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Register a new userA positive operator-valued measure for two-photon detection via sum-frequency generation
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Spontaneous parametric down conversion (PDC), in the perturbative limit, can be considered as a probabilistic splitting of one input photon into two output photons. Conversely, sum-frequency generation (SFG) implements the reverse process of combining two input photons into one. Here we show that a single-photon projective measurement in the temporal-mode basis of the output photon of a two-photon SFG process effects a generalized measurement on the input two-photon state. We describe the positive-operator-valued measure (POVM) associated with such a measurement, and show that its elements are proportional to the two-photon states produced by the time-reversed PDC process. Such a detection acts as a joint measurement on two photons, and is thus an important component of many quantum information processing protocols relying on photonic entanglement. Using the retrodictive approach, we analyze the properties of the two-photon POVM that are relevant for quantum protocols exploiting two-photon states and measurements.
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Evaluating the amount of information obtained from nonorthogonal quantum states is an important topic in the field of quantum information. The commonly used evaluation method is Holevo bound, which only provides an upper bound for quantum measurement. In this paper, we provide a theoretical study of the positive operator-valued measure (POVM) for two nonorthogonal mixed states. We construct a generalized POVM measurement operation, and derive the optimal POVM measurement by Lagrange multiplier method. With simulation, we find that the optimal POVM measurement provides a lower bound for quantum state measurement, which is significantly lower than that predicted by Holevo bound. The derivation of optimal POVM measurement will play an important role in the security research of quantum key distribution.
We tackle the dynamical description of the quantum measurement process, by explicitly addressing the interaction between the system under investigation with the measurement apparatus, the latter ultimately considered as macroscopic quantum object. We consider arbitrary Positive Operator Valued Measures (POVMs), such that the orthogonality constraint on the measurement operators is relaxed. We show that, likewise the well-known von-Neumann scheme for projective measurements, it is possible to build up a dynamical model holding a unitary propagator characterized by a single time-independent Hamiltonian. This is achieved by modifying the standard model so as to compensate for the possible lack of orthogonality among the measurement operators of arbitrary POVMs.
We introduce several notions of random positive operator valued measures (POVMs), and we prove that some of them are equivalent. We then study statistical properties of the effect operators for the canonical examples, obtaining limiting eigenvalue distributions with the help of free probability theory. Similarly, we obtain the large system limit for several quantities of interest in quantum information theory, such as the sharpness, the noise content, and the probability range. Finally, we study different compatibility criteria, and we compare them for generic POVMs.
Tailoring spectral properties of photon pairs is of great importance for optical quantum information and measurement applications. High-resolution spectral measurement is a key technique for engineering spectral properties of photons, making them ideal for various quantum applications. Here we demonstrate spectral measurements and optimization of frequency-entangled photon pairs produced via spontaneous parametric downconversion (SPDC), utilizing frequency-resolved sum-frequency generation (SFG), the reverse process of SPDC. A joint phase-matching spectrum of a nonlinear crystal around 1580 nm is captured with a 40 pm resolution and a > 40 dB signal-to-noise ratio, significantly improved compared to traditional frequency-resolved coincidence measurements. Moreover, our scheme is applicable to collinear degenerate sources whose characterization is difficult with previously demonstrated stimulated difference frequency generation (DFG). We also illustrate that the observed phase-matching function is useful for finding an optimal pump spectrum to maximize the spectral indistinguishability of SPDC photons. We expect that our precise spectral characterization technique will be useful tool for characterizing and tailoring SPDC sources for a wide range of optical quantum applications
Optimization of the mean efficiency for unambiguous (or error free)discrimination among $N$ given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit $N$-sphere for equiprobable states and on an $N$-ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a POVM requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 till $N$ and the Neumarks extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space system + ancilla as the tensor product of a two-dimensional ancilla space and the $N$-dimensional system space.
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