Path integrals and lower bounds for density matrices

David Hollar

William H. Newman, Atsuo Kuki

We generalize the discretized Feynman path integral expansion by replacing the path through pure states with a path through idempotent density matrices. The transformed expression converges to the ordinary path integral, but is computationally more flexible than the ordinary form. By introducing a particular choice of these idempotent density matrices, based on rotational averaging around the two-particle center of mass, we greatly reduce the sign problem for systems of two fermions in three dimensions. In the ordinary path integral Monte Carlo approach, low temperature simulation of fermions is inefficient as the integral decays exponentially faster than the integrand (and its variance) as the temperature decreases. The new rotationally averaged algorithm dramatically retards this relative decay. Fermionic simulations of the model system of Kestner and Sinanoglu, and of the harmonic oscillator, demonstrate this improvement, as integrals calculated at temperatures much lower than the singlet–triplet splitting display negligible decay of the desired averages relative to the partition function of the modified path ensemble.

Osvaldo Civitarese, Manuel Gadella

Cristian Predescu

In this paper I provide significant mathematical evidence in support of the existence of direct short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is "experimental," I only provide a "physicist's" proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in Appendix B, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present paper is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.

K. Husimi, Yoshihiro Kitano, Toshiyuki Nishiyama

Seyed Mohammad Hashemi Rafsanjani, Shantanu Agarwal

By focusing on the X-matrix part of a density matrix of two qubits we provide an algebraic lower bound for the concurrence. The lower bound is generalized for cases beyond two qubits and can serve as a sufficient condition for non-separability for bipartite density matrices of arbitrary dimension. Experimentally, our lower bound can be used to confirm non-separability without performing a complete state tomography.