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 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.
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.
Theory and related designs for enhancing communications and more generally information transfer by using quantum hyper-entanglement are discussed. Quantum hyper-entanglement refers to entanglement in more than one quantum mechanical degree of freedom, e.g. polarization, energy-time, orbital angular momentum, radial quantum number and frequency. Each extra degree of freedom increases the dimensionality d of the underlying Hilbert space. The hyperentangled signal-to-noise ratio (SNR) and signal-to-interference ratio (SIR) are d times the related classical quantity due to the enhancement born of hyper-entanglement. Communication time can be reduced by a factor d or d ⋅ M, where M is the number of message photons used. Errors in parameter estimation related to information stored in the signal experience a factor of d ⋅ M reduction. This paper considers the use of generalized Bell states and discusses circuitry for their transmission, analysis and detection. Holevo bounds and Von-Neumann entropies are calculated for both the hyper-entangled case and non-entangled case. It is shown that hyper-entanglement can increase the maximum information transferrable by more than log2 (d) bits. The Holevo bound results are discussed in the context of the types of generalized Bell states. Generalized measures of effectiveness (MOEs) are introduced that hold for linear combinations of generalized Bell states. Different sets of complete orthogonal projection operators and their applications are discussed. Loss mechanisms due to transmission hardware such as orbital angular moment polarization rotation, and polarizing beam splitter cross-talk are calculated. Propagation and detection loss as well as noise are considered.
A method of generating superdense coding based on quantum hyper-entanglement and facilitated by quantum networks is discussed. Superdense coding refers to the coding of more than one classical bit into each qubit. Quantum hyperentanglement refers to quantum entanglement in more than one degree of freedom, e.g. polarization, energy-time, and orbital angular momentum (OAM). The new superdense coding scheme permits 2L bits to be encoded into each qubit where L is the number of degrees of freedom used for quantum hyper-entanglement. The superdense coding procedure is based on a generalization of the Bell state for L degrees of freedom. Theory describing the structure, generation/transmission, and detection of the generalized Bell state is developed. Circuit models are provided describing the generation/transmission process and detection process. Detection processes are represented mathematically as projection operators. A mathematical proof that that the detection scheme permits the generalized Bell states to be distinguished with 100% probability is provided. Measures of effectiveness (MOEs) are derived for the superdense coding scheme based on open systems theory represented in terms of density operators. Noise and loss related to generation/transmission, detection and propagation are included. The MOEs include various probabilities, quantum Chernoff bound, a measure of the number of message photons that must be transmitted to successfully send and receive a message, SNR and the quantum Cramer Rao’ lower bound. Quantum networks with quantum memory are used to increase the efficiency of the superdense coding scheme.
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.
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.
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.