Separability and entanglement of two qubits density matrices using Lorentz transformations

TL;DR

This paper explores how Lorentz transformations can be used to analyze the separability and entanglement of two-qubit density matrices, providing explicit relations and numerical demonstrations for different cases.

Contribution

It introduces a Lorentz transformation framework for explicit separability analysis of two-qubit density matrices, including generic and non-generic cases.

Findings

Lorentz transformations diagonalize the Hilbert-Schmidt decomposition of two-qubit states.

Explicit relations between density matrix parameters and Lorentz-transformed parameters are derived.

Numerical calculations demonstrate theoretical results and analyze non-generic cases.

Abstract

Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form R(u,v=0,1,2,3) of the Hilbert-Schmidt (HS) decomposition of the density matrix. For the generic case in which Lorentz transformations diagonalize R(u,v=0,1,2,3) (into s(0),s(1),s(2),s(3)) we give relations between the R parameters and the s parameters. In particular we consider two cases: a) Two qubits density matrices with one pair of linear terms in the HS decomposition. b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case a) are analyzed and the non-generic property is related explicitly to Lorentz velocity beta=1 which is not reachable physically