Quantum Network Methods

driada.network.quantum.renyi_divergence(A, B, q)[source]

Calculate the quantum Rényi divergence between two density matrices.

The quantum Rényi divergence generalizes the quantum relative entropy (Kullback-Leibler divergence) and quantifies distinguishability between quantum states.

Parameters:

  • A (numpy.ndarray) – First density matrix (must be square, positive semi-definite, trace 1).

  • B (numpy.ndarray) – Second density matrix (must be same shape as A).

  • q (float) – Order parameter. Must be positive. q=1 gives quantum relative entropy.

Returns:

The quantum Rényi divergence D_q(A||B) in bits.

Return type:

float

Raises:

ValueError – If q <= 0. If A and B have different shapes or are not square matrices.

See also

js_divergence

Quantum Jensen-Shannon divergence.

manual_entropy

Shannon/von Neumann entropy calculation.

Notes

The quantum Rényi divergence is defined as: - For q = 1: D_1(ρ||σ) = Tr(ρ(log₂ ρ - log₂ σ)) - For q ≠ 1: D_q(ρ||σ) = (1/(q-1)) log₂(Tr(ρ^q σ^(1-q)))

Properties: - D_q(ρ||σ) ≥ 0 with equality iff ρ = σ - Not symmetric: D_q(ρ||σ) ≠ D_q(σ||ρ) in general - For classical (diagonal) states, reduces to classical Rényi divergence

References

Müller-Lennert, M., et al. (2013). On quantum Rényi entropies: A new generalization and some properties. J. Math. Phys. 54, 122203.

Examples

>>> rho = np.array([[0.7, 0.1], [0.1, 0.3]])
>>> sigma = np.array([[0.5, 0.0], [0.0, 0.5]])
>>> div = renyi_divergence(rho, sigma, q=0.5)
driada.network.quantum.get_density_matrix(A, t, norm=0)[source]

Compute quantum density matrix from graph adjacency matrix.

Constructs a quantum-like Gibbs state density matrix from a graph using the graph Laplacian, following De Domenico & Biamonte’s formulation.

Parameters:

  • A (numpy.ndarray) – Adjacency matrix of the graph (must be square).

  • t (float) – Inverse temperature parameter β (also interpreted as time). Controls the “quantumness” of the state. Must be positive.

  • norm (int, optional) – If 1, use normalized Laplacian. If 0, use regular Laplacian. Default is 0.

Returns:

Density matrix ρ = exp(-tL) / Z(t), where L is the Laplacian and Z(t) = Tr[exp(-tL)] is the partition function.

Return type:

numpy.ndarray

Raises:

ValueError – If A is not square or t is not positive.

See also

renyi_divergence

Uses density matrices for divergence calculation.

js_divergence

Uses density matrices for network comparison.

get_laplacian

Computes graph Laplacian.

get_norm_laplacian

Computes normalized Laplacian.

Notes

This density matrix is formally proportional to the propagator of a diffusive process on the network. The construction treats the Laplacian as a Hamiltonian in a quantum Gibbs state: ρ = (1/Z) exp(-βH), where H = L

References

De Domenico, M., & Biamonte, J. (2016). Spectral entropies as information-theoretic tools for complex network comparison. Physical Review X, 6(4), 041062.

Examples

>>> A = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]])
>>> rho = get_density_matrix(A, t=1.0)
>>> np.trace(rho)  # Trace should be 1
np.float64(1.0)
driada.network.quantum.manual_entropy(pr)[source]

Calculate Shannon entropy from probability distribution.

Computes entropy while handling numerical issues with zero and near-zero probabilities.

Parameters:

pr (numpy.ndarray) – Probability distribution. Non-negative values that should sum to 1. Can also be eigenvalues of a density matrix for von Neumann entropy.

Returns:

Shannon entropy in bits.

Return type:

float

See also

scipy.stats.entropy

Alternative implementation.

renyi_divergence

Generalized entropy measure.

js_divergence

Uses this for von Neumann entropy calculation.

Notes

The function filters out zero values and very small values (< 1e-15) to avoid numerical issues with logarithms.

For quantum states, this computes the von Neumann entropy when given eigenvalues of a density matrix: S(ρ) = -Σᵢ λᵢ log₂(λᵢ).

References

Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3), 379-423.

Examples

>>> pr = np.array([0.5, 0.5, 0.0])
>>> H = manual_entropy(pr)
>>> np.isclose(H, 1.0)  # Maximum entropy for 2 equiprobable states
np.True_
driada.network.quantum.js_divergence(A, B, t, return_partial_entropies=True)[source]

Calculate quantum Jensen-Shannon divergence between two graphs.

Computes the quantum generalization of JS divergence using von Neumann entropy of density matrices derived from graph Laplacians.

Parameters:

  • A (numpy.ndarray) – Adjacency matrix of first graph (must be square).

  • B (numpy.ndarray) – Adjacency matrix of second graph (must be same shape as A).

  • t (float) – Inverse temperature parameter β for density matrix computation. Must be positive.

  • return_partial_entropies (bool, optional) – If True, return individual entropies along with JS divergence. Default is True.

Returns:

If return_partial_entropies=False: Square root of QJSD value. If return_partial_entropies=True: tuple of (S(ρ_mix), S(ρ_A), S(ρ_B), sqrt(QJSD)).

Return type:

float or tuple

Raises:

ValueError – If A and B have different shapes or are not square matrices.

See also

renyi_divergence

Alternative quantum divergence measure.

get_density_matrix

Used to compute density matrices.

manual_entropy

Used for von Neumann entropy calculation.

Notes

The quantum Jensen-Shannon divergence is defined as: QJSD(ρ_A, ρ_B) = S((ρ_A + ρ_B)/2) - (S(ρ_A) + S(ρ_B))/2

where S(ρ) = -Tr(ρ log₂ ρ) is the von Neumann entropy.

Important: This function returns the SQUARE ROOT of QJSD, which is a proper metric on the quantum state space. To get the divergence itself, square the returned value.

The function quantifies the distinguishability between two network structures in a quantum information context. Returns 0 if calculation results in negative values due to numerical precision issues (which can occur for very similar graphs).

References

Lamberti, P., et al. (2008). Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states. Physical Review A.

Examples

>>> A = np.array([[0, 1], [1, 0]])
>>> B = np.array([[0, 1], [1, 0]])
>>> js_div = js_divergence(A, B, t=1.0, return_partial_entropies=False)
>>> js_div  # Should be 0 for identical graphs
0.0

Quantum-inspired methods for network analysis, including quantum walks and density matrix approaches.