  Thermodynamic uncertainty relations (TURs) place strict bounds on the
fluctuations of thermodynamic quantities in terms of the associated entropy
production. In this work we identify the tightest (and saturable) matrix-valued
TUR that can be derived from the exchange fluctuation theorems describing the
statistics of heat and particle flow between multiple systems. Our result holds
for both quantum and classical systems, undergoing general non-Markovian and
non-stationary processes. Moreover, it provides bounds not only for the
variances, but also for the correlations between thermodynamic quantities. To
demonstrate the relevance of TURs to the design of nanoscale machines, we
consider the operation of a two-qubit SWAP engine undergoing an Otto cycle and
show how our results can be used to place strict bounds on the correlations
between heat and work.
