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In this paper, we investigate the geometric phase (GP) acquired by two-mode mixed squeezed-coherent states (SCSs) during unitary cyclic evolution, examining the influence of squeezing parameter and classical weight. We analyze the GP for three distinct mixed states, each characterized by different configurations of the SCSs. Our results reveal that increasing the squeezing parameters of individual modes compresses the GP contours in different patterns: linearly, hyperbolically, and elliptically, depending on the mixed state configuration. This behavior demonstrates the enhancement of quantum measurement precision in squeezed states through uncertainty adjustment, consistent with the established theoretical predictions.

In a Hamiltonian approach to anomalies, parity and time-reversal symmetries can be restored by introducing suitable impure (or mixed) states. However, the expectation values of observables such as the Hamiltonian diverges in such impure states. Here, we show that such divergent expectation values can be treated within a renormalization group (RG) framework, leading to a set of β-functions in the moduli space of the operators representing the observables. This leads to well-defined expectation values of the Hamiltonian in a phase where the impure state restores the P and T symmetry. We also show that this RG procedure leads to a mass gap in the spectrum. Such a framework may be relevant for long wavelength descriptions of condensed matter systems such as the quantum spin Hall (QSH) effect.

Adding the maximally mixed state with some weight to the entanglement system leads to disentanglement of the latter. For each predefined entangled state there exists a minimal value of this weight for which the system loses its entanglement properties. These values were proposed to be used as a quantitative measure of entanglement called robustness [G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999)]. Using the concurrence, we propose the derivation of this measure for the system of two-qubit. Namely, for a two-qubit pure state, an exact expression of robustness is obtained. Finally, in the same way, the robustness of special cases of mixed two-qubit states is calculated.

The nature of vortex structure in the mixed state of high-temperature superconductors (HTS) is investigated by solving the Bogoliubov-de Gennes equations with consideration of competition between antiferromagnetic (AF) and d-wave superconductivity (DSC) orders. By varying the applied magnetic field and temperature, the geometry of vortex structure can take two different forms: conventional vortex lattice (triangular or square), or vortex stripe phases where all the order parameters including spin density wave, charge density wave and superconducting order exhibit stripe-like behavior. This novel vortex stripe phases may show up at low temperature and adjacent to upper critical field H_c2 Phase diagram of temperature dependence of H_c2 will be presented. Our results may shed light on the understanding of the low-temperature H_c2 anomalies in some HTS. New experiments are proposed to test our predictions.

It is well known that most of the high temperature superconductors (at least hole-type) are in d-wave state. But it is still an unsolved problem whether it is a pure d-wave state or one has some kind of mixed state. Among the candidates for an admixture, there are s- and d-wave states. Existing experiments could not resolve this issue. New possibilities for experimental resolution of this problem are opened via recent observation of the collective modes in UBe₁₃ (heavy fermion superconductor) by microwave impedance technique experiments and in Sr₂RuO₄ (high temperature superconductor) by ultrasound attenuation experiments.

Some theoretical treatments show that the most likely state is a mixture of two d-wave states: d_x²-y² and d_xy with a small admixture of former state. To create the theoretical basis for investigation of possible mixed superconducting state in unconventional superconductors by sound attenuation and microwave absorption experiments, I derive for the first time a full set of equations for collective modes spectrum in d_x²-y²-state with small admixture of d_xy state. These equations allow to calculate the whole collective mode spectrum in mixed d_x²-y²+iεd_xy state and distinguish this state from pure d-wave states (whose collective mode spectrum has been calculated earlier) by ultrasound attenuation and microwave absorption experiments.

The interaction between a monolayer of fine ferromagnetic particles and a semi-infinite superconductor has been investigated in the mixed state. The frozen and diamagnetic images model was employed to calculate the levitation force as a function of the levitation height as well as the temperature of the monolayer under the zero-field-cooled (ZFC) and the field-cooled (FC) conditions. Results showed the well-known monotonic decrease of the levitation force as a function of the levitation height while it increases rapidly as a function of temperature up to saturation. As a result of the first order approximation used in our calculations in which the interaction was represented by the forces between the magnetic dipoles and their images, the levitation force was dominated by the diamagnetic properties rather than the flux pinning effects of the superconductor for small values of levitation heights compared to the initial field cooling height.

Mixed states typically arise when quantum systems interact with the outside world. Evolution of open quantum systems in general are described by quantum operations which are represented by completely positive maps. We elucidate the notion of geometric phase for a quantum system described by a mixed state undergoing unitary evolution and non-unitary evolutions. We discuss parallel transport condition for mixed states both in the case of unitary maps and completely positive maps. We find that the relative phase shift of a system not only depends on the state of the system, but also depends on the initial state of the ancilla with which it might have interacted in the past. The geometric phase change during a sequence of quantum operations is shown to be non-additive in nature. This property can attribute a "memory" to a quantum channel. We explore these ideas and illustrate them with simple examples.

We examine evolutions where each component of a given decomposition of a mixed quantal state evolves independently in a unitary fashion. The geometric phase and parallel transport conditions for this type of decomposition dependent evolution are delineated. We compare this geometric phase with those previously defined for unitarily evolving mixed states, and mixed state evolutions governed by completely positive maps.

An analytical method to calculate the sub-entropies and entanglement for the mixed state as an initial field is presented. Also, we investigate the effects of the atomic motion and the field-mode structure on sub-entropies and phase properties of the coherent superposition state and a statistical mixture of coherent states as initial field states taking into account different forms of the intensity-dependent coupling. The initial state, the atomic motion and the field-mode structure play important roles in the time evolution of the entropies, entanglement and phase properties.

This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and $X$-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.