We address the quantum characterization of photon counters based on transition-edge sensors (TESs) and present the first experimental tomography of the positive operator-valued measure (POVM) of a TES. We provide the reliable tomographic reconstruction of the POVM elements up to 11 detected photons and M = 100 incoming photons, demonstrating that it is a linear detector. 7 References 8

A quantum measurement can be described by a set of matrices, one for each possible outcome, which represents the probability operator-valued measure (POVM) of the sensor. Efficient protocols of POVM extraction for arbitrary sensors are required. We present the first experimental POVM reconstruction that takes explicit advantage of a quantum resource, i.e. nonclassical correlations with an ancillary state. POVM of a photon-number-resolving detector is reconstructed by using strong quantum correlations of twin-beams generated by parametric downconversion. Our reconstruction method is more statistically robust than POVM reconstruction methods that use classical input states.PACS numbers: 42.50. Dv, 42.50.Ar, 03.65.Ta, 85.60.Gz Measurements are at heart of scientific method, because they allow to gauge observables in experimental tests, leading either to the confirmation or to the ruling out of the scientific hypothesis. In quantum mechanics measurements play a critical role because they connect the abstract description of quantum phenomena in Hilbert space to observable events. In the process of measurement, a quantum mechanical object interacts with a measurement device, and a measurement outcome is a result of such interaction. A complete quantum mechanical description of a measurement device is its positive operator-valued measure (POVM). In the quantum realm, sensor calibration corresponds to determining its POVM. In the last decade, the rapid development of innovative quantum technologies promoted POVMs from being an abstract theoretical tool to the experimental realm. In particular, precise and fully quantum characterization techniques for sensors [1-4] play a critical role for the implementation of quantum information processing, metrology and imaging [5][6][7][8][9][10][11][12][13][14][15][16], as well as tomography of states [17][18][19][20][21][22][23][24] and operations [25][26][27][28][29][30]. Quantum sensor characterization can be thus seen as a simultaneous measurement of multiple parameters, therefore the efficiency of such measurement is of utmost importance. However, POVM extraction has been experimentally pursued by brute force methods so far, i.e. by probing sensors with a suitably large set of interrelated input signals, classical states, yielding slow convergence [2,3]. It was shown [1] that taking advantage of quantum resources, e.g. entanglement, can improve convergence beyond the traditional methods. Here, we present the first experimental POVM's reconstruction that explicitly uses a quantum resource, i.e. nonclassical correlations with an ancillary state [31]. Our experiment represents a major step forward towards quantum mechanical treatment of sensors: it demonstrates reconstruction of an inherently quantum measure of an arbitrary detector's performance-its POVM-by realizing for the first time the method of ref. [1].A POVM is defined as a set of operators (matrices) Π n that give the probability of the measurement outcomes via the Born Rule p n = Tr [ Π n ], where is the density opera...

Here we present the first experimental entanglement-assisted quantum characterization of an unknown photon-number-resolving detector, i.e. the reconstruction of its positive operator-valued measure (POVM), obtained by exploiting the quantum correlations of a twin-beam generated by parametric down conversion (PDC). Theoretical frameworkHere we present the first experimental entanglement-assisted quantum characterization of an unknown photon-numberresolving (PNR) detector, obtained by exploiting the quantum correlations of a twin-beam generated by parametric down conversion (PDC): one beam is revealed by the detector under calibration (DUC), whereas the other one is sent to a quantum tomographer (T). The calibration of a quantum apparatus corresponds to determining its positive operatorvalued measure (POVM), i.e. the set of operators Π n providing the probability of the measurement outcomes via the Born rule p n = Tr [ρ Π n ], where ρ is the density operator describing the system being measured. The reconstruction of a detector POVM can be achieved by the inversion of the Born Rule [1], but recently another calibration scheme was proposed [2], based on quantum correlations of a bipartite system whose parts are sent to the DUC and to a tomographer. This scheme, both more reliable and less demanding, is a clear example of ancilla-assisted quantum scheme, where quantum correlations play a key role in order to improve both precision and stability. To perform the tomographic reconstruction needed here [2], we make use of a maximum-likelihood based method [3,4] exploiting on/off detectors with variable quantum efficiency to reconstruct the diagonal elements of the density matrix of quantum optical states. This method has been tested in different regimes [5] and generalized to the bipartite case [6], improved with the addition of a proper energy constraint [7] and extended also to the off-diagonal elements of the density matrix [8,9]: now it will be applied also to this ancilla-assisted POVM reconstruction experiment. Experimental setup and resultsThe experimental setup (Fig. 1) is composed by a 400 nm mode-locked laser pumping a LiIO 3 crystal in order to produce degenerate and non-collinear Type-I Parametric Down Conversion. One of the selected branches of degenerate PDC (at 800 nm) is sent to the Tomographer, composed by a calcite polarizer, a pinhole, an Interference Filter (20 nm of Full Width at Half Maximum) and a fibre coupler connected by a multimode fibre to a Si-Single Photon Avalanche Diode (SPAD). Being the down converted photons characterized by the same polarization, the polarizer is used to change step by step the efficiency of the tomographer. The correlated branch is sent to the PNR DUC, that is constituted by a 50% − 50% multimode fibre beam splitter whose outputs are connected to two Si-SPADs, while the input port is again connected to the same coupling system described above (it is straightforward to notice that our PNR is able to discriminate between 0, 1 and 2-or-more photons detected per pulse). Th...

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