We consider a Squeezed Harmonic Oscillator (SHO) dissipated as a Dekker oscillator and exposed to an environment under Thermal Non-Equilibrium Conditions (TNEC). The eigenvalues and eigenvectors of the system are obtained to determine the thermal distribution. Using Tsallis statistics under TNEC, the oscillator is constrained to its two lowest energy levels as a quasi-qubit and its thermal state is represented as a qutrit based on a simple harmonic oscillator basis. The Hadamard operation on the environmental entanglement with quasi-qubit is analyzed. This analysis examines the impact of temperature, the frequencies of the system and the environment. The fidelity of the Dekker state with the superposition state is compared with that of its environmental entanglement. We find that reducing the entanglement with the environment increases the fidelity up to 99%, and this increase is directly linked to a reduction in system-environment entanglement. These results may be beneficial for the control of single-qubit gates exposed to environmental effects.
We consider a Squeezed Harmonic Oscillator (SHO) dissipated as a Dekker oscillator and exposed to an environment under Thermal Non-Equilibrium Conditions (TNEC). The eigenvalues and eigenvectors of the system are obtained to determine the thermal distribution. Using Tsallis statistics under TNEC, the oscillator is constrained to its two lowest energy levels as a quasi-qubit and its thermal state is represented as a qutrit based on a simple harmonic oscillator basis. The Hadamard operation on the environmental entanglement with quasi-qubit is analyzed. This analysis examines the impact of temperature, the frequencies of the system and the environment. The fidelity of the Dekker state with the superposition state is compared with that of its environmental entanglement. We find that reducing the entanglement with the environment increases the fidelity up to 99%, and this increase is directly linked to a reduction in system-environment entanglement. These results may be beneficial for the control of single-qubit gates exposed to environmental effects.
Abstract We explore how the Wigner-parity operator affects a system of strongly coupled two-harmonic oscillators (SCTHO) interacting with a bosonic bath under thermal non-equilibrium conditions (TNEC). By deforming the Lindblad equation with the Wigner-parity algebra, we model the system’s time evolution, constraining each oscillator to its two lowest energy levels. We use non-Gaussian Tsallis statistics to describe TNEC and examine the time development of concurrence both before and after a CNOT gate operation. Our findings indicate that parity is a strong marker of non-classical behavior in SCTHO exposed to thermal noise. Parity dependent Wigner negativities measure the loss of quantum coherence during CNOT gate operations. This loss is confirmed by Lindblad dynamics. We observe that the parity framework increases concurrence between oscillators after the CNOT, and we analyze how concurrence depends on Wigner parity and system frequency across different Tsallis q -regions. Our analysis reveals that Wigner parity deformation dramatically suppresses decoherence, enhancing concurrence longevity, while Tsallis statistics introduce a distinct hierarchy in thermal resilience. The CNOT gate’s operation induces a frequency dependent entanglement crossover, with parity protection proving most effective at high frequencies. This model offers potential for optimizing thermal solid-state quantum gates (QGs) and provides insights into fault-tolerant encoding in noisy oscillator-based qubits. Furthermore, by extending the analysis to a microscopic model with temperature-dependent decay rates derived from a Tsallis-distributed bath, we demonstrate how non-Gaussian environmental statistics directly shape the decoherence pathways and concurrence dynamics.
