qics.cones.SandRenyiEntr¶

class qics.cones.SandRenyiEntr(n, alpha, iscomplex=False)[source]¶

A class representing the epigraph of the (homogenized) sandwiched Renyi entropy, i.e., for some \(\alpha\in[1/2, 1)\),

\[\mathcal{SRE}_{n} = \text{cl}\{ (t,u,X,Y) \in \mathbb{R} \times \mathbb{R}_{++} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq u \hat{D}_\alpha(u^{-1}X \| u^{-1}Y) \},\]

where

\[\hat{D}_\alpha(X \| Y) = \frac{1}{\alpha-1} \log(\text{tr}[ ( Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}} )^\alpha ]),\]

is the sandwiched \(\alpha\)-Renyi divergence.

Parameters:

nint: Dimension of the matrices \(X\) and \(Y\).
alphafloat: The exponent \(\alpha\) used to parameterize the sandwiched Renyi entropy.
iscomplexbool: Whether the matrices \(X\) and \(Y\) are defined over \(\mathbb{H}^n\) (True), or restricted to \(\mathbb{S}^n\) (False). The default is False.