qics.cones.SandQuasiEntr

class qics.cones.SandQuasiEntr(n, alpha, iscomplex=False)

A class representing the epigraph or hypograph of the sandwiched quasi-relative entropy, i.e.,

\[\mathcal{SQE}_{n, \alpha} = \text{cl} \{ (t, X, Y) \in \mathbb{R} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq -\text{tr}[ ( Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}} )^\alpha ] \},\]

when \(\alpha\in[1/2, 1]\), and

\[\mathcal{SQE}_{n, \alpha} = \text{cl}\{ (t, X, Y) \in \mathbb{R} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq \text{tr}[ ( Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}} )^\alpha ] \},\]

when \(\alpha\in[1, 2]\).

Parameters:

nint

Dimension of the matrices \(X\) and \(Y\).

alphafloat

The exponent \(\alpha\) used to parameterize the sandwiched quasi-relative 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.

See also

QuasiEntr

Renyi entropy

SandRenyiEntr

Sandwiched Renyi entropy

QuantRelEntr

Quantum relative entropy

Notes

The sandwiched Renyi entropy is defined as the function

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

where \(\hat{\Psi}_\alpha\) is the sandwiched quasi-relative entropy, defined as

\[\hat{\Psi}_\alpha(X, Y) = \text{tr}\!\left[ \left(Y^\frac{1-\alpha}{2\alpha} X Y^\frac{1-\alpha}{2\alpha} \right)^\alpha \right].\]

Note that \(\hat{\Psi}_\alpha\) is jointly concave for \(\alpha\in[1/2,1]\), and jointly convex for \(\alpha\in[1, 2]\), whereas \(\hat{D}_\alpha\) is jointly convex for \(\alpha\in[1/2, 1)\), but is neither convex nor concave for \(\alpha\in(1, 2]\).

Note that due to monotonicity of \(x \mapsto \log(x)\), we can minimize the sandwiched Renyi entropy by using the identities

\[\min_{(X,Y)\in\mathcal{C}} \hat{D}_\alpha(X \| Y) = \frac{1}{\alpha - 1} \log\left( \max_{(X,Y)\in\mathcal{C}} \hat{\Psi}_\alpha(X, Y) \right),\]

if \(\alpha\in[1/2, 1)\), and

\[\min_{(X,Y)\in\mathcal{C}} \hat{D}_\alpha(X \| Y) = \frac{1}{\alpha - 1} \log\left( \min_{(X,Y)\in\mathcal{C}} \hat{\Psi}_\alpha(X, Y) \right),\]

if \(\alpha\in(1, 2]\).