  The wave-particle duality demonstrates a competition relation between wave
and particle behavior for a particle going through an interferometer. This
duality can be formulated as an inequality, which upper bounds the sum of
interference visibility and path information. However, if the particle is
entangled with a quantum memory, then the bound may decrease. Here, we find the
duality relation between coherence and path information for a particle going
through a multipath interferometer in the presence of a quantum memory,
offering an upper bound on the duality relation which is directly connected
with the amount of entanglement between the particle and the quantum memory.
