toqito.channel_metrics.channel_measured_relative_entropy
========================================================

.. py:module:: toqito.channel_metrics.channel_measured_relative_entropy

.. autoapi-nested-parse::

   Measured relative entropy (channel) is how well two channels can be distinguished by measuring them individually.





Module Contents
---------------

.. py:function:: channel_measured_relative_entropy(channel_1, channel_2, in_dim, m, k, hamiltonian, energy)

   Compute the measured relative entropy of two quantum channels [@Huang_2025_Msrd_Rel_Entr].

   Given a quantum channel $\mathcal{N}_{A \to B}$, a completely positive map $\mathcal{M}_{A \to B}$,
   a Hamiltonian $H_A$ (Hermitian operator acting on system $A$), and an energy constraint
   $E \in \mathbb{R}$, the energy-constrained measured relative entropy of channels is defined as:

   $$
       D^{M}_{H,E}(\mathcal{N}\Vert\mathcal{M}) :=
       \sup_{\substack{d_{R'} \in \mathbb{N},\\ \rho_{R'A} \in \mathcal{D}(\mathcal{H}_{R'A})}}
       \left\{D^{M}\!\left(\mathcal{N}_{A \to B}(\rho_{R'A}) \middle\Vert \mathcal{M}_{A \to B}(\rho_{R'A})\right):
       \operatorname{Tr}[H_A \rho_A] \le E\right\}.
   $$

   When their Choi operators $\Gamma^{\mathcal{N}}$ and $\Gamma^{\mathcal{M}}$ are
   $d_A d_B \times d_A d_B$ matrices, the quantity $D^{M}_{H,E}(\mathcal{N}\Vert\mathcal{M})$ can be
   efficiently calculated by means of a semi-definite program up to an additive error $\varepsilon$,
   by means of $O(\sqrt{\ln(1/\varepsilon)})$ linear matrix inequalities, each of size
   $2d_A d_B \times 2d_A d_B$. Specifically, there exist $m, k \in \mathbb{N}$ such that
   $m + k = O(\sqrt{\ln(1/\varepsilon)})$ and the following inequality holds:

   $$
       \left|D^{M}_{H,E}(\mathcal{N}\Vert\mathcal{M})
       - D^{M}_{H,E,m,k}(\mathcal{N}\Vert\mathcal{M})\right| \le \varepsilon,
   $$

   where

   $$
       D_{H,E,m,k}^{M}(\mathcal{N} \| \mathcal{M}) :=
       \sup\limits_{\substack{\Omega, \rho > 0, \Theta \in \mathbb{H}, \\
       T_1, \dots, T_m \in \mathbb{H}, \\ Z_0, \dots, Z_k \in \mathbb{H}}}
       \left\{ \begin{gathered}
       \operatorname{Tr}[\Theta \Gamma^{\mathcal{N}}]
       - \operatorname{Tr}[\Omega \Gamma^{\mathcal{M}}] + 1 : \\
       \operatorname{Tr}[\rho] = 1, \operatorname{Tr}[H\rho] \leq E, \\
       Z_0 = \Omega, \sum_{j=1}^m w_j T_j = 2^{-k} \Theta, \\
       \left\{ \begin{bmatrix} Z_i & Z_{i+1} \\ Z_{i+1} & \rho \otimes I \end{bmatrix}
       \geq 0 \right\}_{i=0}^{k-1}, \\
       \left\{ \begin{bmatrix} Z_k - \rho \otimes I - T_j & -\sqrt{t_j} T_j \\
       -\sqrt{t_j} T_j & \rho \otimes I - t_j T_j \end{bmatrix} \geq 0 \right\}_{j=1}^m
       \end{gathered} \right\}
   $$

   and, for all $j \in \{1, \dots, m\}$, $w_j$ and $t_j$
   are the weights and nodes, respectively, for the $m$-point Gauss--Legendre quadrature
   on the interval $[0, 1]$.

   .. rubric:: Examples

   We can find the measured relative entropy between a depolarizing channel of dimension 2
   and the identity channel, constrained by a Hamiltonian and energy, as follows:

   ```python exec="1" source="above"
   from toqito.channel_metrics import channel_measured_relative_entropy
   from toqito.channels import depolarizing
   import numpy as np
   channel_1 = depolarizing(2, 0.2)
   channel_2 = np.eye(4)
   in_dim = 2
   m = 5
   k = 5
   hamiltonian = np.zeros((2, 2))
   energy = 100
   print(channel_measured_relative_entropy(channel_1, channel_2, in_dim, m, k, hamiltonian, energy))
   ```

   :raises ValueError: If `channel_1` is not a quantum channel or `channel_2` is not completely positive.

   :param channel_1: Choi matrix for the first channel.
   :param channel_2: Choi matrix for the second channel.
   :param in_dim: The dimension of the input of the quantum channels.
   :param m: One of the optimization parameters.
   :param k: The other optimization parameter.
   :param hamiltonian: The Hamiltonian.
   :param energy: The energy constraint.

   :returns: The measured relative entropy between `channel_1` and `channel_2`.