With the goal in mind of designing radars, interferometers and other sensors based on quantum entanglement the virtues of N00N states, plain M and M states (PMMSs) and linear combinations of M and M states (LCMMS) are considered. A derivation of the closed form expression for the detection operator that is optimal subject to constraints is provided. The raising and lowering properties of the detection operator and its square are developed. The expectations of the optimal detection operator and its square are derived. The expression for the visibility, the maximum expectation of the optimal detection operator, is developed. From the expectation of the square of the detection operator and the visibility, the phase error and the minimum phase error for the detection operator are derived. The optimal resolution for the maximum visibility and minimum phase error are found. For the visibility, comparisons between PMMSs, LCMMS and N00N states are provided. For the minimum phase error comparisons between LCMMS, PMMSs, N00N states, separate photon states (SPSs), the shot noise limit (SNL), and the Heisenberg limit (HL) are provided. A representative collection of computational results illustrating the superiority of LCMMS when compared to PMMSs and N00N states is given. It is found for a resolution 12 times the classical result LCMMS has visibility 11 times that of N00N states and four times that of PMMSs. For the same case, the minimum phase error for LCMMS is 10.7 times smaller than that of PMMS and 29.7 times smaller than that of N00N states.
With the goal in mind of designing radars, interferometers and other sensors based on quantum entanglement the virtues of N00N states, plain M and M states (PMMSs) and linear combinations of M and M states (LCMMS) are considered. A derivation of the closed form expression for the detection operator that is optimal subject to constraints is provided. The raising and lowering properties of the detection operator and its square are developed. The expectations of the optimal detection operator and its square are derived. The expression for the visibility, the maximum expectation of the optimal detection operator, is developed. From the expectation of the square of the detection operator and the visibility, the phase error and the minimum phase error for the detection operator are derived. The optimal resolution for the maximum visibility and minimum phase error are found. For the visibility, comparisons between PMMSs, LCMMS and N00N states are provided. For the minimum phase error comparisons between LCMMS, PMMSs, N00N states, separate photon states (SPSs), the shot noise limit (SNL), and the Heisenberg limit (HL) are provided. A representative collection of computational results illustrating the superiority of LCMMS when compared to PMMSs and N00N states is given. It is found for a resolution 12 times the classical result LCMMS has visibility 11 times that of N00N states and four times that of PMMSs. For the same case, the minimum phase error for LCMMS is 10.7 times smaller than that of PMMS and 29.7 times smaller than that of N00N states.
With the goal of designing interferometers and interferometer sensors, e.g., LADARs with enhanced sensitivity, resolution, and phase estimation, states using quantum entanglement are discussed. These states include N00N states, plain M and M states (PMMSs), and linear combinations of M and M states (LCMMS). Closed form expressions for the optimal detection operators; visibility, a measure of the state’s robustness to loss and noise; a resolution measure; and phase estimate error, are provided in closed form. The optimal resolution for the maximum visibility and minimum phase error are found. For the visibility, comparisons between PMMSs, LCMMS, and N00N states are provided. For the minimum phase error, comparisons between LCMMS, PMMSs, N00N states, separate photon states (SPSs), the shot noise limit (SNL), and the Heisenberg limit (HL) are provided. A representative collection of computational results illustrating the superiority of LCMMS when compared to PMMSs and N00N states is given. It is found that for a resolution 12 times the classical result LCMMS has visibility 11 times that of N00N states and 4 times that of PMMSs. For the same case, the minimum phase error for LCMMS is 10.7 times smaller than that of PMMS and 29.7 times smaller than that of N00N states.
Novel multiphoton entangled states have been developed, which offer utility for sensors. These states are linear combinations of M&N states (LCMNS). Closed form expressions for the coefficients have been derived that result in states offering sensitivities far greater than those found in the literature. Adaptive optics procedures have been conceived that permit these coefficients to be determined. LCMNS can also be used to generate high quality hyperentangled states. Various states generated by this process are developed. The utility of these procedures for quantum interferometry and RF signal detection is examined. The susceptibility to loss of the various interferometry and RF detection approaches is examined using an open systems analysis. The novel quantum interferometer uses adaptive optics, which offers a huge improvement in phase sensitivity over previous interferometers, while greatly reducing the number of photons used. Large numbers of photons can contribute to vibration reducing sensitivity. The approach based on entanglement presented here, even in the presence of loss can reduce the number of photons used significantly, while enhancing phase sensitivity. The new interferometer simultaneously minimizes phase error, maximizes visibility and greatly reduces the number of photons used. When the interferometer employing LCMNS is combined with an atomic RF antenna, E-field sensitivity is increased to a value nearly 400,000 times that found in the literature. Extensive numerical results are provided comparing phase sensitivity, visibility and E-field sensitivity for LCMNS, N00N states, and non-entangled states. LCMNS offer huge improvements in sensitivity compared to other approaches.
