \). The solid curve represents the theoretical prediction for a measurement without experimental imperfections, the broken line was calculated for an interferomter visibility of \( V _ { P M } = 0 . 9 3 \).

The conditional average \( A _ { \mathrm { o p t . } } ( m _ { 1 } , m _ { 2 } ) \) is obtained from the correlations between \( \hat { S } _ { P M } \) and the two measurement results \( m _ { 1 } \) and \( m _ { 2 } \) that originate from the statistics of the initial state \( \psi \). Specifically, the estimate is obtained by updating the initial statistics of \( \psi \) based on the outcomes \( m _ { 1 } \) and \( m _ { 2 } \), where the measurement strength controls the relative statistical weights of the information obtained from \( m _ { 1 } \) and \( m _ { 2 } \). At a maximal measurement strength of \( \theta \: = \: 2 2 . 5 ^ { \circ } \), the PM-measurement completely randomizes the HV-polarization, so that the conditional average \( A _ { \mathrm { o p t . } } ( m _ { 1 } , m _ { 2 } ) \) is independent of \( m _ { 2 } \) and the estimation procedure is based on the classical correlations between \( m _ { 1 } \) and \( \hat { S } _ { P M } \). As the measurement strength is weakend, a small contribution of non-classical correlations emerges as the conditional averages for \( m _ { 2 } = + 1 \) and for \( m _ { 2 } = - 1 \) split, with the estimates for the more likely \( m _ { 2 } \)-outcomes dropping towards zero and the estimates for the less likely \( m _ { 2 } \)-outcomes diverging to values greater than +1 for \( m _ { 1 } = + 1 \) and more negative than −1 for \( m _ { 1 } = - 1 \). Even small contributions of non-classical correlations therefore result in estimates that cannot be reproduced by classical statistics. Due to experimental imperfections, the anomalous values of \( A _ { \mathrm { o p t . } } ( + 1 , + 1 ) > 1 \) are easier to observe than the anomalous values of \( A _ { \mathrm { o p t . } } ( - 1 , + 1 ) < - 1 \). Specifically, the

small probabilities of the result (−1, +1) are significantly enlarged by the noise background associated with limited visibilities. As the measurement strength drops, the initial bias in favor of P-polarization in the input state \( \psi \) begins to outweigh the effect of the measurement result of \( m _ { 1 } \, = \, - 1 \) that would indicate M-polarization. Of particular interest is the crossing point around \( \theta = 1 2 . 3 ^ { \circ } \), where the initial information provided by \( \psi \) and the measurement information \( m _ { 1 } \) become equivalent and the estimate is \( A _ { \mathrm { o p t . } } ( - 1 , m _ { 2 } ) \, = \, 0 \) for both \( m _ { 2 } \, = \, + 1 \) and \( m _ { 2 } \, = \, - 1 \). For measurement strengths below this crossing point, the initial bias provided by the initial state towards P-polarization clearly dominates the estimate, resulting in positive values of \( A _ { \mathrm { o p t . } } ( - 1 , m _ { 2 } ) \). Significantly, the increase of the estimate with reduction in measurement strength is much faster for \( m _ { 2 } = + 1 \) than for \( m _ { 2 } = - 1 \), since the lower probability of the outcome \( m _ { 2 } = + 1 \) effectively enhances the statistical weight of the information. For \( \theta \approx 1 1 ^ { \circ } \), this enhancement of the estimate even results in a crossing between \( A _ { \mathrm { o p t . } } ( - 1 , + 1 ) \) and \( A _ { \mathrm { o p t . } } ( + 1 , + 1 ) \), so that the value estimated for an outcome of \( m _ { 1 } = - 1 \) actually exceeds the value estimated for an outcome of \( m _ { 1 } = + 1 \) at measurement strengths of \( \theta < 1 1 ^ { \circ } \). This counter-intuitive difference between the outcome of the PM-measurement and the estimated value of PM-polarization appears due to the effects of the measurement outcome \( m _ { 1 } \) on the quantum correlations between \( m _ { 2 } \) and the target observable \( \hat { S } _ { H V } \) in the initial state. Specifically, low probability outcomes always enhance the correlations between measurement results and target observable. Therefore, the low probability outcome \( m _ { 1 } = - 1 \) enhances the correlation between \( m _ { 2 } \, = \, + 1 \) and \( \bar { S } _ { H V } \), which favours the P-polarization. On the other hand, the much higher probability of \( m _ { 1 } = + 1 \) does not result in a comparative enhancement of this correlation, so that the estimated value \( A _ { \mathrm { o p t . } } ( + 1 , + 1 ) \) for an outcome of \( m _ { 1 } = + 1 \) 1 is actually lower than the estimated value \( A _ { \mathrm { o p t . } } ( - 1 , + 1 ) \) for an outcome of \( m _ { 1 } \, = \, - 1 \). These non-classical aspects of correlations between measurement results and target observable highlight the importance of the relation between the two measurement outcomes: it is impossible to isolate the measurement result \( m _ { 1 } \) from the context established by both \( \psi \) and \( m _ { 2 } \). Since the estimated values \( A _ { \mathrm { o p t . } } ( m _ { 1 } , m _ { 2 } ) \) correspond to weak values, this observation may also provide a practical example of the relation between weak values and contextuality [20].

