  A cost-precision trade-off relationship, the so-called thermodynamic
uncertainty relation (TUR), has been recently discovered in stochastic
thermodynamics. It bounds certain thermodynamic observables in terms of the
associated entropy production. In this work, we experimentally study the TUR in
a two-qubit system using an NMR setup. Each qubit is prepared in an equilibrium
state, but at different temperatures. The qubits are then coupled, allowing
energy exchange (in the form of heat). Using the quantum state tomography
technique we obtain the moments of heat exchange within a certain time interval
and analyze the relative uncertainty of the energy exchange process. We find
that generalized versions of the TUR, which are based on the fluctuation
relation, are obeyed. However, the specialized TUR, a tighter bound that is
valid under specific dynamics, is violated in certain regimes of operation, in
excellent agreement with analytic results. Altogether, this experiment-theory
study provides a deep understanding of heat exchange in quantum systems,
revealing favorable noise-dissipation regimes of operation.
