2 00 3 Unitary realizations of the ideal phase measurement

We explicitly construct a large class of unitary transformations that allow to perform the ideal estimation of the phase-shift on a single-mode radiation field. The ideal phase distribution is obtained by heterodyne detection on two radiation modes after the interaction. The quantum estimation of an unknown phase shift—the so called quantum phase measurement—is the essential problem of high sensitive interferometry, and has received much attention in quantum optics [1]. For a single-mode electromagnetic field, the measurement cannot be achieved exactly, even in principle, due to the lack of a unique self-adjoint operator [2]. In fact, the absence of a proper self-adjoint operator is mainly due to the semi-boundedness of the spectrum of the number operator [3,4], which is canonically conjugated to the phase in the sense of a Fourier-transform pair [5]. This observation opened the route for an exact phase measurement in terms of two-mode fields, where a phase-difference operator becomes conjugated to an unbounded number-difference operator [6]. In fact, a concrete experimental setup using unconventional heterodyne detection has been suggested [7] for this kind of measurement. However in the single-mode case, no feasible scheme that can provide the optimal phase measurement has been devised yet. The most general and concrete approach to the problem of the phase measurement is quantum estimation theory [8], a framework that has become popular only in the last ten years in the field of quantum information. Quantum estimation theory provides a more general description of quantum statistics in terms of POVM’s (positive operator-valued measures) and gives the theoretical definition of an optimized phase measurement. The most powerful method for deriving the optimal phase measurement was given by Holevo [9] in the Preprint submitted to Elsevier Science 1 February 2008 covariant case. In this way the optimal POVM for phase estimation has been derived for a single-mode field. More generally, the problem of estimating the phase shift has been addressed in Ref. [10] for any degenerate shift operator with discrete spectrum, either bounded, bounded from below, or unbounded, extending the Holevo method for the covariant estimation problem. As already stated, quantum estimation theory provides the optimal POVM for the phase measurement. This writes in terms of projectors on SusskindGlogower states [11] dμ(φ) = dφ 2π |e〉〈e| , (1) where |eiφ〉 = ∑n=0 eiφn|n〉. Notice that the states |eiφ〉 are not normalizable, neither orthogonal, however they provide a resolution of the identity, and thus guarantee the completeness of the POVM, namely ∫ 2π 0 dμ(φ) = I (2) For a system in state ρ, the POVM in Eq. (1) gives the ideal phase distribution p(φ) according to Born’s rule p(φ) = Tr[dμ(φ) ρ] = dφ 2π 〈e|ρ|e〉 . (3) In this Letter we will explicitly construct some unitary transformations that allows to perform the ideal phase measurement described by the POVM in Eq. (1). First, we will introduce an isometry Ṽ which enlarges the Hilbert space of the system (say Ha for mode a) to the tensor product Ha ⊗ Hb for two modes a and b. Then, we will prove that the exact measurement of the complex photocurrent Z = a − b† provides through its marginal distribution the ideal probability density p(φ) of Eq. (3). Finally, we will construct a large class of unitary operators on Ha ⊗ Hb ⊗ Hc, where Hc denotes the Hilbert space of an ancillary arbitrary system, such that the isometry Ṽ is realized with unit probability. We start by introducing the eigenstates of the heterodyne photocurrent Z = a− b† [12,13,7] Z|D(z)〉〉ab = z|D(z)〉〉ab , (4) where D(z) = exp(za† − z∗a) denotes the displacement operator. Here and in the following we use the notation [14] for bipartite pure states on Ha ⊗Hb