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International Journal of Modern Physics B

GIBBS ENTROPY APPROACH TO THE REALIZATION OF QUBITS AND QUTRITS

Claudia M. Sarris, Araceli N. Proto, F. Bary Malik

Published Oct 2, 2013• 0 citations
Publisher

Abstract

We investigate, in this paper, the possibilities of generating qubits and qutrits in strongly correlated systems described by a modified of Hubbard Hamiltonian. Out of the complete set of commutating operators that form a close Lie algebra with this Hamiltonian, one can generate a particular operator, the expectation values of which, with respect to the density matrix generated from the Gibbs entropy by Maximum Entropy Principle (MEP), are 0 and ±1 near a particular temperature. This density matrix is generated by the superposition of highly coherent two-electronic states, analogous to the BCS ones. The concurrent existence of the expectation values of 0, +1 and -1 of this operator with respect to the density matrix occurs near the phase transition of aligned states to anti-aligned states. These are qutrits, which in the absence of a magnetic field reduces to qubits. We also present the general uncertainty principle (GUP) valid for the set of these operators, evaluate its value for specific heat, and examine the behavior of the specific heat and the related GUP as a function of the temperature. This temperature dependence of the specific heat, exhibits the expected trend of phase transition near the transition temperature. For the chosen Hamiltonian, we present the derivation of the postulate of Weiss' mean field theory. This relation points to the fact that to generate qubits and qutrits for the system investigated here, it must have an intrinsic magnetic field and be a strongly correlated system such as manganites. This investigation further points to the fact that the qutrits gate may be a suitable quantum computing algorithm for systems with intrinsic magnetic and applied electromagnetic fields, since in the presence of such fields z-projections of the state with spin-1 are no longer degenerate. This investigation establishes that the thermodynamical evolution of fermion pair in the presence of an interaction with its environment are different for qubits and qutrits, particularly in the presence of an internal and external magnetic field and possibly for the general case of electro-magnetic field.

