Methods of generation, evaluation and applications of multiphoton entanglement, hyperentanglement, hybrid entangled states, and wavefront design are discussed. Measures of effectiveness from quantum information and estimation theory are used to determine the effectiveness of the various approaches. Lie optics, i.e. Lie algebra combined with optics results are discussed for generation of wavefronts that are non-diffracting, self-healing, and self-accelerating. A comparison of the effectiveness of using different wavefronts is provided. Non-diffracting Bessel beams and Laguerre-Gaussian beams are discussed in cylindrical coordinates. Closed form results for creating a transmittance function that permits wavefront generation using an axicon, spatial light modulator, or metasurfaces are provided. These results include closed form expressions for the transmittance coefficients. Methods of using wavefront generation techniques to produce knotted light are included, as well as applications of knots in light. The Bessel beam expansion method is shown to be more robust in the presence of turbulence than the Laguerre-Gaussian expansion approach. A derivation of maximally uniform Bessel beams with respect to a Cramer Rao criterion is included. Applications of wavefront generation techniques are discussed for sensing, imaging in the atmosphere and other media, communication, manipulation of small particles and microscopy. Entanglement generation procedures discussed include use of third order nonlinear susceptibility to generate linear combination of M&N states; generation of entanglement using nonlinear crystals and how this can be generalized using Schrodinger kitten states; devices for generation of hyperentanglement; and designs for generating hybrid states combining hyperentanglement and multiphoton entanglement. Numerical results show the effectiveness of the combined procedures.
Methods of generation, evaluation and applications of multiphoton entanglement, hyperentanglement, hybrid entangled states, and wavefront design are discussed. Measures of effectiveness from quantum information and estimation theory are used to determine the effectiveness of the various approaches. Lie optics, i.e. Lie algebra combined with optics results are discussed for generation of wavefronts that are non-diffracting, self-healing, and self-accelerating. A comparison of the effectiveness of using different wavefronts is provided. Non-diffracting Bessel beams and Laguerre-Gaussian beams are discussed in cylindrical coordinates. Closed form results for creating a transmittance function that permits wavefront generation using an axicon, spatial light modulator, or metasurfaces are provided. These results include closed form expressions for the transmittance coefficients. Methods of using wavefront generation techniques to produce knotted light are included, as well as applications of knots in light. The Bessel beam expansion method is shown to be more robust in the presence of turbulence than the Laguerre-Gaussian expansion approach. A derivation of maximally uniform Bessel beams with respect to a Cramer Rao criterion is included. Applications of wavefront generation techniques are discussed for sensing, imaging in the atmosphere and other media, communication, manipulation of small particles and microscopy. Entanglement generation procedures discussed include use of third order nonlinear susceptibility to generate linear combination of M&N states; generation of entanglement using nonlinear crystals and how this can be generalized using Schrodinger kitten states; devices for generation of hyperentanglement; and designs for generating hybrid states combining hyperentanglement and multiphoton entanglement. Numerical results show the effectiveness of the combined procedures.
Quantum systems combining various forms of quantum entanglement, quantum networks and wavefront design via Lie optics are discussed. The forms of entanglement include multiphoton entanglement and hyperentanglement, with an emphasis on hybrid states born of combining these two forms of entanglement. Lie optics refers to the use of Lie algebra techniques in optics. Improvements born of combining these techniques are discussed in terms of various measures of effectiveness (MOE) including: signal-to-noise ratio (SNR); signal-to-interference ratio (SIR); quantum Cramer Rao lower bound (QCRLB); classical Cramer Rao lower bound (CCRLB); measurement time (MT); quantum Chernoff bound; sensing range; communications range; phase and angular resolution; size, weight, power and costs (SWAPC); various measures associated with information content of signals; superdense coding; and the Holevo bound. Quantum networks coupled with hyperentanglement increase the likelihood that samples are statistically independent improving both classical and quantum estimation results. This can reduce both estimation uncertainty and measurement time. Quantum networks also offer opportunities for additional processing of signals exploiting signal processing, information fusion, adaptive optics and other techniques. Wavefronts designed using Lie optics offer additional advantages in terms of reduced diffraction, the ability to design self-accelerating curves to enhance sensing and communications, and improved MOE values in all categories. The utility of three different types of designed wavefronts in terms of MOEs are discussed. Combining hybrid states with wavefronts designed using Lie optics offers significant MOE improvements greatly expanding communication range, sensing range and significantly improving the information carrying content of signals. Extensive numerical results are provided.
