The uncertainty principle lies at the heart of quantum theory,
illuminating a dramatic difference with classical mechanics. The
principle bounds the uncertainties of the outcomes of any two
observables on a system in terms of the expectation value of their
commutator. It implies that an observer cannot predict the outcomes
of two incompatible measurements to arbitrary precision. However, this
implication is only valid if the observer does not possess a quantum
memory, an unrealistic assumption in light of recent technological
advances. In this work we strengthen the uncertainty principle to one
that applies even if the observer has a quantum memory. We provide a
lower bound on the uncertainty of the outcomes of two measurements which
depends on the entanglement between the system and the quantum memory.
We expect our uncertainty principle to have widespread use in quantum
information theory, and describe in detail its application to quantum
cryptography. The talk is based on joint work with Mario Berta, Roger
Colbeck, Joe Renes and Renato Renner (http://arxiv.org/abs/0909.0950).