Quantifying precision loss in local quantum thermometry via diagonal discord

When quantum information is spread over a system through nonclassical correlation, it makes retrieving information by local measurements difficult—making global measurement necessary for optimal parameter estimation. In this paper, we consider temperature estimation of a system in a Gibbs state and quantify the separation between the estimation performance of the global optimal measurement scheme and a greedy local measurement scheme by diagonal quantum discord. In a greedy local scheme, instead of global measurements, one performs sequential local measurement on subsystems, which is potentially enhanced by feed-forward communication. We show that, for finite-dimensional systems, diagonal discord quantifies the difference in the quantum Fisher information quantifying the precision limits for temperature estimation of these two schemes, and we analytically obtain the relation in the high-temperature limit. We further verify this result by employing the examples of spins with Heisenberg's interaction.

Figure 1

Greedy local scheme: We first measure a subsystem $A$ and then measure the other subsystem $B$. The measurement on $A$ is optimum in the sense of local QFI. The constrained QFI of this greedy local scheme is given as ${\mathcal{F}}_{A\to B}\left(T\right)={\mathcal{F}}_A\left(T\right)+{\mathcal{F}}_{B|A}\left(T\right)$, and we explore how the quantum discord $D_{A\to B}$ affects the loss of precision loss, i.e., ${\mathcal{F}}_{AB}\left(T\right)-{\mathcal{F}}_{A\to B}\left(T\right)$.

Figure 2

$\mathrm{\Delta }\mathcal{F}$ and $-(1/T){\partial }_T{\mathcal{D}}_{A\to B}$, for a Heisenberg system with two qubits at (a) $B_1/J_x=3$, $B_2/J_x=1$, $J_z/J_x=2$, $J_y/J_x=1$ and (b) $B_1=B_2=0$, $J_z/J_x=2$, $J_y=0$.

Figure 3

$\mathrm{\Delta }{\mathcal{F}}_{123}$ and $-(1/T){\partial }_T{\mathcal{D}}_{123}$, for a Heisenberg system with three qubits at (a) $B/J=1$, $\alpha =0.3$ and (b) $B/J=2$, $\alpha =0.3$. Note that the paths denoted by subscripts 132 and 213 have the same results.

Figure 4

$|\left(\mathrm{\Delta }\mathcal{F}+(1/T){\partial }_T{\mathcal{D}}_{A\to B}\right)/\left(\mathrm{\Delta }\mathcal{F}-(1/T){\partial }_T{\mathcal{D}}_{A\to B}\right)|$. (a) $T/J=0.4$. Note that the increase of relative error at the edges is due to larger coupling amplitude making $T/|J\pm \lambda |$ smaller. (b) $T/J=2$.