For a simple example, consider two plates near each other, one with a positive charge and the other negative close together. That's a capacitor. Now physically pull the plates a bit away. Voila! Every individual piece is in the same state yet you now have more energy stored!
Take the same capacitor, but make each plate into 2 layers. Pull the outer layers away and attach your electrodes to the inner layers. You can arrange to have more energy, at the same voltage. As the capacitor discharges, bring the outer layers in to keep the voltage across the inner plates constant. Reverse the process as it charges.
Please be clear, I'm not claiming that their battery works this way, or even that it works at all. I'm only making the claim that it is actually possible to design a device where what looks impossible here is not necessarily impossible.
From the paper:
> At the cathode, the build-up of plated metallic lithium changes the morphology of the cathode to increase C_C and create a very slow fade of V_cell with time of discharge
How much more energy are we talking about?
(This is somewhat circular reasoning. The point is that the charges on capacitor plates attract each other so it takes work to separate the plates. Integrating the force along the path the plate takes during the separation gives the extra energy stored in the capacitor.)
A similar thing is a static generator made using a pie tin. You put a little charge on the pie tin by some means, like rubbing on styrofoam, and then you separate the tin from the styrofoam, which gives more energy to the tin (increases the voltage), which then you can transfer to a Leyden jar.
I also discovered in high school that you can use a pair of plastic chairs and two willing participants to generate high voltages when there is low humidity. Person one stands up off the chair, touches person two, then sits down. Person two then stands up off the chair, touches person one, then sits down. Repeat. Eventually, you will generate fairly high voltages that sort of hurt. You can hear the excess static charge on the chair squeal into the atmosphere after standing up.