Universal Properties of Critical Mixed-States from Measurement and Feedback

We explore the universal properties of mixed quantum matter obtained from “single-shot” adaptive evolution, in which a quantum-critical ground-state is manipulated through a single round of local measurements and local unitary operations conditioned on spatially distant measurement outcomes. The resulting mixed quantum states are characterized by altered long-distance correlations between local observables, mixed-state entropy, and entanglement negativity. By invoking a coarse-grained, continuum description of single-shot adaptation in (1+1) dimensions, we find that the extensive mixed-state entropy exhibits a subleading, constant correction ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>γ</a:mi> </a:math> ), while the entanglement negativity can grow logarithmically with subregion size, with a coefficient ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>α</c:mi> </c:math> ); both constants can attain universal values which are distinct from the expected behavior in any quantum-critical ground-state. We investigate these properties in single-shot adaptation on (1) the critical point between a one-dimensional <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>Z</e:mi> <e:mn>2</e:mn> </e:msub> <e:mo>×</e:mo> <e:msub> <e:mi>Z</e:mi> <e:mn>2</e:mn> </e:msub> </e:math> symmetry-protected topological (SPT) order and a symmetry-broken state, and (2) a spinful Tomonaga-Luttinger liquid. In the former case, adaptive evolution that decoheres one sublattice of the SPT can yield a critical mixed-state in which <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>α</g:mi> </g:math> attains a universal value, which is half of that in the original state. In the latter case, we show how adaptation—involving feedback on the spin degrees of freedom, after measuring the local charge—modifies long-distance correlations, and determine via an exact replica field-theoretic calculation that <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>α</i:mi> </i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>γ</k:mi> </k:math> vary continuously with the strength of feedback. Numerical studies confirm these results.