High-dimensional private quantum channels and regular polytopes

TL;DR

This paper explores the geometric relationship between private quantum channels and regular polytopes in higher dimensions, extending known connections from qubits to qutrits and beyond, with potential implications for quantum communication security.

Contribution

It establishes a link between private quantum channels and regular polytopes in higher dimensions, expanding the geometric understanding of quantum information protocols.

Findings

Connection between qutrit PQCs and 4-polytopes

Use of generalized Gell-Mann matrices and quantum Fourier transform

Proposed formula for higher-dimensional cases

Abstract

As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytope. Finally, we attempt to devise a formula for connections on higher dimensional cases.