  We present the theory of how to achieve phase measurements with the minimum
possible variance in ways that are readily implementable with current
experimental techniques. Measurements whose statistics have high-frequency
fringes, such as those obtained from NOON states, have commensurately high
information yield. However this information is also highly ambiguous because it
does not distinguish between phases at the same point on different fringes. We
provide schemes to eliminate this phase ambiguity in a highly efficient way,
providing phase estimates with uncertainty that is within a small constant
factor of the Heisenberg limit, the minimum allowed by the laws of quantum
mechanics. These techniques apply to NOON state and multi-pass interferometry,
as well as phase measurements in quantum computing. We have reported the
experimental implementation of some of these schemes with multi-pass
interferometry elsewhere. Here we present the theoretical foundation, and also
present some new experimental results. There are three key innovations to the
theory in this paper. First, we examine the intrinsic phase properties of the
sequence of states (in multiple time modes) via the equivalent two-mode state.
Second, we identify the key feature of the equivalent state that enables the
optimal scaling of the intrinsic phase uncertainty to be obtained. This enables
us to identify appropriate combinations of states to use. The remaining
difficulty is that the ideal phase measurements to achieve this intrinic phase
uncertainty are often not physically realizable. The third innovation is to
solve this problem by using realizable measurements that closely approximate
the optimal measurements, enabling the optimal scaling to be preserved.
