We investigated quantum critical behaviours in the non-equilibrium steady
state of a XXZ spin chain with boundary Markovian noise using the Fisher
information. The latter represents the distance between two infinitesimally
close states, and its superextensive size scaling witnesses a critical
behaviour due to a phase transition, since all the interaction terms are
extensive. Perturbatively in the noise strength, we found superextensive Fisher
information at anisotropy ∣Δ∣⩽1 and irrational
πarccosΔ irrespective of the order of two non-commuting
limits, i.e. the thermodynamic limit and the limit of sending
πarccosΔ to an irrational number via a sequence of rational
approximants. From this result we argue the existence of a non-equilibrium
quantum phase transition with a critical phase ∣Δ∣⩽1. From the
non-superextensivity of the Fisher information of reduced states, we infer that
this non-equilibrium quantum phase transition does not have local order
parameters but has non-local ones, at least at ∣Δ∣=1. In the
non-perturbative regime for the noise strength, we numerically computed the
reduced Fisher information which lower bounds the full state Fisher
information, and is superextensive only at ∣Δ∣=1. Form the latter
result, we derived local order parameters at ∣Δ∣=1 in the
non-perturbative case. The existence of critical behaviour witnessed by the
Fisher information in the phase ∣Δ∣<1 is still an open problem. The
Fisher information also represents the best sensitivity for any estimation of
the control parameter, in our case the anisotropy Δ, and its
superextensivity implies enhanced estimation precision which is also highly
robust in the presence of a critical phase