Theorem 4: relative Renyi entanglement of pure states

For all positive natural numbers $a,b$, every complex Euclidean vector $\psi$ on the $a\times b$ bipartite system with $\|\psi\|=1$, and every real $\alpha\in[0,2]$, put $\rho=\psi\psi^\dagger$. A separable state here is a positive semidefinite trace-one matrix admitting a finite decomposition $\sum_jw_jA_j\otimes B_j$, with $w_j\ge0$ and positive semidefinite trace-one factors $A_j,B_j$, without a separately imposed sum-of-weights condition. For positive semidefinite $X$, define $S_t(X)$ by spectral functional calculus using the scalar function that is $0$ at $0$ and $x^t$ at nonzero $x$. For density matrices $R,S$, define $D_\alpha(R\|S)\in[0,\infty]$ as follows: at $\alpha=1$, it is $+\infty$ if $\ker S\not\subseteq\ker R$, and otherwise it is $\max\{\operatorname{Re}\operatorname{tr}(R\log R-R\log S),0\}/\log2$, with $R\log R$ interpreted through the scalar function $x\log x$; at $\alpha\ne1$, it is $+\infty$ if $\alpha>1$ and $\ker S\not\subseteq\ker R$, and otherwise, writing $z=\operatorname{Re}\operatorname{tr}(S_\alpha(R)S_{1-\alpha}(S))$, it is $+\infty$ when $z\le0$ and $\max\{\log_2z/(\alpha-1),0\}$ when $z>0$. The definition assigns $+\infty$ if either argument is not a density matrix. The asserted equality is

Here the infimum is in $[0,\infty]$, and, if $\lambda_0,\ldots,\lambda_{a-1}$ are the descending eigenvalues with multiplicity of $(\sum_{j<b}\psi(i,j)\overline{\psi(i',j)})_{i,i'}$, then $H_\infty(\psi)=-\log_2\lambda_0$, $H_1(\psi)=-\sum_i\lambda_i\log_2\lambda_i$, and $H_\gamma(\psi)=\log_2(\sum_i\lambda_i^\gamma)/(1-\gamma)$ for finite $\gamma\ne0,1$. All logarithms and powers use their total real operations, including $\log 0=0$; the maximum on the right is embedded as a finite element of $[0,\infty]$.