DOI
10.1142/s0217979213470061

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We investigate, in this paper, the possibilities of generating qubits and qutrits in strongly correlated systems described by a modified of Hubbard Hamiltonian. Out of the complete set of commutating operators that form a close Lie algebra with this Hamiltonian, one can generate a particular operator, the expectation values of which, with respect to the density matrix generated from the Gibbs entropy by Maximum Entropy Principle (MEP), are 0 and ±1 near a particular temperature. This density matrix is generated by the superposition of highly coherent two-electronic states, analogous to the BCS ones. The concurrent existence of the expectation values of 0, +1 and -1 of this operator with respect to the density matrix occurs near the phase transition of aligned states to anti-aligned states. These are qutrits, which in the absence of a magnetic field reduces to qubits. We also present the general uncertainty principle (GUP) valid for the set of these operators, evaluate its value for specific heat, and examine the behavior of the specific heat and the related GUP as a function of the temperature. This temperature dependence of the specific heat, exhibits the expected trend of phase transition near the transition temperature. For the chosen Hamiltonian, we present the derivation of the postulate of Weiss' mean field theory. This relation points to the fact that to generate qubits and qutrits for the system investigated here, it must have an intrinsic magnetic field and be a strongly correlated system such as manganites. This investigation further points to the fact that the qutrits gate may be a suitable quantum computing algorithm for systems with intrinsic magnetic and applied electromagnetic fields, since in the presence of such fields z-projections of the state with spin-1 are no longer degenerate. This investigation establishes that the thermodynamical evolution of fermion pair in the presence of an interaction with its environment are different for qubits and qutrits, particularly in the presence of an internal and external magnetic field and possibly for the general case of electro-magnetic field.
Abstract We investigate a hybrid qubit–qutrit system exposed to both a magnetic field and classical dephasing noise. The quantum system’s characteristics encompass diverse parameters, including spin-exchange interaction, dephasing, and the magnetic field. To incorporate thermal effects, we employ the system’s Hamiltonian to generate an initial qubit–qutrit density matrix within the framework of the Gibbs density operator. Furthermore, we model dephasing effects on the initial thermal state of the system using an Ornstein–Uhlenbeck process. We employ geometric discord, negativity, and entropic coherence functions to depict the quantum correlations across various parameter settings. Our results reveal that initially, quantum correlations attain non-maximal values, with their dynamics intricately reliant on the underlying system parameters. Specifically, when the system is primarily characterized by the magnetic field, we observe heightened levels of quantum correlations. Additionally, temperature-based characterization is found to have the most detrimental effect on the state. Geometric discord is observed to capture a higher degree of quantum correlations, albeit saturating rapidly at zero compared to entanglement and coherence. Finally, we investigated the effects of common environmental coupling and more pronounced non-Markovian dynamics in the system, revealing an enhanced preservation of quantum correlations. These modifications allow for prolonged coherence and entanglement, underscoring the potential of structured environmental interactions to mitigate decoherence effects and sustain quantum correlations over time.
Quantum information and quantum computing (QIQC) systems, relying on the phenomena of superposition and entanglement, offer the potential for vast improvements in certain computations. A practical QC realization requires maintaining the stored information for time-scales long enough to implement algorithms. One primary cause of information loss is decoherence, i.e., the loss of coherence between two energy levels in a quantum system. This work attributes decoherence to dissipation occurring as the system evolves and uses steepest-entropy-ascent quantum thermodynamics (SEAQT) to predict the evolution of system state. SEAQT asserts that, at any instant of time, the system state evolves such that the rate of system entropy change is maximized while conserving system energy. With this principle, the SEAQT equation of motion is applicable to systems in any state, near or far from stable equilibrium, making SEAQT particularly well suited for predicting the dissipation occurring as quantum algorithms are implemented. In the present research, the dynamics of qubits (quantum-bits) using the SEAQT framework are first examined during common quantum gates (combinations of which form algorithms). This is then extended to modeling a system of multiple qubits implementing Shor's algorithm on a nuclear-magnetic-resonance (NMR) QC. Additionally, the SEAQT framework is used to predict experimentally observed dissipation occurring in a two-qubit NMR QC undergoing a so called ``quenching'' process. In addition, several methods for perturbing the density or so-called ``state'' operator used by the SEAQT equation of motion subject to an arbitrary set of expectation value constraints are presented. These are then used as the basis for randomly generating states used in analyzing the dynamics of entangled, non-interacting systems within SEAQT. Finally, a reservoir interaction model is developed for general quantum systems where each system locally experiences a heat interaction with an external reservoir. This model is then used as the basis for developing a decoherence control scheme, which effectively transfers entropy out of the QIQC system as it is generated, thus, reducing the decoherence. Reservoir interactions are modeled for single qubits and the control scheme is employed in modeling an NMR QC and shown to eliminate nearly all of the noise caused by decoherence/dissipation.
The dynamics of a central spin-1/2 in presence of a local magnetic field and a bath of N spin-1/2 particles is studied in the thermodynamic limit. The interaction between the spins is Heisenberg XY type and the bath is considered to be a perfect thermal reservoir. In this case, the evolution of the populations of the reduced density matrix are obtained for different temperatures. A Born approximation is made but not a Markov approximation resulting a non-Markovian dynamics. The measure of the way that the system mixes is obtained by means of the von Neumann entropy. For low temperatures, results show that there are oscillations of populations and of the von Neumann entropy, indicating that the central spin becomes a pure state with characteristic time periods in which it is possible to extract or recuperate information. In the regime of high temperatures, the evolution shows a final maximum mixed state with entropy S=ln 2 as it is expected for a two level system.
This paper is a contribution to the general theory of quantum spin systems. We deal with a general theory in the sense that no concrete interaction shows up but an arbitrary relatively short-range interaction is chosen. It is well known that the mean entropy plays an important role in the thermodynamics of quantum spin systems: it is one of the ingredients of the Lanford–Ruelle–Robinson variational principle. We show that in the background of the existence of the mean entropy, there is an operator convergence which resembles the McMillan theorem from information theory. Asymptotic equipartition property and several entropy densities are investigated, in particular, the relation of the Gibbs condition to relative entropies. Although this paper is not intended to be a review, we try to give an overview of the theory of quantum Gibbs states in the mathematical sense. The paper is organized as follows. Section 1 provides an introduction and further sections contain our new results. Each section is closed with a short discussion on sources and references.
The dynamics of a central spin-1/2 in presence of a local magnetic field and a bath of N spin-1/2 particles is studied in the thermodynamic limit. The interaction between the spins is Heisenberg XY type and the bath is considered to be a perfect thermal reservoir. In this case, the evolution of the populations of the reduced density matrix is obtained for different temperatures. A Born approximation is made but not a Markov approximation resulting a non-Markovian dynamics. The measure of the entanglement between the spin and the bath is obtained by means of the von Neumann entropy. For low temperatures, results show that there are oscillations of populations and of the von Neumann entropy as measure of entanglement, indicating that the central spin becomes disentangled of the spins of the bath with characteristic time periods in which it is possible to extract or recuperate information. In the regime of high temperatures, the evolution shows a final maximum entangled state with entropy S = ln 2 as it is expected for a two level system. I
The exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg XYXYXY spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using an operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming-like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment, and the interaction between the constituents of the spin environment. We also investigate the effect of detuning, which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two-spin qubits.