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For two $d$-dimensional systems, $d\ge 2$, an isotropic quantum state has the formThe eigenvalue on the maximally entangled state is $[1+(d^2-1)p]/d^2$; all orthogonal directions have eigenvalue $(1-p)/d^2$. Thus positive semidefiniteness requiresNegative $p$ can therefore describe a valid density matrix; the interval $0\le p\le 1$ is only its convex-mixture subfamily.

The defining symmetry is invariance under $U\otimes \overline{U}$ for every unitary operator $U$. It differs from the $U\otimes U$ invariance of a Werner state. The overlap $F=⟨{\Phi }^{+}|{\rho }_p|{\Phi }^{+}⟩$ equals $[1+(d^2-1)p]/d^2$.

Within the physical range, the state is a separable quantum state exactly when $p\le 1/(d+1)$, equivalently $F\le 1/d$. Indeed, partial transpose sends $\Phi$ to the swap operator divided by $d$; on its antisymmetric subspace the resulting eigenvalue is $[1-(d+1)p]/d^2$. Above the threshold it is negative, proving entanglement. Below it, an isotropic twirling channel averages product states using correlated local unitary operators and Haar measure. Product-state overlaps cover $0\le F\le 1/d$, so these averages give separable quantum states throughout that interval.^[ref]

For $d=2$, the valid range is $-1/3\le p\le 1$, and entanglement begins above $p=1/3$. The point $p=1$ is a Bell state; $p=0$ is maximally mixed.

The isotropic twirling channel averages a bipartite density matrix over matched local unitaries:where $dU$ is normalized Haar measure on the $d$-dimensional unitary group. Complex conjugation is taken in the basis defining $|{\Phi }_d⟩=d^{-1/2}{\sum }_j|jj⟩$.

Writing $P=|{\Phi }_d⟩⟨{\Phi }_d|$ and $F=\mathrm{Tr}(P\rho )$, the result is $FP+(1-F)(I-P)/(d^2-1)$ for $d\ge 2$. The averaging preserves $F$, because $U\otimes \overline{U}$ fixes $|{\Phi }_d⟩$, and erases distinctions within its orthogonal complement. Shared classical randomness and local unitaries implement the operation; the resulting isotropic quantum state need not be separable.