Quantum Charging Advantage Cannot Be Extensive without Global Operations

Quantum batteries are devices made from quantum states, which store and release energy in a fast and efficient manner, thus offering numerous possibilities in future technological applications. They offer a significant charging speedup when compared to classical batteries, due to the possibility of using entangling charging operations. We show that the maximal speedup that can be achieved is extensive in the number of cells, thus offering at most quadratic scaling in the charging power over the classically achievable linear scaling. To reach such a scaling, a global charging protocol, charging all the cells collectively, needs to be employed. This concludes the quest on the limits of charging power of quantum batteries and adds to other results in which quantum methods are known to provide at most quadratic scaling over their classical counterparts.

Figure 1

(a) Maximum power $P_{\max }$ and maximum operator norm of the commutator $\parallel [\widehat{H},\widehat{V}]\parallel$ as a function of $L$ (maximized over all time and 500 realizations of disorder for each value of $L$), starting from the ground state of initial Hamiltonian $\widehat{H}={\sum }_{l=1}^{L}h{\widehat{\sigma }}_l^z$ and charged by driving Hamiltonian as in Eq. (15). For this example, we fixed $h=1$. We observe a slight decrease in $P_{\max }$ and $\parallel [\widehat{H},\widehat{V}]\parallel$ with growing $L$, which is a result of increasing dimensionality of the system, resulting in the lower chance of the of the initial state to be optimal. While keeping the number of realizations of different driving fixed at 500, which means that upper the bound is more difficult to reach. (b) The same as (a) but for a driving given by Eq. (16). For this example, we additionally fixed $V=1$.