  The uncertainty principle can be understood as constraining the probability
of winning a game in which Alice measures one of two conjugate observables,
such as position or momentum, on a system provided by Bob, and he is to guess
the outcome. Two variants are possible: either Alice tells Bob which observable
she measured, or he has to furnish guesses for both cases. Here I derive new
uncertainty relations for both, formulated directly in terms of Bob's guessing
probabilities. For the former these relate to the entanglement that can be
recovered by action on Bob's system alone. This gives a condition for
approximate quantum error correction in terms of the recoverability of
"amplitude" and "phase" information, implicitly used in the recent construction
of efficient quantum polar codes. I also find a tight relation on the guessing
probabilities for the latter game, which has application to wave-particle
duality relations.