The quantum oscillator model plays a significant role in quantum optics and quantum information and has been one of the hot topics in related research fields. Inspired by the single-mode linear harmonic oscillator and the two-mode entangled state representation, we construct a two-mode coupled harmonic oscillator in this work. Different from the quantum transformation method used in previous literature, the entangled state representation is directly used in this work to solve its energy eigenvalues and eigenfunctions easily. The energy eigenvalues and eigenfunctions of this two-mode coupled harmonic oscillator are continuous compared with those of the one-mode harmonic oscillator. Using the matrix theory of quantum operators, we derive the transformation and inverse transformation of the time evolution operator corresponding to the two-mode coupled harmonic oscillator. In addition, using the entangled state representation, the specific form of the time evolution of the two-mode vacuum state under the action of the oscillator is obtained. Through the analysis of quantum fidelity, it is found that the fidelity of the output quantum state decreases with the oscillator frequency increasing, and the fidelity eventually tends to zero with the increase of time. When analyzing the orthogonal squeezing properties of the output quantum state, this type of two-mode oscillator does not have the orthogonal squeezing effect, but it has a strong quantum dissipation effect instead. This conclusion is further verified by the quasi-probability distribution Q function of the quantum state phase space. Therefore, the two-mode coupled harmonic oscillator has a major reference value in quantum control such as quantum decoherence and quantum information transmission. Like the two-mode squeezed vacuum state, the photon distribution of the output quantum light field corresponding to the two-mode harmonic oscillator presents a super-Poisson distribution, and the photons exhibit a strong anti-bunching effect. Using the three-dimensional discrete plot of the photon number distribution, the super-Poisson distribution and quantum dissipation effect of the output quantum state are intuitively demonstrated. Finally, the SV, which is an entanglement criterion, is used to determine that the output quantum state has a high degree of entanglement. Further numerical analysis shows that the degree of entanglement increases with the action time and the oscillator frequency. In summary, the two-mode coupled harmonic oscillator constructed in this work can be used to prepare highly entangled quantum states through a complete quantum dissipation process. This provides theoretical support for experimental preparing quantum entangled states based on dissipative mechanisms.
A meaningful description of the low-energy dynamics of multipartite entanglement is provided\nfor harmonic oscillator systems in general dissipative scenarios. Without doing any\ncentral approximation, this mainly relies on a set of reasonable physical statements on the\nenvironment and system-environment interaction consistent with a linear analysis of the\nopen-system dynamics.\nWe rst investigate the inseparability properties of arbitrary entangled states, which\ngenerally entails an optimization procedure of certain functional de ned in the in nitedimensional\nHilbert space of continuous-variable systems. Using a Gaussian-like assumption\nanalogous to that used in the derivation of Gaussian entanglement of formation, we\nderive a computational-e cient form of such functional which considerable simpli es the\noptimization task. This consists on a hierarchy of separability criteria that permit in a\nuni ed way to characterize k-partite entanglement of broad classes of Gaussian and non-\nGaussian states. The strength of the criteria is proven by showing that they satisfactorily\nreproduce previous results from the theory of entanglement, like PPT criterion applied to\narbitrary two-mode and pure three-mode Gaussian states.\nThe separability criteria allow to monitor the transient evolution of multipartite entanglement\nunder the environmental in\nuence, this permitted to show that non-Gaussian\nentanglement may be as robust against harmful dissipative e ects and thermal noise as\nGaussian one. We devote special attention to the stationary properties of tripartite entanglement\nwhen the system oscillators are in contact with either a common environment\nor independent environments at initial di erent temperatures. For the former dissipative\nscheme, it is shown that the environment mediates an e ective many-party interaction with\na spatial long-range feature between system oscillators, which is able to generate tripartite\nentanglement whereas two-mode entanglement is degraded. Regarding the second scheme,\nit is illustrated that thermal non-equilibrium conditions result in a rich variety of quantum\ncorrelations, however a temperature gradient eventually destroys the entanglement shared\nby the system oscillators as a consequence of the growth of thermal noise. Finally, the\nstationary entanglement and energy current across a nite harmonic chain are studied in\ndetail by doing an extensive numerical analysis of both phenomena for a broad range of the\nparameters (i.e., temperatures, oscillator frequencies, and oscillator coupling strengths).\nOur numerical ndings are discussed in terms of the derived energy current expressions,\nthese show an explicit dependence on the two-time correlation functions (between oscillator\noperators) which carry quantum correlations.
