  Heisenberg's uncertainty principle, exemplified by the gamma ray thought
experiment, suggests that any finite precision measurement disturbs any
observables noncommuting with the measured observable. Here, it is shown that
this statement contradicts the limit of the accuracy of measurements under
conservation laws originally found by Wigner in 1950s, and should be modified
to correctly derive the unavoidable noise caused by the conservation law
induced decoherence. The obtained accuracy limit leads to an interesting
conclusion that a widely accepted, but rather naive, physical encoding of
qubits for quantum computing suffers significantly from the decoherence induced
by the angular momentum conservation law.
