Finite Pauli dimer spectrum and exact separability threshold

Given: The Pauli matrices and exchange operator, the singlet/triplet eigenspaces, the finite density and separability definitions, and the dimensionless parameter $x\ge 0$.

Spectrum of the two-qubit Pauli exchange matrix and Pauli matrices and the two-qubit exchange operator: the singlet $s$ is a unit vector, the three triplets form an orthonormal basis of $s^{\perp }$, and $C$ has eigenvalue $-3$ on $s$ and eigenvalue $1$ on that triplet space. The rank-one convention gives $P_{s}x=⟨x,s⟩s$, so $P_s$ is the orthogonal projection onto $\mathbb{C}s$ and $P_t=I_4-P_s$ is the orthogonal projection onto the triplet space; their ranks are one and three.

Finite two-qubit density matrices and separability: a density matrix is Hermitian, positive semidefinite, and has trace one.

Bloch-ball representation of a qubit density matrix: the Bloch-coordinate matrix with vector $a=0$ is the positive trace-one matrix $I_2\mathrm{/}2$.

Finite two-qubit density matrices and separability: a separable two-qubit density matrix is a finite convex combination of product qubit density matrices.

Pauli exchange expectation bound for separable states: every separable two-qubit density matrix has $\mathrm{tr}(\rho C)\ge -1$.

Explicit separable state at the Pauli dimer boundary: ${\rho }_{\ast }=\frac{1}{4}(I_4-C\mathrm{/}3)$ is separable.

The natural logarithm as the inverse of the exponential function, The real exponential function and the number $e$ by a power series, and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is the inverse of the strictly increasing real exponential on positive reals, so $e^{4x}>3$ is equivalent to $x>(\log 3)\mathrm{/}4$.