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We propose a general framework for using local measurements, local unitaries,and non-local classical communication to construct quantum channels which canefficiently prepare mixed states with long-range quantum order or quantumcriticality. As an illustration, symmetry-protected topological (SPT) phasescan be universally converted into mixed-states with long-range entanglement,which can undergo phase transitions with quantum critical correlations of localoperators and a logarithmic scaling of the entanglement negativity, despitecoexisting with volume-law entropy. Within the same framework, we present twoapplications using fermion occupation number measurement to convert (i) spinfulfree fermions in one dimension into a quantum-critical mixed state withenhanced algebraic correlations between spins and (ii) Chern insulators into amixed state with critical quantum correlations in the bulk. The latter is anexample where mixed-state quantum criticality can emerge from a gapped state ofmatter in constant depth using local quantum operations and non-local classicalcommunication.

Mixed-state long-range order and criticality from measurement and feedback