Secret Entanglement, Public Geometry. Quantum Cryptography from a Geometric Perspective

TL;DR

This paper introduces a geometric approach to quantum cryptography using projective Hilbert space, entanglement measures, and state trajectories, where secret information is encoded in the choice of entanglement foliation.

Contribution

It proposes a novel geometric framework for quantum cryptography that encodes secrets in the hidden entanglement foliation within a publicly known geometric structure.

Findings

Entanglement foliations can serve as secret keys.

State trajectories encode information via their relation to entanglement levels.

Toy models demonstrate the feasibility of geometric entanglement coding.

Abstract

Can a secret be hidden not in which quantum state is prepared, but in the way that state \emph{moves} through its space of possibilities? Motivated by this question, we propose an essential geometric perspective on quantum cryptography in which projective Hilbert space and its entanglement foliations play a central role. The basic ingredients are: (a) the Fubini-Study metric on the manifold of pure states, (b) a family of entanglement measures viewed as scalar functions on this manifold, and (c) controlled trajectories generated by unitary operations. The geometric structure -- state manifold, metric, and allowed moves -- is fully public, as is the functional form of the entanglement family. What remains secret is the choice of parameter $\theta$ that selects a specific entanglement functional $E_\theta$ and the corresponding foliation into constant-entanglement hypersurfaces. In this setting, classical messages are encoded not only in the sequence of states but also in the pattern of upward, downward, or tangential steps with respect to the hidden foliation. We formalize this idea in terms of geometric entanglement codes and illustrate it with two toy constructions in which incompatible foliations play the role of mutually unbiased bases.