qics.cones.QuasiEntr

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

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

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

when \(\alpha\in[0, 1]\), and

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

when \(\alpha\in[-1, 0] \cup [1, 2]\).

Parameters:

nint

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

alphafloat

The exponent \(\alpha\) used to parameterize the 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

RenyiEntr

Renyi entropy

SandQuasiEntr

Sandwiched quasi-relative entropy

QuantRelEntr

Quantum relative entropy

Notes

The Renyi entropy is defined as the function

\[D_\alpha(X \| Y) = \frac{1}{\alpha - 1} \log(\Psi_\alpha(X, Y)),\]

where \(\Psi_\alpha\) is the quasi-relative entropy defined as

\[\Psi_\alpha(X, Y) = \text{tr}[ X^\alpha Y^{1-\alpha} ].\]

Note that \(\Psi_\alpha\) is jointly concave for \(\alpha\in[1/2, 1]\), and jointly convex for \(\alpha\in[-1, 0] \cup [1, 2]\), whereas \(D_\alpha\) is jointly convex for \(\alpha\in[0, 1)\), but is neither convex nor concave for \(\alpha\in[-1, 0) \cup (1, 2]\).

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

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

if \(\alpha\in[0, 1)\), and

\[\min_{(X,Y)\in\mathcal{C}} D_\alpha(X \| Y) = \frac{1}{\alpha - 1} \log\left( \min_{(X,Y)\in\mathcal{C}} \Psi_\alpha(X, Y) \right),\]

if \(\alpha\in(1, 2]\). Similarly, we can maximize the Renyi entropy by using the identities

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

for \(\alpha\in[-1, 0]\).