We study the role of quantum entanglement (particle entanglement and mode entanglement) in optical phase estimation by employing the mode description and the particle description of bosonic probe states. To determine the physical resources responsible for Heisenberg scaling (attainable only in the noiseless case), we restrict our analysis to noiseless quantum-limited optical phase estimation. The quantum Fisher information (QFI) is expressed as a function of the first- and second-order optical coherence functions. The resulting form of the QFI elucidates the deriving metrological resources for quantum phase estimation: field intensity and photon detection correlations. Our analysis confirms that mode entanglement is not required for quantum-enhanced interferometry, whereas particle entanglement is a necessary requirement. Furthermore, the derived forms of the QFI equations are summations of two terms: the classical (shot-noise-limit scaling) term and the quantum (Heisenberg scaling) term. This allows us to clearly identify the physical resources responsible for quantum enhancement in optical phase estimation.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
We compare the basic resource requirements for first and second quantized bosonic mappings in a system consisting of
N particles in
M modes. In addition to the standard binary first quantized mapping, we investigate the unary first quantized mapping. Our comparison focuses on the
k-body reduced density matrix (
k-RDM) and two standard bosonic Hamiltonians. The first quantized mappings use less resources for off-diagonal terms of the
k-RDM by a factor of $ \sim N^k$, compared to the second quantized mappings. The number of gates for the first quantized binary mapping increases faster with
M compared to the other mappings. Nevertheless, a detailed numeric analysis reveals that the binary first quantized mapping still requires fewer gates than the binary and unary second quantized ones for realistic combinations of
N and
M, while requiring exponentially fewer qubits than the unary mappings. Additionally, the number of CNOT and
Rz(φ) gates necessary to express a single Trotter step of the Hamiltonian in the binary first quantized mapping is comparable to the (most efficient for a single Trotter step) unary first quantized one when
M=2n for both the Bose-Hubbard model and the harmonic trap with short-range interactions. Additionally the binary mapping leads to lower one-norms than the unary mapping making it the overall most efficient choice for qubitization-based quantum phase estimation.