Sequential Generation of Two-Dimensional Super-Area-Law States with Local Parent Hamiltonian

We construct examples of highly entangled two-dimensional states by exploiting a correspondence between stochastic processes in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>d</a:mi> </a:math> dimensions and quantum states in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>d</c:mi> <c:mo>+</c:mo> <c:mn>1</c:mn> </c:math> dimensions. The entanglement structure of these states, which we explicitly calculate, can be tuned between area law, sub-volume law, and volume law. This correspondence also enables a sequential generation protocol: the states can be prepared through a series of unitary transformations acting on an auxiliary system. We also discuss the conditions under which these states have local, frustration-free parent Hamiltonians.