Complementarity in the Einstein-Bohr photon box

The Bohr-Einstein photon box thought experiment is a forerunner of the EPR experiment: a packet of radiation escapes from a box, and the box-plus-radiation state remains entangled. Hence, a measurement on the box makes a difference for the state of the far-away radiation long after its escape. This consequence is analogous to the EPR experiment, but an additional aspect of the Bohr-Einstein box is that complementary magnitudes of the escaped radiation seem correlated to jointly measurable box quantities, which appears to imply a violation of the uncertainty relations. We show how the details of the dynamics make this latter paradox disappear. This calculation makes Bohr's qualitative arguments in defense of the uncertainty relations precise and reinforces the EPR-like behavior that Einstein later claimed to have had in mind. The calculation reveals that the quantum description of the far-away radiation depends on exactly when the box measurement is performed.