  Bulk-boundary correspondence, a central principle in topological matter
relating bulk topological invariants to edge states, breaks down in a generic
class of non-Hermitian systems that have so far eluded experimental effort.
Here we theoretically predict and experimentally observe non-Hermitian
bulk-boundary correspondence, a fundamental generalization of the conventional
bulk-boundary correspondence, in discrete-time non-unitary quantum-walk
dynamics of single photons. We experimentally demonstrate photon localizations
near boundaries even in the absence of topological edge states, thus confirming
the non-Hermitian skin effect. Facilitated by our experimental scheme of
edge-state reconstruction, we directly measure topological edge states, which
match excellently with non-Bloch topological invariants calculated from
localized bulk-state wave functions. Our work unequivocally establishes the
non-Hermitian bulk-boundary correspondence as a general principle underlying
non-Hermitian topological systems, and paves the way for a complete
understanding of topological matter in open systems.
