Trade-off between information and disturbance in qubit thermometry

We address the trade-off between information and disturbance in qubit thermometry from the perspective of quantum estimation theory. Given a quantum measurement, we quantify information via the Fisher information of the measurement and disturbance via four different figures of merit, which capture different aspects (statistical, thermodynamical, geometrical) of the trade-off. For each disturbance measure, the efficient measurements, i.e., the measurements that introduce a disturbance not greater than any other measurement extracting the same amount of information, are determined explicitly. The family of efficient measurements varies with the choice of the disturbance measure. On the other hand, commutativity between the elements of the probability operator-valued measure (POVM) and the equilibrium state of the thermometer is a necessary condition for efficiency with respect to any figure of disturbance.

Figure 1

Graphical representation of the set of binary measurements on a qubit. Each measurement scheme $\mathcal{M}$ is described by a POVM $({\mathrm{\Pi }}_0,{\mathrm{\Pi }}_1)$. Since ${\mathrm{\Pi }}_1={\mathbb{I}}_2-{\mathrm{\Pi }}_0$, it is sufficient to specify ${\mathrm{\Pi }}_0$, e.g., via its cartesian coordinates $(w,x,y,z)$. Any binary POVM is thus associated with a point of the cone of equations $w^2-x^2-y^2-z^2\ge 0$ and $0<w\le 1/2$. The spherical caps are the loci of points with fixed value of the POVM purity $\gamma$.

Figure 2

Upper panel: The Fisher information ${\mathcal{F}}_{\beta }$ as a function of the measurement noncommutativity $\chi$. Lower panel: The Fisher information ${\mathcal{F}}_{\beta }$ as a function of the measurement purity $\gamma$. Both plots are obtained upon fixing $\beta =\delta =1$.

Figure 3

Upper panel: Range of the $\mathrm{\Delta }$-disturbance ${\mathfrak{D}}_{\beta }^{\left(\mathrm{\Delta }\right)}$ as a function of the measurement noncommutativity $\chi$. Lower panel: A similar plot as a function of the measurement purity $\gamma$. Both plots are obtained upon fixing $\beta =\delta =1$.

Figure 4

Upper panel: Information and disturbance trade-off region for ${\mathfrak{D}}_{\beta }^{\left(\mathrm{\Delta }\right)}$. The region is foliated into line segments by fixing the value of the noncommutativity $\chi$. Lower panel: The quantity $\kappa \left(\chi \right)$ (see Proposition t2 for details) for different values of the product $\beta \delta$.

Figure 5

Range of the $F$-disturbance ${\mathfrak{D}}_{\beta }^{\left(F\right)}$ as a function of the measurement noncommutativity $\chi$ (upper panel) or the measurement purity $\gamma$ (lower panel). Both plots are obtained upon fixing $\beta \delta =1$.

Figure 6

Upper panel: Trade-off region for the $F$-disturbance measure ${\mathfrak{D}}_{\beta }^{\left(F\right)}$. The $F$-efficient measurements (dashed) correspond to $\theta =0$, while $\lambda ={\lambda }_{\mathcal{F}}^{\text{opt}}$ and $w=w_{\mathcal{F}}^{\text{opt}}$ are determined numerically. Lower panel: Parameters ${\lambda }_{\mathcal{F}}^{\text{opt}}$ and $w_{\mathcal{F}}^{\text{opt}}$ of the $F$-efficient POVMs as a function of the extracted information.

Figure 7

Range of the $\tau$-disturbance ${\mathfrak{D}}_{\beta }^{\left(\tau \right)}$ as a function of the measurement noncommutativity $\chi$ (upper panel) or the measurement purity $\gamma$ (lower panel). Both regions are unbounded.

Figure 8

Upper panel: The trade-off region for the $\tau$-disturbance measure ${\mathfrak{D}}_{\beta }^{\left(\tau \right)}$. The $\tau$-efficient measurements correspond to $\theta =0$, while $\lambda ={\lambda }_{\mathcal{F}}^{\text{opt}}$ and $w=w_{\mathcal{F}}^{\text{opt}}$ are determined numerically. Lower panel: Behavior of the optimal parameters ${\lambda }_{\mathcal{F}}^{\text{opt}}$ and $w_{\mathcal{F}}^{\text{opt}}$ as a function of the FI ${\mathcal{F}}_{\beta }$ for $\tau$-efficient measurements.

Figure 9

Range of the $\pi$-disturbance ${\mathfrak{D}}_{\beta }^{\left(\pi \right)}$ as either $\chi$ or $\gamma$ are varied. Dots correspond to performances of randomly generated POVMs with parameters $(\lambda ,\theta ,w)$ chosen uniformly.

Figure 10

Upper panel: Trade-off region for the $\pi$-disturbance measure ${\mathfrak{D}}_{\beta }^{\left(\pi \right)}$. Dots correspond to the performances of random measurements, whose POVMs have been generated with uniformly randomized parameters $(\lambda ,\theta ,w)$. The $\pi$-efficient measurements (dashed) correspond to $\theta =0$, while $\lambda ={\lambda }_{\mathcal{F}}^{\text{opt}}$ and $w=w_{\mathcal{F}}^{\text{opt}}$ are determined numerically. Bottom: Behavior of the optimal parameters ${\lambda }_{\mathcal{F}}^{\text{opt}}$ and $w_{\mathcal{F}}^{\text{opt}}$ as a function of the FI ${\mathcal{F}}_{\beta }$ for the $\pi$-efficient measurements.