  According to Landauer's principle, erasing one bit of information incurs a
minimum energy cost. Recently, Vaccaro and Barnett (VB) explored information
erasure within the context of generalized Gibbs ensembles and demonstrated that
for energy-degenerate spin reservoirs, the cost of erasure can be solely in
terms of a minimum amount of spin angular momentum and no energy. As opposed to
the Landauer case, the cost of erasure in this case is associated with the
discrete variable.
  Here we study the {\it discrete} fluctuations in this cost and the
probability of violation of the VB bound. We also obtain a Jarzynski-like
equality for the VB erasure protocol. We find that the fluctuations below the
VB bound are exponentially suppressed at a far greater rate and more tightly
than for an equivalent Jarzynski expression for VB erasure. We expose a
trade-off between the size of the fluctuations and the cost of erasure. We find
that the discrete nature of the fluctuations is pronounced in the regime where
reservoir spins are maximally polarized. We also state the first laws of
thermodynamics corresponding to the conservation of spin angular momentum for
this particular erasure protocol. Our work will be important for novel heat
engines based on information erasure schemes that do not incur an energy cost.
