  Distributed quantum sensing enables the estimation of multiple parameters
encoded in spatially separated probes. While traditional quantum sensing is
often focused on estimating a single parameter with maximum precision,
distributed quantum sensing seeks to estimate some function of multiple
parameters that are only locally accessible for each party involved. In such
settings it is natural to not want to give away more information than is
necessary. To address this, we use the concept of privacy with respect to a
function, ensuring that only information about the target function is available
to all the parties, and no other information. We define a measure of privacy
(essentially how close we are to this condition being satisfied), and show it
satisfies a set of naturally desirable properties of such a measure. Using this
privacy measure, we identify and construct entangled resources states that
ensure privacy for a given function under different resource distributions and
encoding dynamics, characterized by Hamiltonian evolution. For separable and
parallel Hamiltonians, we prove that the GHZ state is the only private state
for certain linear functions, with the minimum amount of required resources, up
to SLOCC. Recognizing the vulnerability of this state to particle loss, we
create families of private states, that remain robust even against loss of
qubits, by incorporating additional resources. We then extend our findings to
different resource distribution scenarios and Hamiltonians, resulting in a
comprehensive set of private and robust states for distributed quantum
estimation. These results advance the understanding of privacy and robustness
in multi-parameter quantum sensing.
