Creating large NOON states with imperfect phase control

Optical NOON states ${{\left( {\left| {\left. {N,0} \right\rangle + } \right|\left. {0,N} \right\rangle } \right)} \mathord{\left/ {\vphantom {{\left( {\left| {\left. {N,0} \right\rangle + } \right|\left. {0,N} \right\rangle } \right)} {\sqrt 2 }}} \right. \kern-\nulldelimiterspace} {\sqrt 2 }}${{\left( {\left| {\left. {N,0} \right\rangle + } \right|\left. {0,N} \right\rangle } \right)} \mathord{\left/ {\vphantom {{\left( {\left| {\left. {N,0} \right\rangle + } \right|\left. {0,N} \right\rangle } \right)} {\sqrt 2 }}} \right. \kern-\nulldelimiterspace} {\sqrt 2 }} are an important resource for Heisenberg-limited metrology and quantum lithography. The only known methods for creating NOON states with arbitrary N via linear optics and projective measurements seem to have a limited range of application due to imperfect phase control. Here, we show that bootstrapping techniques can be used to create high-fidelity NOON states of arbitrary size.