  In tensor networks, a geometric operation of pushing a bond cut surface
toward a minimal surface corresponds to entanglement distillation. Cutting
bonds defines a reduced transition matrix on the bond cut surface and the
associated quantum state naturally emerges from it. We justify this picture
quantitatively by evaluating the trace distance between the maximally entangled
states and the states on bond cut surfaces in the multi-scale entanglement
renormalization ansatz (MERA) and matrix product states in a canonical form.
Our numerical result for the random MERA is in a reasonable agreement with our
proposal. The result sheds new light on a deeper understanding of the
Ryu-Takayanagi formula for entanglement entropy in holography and the emergence
of geometry from the entanglement structure.
