  Shared entanglement can significantly amplify classical correlations between
systems interacting over a limited quantum channel. A natural avenue is to use
entanglement of the same dimension as the channel because this allows for
unitary encodings, which preserve global coherence until a measurement is
performed. Contrasting this, we here demonstrate a distributed task based on a
qubit channel, for which irreversible encoding operations can outperform any
possible coherence-preserving protocol. This corresponds to using
high-dimensional entanglement and encoding information by compressing one of
the subsystems into a qubit. Demonstrating this phenomenon requires the
preparation of a four-dimensional maximally entangled state, the compression of
two qubits into one and joint qubit-ququart entangled measurements, with all
modules executed at near-optimal fidelity. We report on a proof-of-principle
experiment that achieves the advantage by realizing separate systems in
distinct and independently controlled paths of a single photon. Our result
demonstrates the relevance of high-dimensional entanglement and non-unitary
operations for enhancing the communication capabilities of standard qubit
transmissions.
