A sensor based on quantum multiphoton entanglement (ME) combined with hyper-entanglement (HE) is described. Measures of effectiveness (MOEs) are derived describing the benefits of forming hybrid states between ME and HE. An open systems approach is taken with both environmental noise and loss factors included. Methods of generating the ME and HE states are described with schematics provided. Networks are introduced yielding additional performance improvements. Waveforms for effective atmospheric propagation are discussed. A new parameterized mode is derived using Lie algebra techniques. The waveform is shown to be a scaled and translated form of the Airy wave. The additional scaling and translation parameters derived from Lie algebra theory show promise for selecting the best performing waveform or linear combination of such waveforms. Alternatively, parameters for single waveforms or linear combinations of parameterized Airy waves can be selected through optimization. An MOE, the Holevo bound, is maximized ultimately yielding the Holevo-Shumacher-Westmoreland capacity. Analysis of the translated and scaled Airy function show that its asymptotic form yields much less loss during atmospheric propagation. It is robust under turbulence and atmospheric inhomogeneities. Additional parameters in the argument of the Airy function that occur as a natural result of the derivation show promise for using the waveform to facilitate imaging around corners. Second quantization is applied resulting in a version of the waveform when only one signal photon is present for entangled or non-entangled systems. Even when only one signal photon is propagating the waveform is shown to have advantages for sensing and imaging.
A sensor based on quantum multiphoton entanglement (ME) combined with hyper-entanglement (HE) is described. Measures of effectiveness (MOEs) are derived describing the benefits of forming hybrid states between ME and HE. An open systems approach is taken with both environmental noise and loss factors included. Methods of generating the ME and HE states are described with schematics provided. Networks are introduced yielding additional performance improvements. Waveforms for effective atmospheric propagation are discussed. A new parameterized mode is derived using Lie algebra techniques. The waveform is shown to be a scaled and translated form of the Airy wave. The additional scaling and translation parameters derived from Lie algebra theory show promise for selecting the best performing waveform or linear combination of such waveforms. Alternatively, parameters for single waveforms or linear combinations of parameterized Airy waves can be selected through optimization. An MOE, the Holevo bound, is maximized ultimately yielding the Holevo-Shumacher-Westmoreland capacity. Analysis of the translated and scaled Airy function show that its asymptotic form yields much less loss during atmospheric propagation. It is robust under turbulence and atmospheric inhomogeneities. Additional parameters in the argument of the Airy function that occur as a natural result of the derivation show promise for using the waveform to facilitate imaging around corners. Second quantization is applied resulting in a version of the waveform when only one signal photon is present for entangled or non-entangled systems. Even when only one signal photon is propagating the waveform is shown to have advantages for sensing and imaging.
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 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.
An atmospheric imaging system based on quantum hyper-entanglement has been developed. Hyper-entanglement refers to entanglement in more than one degree of freedom, e.g. polarization, energy-time and orbital angular momentum (OAM). The system functions at optical or infrared frequencies. Only the signal photon propagates in the atmosphere, the ancilla photon is retained within the detector. This results in loss being essentially classical, giving rise to stronger forms of entanglement. Bell state generation and Bell state measurement, i.e. the ability to distinguish the various Bell states is discussed. A mathematical representation of Bell state generation and the Bell state analyzer, including a projection operator describing the measurement process is provided. Signatures for unique detection of the various Bell states are developed. A method and design for creating states hyper-entangled in polarization, energy-time, OAM and the radial quantum number is examined. Hermite-Gaussian (HG) modes, Laguerre-Gaussian (LG) modes, OAM dependence of the LG modes and a method of mode conversion are discussed. A projection operator that represents the combined measurements between the different degrees of freedom is provided. A model of generation and detection efficiencies for the different degrees of freedom and the implications for signal to noise ratio (SNR), signal to interference ratio (SIR), the quantum Cramer-Rao lower bound, and the measurement time are provided in closed form. A formula describing how hyper-entanglement greatly improves the maximum detection range of the system is derived. The formalism permits random noise and entangled or non-entangled sources of interference to be modeled.
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.
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.
A hyper-entanglement based atmospheric imaging/detection system involving only a signal and an ancilla photon will be considered for optical and infrared frequencies. Only the signal photon will propagate in the atmosphere and its loss will be classical. The ancilla photon will remain within the sensor experiencing low loss. Closed form expressions for the wave function, normalization, density operator, reduced density operator, symmetrized logarithmic derivative, quantum Fisher information, quantum Cramer-Rao lower bound, coincidence probabilities, probability of detection, probability of false alarm, probability of error after M measurements, signal to noise ratio, quantum Chernoff bound, time-on-target expressions related to probability of error and resolution will be provided. The effect of noise in every mode will be included as well as loss. The system will provide the basic design for an imaging/detection systems functioning at optical or infrared frequencies that offer better than classical angular and range resolution. Optimization for enhanced resolution will be included. The signal to noise ratio will be increased by a factor equal to the number of modes employed during the hyper-entanglement process. Likewise, the measurement time can be reduced by the same factor. The hyper-entanglement generator will typically make use of entanglement in polarization, energy-time, orbital angular momentum, etc. Mathematical results will be provided describing the system’s performance as a function of loss mechanisms and noise.