  In the context of holographic duality with AdS3 asymptotics, the
Ryu-Takayanagi formula states that the entanglement entropy of a subregion is
given by the length of a certain bulk geodesic. The entanglement entropy can be
operationalized as the entanglement cost necessary to transmit the state of the
subregion from one party to another while preserving all correlations with a
reference party. The question then arises as to whether the lengths of other
bulk curves can be interpreted as entanglement costs for some other information
theoretic tasks. Building on recent results showing that the length of more
general bulk curves is computed by the differential entropy, we introduce a new
task called constrained state merging, whereby the state of the boundary
subregion must be transmitted using operations restricted in location and scale
in a way determined by the geometry of the bulk curve. Our main result is that
the cost to transmit the state of a subregion under the conditions of
constrained state merging is given by the differential entropy and hence the
signed length of the corresponding bulk curve. When the cost is negative,
constrained state merging distills entanglement rather than consuming it. This
demonstration has two parts: first, we exhibit a protocol whose cost is the
length of the curve and second, we prove that this protocol is optimal in that
it uses the minimum amount of entanglement. In order to complete the proof, we
additionally demonstrate that single-shot smooth conditional entropies for
intervals in 1+1-dimensional conformal field theories with large central charge
are well approximated by their von Neumann counterparts. We also revisit the
relationship between the differential entropy and the maximum entropy among
locally consistent density operators, demonstrating large quantitative
discrepancy between the two quantities in conformal field theories.
