Lorentzian geometry of qubit entanglement

TL;DR

This paper explores the connection between qubit entanglement and Lorentzian geometry, introducing a framework based on Partial Lorentz Transformations and energy conditions to detect and visualize entanglement.

Contribution

It provides a theoretical foundation for a Lorentzian geometric approach to qubit entanglement detection, including a new visualization method of state space.

Findings

All states satisfy a Dominant Energy Condition (DEC).

Separable states satisfy the Strong Energy Condition (SEC).

Entangled states violate the SEC.

Abstract

We study the relation between qubit entanglement and Lorentzian geometry. In an earlier paper, we had given a recipe for detecting two qubit entanglement. The entanglement criterion is based on Partial Lorentz Transformations (PLT) on individual qubits. The present paper gives the theoretical framework underlying the PLT test. The treatment is based physically, on the causal structure of Minkowski spacetime, and mathematically, on a Lorentzian Singular Value Decomposition. A surprising feature is the natural emergence of "Energy conditions" used in Relativity. All states satisfy a "Dominant Energy Condition" (DEC) and separable states satisfy the Strong Energy Condition(SEC), while entangled states violate the SEC. Apart from testing for entanglement, our approach also enables us to construct a separable form for the density matrix in those cases where it exists. Our approach leads to a simple graphical three dimensional representation of the state space which shows the entangled states within the set of all states.