  Wave-particle duality is one of the basic features of quantum mechanics,
giving rise to the use of complex numbers in describing states of quantum
systems, their dynamics, and interaction. Since the inception of quantum
theory, it has been debated whether complex numbers are actually essential, or
whether an alternative consistent formulation is possible using real numbers
only. Here, we attack this long-standing problem both theoretically and
experimentally, using the powerful tools of quantum resource theories. We show
that - under reasonable assumptions - quantum states are easier to create and
manipulate if they only have real elements. This gives an operational meaning
to the resource theory of imaginarity. We identify and answer several important
questions which include the state-conversion problem for all qubit states and
all pure states of any dimension, and the approximate imaginarity distillation
for all quantum states. As an application, we show that imaginarity plays a
crucial role for state discrimination: there exist real quantum states which
can be perfectly distinguished via local operations and classical
communication, but which cannot be distinguished with any nonzero probability
if one of the parties has no access to imaginarity. We confirm this phenomenon
experimentally with linear optics, performing discrimination of different
two-photon quantum states by local projective measurements. These results prove
that complex numbers are an indispensable part of quantum mechanics.
