High-dimensional Private Quantum Channels and Regular Polytopes

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-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.

Platonic Bell inequalities for all dimensions

“…[13], different types of single qubit private quantum channels have been constructed and linked to threedimensional Platonic solids. In a more recent work [14] four-dimensional Platonic solids were connected to qutrit-based private quantum channels. Such regular polyhedra were also used in reference frame alignment [15] and quantum hashing protocols [16].…”