Lorentzian geometry of qubit entanglement. (arXiv:1801.00611v1 [quant-ph])
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.