PHYSICAL REVIEW A VOLUME 51, NUMBER 3 Measurement of the Bell operator and quantum teleportation MARCH 1995 Samuel L. Braunstein and A. Mann 'Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel Department of Physics, Technion Isra— el Institute of Technology, 32 000 Haifa, Israel (Received 14 March 1994) We propose a scheme for measuring an optical version of the Bell operator, using a generalization of the Hong-Ou-Mandel interferometer. Discrete-mode calculations show this to be sufficient to allow teleportation of the state of polarization of a photon with a conditional efficiency approaching 100%. The feasibility of the scheme is investigated through full broadband calculations. PACS number(s): 03.65. Bz, 42.50. Dv, 89. 70. +c Ever since the violation of Bell inequalities was discov- ered [1 — 3], we have understood that the transfer of informa- tion [4] in quantum systems can be quite unintuitive. One striking example is teleportation [5], the recently developed scheme for the "disembodied" transport of an unknown quantum state. It is disembodied in the sense that the state is transported through multiple channels connecting the sender of the state and its receiver, but any channel taken separately contains no extractable information about the state being transported. By contrast, under purely unitary evolution, such as one would expect in evolution from the Schrodinger equation alone, at least one of the channels must contain some information. In the case studied here, two channels are used: one quantum and one classical; the classical signal is derived from a non- unitary collapse due to a measurement. The proof, in principle, in Ref. [5] for quantum mechanics to allow such disembodied transport was based on discrete- mode calculations, which are most appropriate for well sepa- rated modes, such as optical modes in a high Q cavity. Such discrete-mode calculations can be "mapped" into what is performed in the laboratory in many different ways, but with each such implementation comes inherent subtleties that may make a particular mapping invalid (i.e. , the sought after ef- fect is not seen). Here we describe a proposal for performing this disembodied transport of the state of polarization of a photon (a two-state system). We perform broadband calcula- tions to determine the likely difficulties with implementing our proposal. We find our implementation to be robust with respect to low detector efficiency, phase jiggle, and unavoid- able differences between the multiple optical components needed. Our numbers suggest that successful teleportation events could be generated at the rate of about one per second, though very narrow bandwidth down-convertors are essen- tial. The difficult part for any implementation of teleportation is the generation of the classical signal, which requires that a measurement be made on a pair of particles, e. g. , photons. This measurement must effectively collapse the incoming particles' wave packets into one of four strongly superposed (entangled) states and produce a unique signal for each. Our scheme fails to achieve this ideal, but does work for two of the four required cases — yielding at most a 50% absolute efficiency. Here we only present detailed calculations for generating one of the four signals — yielding at most a 25% absolute efficiency. However, this signal is uniquely charac- terized by a four-way coincidence, so even in the presence of low detector efficiency, a conditional efficiency approaching 100% should be achievable. Our scheme for generating the classical signals general- izes the Hong-Ou-Mandel interferometer [6] by allowing for states with arbitrary polarizations. We show that the descrip- tion of this device in terms of measurement of eigenstates of Bell operators [7] yields an intuitive understanding of its operation. Our implementation requires interference between independent sources as recently suggested by Zukowski et al. [8]. We modify their work to allow for differences between the independent light sources and we derive their continuous mode wave-packet collapse language rigorously from Glauber photodetection theory [9]. We start by reviewing the discrete-mode description of teleportation in terms of photon polarization and discuss our detection scheme based on the generalized Hong-Ou-Mandel interferometer. We then show how wave-packet collapse may be used, to a limited extent, at the amplitude level to describe photodetection of light. Finally, we calculate the amplitude for successfully teleporting the state of polarization of a single-photon state and discuss typical numbers for our scheme's performance. Quantum states can be more highly correlated than would be allowed by any local classical theory — this being one consequence of the violation of the Bell inequalities. The most highly correlated quantum states are eigenstates of some Bell operator [7].A suitable version of the Bell opera- tor, given in Ref. [7], has a complete set of eigenstates for two-photon states in modes a and b, where I I ) and I~ ) are single-photon states with vertical and horizontal polarization. As pointed out in Ref. [5] a measuring device that "collapses" the wave packet of a two-particle state into this Il'-4 correlated basis can be used to teleport a particle. Suppose we have an arbitrary (but unknown) state IP), =uII ), +PI~ ), in mode 1. Teleportation is based on the following decomposition: 1050-2947/95/51(3)/1727(4)/$06. 00 51 R1727 1995 The American Physical Society