Enhancing the Amplitude of Cat and Compass States and Their Use in Metrology

Arman, . (2025) Enhancing the Amplitude of Cat and Compass States and Their Use in Metrology. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

The cat state was initially introduced by Schrödinger in 1926, in the context of quantum superposition and entanglement, to explain fundamental microscopic phenomena, via relating these to macroscopic scales, with an instance of a dead and alive real-world cat. Since then, these states and similar compass state have found their usefulness in quantum metrology as parameter estimation and continuous variable (CV) quantum error correction (cat and gkp codes) via aniclla by considering them as qubits. The coherent and incoherent changes, as small perturbations to the system, cause either a small phase space translation or a rotation shift. These states in phase space have sub-Planck structures which are extremely sensitive to external or small changes, leading to large changes in distribution and thereby losing quantum information largely. This opens up the prospects of metrology for estimating parameters such as displacement and phase shift, which affect the probe state significantly. Similarly, correction to the logical qubit suffering from different types of noises requires such states that are sensitive to the present noise on information processing platforms, and errors should be easily reversible. This may lead to accessible and fault-tolerant error correction. Even in the discrete variable systems, it requires a large dimensionality of Hilbert space via having more than one physical qubit to make a successful one logical qubit correction. This is inherent to the bosonic system with infinite Hilbert space dimensionality. Both quantum metrology as well as quantum error correction are impactful in terms of the sensitivity of non-classical states in CV systems, where cat and compass states perform well with an increase in their amplitude or average energy. The larger the amplitude or average energy, the more dimensions these states access with a higher success probability in Hilbert space. Thereby, investigation of models and operations is required to obtain a large amplitude of cat and compass states. We study this with the help of two non-Gaussian operations, including squeezing, displacement, Fock state, and photon additions. These operations are motivated by already present literature for enhancing non-classicality or preparing non-classical states. This analysis involves finding regions of high fidelity and high Fisher information. This Fisher information is evaluated only for displacement changes in the phase space in this thesis, while there has been an extensive study of the phase shift parameter for these states. Furthermore, the preparation models are discussed in small detail to complete the investigation for obtaining large amplitudes. We study metrological applications of these states in analyzing decoherence channels. Both decoherence channels in the CV system, dissipation and dephasing, have been investigated in parameter estimation applications. The parameters, such as coherent phase shift and displacement, and their estimation studies are largely discussed with and without decoherence channels. We examined the metrological advantages of the cat and compass states when undergoing the aforementioned decoherence channels. A quantum system suffering from decoherence becomes parameter-dependent. The decoherence rates as parameters affect the initially prepared state of the quantum system, leading to the decay of quantum information, specifically the amplitude of the coherent basis in the dissipation channel and off-diagonal terms in the Fock basis for the dephasing channel. Tracking these amplitudes in the dissipation or off-diagonal Fock basis terms enables the estimation of the decoherence rates. We use the Fisher information analysis, including a special case of channel purification, followed by exact numerical evaluation. This process leads to selecting the better non-Gaussian states for estimating the damping and dephasing channels. The cat and compass state with phase-space squeezing show better estimation precision locally as compared to the superposition of squeezed states, having global metrological potential for estimating the diffusive constant. In summary, we discuss the future outlook in the field of quantum metrology as well as error correction in both discrete variable and continuous variable systems. There are other codes in the error correction that use similar states having the same phase space sub-Planck structures, but robust against noises other than photon loss. Therefore, in analogy to these states, it becomes pertinent to have quantum information investigation under diferent types of noises as well as a comparative analysis in the context of metrology in different platforms, considering their model of preparations.

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