  The interplay of interactions and strong disorder can lead to an exotic
quantum many-body localized (MBL) phase. Beyond the absence of transport, the
MBL phase has distinctive signatures, such as slow dephasing and logarithmic
entanglement growth; they commonly result in slow and subtle modification of
the dynamics, making their measurement challenging. Here, we experimentally
characterize these properties of the MBL phase in a system of coupled
superconducting qubits. By implementing phase sensitive techniques, we map out
the structure of local integrals of motion in the MBL phase. Tomographic
reconstruction of single and two qubit density matrices allowed us to determine
the spatial and temporal entanglement growth between the localized sites. In
addition, we study the preservation of entanglement in the MBL phase. The
interferometric protocols implemented here measure affirmative correlations and
allow us to exclude artifacts due to the imperfect isolation of the system. By
measuring elusive MBL quantities, our work highlights the advantages of phase
sensitive measurements in studying novel phases of matter.
