Mixed-state long-range entanglement from adaptive circuits

Quantum simulators have advanced significantly in their controllability, and many platforms can now perform mid-circuit measurements. I will show how measurements and unitary evolution conditioned on the measurement outcomes ("adaptive quantum circuits") can produce mixed state long-range order and criticality, despite coexisting with extensive entropy. As an illustration, I will show how symmetry-protected topological order can be universally converted into mixed state long-range order, which can undergo a mixed state phase transition with logarithmic scaling of entanglement negativity, a measure of quantum correlations in mixed states. Further applications include fermion occupation measurements and feedback, which are possible in optical lattices, and I will show how these can efficiently transform a gapped pure ground state into a quantum critical mixed state.