Optimal work extraction from quantum batteries based on the expected utility hypothesis

Work extraction in quantum finite systems is an important issue in quantum thermodynamics. The optimal work extracted is called ergotropy, and it is achieved by maximizing the average work extracted over all the unitary cycles. However, an agent that is non-neutral to risk is affected by fluctuations and should extract work by following the expected utility hypothesis. Thus, we investigate the optimal work extraction performed by a risk non-neutral agent by maximizing the average utility function over all the unitary cycles. We mainly focus on initial states that are incoherent with respect to the energy basis, achieving a probability distribution of work. In this case we show how the optimal work extraction will be performed with an incoherent unitary transformation, namely a permutation of the energy basis, which depends on the risk aversion of the agent. We give several examples, in particular also the work extraction from an ensemble of quantum batteries is examined. Furthermore, we also investigate how work extraction is affected by the presence of initial quantum coherence in the energy basis by considering a quasiprobability distribution of work.

Figure 1

The average work $\langle w\rangle$ and $\langle e^{-rw}\rangle$ corresponding to the optimal unitary operator in the function of $r$ for a single battery with $d=3$. We put $p_1=0.1, p_2=0.4, p_3=0.5, {\epsilon }_1=0, {\epsilon }_2=0.579$, and ${\epsilon }_3=1$. We calculate the optimal unitary operator by selecting the unitary operator giving the minimum value of $\langle e^{-rw}\rangle$.

Figure 2

Optimal permutation in the plane $(p,r)$. In the orange region we get the NOT (2,1) and in the white region the identity (1,2). The critical value $r=r^{*}$ is the bottom boundary of the white region.

Figure 3

In the top left panel, we plot the probability $P_k$ in the function of $r$. $P_k$ is the probability to get a permutation $k$, where $k=(1,2,3),(1,3,2),...,(3,2,1)$, by choosing randomly $p_1$ and $p_2$, e.g., by flipping a coin. In detail, $P_k$ is calculated as the frequency to get a certain permutation in the plane $(p_1,p_2)$. In the remaining panel, the optimal permutation in the plane $(p_1,p_2)$ for different value of $r$. In detail, we put $r=0$ (top right panel), $r=-1$ (bottom left panel), and $r=1$ (bottom right panel). We consider the energies ${\epsilon }_k=k$.

Figure 4

The optimal certainty equivalent ${\mathcal{E}}_{\mathrm{CE}}\left({\rho }^{\otimes n}\right)$ versus the number $n$ of batteries (blue points). We put $r=1, p_1=0.538, p_2=0.237, p_3=0.225, {\epsilon }_1=0, {\epsilon }_2=0.579$, and ${\epsilon }_3=1$. For the asymptotic value in Eq. (75) (red dashed line) we get $f\left[{\tilde{p}}^{*}\right]\approx 0.0043$.