Abstract In the present paper, we study the influence of non-commutativity on entanglement in a system of two oscillators-modes in interaction with its environment. The considered system is a two-dimensional harmonic oscillator in non-commuting spatial coordinates coupled to its environment. The dynamics of the covariance matrix, the separability criteria for two Gaussian states in non-commutative space coordinates, and the logarithmic negativity are used to evaluate the quantum entanglement in the system, which is compared to the commutative space coordinates case. The result is applied for two initially entangled states, namely the squeezed vacuum and squeezed thermal states. It can be observed that the phenomenon of entanglement sudden death appears more early in the system for the case of squeezed vacuum state than in the case of squeezed thermal state. Thereafter, it is also observed that non-commutativity effects lead to an increasing of entanglement of initially entangled quantum states, and reduce the separability in the open quantum system. It turns out that a separable state in the usual commutative quantum mechanics might be entangled in non-commutative extension.
Based on the exact dynamics of a two-qubit system and environment, we investigate system-environment (SE) quantum and classical correlations. The coupling is chosen to represent a dephasing channel for one of the qubits and the environment is a proper thermal bath. First we discuss the general issue of dilation for qubit phase damping. Based on the usual thermal bath of harmonic oscillators, we derive criteria of separability and entanglement between an initial X state and the environment. Applying these criteria to initial Werner states, we find that entanglement between the system and environment is built up in time for temperatures below a certain critical temperature Tcrit. On the other hand, the total state remains separable during those short times that are relevant for decoherence and loss of entanglement in the two-qubit state. Close to Tcrit the SE correlations oscillate between separable and entangled. Even though these oscillations are also observed in the entanglement between the two qubits, no simple relation between the loss of entanglement in the two-qubit system and the build-up of entanglement between the system and environment is found.
Abstract This work develops a theoretical framework for scalable noise resilience in coupled-oscillator quantum processors by integrating non-local Lechner–Hauke–Zoller (LHZ) interactions with non-extensive Tsallis statistics. The study focuses on a system of three strongly coupled harmonic oscillators (TSCHO), each interacting with a bosonic bath, and truncating each oscillator to its two lowest energy levels, obtaining an effective eight-dimensional Hilbert space. The Hamiltonian is diagonalized, yielding eigenstates that include dark states, and squeezed parity-mixed states. Thermal non-equilibrium conditions are then introduced using Tsallis statistics, constructing a generalized initial density matrix parameterized by the entropic index q . The dissipative dynamics is described by a Lindblad master equation derived under the Born–Markov approximation, with bath-induced decay rates that incorporate a q -deformed Bose–Einstein distribution. The concurrence is computed for two and three-oscillator subsystems and compared as a function of time and temperature. Next, the LHZ architecture is incorporated by adding genuine three-body non-local interactions, and the corresponding dissipative coefficients are derived. The time evolution of entanglement and fidelity is simulated within the LHZ-protected subspace. Finally, a CNOT gate is applied to one of the subsystems which exhibits significant concurrence and its fidelity is evaluated across different q regimes and temperatures. The results demonstrate that scaling from two to three oscillators spontaneously generates asymmetric dissipation channels that protect specific subsystems. A counter-intuitive temperature assisted stabilization effect is validated: for q ≠ 1 , higher bath temperatures do not destroy entanglement but rather preserve it. In the LHZ architecture for TSCHO quantitative analysis shows that at low frequencies and elevated temperatures, the non-extensive regimes outperform the extensive one; however, at high frequencies, thermal effects are suppressed and the q = 1 regime remains dominant. The dissipatively protected CNOT gate achieves meaningful fidelity without the massive qubit overhead of conventional quantum error correction. While the numerical analysis is performed for three oscillators, the proposed framework is theoretically scalable to larger systems, and the underlying mechanisms—non-local parity protection and non-extensive thermodynamics—are general. Overall, this approach suggests a potential paradigm shift from fighting decoherence to engineering intrinsic noise resilience at the hardware level, establishing non-locality and non-extensive thermodynamics as physical resources for robust quantum information processing.