Quantum mechanics cements the intimate relationship between the nature of light and its detection. Historically, quantum theories of photodetection have generally fallen into two categories: the first tries to determine what quantum field observable is measured when photoelectrons are detected, laying the theoretical groundwork for photodetection being possible. The second type are phenomenological theories, which take great care to model the details of specific photodetectors. In this dissertation, we fill in the gap between these two models in the modern literature on photodetection by constructing a fully quantum-mechanical and sufficiently realistic model that includes all stages of the photodetection process. We accomplish this within the framework of quantum information theory using the language of positive operator valued measures (POVMs). \n\nA POVM provides the most general description of a quantum measurement. In the context of single-photon detection, the photodetector POVM provides a complete characterization of the single-photon detector (SPD) from which all figures of merit can be calculated. Each element of the POVM is comprised of a weighted sum over single-photon state projectors. To construct SPD POVMs, we identify the states projected onto by a measurement and the associated weights. We first accomplish this for a simple time-dependent system, laying out how to efficiently project onto arbitrary single-photon states. Then, we describe the three stages of a realistic SPD: transmission, amplification, and measurement.\n\nIn the transmission stage of photodetection a photon enters a network of discrete energy levels, its energy propagating through that network and finally exiting into another output continuum of modes, a process entirely described by a single complex function. In the amplification stage of photodetection, a single excitation is amplified into a macroscopic signal via a nonlinear amplification mechanism. In the final stage, an inefficient ``classical'' measurement is made on the macroscopic signal. By combining these three stages we form a chain of inference from the final ``click'' to the input photon and construct a POVM projecting onto the input Hilbert space. Lastly, we discuss limits and tradeoffs that arise at each stage and implications for photodetection applications.\n\nThis dissertation contains previously published and unpublished material.
The time-frequency degree of freedom of the electromagnetic field is the final frontier for single-photon measurements. The temporal and spectral distribution a measurement retrodicts (that is, the state it projects onto) is determined by the detector's intrinsic resonance structure. In this paper, we construct ideal and more realistic positive operator-valued measures (POVMs) that project onto arbitrary single-photon wave packets with high efficiency and low noise. We discuss applications to super-resolved measurements and quantum communication. In doing so we will give a fully quantum description of the entire photodetection process, give prescriptions for (in principle) performing single-shot Heisenberg-limited time-frequency measurements of single photons, and discuss fundamental limits and trade-offs inherent to single-photon detection.
Identifying spatial quantum correlations in mixed states is challenging because thermal mixed-state contributions obscure the entanglement encoded in subsystem entropy. Here, we introduce the entanglement projected entropy (EPE), a purification-independent Gaussian spatial filter for mixed fermionic states. By resolving subsystem entropy into Gaussian entropy channels and projecting their purification partners onto the physical complement, we obtain a closed-form expression in terms of the physical covariance matrix. In a one-dimensional free-fermion chain, it removes the volume-law mixed-state background and recovers the zero-temperature conformal scaling with the
c/3 coefficient. In a two-dimensional half-filled
π-flux model, it reveals a universal finite-temperature scaling collapse governed by a Dirac infrared length fixed by the low-energy velocity. These results establish EPE as an entropy-channel filter that exposes boundary-sensitive universal scaling hidden beneath mixed-state entropy.