Measurement connects the world of quantum phenomena to the world of classical events. It has both a passive role-in observing quantum systems-and an active one, in preparing quantum states and controlling them. In view of the central status of measurement in quantum mechanics, it is surprising that there is no general recipe for designing a detector that measures a given observable 1 . Compounding this, the characterization of existing detectors is typically based on partial calibrations or elaborate models. Thus, experimental specification (that is, tomography) of a detector is of fundamental and practical importance. Here, we present the realization of quantum detector tomography 2-4 . We identify the positive-operator-valued measure describing the detector, with no ancillary assumptions. This result completes the triad, state 5-11 , process 12-17 and detector tomography, required to fully specify an experiment. We characterize an avalanche photodiode and a photon-number-resolving detector capable of detecting up to eight photons 18 . This creates a new set of tools for accurately detecting and preparing non-classical light.The reduction of the quantum state of a system by measurement, as postulated by von Neumann, is now generally accepted to be a limiting case of a more general theory of quantum measurement that involves state reduction on an extended Hilbert space encompassing the system and an (possibly fictional) ancilla. However, even within this general theory, it is not known how to incorporate the complete chain of apparatus components in a derivation of the actual measurement: as Braginsky has written, ''the Schrödinger equation cannot tell us the connection between the design of the measuring device and the nature of the measurement'' 1 . Measurement is increasingly becoming a driving component in quantum technologies such as super-resolution metrology 19 , Heisenberg-limited sensitivity 20 and quantum computing 21 . Input states and dynamical processes are accepted as resources for quantum technologies and therefore the techniques of quantum state tomography [5][6][7][8][9][10][11]22 and quantum process tomography [12][13][14][15][16][17]23 have been developed to measure them. A distinct omission is that of the experimental tomography of detectors, which would enable more accurate classification of measurement types, objective comparison of competing devices and precise design of new detectors. This omission is even more striking given that the tomography of states and processes are predicated on a well-characterized detector. Here, we extend previous theoretical descriptions of detector tomography 2-4 to include regularization and to accommodate the classical uncertainties of the experimental apparatus. We apply this theory to the characterization of two quantum detectors.Characterizing a detector consists of determining its corresponding positive-operator-valued measure (POVM). Given an input
We demonstrate phase super-resolution in the absence of entangled states. The key insight is to use the inherent time-reversal symmetry of quantum mechanics: our theory shows that it is possible to measure, as opposed to prepare, entangled states. Our approach is robust, requiring only photons that exhibit classical interference: we experimentally demonstrate high-visibility phase super-resolution with three, four, and six photons using a standard laser and photon counters. Our six-photon experiment demonstrates the best phase super-resolution yet reported with high visibility and resolution.Common wisdom holds that entangled states are a necessary resource for many protocols in quantum information. An example is quantum metrology, which promises super-precise measurement, surpassing that possible with classical states of light and matter [1,2]. In the last 20 years quantum metrology schemes have been proposed for improved optical [3][4][5][6][7][8] and matter-wave [9] interferometry, atomic spectroscopy [10], and lithography [11][12][13]. The entangled states in these schemes give rise to phase super-resolution, where the interference oscillation occurs over a phase N-times smaller than one cycle of classical light [14,15] and phase super-sensitivity, a reduction of phase uncertainty.Many quantum metrology schemes are based on pathentangled number states. The canonical example is the noon-state [1], a two-mode state with either N particles in one mode and 0 in the other or vice-versa, i.e., (|N0 +|0N )/ √ 2. A deterministic optical source of path-entangled states is yet to be realised, requiring optical nonlinearities many orders of magnitude larger than those currently possible. However, entangled states can be made non-deterministically using single-photon sources, linear optics, and photon-resolving detectors [16]: leading to a flurry of proposals to generate pathentangled states [17][18][19][20]. While phase super-resolution with two-photons has been demonstrated often since 1990 [21][22][23][24], phase super-resolution was experimentally demonstrated for 3-photon [14] and 4-photon [15] states only recently. As efficient photon sources and photonnumber resolving detectors do not yet exist, all demonstrations to date necessarily used multiphoton coincidence post-selection [25]. Problematically, current photon sources are extremely dim and true photon-number resolving detectors are expensive and uncommon. In this paper we introduce a time-reversal technique that eliminates the need for exotic sources and detectors, achieving high-visibility phase super-resolution with a standard laser and photon detectors.
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