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International Journal of Modern Physics B

MAXENT APPROACH TO QUBITS

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

Published May 30, 2007• 1 citation
Publisher

Abstract

We consider the Hubbard-like Hamiltonian used in Physica B 352 (2004) 269-279, using the MaxEnt approach and a CSCO, as the set of relevant operators which, obviously, close a partial Lie algebra with the Hamiltonian. So, it is possible to define a positive definite metric on the space spanned by the mean values of the CSCO. Particularly, the selected operators represent the occupation number, the aligned and antialigned pairs per site, respectively. The interesting question to be examined here is whether the above mentioned Hamiltonian have the property that the expectation values of a suitable operator are indeed zero and one (or a constant) qubits, and zero and +1 and -1, i.e., qutrits.

DOI
10.1142/s0217979207043853

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We consider the Hubbard-like Hamiltonian used in Physica B 352 (2004) 269-279, using the MaxEnt approach and a CSCO, as the set of relevant operators which, obviously, close a partial Lie algebra with the Hamiltonian. So, it is possible to define a positive definite metric on the space spanned by the mean values of the CSCO. Particularly, the selected operators represent the occupation number, the aligned and antialigned pairs per site, respectively. The interesting question to be examined here is whether the above mentioned Hamiltonian have the property that the expectation values of a suitable operator are indeed zero and one (or a constant) qubits, and zero and +1 and -1, i.e., qutrits.
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.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.
This chapter aims to develop an appropriate notation and to facilitate the subsequent discussion of the role played by the Lie algebras. It is unlikely that one manages to express a sufficiently realistic model only in terms of the operators that span a finite-dimensional Lie algebra, but it may be possible to simulate most of the relevant features of many physical phenomena by means of such simple models. The chapter discusses well-known approximate methods for the treatment of the time-evolution equation. If the Hamiltonian operator belongs to a finite-dimensional Lie algebra, then the differential equation for the time-evolution operator is easier to solve, and even the calculation of transition probabilities and expectation values of observables may be straightforward. The most relevant feature of this application of the Lie algebraic method is the occurrence of a matrix equation of finite dimension in contrast with the expansion in a basis set of state vectors, which typically leads to matrices of infinite dimension.