toqito.state_metrics.measured_relative_entropy

Measured relative entropy quantifies how well two states can be distinguished by measuring individual copies.

Module Contents

toqito.state_metrics.measured_relative_entropy.measured_relative_entropy(rho, sigma, eps=1e-05)

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

Given a quantum state (rho) and a positive semi-definite operator (sigma), the measured relative entropy is defined by optimizing the relative entropy over all possible measurements:

[

D^M(rho | sigma) := sup_{mathcal{X}, (Lambda_x)_{x in mathcal{X}}} sum_{x in mathcal{X}} operatorname{Tr}[Lambda_x rho] ln left( frac{operatorname{Tr}[Lambda_x rho]}{operatorname{Tr}[Lambda_x sigma]} right),

]

where the supremum is over every finite alphabet (mathcal{X}) and every positive-operator valued measure (POVM) ((Lambda_x)_{x in mathcal{X}}) (i.e., satisfying (Lambda_x geq 0) for all (x in mathcal{X}) and (sum_{x in mathcal{X}}Lambda_x = I)).

When (rho) and (sigma) are (d times d) matrices, the quantity (D^M(rho | sigma)) 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 times 2d). Specifically, there exist (m, k in mathbb{N}) such that (m+k = O(sqrt{ln(1/varepsilon)})) and the following inequality holds:

[

|D^M(\rho \| \sigma) - D_{m,k}^M(\rho \| \sigma)| leq varepsilon,

]

where

[

D_{m,k}^M(rho | sigma) := mathop{sup}limits_{substack{

omega > 0,; theta in mathbb{H},\ T_1,dots,T_m in mathbb{H},\ Z_0,dots,Z_k in mathbb{H}}}

left{ begin{array}{c} operatorname{Tr}[theta rho] - operatorname{Tr}[omega sigma] + 1 : \[6pt] Z_0 = omega, qquad sum_{j=1}^m w_j T_j = 2^{-k} theta, \[6pt] left{begin{bmatrix} Z_i & Z_{i+1}\ Z_{i+1} & I end{bmatrix} ge 0 right}_{i=0}^{k-1}, \[10pt] left{begin{bmatrix} Z_k - I - T_j & -sqrt{t_j}T_j \ -sqrt{t_j}T_j & I - t_jT_j end{bmatrix} ge 0 right}_{j=1}^{m} end{array} 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

Consider the following quantum state (rho = frac{1}{2}(I + r cdot mathbf{sigma})) and the PSD operator (sigma = frac{1}{2}(I + s cdot mathbf{sigma})), where (r = (0.9, 0.05, -0.02)), (s = (-0.8, 0.1, 0.1)), and (mathbf{sigma} = (sigma_x, sigma_y, sigma_z)) are the Pauli operators.

Calculating the measured relative entropy can be done as follows.

```python exec=”1” source=”above” from toqito.matrices import pauli from toqito.state_metrics import measured_relative_entropy import numpy as np

r = np.array([0.9, 0.05, -0.02]) s = np.array([-0.8, 0.1, 0.1]) rho = 0.5 * (pauli(“I”) + r[0] * pauli(“X”) + r[1] * pauli(“Y”) + r[2] * pauli(“Z”)) sigma = 0.5 * (pauli(“I”) + s[0] * pauli(“X”) + s[1] * pauli(“Y”) + s[2] * pauli(“Z”)) print(measured_relative_entropy(rho, sigma, 1e-5)) ```

Raises:

ValueError – If rho if not a density operator or if sigma is not positive semi-definite.

Parameters:

• rho (numpy.ndarray) – Density operator.

• sigma (numpy.ndarray) – Positive semi-definite operator.

• eps (float) – Tolerance level.

Returns:

The measured relative entropy between rho and sigma.

Return type:

float