  We develop a synthesis of Turing's paradigm of computation and von Neumann's
quantum logic to serve as a model for quantum computation with recursion, such
that potentially non-terminating computation can take place, as in a quantum
Turing machine. This model is based on the extension of von Neumann's quantum
logic to partial states, defined here as sub-probability measures on the
Hilbert space, equipped with the natural pointwise partial ordering. The
sub-probability measures allow a certain probability for the non-termination of
the computation. We then derive an extension of Gleason's theorem and show
that, for Hilbert spaces of dimension greater than two, the partial order of
sub-probability measures is order isomorphic with the collection of partial
density operators, i.e., trace class positive operators with trace between zero
and one, equipped with the usual partial ordering induced from positive
operators. We show that the expected value of a bounded observable with respect
to a partial state can be defined as a closed bounded interval, which extends
the classical definition of expected value.
