  Collisions between heavy atomic nuclei at ultra-relativistic energies are
carried out at particle colliders to produce the quark-gluon plasma, a state of
matter where quarks and gluons are not confined into hadrons, and colour
degrees of freedom are liberated. This state is thought to be produced as a
transient phenomenon before it fragments into thousands of particles that reach
the particle detectors. Despite two decades of investigations, one of the big
open questions is to obtain an experimental determination of the temperature
reached in a heavy-ion collision, and a simultaneous determination of another
thermodynamic quantity, such as the entropy density, that would give access to
the number of degrees of freedom. Here we obtain the first such determination,
utilizing state-of-the-art hydrodynamic simulations. We define an effective
temperature, averaged over the space-time evolution of the medium. Then, using
experimental data, we determine this temperature, the corresponding entropy
density and speed of sound in the matter created in lead-lead collisions at the
Large Hadron Collider. Our results agree with first-principles calculations
from lattice quantum chromodynamics and confirm that a deconfined phase of
matter is indeed produced.
