Quantum dot based Kitaev chains: Majorana quality measures and scaling with increasing chain length

Realizing Majorana bound states (MBSs) in short, well-controllable chains of coupled quantum dots sidesteps the problem of disorder but requires finetuning and does not give the true topological protection inherent to long chains. Here, we introduce a quality measure that is also applicable in the presence of strong electron-electron interactions and that quantifies the closeness to topological protection of finetuned MBSs in short quantum dot chains. We call this measure local distinguishability (LD) because it puts a bound to the degree an arbitrary local measurement can distinguish between two states. We study the LD for quantum dot chains of different length. The three-dot chain is studied in detail, and we find that it may not always be an improvement over the two-dot case, a fact that can be understood within an effective model derived from perturbation theory. For longer chains, the LD vanishes exponentially, signaling a transition to a topological phase with two ground states that cannot be distinguished by any local measurement.