New results have been derived for wavefronts for single and multiple photons and combined with quantum hyperentanglement, multiphoton entanglement, and network properties. These new results arise from use of Lie algebra techniques for disentanglement of certain operators related to propagation. The new results include closed form exact expressions for functions related to the disentanglement process. These expressions permit many different solutions to the disentanglement process. These new expressions offer much higher order programmable self-accelerating curves to be created. This in turn facilitates more effective quantum-based sensing and communication around corners and other obstacles, with reduced loss. This approach significantly increases the information that can be communicated or stored when using a recently derived extension of superdense coding. The resulting wavefronts are shown to reduce diffraction and have the self-healing property, i.e. automatically reobtain their original form when initially damaged by turbulence. Each node of the network will transmit at least one signal and one ancilla photon in a hyper-entangled state. Parameters used for hyper-entanglement will include photon polarization, energy-time, orbital angular momentum, radial quantum number, etc. as well as up to 12 parameters characterizing the wavefront properties. Eigenfunctions of operators associated with each 2D paraxial wave equation will be determined each being a function of up to six parameters. The same wavefront or combinations of wavefronts can be applied to multiple photons to combine hyper-entanglement with multiphoton entanglement offering additional improvements. Measures of effectiveness such as the signal-to-noise ratio (SNR), measurement time, resolution measures, and Holevo bound are considered.
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.
Methods of improving quantum LADARs and related sensors are developed based on quantum entanglement and hyperentanglement. Multiple single photon states are used to obtain a multiphoton entangled state. These states can be N00N states, M&N states (M&N), or a linear combination of M&N states (LCMNS) or generalized states kindred to LCMNS. The procedure for doing this derives from the fundamental theorem of algebra. Various states generated by this process are developed. A diagram of a device for producing such states is examined. They are related to a simpler version of this concept introduced in the seminal experiment by Mitchell-Lundeen and Steinberg. A certain class of states obtained through the Schrodinger kitten process are shown to be effective for generating states hyper-entangled in polarization and energy-time. A diagram of a device for producing such states is considered. Alternate methods of generating hyperentanglement in polarization and energy-time are discussed. Improvements offered by networks are discussed. The utility of these procedures for sensing and communications is examined. An open systems analysis based on density operator theory is conducted including both noise and loss mechanisms. The susceptibility to noise and loss of the various hyper-entanglement procedures is examined. Various measures of effectiveness (MOEs) are derived to quantize system performance. MOEs include but are not limited to, SNR, signal-to-interference ratio, quantum Cramer Rao’ lower bound, quantum Chernoff bound, measurement time, the Holevo bound, sensing range, and resolution. A summary table with expanded MOEs results drawing from multiple papers is provided.
Wavefronts for single and multiple photons are derived and combined with quantum hyper-entanglement, multiphoton entanglement, and network properties. Each node of the network will transmit at least one signal and one ancilla photon in a hyper-entangled state. Parameters used for hyper-entanglement will include photon polarization, energy-time, orbital angular momentum (OAM), radial quantum number, etc. as well as up to 12 parameters characterizing the wavefront properties. These parameters arise from using Lie algebra techniques to solve the paraxial wave equation. Eigenfunctions of operators associated with each 2D paraxial wave equation will be determined each being a function of up to six parameters. The resulting wavefronts will be shown in some cases to be propagation invariant and have the self-healing property. The free parameters of the derived wavefront will offer a programmable wavefront that will permit better imaging around obstacles and reduced propagation loss. Unlike the results found in the literature the free parameters will permit photonic trajectories to be programmed for better performance. Parabolic trajectories, i.e. ones with quadratic terms will emerge as well as those with cubic or higher degree terms. This will further facilitate imaging around obstacles. The same wavefront or combinations of wavefronts can be applied to multiple signal photons to combine hyper-entanglement with multiphoton entanglement offering additional performance improvements. Wavefront engineering can be applied to a single ancilla or multiple ancilla photons. Enhanced superdense coding will be considered. Measures of effectiveness (MOEs) such as the SNR, measurement time, resolution measures, and Holevo bound will be derived or provided.
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.