Universality Classes for Purification in Nonunitary Quantum Processes

We consider the universal aspects of two problems: (i) the singular value structure of a product <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>M</a:mi> <a:mi>t</a:mi> </a:msub> <a:mo>=</a:mo> <a:msub> <a:mi>m</a:mi> <a:mi>t</a:mi> </a:msub> <a:msub> <a:mi>m</a:mi> <a:mrow> <a:mi>t</a:mi> <a:mo>−</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msub> <a:mo>…</a:mo> <a:msub> <a:mi>m</a:mi> <a:mn>1</a:mn> </a:msub> </a:math> of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>m</c:mi> <c:mi>i</c:mi> </c:msub> </c:math> , representing time steps in the evolution, but the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>m</e:mi> <e:mi>i</e:mi> </e:msub> </e:math> are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.