vprusso
Implement the conditional Rényi entropies
Note: Unless otherwise specified, all the definitions and properties are taken from 1504.00233.
Is your feature request related to a problem? Please describe. Conditional Rényi entropies gained a lot of traction in quantum cryptography recently. Toqito could benefit from having them implemented.
There are two "main" types of conditional Rényi entropies: the Petz ones and the sandwiched ones, each coming in two flavours.
The Petz-Rényi conditional entropies are defined via the Petz-Rényi divergence, which is defined by, for $\alpha\in(0, 1)\cup(1,+\infty)$ and $\rho$ and $\sigma$ being PSD with $\rho\neq0$ as
$$\overline{D}_{\alpha}(\rho\parallel\sigma)=\frac{1}{\alpha-1}\log_2\left(\frac{\mathrm{Tr}\left[\rho^\alpha\sigma^{1-\alpha}\right]}{\mathrm{Tr}[\rho]}\right)$$
if we either have $\alpha<1\wedge\rho\not\perp\sigma$ or $\ker(\sigma)\subseteq\ker(\rho)$, and is defined as $+\infty$ if neither of these conditions are satisfied (Equation 4.86).
We can then define the "downarrow" Petz-Rényi conditional entropy of a PSD bipartite state $\rho_{AB}$ as
$$\overline{H}^{\downarrow}_{\alpha}(A|B)_{\rho_{AB}}=-\overline{D}_{\alpha}\left(\rho_{AB}\parallel I_A\otimes\rho_B\right)$$
(Equation 5.16) while the uparrow version is defined as
$$\overline{H}^{\uparrow}_{\alpha}(A|B)_{\rho_{AB}}=\sup_{\substack{\sigma_B\succeq0\\\mathrm{Tr}[\sigma]=1}}-\overline{D}_{\alpha}\left(\rho_{AB}\parallel I_A\otimes\sigma_B\right)$$
(Equation 5.17). As it turns out, this optimization has a closed-form which will be handy for computing it, namely
$$\overline{H}^{\uparrow}_{\alpha}(A|B)_{\rho_{AB}}=\frac{\alpha}{1-\alpha}\log_2\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_A\left[\rho_{AB}^{\alpha}\right]\right)^{\frac{1}{\alpha}}\right]\right)$$
(Equation 5.28). The sandwiched version is also defined via a divergence, namely we have for $\alpha\in(0, 1)\cup(1,+\infty)$ and $\rho$ and $\sigma$ being PSD with $\rho\neq0$ as
$$\tilde{D}_{\alpha}(\rho\parallel\sigma)=\frac{1}{\alpha-1}\log_2\left(\frac{\mathrm{Tr}\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right]}{\mathrm{Tr}[\rho]}\right)$$
if we either have $\alpha<1\wedge\rho\not\perp\sigma$ or $\ker(\sigma)\subseteq\ker(\rho)$, and is defined as $+\infty$ if neither of these conditions are satisfied (Equations 4.44 and 4.45). The downarrow version is defined as
$$\tilde{H}^{\downarrow}_{\alpha}(A|B)_{\rho_{AB}}=-\tilde{D}_{\alpha}\left(\rho_{AB}\parallel I_A\otimes\rho_B\right)$$
while the uparrow version is defined as
$$\tilde{H}^{\uparrow}_{\alpha}(A|B)_{\rho_{AB}}=\sup_{\substack{\sigma_B\succeq0\\\mathrm{Tr}[\sigma]=1}}-\tilde{D}_{\alpha}\left(\rho_{AB}\parallel I_A\otimes\sigma_B\right)$$
(Equation 5.19). Unfortunately, no closed-form expression is known for this one, so solving the optimization problem is the way to go.
Ideally, the functions implementing these conditional entropies may also be called with $\alpha=1$, in which case it would return the conditional Von Neumann entropy. Additionally, the uparrow sandwiched conditional entropy can be generalized for $\alpha=0$ by defining
$$\tilde{H}^{\uparrow}_{0}(A|B)_{\rho_{AB}}=\sup_{\substack{\sigma_B\succeq0\\\mathrm{Tr}[\sigma]=1}}\log_2\left(\mathrm{Tr}\left[\Pi_{\rho_{AB}}\left(I_A\otimes\sigma_B\right)\right]\right)$$
where $\Pi_{\rho_{AB}}$ is the projector onto the support of $\rho_{AB}$ (1311.3887, Equation 22). Finally, both the up and downarrow of the sandwiched conditional entropy can be generalized to $\alpha=+\infty$, generalizing the conditional min-entropy by
$$\tilde{H}^{\downarrow}_{+\infty}(A|B)_{\rho_{AB}}=\sup\left\{\lambda\in\mathbb{R}\middle|2^{-\lambda}I_A\otimes\rho_B\succeq\rho_{AB}\right\}$$
$$\tilde{H}^{\uparrow}_{+\infty}(A|B)_{\rho_{AB}}=\sup_{\substack{\sigma_B\succeq0\\\mathrm{Tr}[\sigma]=1}}\left\{\lambda\in\mathbb{R}\middle|2^{-\lambda}I_A\otimes\sigma_B\succeq\rho_{AB}\right\}$$
I don't know how much these edge cases are used in the litterature, but implementing these conditional entropies (and in particular the uparrow sandwiched one) in most cases (like $\alpha\geqslant\frac12$ with $\alpha\neq1$) may already be interesting!
Additional functions on conditional entropies could be added. They are described in https://arxiv.org/abs/2410.21976
Replicating the comment from https://github.com/vprusso/toqito/pull/1102#issuecomment-2851602006, as this is likely relevant to this issue:
Some of the Renyi entropy stuff may be better handled by the recent work done in the qics conic quantum information solver package: https://qics.dev/examples/qrep/renyi.html