toqito.channel_metrics.channel_measured_relative_entropy

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

Module Contents

toqito.channel_metrics.channel_measured_relative_entropy.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}Vertmathcal{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}) middleVert mathcal{M}_{A to B}(rho_{R’A})right): operatorname{Tr}[H_A rho_A] le Eright}.

$$

When their Choi operators $Gamma^{mathcal{N}}$ and $Gamma^{mathcal{M}}$ are $d_A d_B times d_A d_B$ matrices, the quantity $D^{M}_{H,E}(mathcal{N}Vertmathcal{M})$ 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}Vertmathcal{M}) - D^{M}_{H,E,m,k}(mathcal{N}Vertmathcal{M})right| le varepsilon,

$$

where

$$

D_{H,E,m,k}^{M}(mathcal{N} | mathcal{M}) := suplimits_{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}[Hrho] 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 $[0, 1]$.

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.

Parameters:

• channel_1 (numpy.ndarray) – Choi matrix for the first channel.

• channel_2 (numpy.ndarray) – Choi matrix for the second channel.

• in_dim (int) – The dimension of the input of the quantum channels.

• m (int) – One of the optimization parameters.

• k (int) – The other optimization parameter.

• hamiltonian (numpy.ndarray) – The Hamiltonian.

• energy (float) – The energy constraint.

Returns:

The measured relative entropy between channel_1 and channel_2.

Return type:

float