J. Phys. A: Math. Gen. 32 (1999) 7791–7801. Printed in the UK PII: S0305-4470(99)01931-9 Vector phase measurement in multipath quantum interferometry Barry C Sanders†¶, Hubert de Guise‡, D J Rowe§ and A Mann‖ † Department of Physics, Macquarie University, Sydney, New South Wales 2109, Australia ‡ Centre de Recherches Math´ ematiques, Universit´ e de Montr´ eal, CP 6128 Succ. Centre-Ville, Montr´ eal, Qu´ ebec, Canada H3C 3J7 § Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 ‖ Department of Physics, Technion—Israel Institute of Technology, Haifa 32000, Israel Received 17 February 1999, in final form 12 August 1999 Abstract. We introduce vector phase states for multipath quantum interferometry and construct the vector phase positive operator-valued measure. We calculate SU(3) phase distributions for three-path quantum interferometry and discuss measurement limits. 1. Introduction Precise interferometric measurements of phase shifts are important for many applications, yet complementarity between particle number and phase limits the amount of information which can be extracted from the interferometer [1]. These limits are well understood in the context of two-path interferometry, but the development of multipath quantum interferometers (MQI) [2] raises issues about the bounds to estimating simultaneous multiple phase shifts [3, 4]. Our aim is to establish rigorous bounds on estimating this multiple phase shift. Specifically, we (1) employ the SU(N) group to describe the interferometer and identify the Fock basis for the input state with the (Cartan) weight basis, (2) develop the SU(N) ‘vector phase’ state (VPS) as the dual basis to the weight states [5], (3) present a class of MQI designs for which a ‘rotated’ VPS basis is translated by MQI, (4) determine ‘vector phase’ distributions for states which can be studied via parametric estimation theory, (5) establish the relation between bounds on vector phase measurement in connection with the Fisher information matrix [6], and (6) calculate and plot SU(3) vector phase distributions. Lie group theory provides the natural language for describing interferometry as a set of unitary transformations. For a single-mode field, it is sufficient to introduce the annihilation and creation operators, a and a † , plus the identity operator, which together span the Heisenberg– Weyl (HW) algebra. The Fock number states {|n〉} are eigenstates of the unitary phase-shift operator exp(iφa † a). (1.1) Whereas the Fock state is an eigenstate of the phase-shift operator (1.1), the unnormalized phase state [1, 7] |θ 〉= ∞  n=0 e inθ |n〉 (1.2) ¶ Web page: http://www.physics.mq.edu.au/∼barry/ 0305-4470/99/447791+11$30.00 © 1999 IOP Publishing Ltd 7791