Single-qubit thermometry

Distinguishing hot from cold is the most primitive form of thermometry. Here we consider how well this task can be performed using a single qubit to distinguish between two different temperatures of a bosonic bath. In this simple setting, we find that letting the qubit equilibrate with the bath is not optimal, and depending on the interaction time it may be advantageous for the qubit to start in a state with some quantum coherence. We also briefly consider the case that the qubit is initially entangled with a second qubit that is not put into contact with the bath and show that entanglement allows for even better thermometry.

Figure 1

Variation of the Euclidean distance $\Delta$, normalized to ${\Delta }_{\infty }$, vs $t$ (dimensionless) and $\theta$ for $n_1=12,n_2=20$. (top) Contours of $\Delta (t,\theta |T_1,T_2)/{\Delta }_{\infty }$. The blue dashed line indicates the value of $\theta$ that maximizes $\Delta$ for each $t$. (bottom) $\Delta /{\Delta }_{\infty }$ for values of various $\theta$.

Figure 2

Normalized distance vs time $t$ (dimensionless) between the transient states for the case with the qubit initially in a maximally entangled state [blue (top) curve] and the best achievable distance for a single qubit [bottom (red) curve; i.e., the envelope of Fig. 1, bottom].