Mathematical analysis is presented related to novel quantum entangled states useful for interferometry. For interferometry applications, the approach based on entanglement presented here, even in the presence of loss offers a quadratic reduction in the number of photons required while maximizing phase sensitivity and visibility. Reducing the number of photons decreases the effects of certain mechanisms that degrade results. The new interferometer is an improvement on a previous interferometer based on linear combination of M&N states (LCMNS). LCMNS have been shown to be much more robust in the presence of environmental effects than plain M&N states or N00N states. Adaptive optics techniques are used to adjust the coefficients of the LCMNS. Closed form expressions for the minimum phase error, visibility, and coefficients are derived. Environmental effects are included using open systems theory. Numerical results are provided.
A factorization scheme will be introduced that potentially permits any N00N or M&M state to be generated and detected. Explicit results for N00N states are presented for N=3. Internal loss within the entanglement generator and external loss due to atmosphere, detectors and targets are modeled. A method using these approaches for quantum entanglement based imaging is provided that gives N times classical resolution, where N is the number photons entangled with explicit results exhibited for N=3. Closed form expressions for the wave function, normalization, density matrix, reduced density matrix, visibility, and probabilities of detection of one through three photons using detectors with general polarization characteristics are provided. Explicit entanglement generator and detector designs are provided in terms of linear and nonlinear photonics devices. The fundamental role of post-selection measurement for generating entanglement is included. A general factorization scheme for M&M states is provided. Discussions of entanglement devices that will produce general M&M states at near visible frequencies are given. A discussion of a bearing measurement device that exhibits both super sensitivity and resolution is provided. Computational results are provided that compare probabilities of detection for three single photon detectors with -45, 45, and 45 degree linear polarization. Results for detecting one to three photons or the vacuum state are compared. Computational results for detecting three photons with these detectors for various values of internal and atmospheric loss are provided. Resolution improvements born of quantum entanglement are shown not to degrade with loss. Loss degrades probability of detection not resolution.
Methods of generating N00N, M&N, linear combinations of M&N states as well as more complicated quantum entangled and quantum hyper-entangled states will be considered. Quantum hyper-entanglement refers to quantum entanglement in more than one degree of freedom, e.g. energy-time, polarization, orbital angular momentum, etc. Internal noise and loss within the entanglement or hyper-entanglement generators and external noise and loss due to atmospheric effects and detectors are modeled. Analysis related to the devices that generate these entangled or hyperentangled states will be provided. The following will be derived: closed form expressions for wave function normalization, wave function, density operator, reduced density operator, phase error bound, the symmetrized logarithmic derivative, the quantum Fisher information, the quantum Cramer-Rao lower bound, the relevant projection operators and the related probability of detection expressions. Generation and detection of the entangled or hyperentangled states will be considered. The entanglement generators will use linear and nonlinear optical devices. Optimization criteria for the quantum states, generation and detection schemes and designs optimal with respect to the criteria will be discussed. Applications of the generated states for producing super sensitivity and super resolution will be discussed. The fundamental role of coincidence measurement for generating entanglement is included. Hyperentanglement offers N times classical resolution, where N is a quantum number associated with the system.
This article presents a comprehensive analysis of two classes of quantum radars, including quantum direct-detection and quantum-entangled noise radars. In the first case, inspired by the well-established concept of single-photon LiDARs, we investigated the performance of single-photon radars, in which state-of-the-art single microwave-photon detectors are employed to enhance the detection sensitivity and enable the detection of weaker signals. We derived analytical expressions for the maximum detection range of both classes of quantum radars in terms of the Lambert W function, by considering all relevant system, target, and environmental parameters. Our formulation facilitates direct comparison of noise radars with direct-detection radars and suggests that a quantum-entangled noise radar can be regarded as an enhanced direct-detection radar with an effective threshold signal-to-noise ratio. Furthermore, we applied this framework to classical-correlated noise radars and defined the parameter range enhancement factor (REF) to quantify the superiority of quantum-entangled noise radars over their classical counterparts. Moreover, we introduced a rule-of-thumb for approximating the REF. We also examined the influence of limitations imposed by various microwave detection technologies. Our analysis shows that the conventional antennas limit the potential benefits of quantum-entangled noise radar systems. We also demonstrated that the optimal detection method for these radars is a microwave detector based on a quantum transducer combined with a single optical-photon detector. We showed that, with the current technology, implementing a quantum-entangled noise radar with the maximum detection range on the order of few kilometers is possible. Finally, we explored the potential applications of quantum-entangled noise radars.