Overview
This work presents a path integral formulation designed for the dynamics of quantum systems characterized by a finite-dimensional Hilbert space. The formulation is developed entirely within a discrete phase space context. It builds upon the discrete Wigner function defined on ${\mathbb{Z}_d} \times {\mathbb{Z}_d}$, where $d$ is an odd prime, and utilizes the associated Weyl transform, which is constructed from generalized displacement operators.
Research Context
The study addresses the need for a path integral representation for quantum system dynamics in finite-dimensional settings. Previous work by Marinov introduced a functional, and this research aims to develop a finite-dimensional counterpart. The discrete Wigner function serves as a foundational element within this discrete phase space framework.
The use of generalized displacement operators is central to constructing the Weyl transform within this discrete phase space. This approach facilitates the derivation of a kernel that propagates the discrete Wigner function over time.
Approach
The research begins by establishing the discrete Wigner function on ${\mathbb{Z}_d} \times {\mathbb{Z}_d}$, with $d$ specified as an odd prime. Concurrently, it employs a Weyl transform built from generalized displacement operators. From these elements, an exact kernel is derived. This kernel's function is to propagate the discrete Wigner function across time.
An iterative composition of this kernel, combined with a short-time approximation, leads to a sum-over-paths expression. This expression is weighted by a discrete phase-space action, which is identified as the finite-dimensional analog of Marinov's functional.
For specific conditions—namely, Hamiltonians that are linear in the phase-space coordinates and at times strictly commensurate with the lattice—the fluctuation sum undergoes a collapse. This collapse results in a deterministic shift, mirroring the discrete equivalent of classical Hamiltonian flow. The formalism was applied to examples involving one and two interacting qutrits, where $d=3$.
Findings
- An exact kernel was derived that propagates the discrete Wigner function in time within a discrete phase space for finite-dimensional quantum systems.
- A sum-over-paths expression was obtained through iterative composition of the kernel and a short-time approximation, weighted by a discrete phase-space action analogous to Marinov's functional.
- For Hamiltonians linear in phase-space coordinates and at times commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing a discrete classical Hamiltonian flow.
- When applied to one and two interacting qutrits ($d=3$), the full entanglement dynamics required the coherent contribution of all fluctuation sectors, with the boundary-term (mean-field) sector alone being insufficient to reproduce it.
- For a non-stabilizer Hamiltonian, where the short-time kernel provides only an approximation, the time-sliced path integral was observed to converge to the exact dynamics.
- This convergence for non-stabilizer Hamiltonians also included the dynamical generation of Wigner negativity.
Why This Matters
The implications of this formalism extend to the semiclassical simulation of many-body spin systems. It also has relevance for the characterization of non-classicality, particularly through the phenomenon of Wigner negativity.