Quantum Lunch: The martingale method meets the detectability lemma

Speaker: Angelo Lucia

The martingale method meets the detectability lemma


When we analyse quantum many-body Hamiltonians for spin lattice systems, one of the key question is whether or not these operator have a spectral gap: a non-vanishing separation between the two lowest energy levels. When we consider frustration-free models, one useful tool for proving that such a gap exists is the so-called martingale method, which requires some knowledge of the groundstate structure of the Hamiltonian to be applied. It is a powerful tool in its simplicity, and was at the hearth of most results about gap for Matrix Products States Hamiltonians, since in 1D its assumptions can be essentially seen as a decay-of-correlations property of the groundstate of the model.

The detectability lemma is a tool developed in the context of quantum Hamiltonian complexity theory, with a very strong combinatorics flavour, and has used between other things to improve the area law bound for gapped Hamiltonians. Recently a converse detectability lemma has appeared, used for spectral gap amplification.

I would like to present the connection between these two lemmas, by presenting a different proof of the martingale method, in order to see it from a more "information theoretic" point of view, and making it more slightly natural for higher dimensions. Under this reformulation, it will be clear that the converse detectability implies a condition similar, but weaker, than the assumption of the martingale method. Moreover, an the detectability lemma implies a converse result than the martingale method: if we do have a spectral gap, then the conditions of the martingale method have to hold.