On the validity of intermediate tracing in multiple quantum interactions

TL;DR

The paper addresses how to manage intermediate tracing in sequential quantum interactions by proving that a system need only be present during its interaction and can be traced out afterward. The analytic proof uses a two-system decomposition with $H_S=H_A\otimes U$ and shows that the evolution of the target system $A$ is unchanged by the presence of a non-interacting incident system $B$, i.e., $d\rho_A/dt=i[\rho_A,H_A]$ while $d\rho_B/dt=0$. A three-qubit numerical example with $H=\boldsymbol{\sigma}\cdot\boldsymbol{\sigma}$ demonstrates three tracing policies yielding identical results, with substantial efficiency gains: Policy 2 ~30% faster than Policy 1 and Policy 3 ~60% faster than Policy 1. The findings have practical impact for simulations of FEBERI-like interactions and radiative-mode dynamics, enabling correct multi-system evolution with reduced computational resources when the interaction region contains at most one active particle at a time.

Abstract

Interactions between many (initially separate) quantum systems raise the question on how to prepare and how to compute the measurable results of their interaction. When one prepares each system individually and let them interact, one has to tensor multiply their density matrices and apply Hamiltonians on the composite system (i.e. the system which includes all the interacting systems) for definite time intervals. Evaluating the final state of one of the systems after multiple consecutive interactions, requires tracing all other systems out of the composite system, which may grow up to immense dimensions. For computation efficiency during the interaction(s) one may consider only the contemporary interacting partial systems, while tracing out the other non interacting systems. In concrete terms, the type of problems to which we direct this formulation is a ``target'' system interacting {\bf succesively} with ``incident'' systems, where the ``incident'' systems do not mutually interact. For example a two-level atom, interacting succesively with free electrons, or a resonant cavity interacting with radiatively free electrons, or a quantum dot interacting succesively with photons. We refer to a ``system'' as one of the components before interaction, while each interaction creates a ``composite system''. A new interaction of the ``composite system'' with another ``system'' creates a ``larger composite system'', unless we trace out one of the systems before this interaction. The scope of this work is to show that under proper conditions one may add a system to the composite system just before it interacts, and one may trace out this very system after it finishes to interact. We show in this work a mathematical proof of the above property and give a computational example.

Figures (2)

• Figure 1: The initial configuration of the qubits A, B and C: in the positive eigenstate of $\sigma_x$, $\sigma_y$ and $\sigma_z$ respectively.
• Figure 2: The 3 qubits after the completion of all interactions, their states are marked by the red arrows. For reference, the white arrows show the states of the qubits before interaction.