Parameter estimation for mixed states from a single copy

Given a single copy of a mixed state of the form $\rho =\lambda {\rho }_1+(1-\lambda ){\rho }_2$, what is the optimal measurement to estimate the parameter $\lambda$ if ${\rho }_1$ and ${\rho }_2$ are known? We present a general strategy to obtain the optimal measurements employing a Bayesian estimator. The measurements are chosen to minimize the deviation between the estimated value and the true value of $\lambda$. We explicitly determine the optimal measurements for a general two-dimensional system and for important higher dimensional cases.

Figure 1

(Color online) Schematic figure of the Bloch representation for the case where ${\rho }_1$ and ${\rho }_2$ are pure. The Bloch vectors ${\vec{r}}_b$ and $\Delta \vec{r}\propto {\vec{r}}_1-{\vec{r}}_2$ are orthogonal because of the circle of Thales. $\vec{r}$ denotes the direction of the considered measurement. In the calculation, it turns out that for the optimal measurement $\vec{r}$ is parallel to ${\vec{r}}_1-{\vec{r}}_2$. See text for further details.

Figure 2

(Color online) Schematic figure of the Bloch representation for the case where ${\rho }_1$ is pure, but ${\rho }_2$ is not. Now, the optimal measurement is a measurement in the direction of $\vec{r}$ which is tilted towards ${\vec{r}}_1$.

Figure 3

(Color online) Deviation of optimal measurement direction $\vec{r}$ from direction of ${\vec{r}}_1-{\vec{r}}_2$ specified by the angle ${\alpha }_0$ as a function of angle $\gamma$ between $\Delta \vec{r}$ and ${\vec{r}}_b$ for $r_b=0.8$. For states ${\rho }_1,{\rho }_2$ with the same entropy $(\gamma =\pm \pi ∕2)$ the deviation is zero. If, say, state ${\rho }_1$ corresponds to less entropy $\vec{r}$ is tilted towards the Bloch vector of ${\rho }_1$. For example, compare the case depicted in Fig. 2, where $\pi ∕2<\gamma <0$ and thus $0<{\alpha }_0<\pi ∕2$, indicating the named tilting.