In the limit of zero measurement strength (\( \theta \)= 0), the estimated values depend only on \( m _ { 2 } \), with the unlikely measurement outcome of \( m _ { 2 } = + 1 \) resulting in an anomalous weak value of \( A _ { \mathrm { o p t . } } ( m _ { 1 } , + 1 ) = \sqrt { 2 } + 1 \) and the likely outcome of \( m _ { 2 } = - 1 \) resulting in a weak value estimate of \( A _ { \mathrm { o p t . } } ( m _ { 1 } , - 1 ) = \sqrt { 2 } \! - \! 1 \). Since these estimates are based only on the outcomes of precise measurements of HV-polarization, they provide a direct illustration of the non-classical correlation between \( \hat { S } _ { P M } \) and \( \hat { S } _ { H V } \) in \( \psi \). Due to the specific choice of initial state, \( A _ { \mathrm { o p t . } } ( m _ { 1 } , + 1 ) \) is larger than \( A _ { \mathrm { o p t . } } ( m _ { 1 } , - 1 ) \), which means that the detection of H-polarization makes P-polarization more likely, while the detection of V-polarization increases the likelihood of M-polarization. If we disregard for a moment that the estimated values for \( m _ { 2 } = + 1 \) lie outside the range of possible eigenvalues, we can give a fairly intuitive characterization of this non-classical correlation. Clearly, the lowest likelihood is assigned to the combination of H-polarization and M-polarization, which are the least likely polarization results obtained in separate measurements of HV-polarization and PM-polarization for the input state \( \psi \). We can therefore summarize the result by observing that quantum correlations between Bloch vector components strongly suppress thej oint contributions of the least likely results, to the point where the correlation can exceed positive probability boundaries, corresponding to an implicit assignment of negative values to the combination of the two least likely outcomes [18].

The results presented in this section clearly show that the final HV-measurement provides additional information about the target observable \( \hat { A } = \hat { S } _ { P M } . \). We can therefore expect that the measurement error will be reduced signifi-cantly if we use \( A _ { m _ { 1 } , m _ { 2 } } = A _ { \mathrm { o p t . } } ( m _ { 1 } , m _ { 2 } ) \) as measurement result assigned to thej oint outcome \( ( m _ { 1 } , m _ { 2 } ) \). In the final section of our discussion, we will therefore take a look at the measurement errors obtained at different measurement strengths \( \theta \) and identify the amount of PM-information obtained from the measurement of HV-polarization.

# VI. EVALUATION OF MEASUREMENT ERRORS

According to Eq. (6), the measurement errors for optimized measurement outcomes \( A _ { m } = A _ { \mathrm { o p t . } } ( m ) \) can be evaluated directly by subtracting the statistical fluctuations of \( A _ { m } \) from the initial fluctuations of the target observable \( \hat { A } \) in the initial state \( \psi \). We can therefore use the results of the previous sections to obtain the measurement errors \( \varepsilon ^ { 2 } ( A ) \) for the measurement outcomes \( m _ { 1 } \) and for the combined measurement outcomes \( ( m _ { 1 } , m _ { 2 } ) \). The results are shown in Fig. 6, together with the measurement error given by Eq. (11), which is obtained by assigning values of \( A _ { m _ { 1 } } = \pm 1 \) to the measurement outcomes \( m _ { 1 } \).

Not surprisingly, the sub-optimal assignment of eigenvalues to the measurement outcomes results in much avoidable extra noise. In fact, the error for this assignment exceeds the uncertainty of \( \Delta A ^ { 2 } \, = \, 0 . 5 \) for the initial state \( \psi \) at measurement strengths of \( \theta < 1 3 . 5 ^ { \circ } \), indicating that one can obtain a better estimate of PM-polarization from the expectation value of the input state. This never happens in the case of the errors \( \varepsilon _ { \mathrm { o p t . } } \) associated with the optimal estimates of the target observable, since the optimized estimates based on the conditional averages for the different