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

TL;DR

This work reframes quantum cryptography through the geometry of quantum state space, treating projective Hilbert space as a public manifold endowed with the Fubini-Study metric and a family of entanglement functionals $E_\theta$. The secret key is the specific $\theta$, which selects a foliation into constant-entanglement leaves; information is encoded not only in prepared states but in how trajectories cross these leaves under unitary moves. The authors formalize geometric entanglement codes and illustrate them with a two-qubit toy model featuring a twisted entanglement measure and a geometric BB84-like protocol leveraging incompatible foliations to mimic basis incompatibility and the information-disturbance tradeoff. While elementary, these constructions provide a clean framework to analyze security questions and to extend to higher-dimensional and multipartite settings, potentially enabling new cryptographic primitives rooted in entanglement geometry.

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 $θ$ that selects a specific entanglement functional $E_θ$ 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.