How Efficiently can Measurement Induce Entanglement?

It is known that measurements, despite being decohering processes, can teleport entanglement. A prime example is Bell teleportation. Such phenomena are not restricted to simple systems; they also manifest in many-body quantum states, termed as 'measurement-induced entanglement' (MIE). While MIE offers a potent method to investigate quantum systems, the efficiency of its generation remains unknown. Specifically, it's unclear how many bits need to be measured to produce a given amount of bits of entanglement. In this work, we establish a strict upper bound on the efficiency of MIE generation: an increase of at most one bit of MIE per bit of measurement, saturated by Bell teleportation. Alongside this proof, we introduce a framework for characterizing MIE using quantum conditional mutual information. This novel approach not only builds upon existing definitions but also distinguishes classical correlations from MIE. To understand concrete examples, we explore the efficiency of MIE generation in states prepared by random unitary circuits. Our numerical analyses reveal that circuits with greater depth produce MIE more efficiently. We explain this relationship through the statistical mechanics model.