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In optimal control problems with nonlinear time-dependent three-dimensional (3D) PDEs, full four-dimensional (4D) discretizations are usually prohibitive due to the storage requirement. For this reason gradient- and Newton-type methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose to use lossy compression algorithms, using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression maintaining a certain error bound turns out to be sufficient. We provide an error analysis, derive expected compression rates, and present numerical examples validating the results.