Quantum Information Theory

Von Neumann Entropy

The von Neumann entropy generalizes Shannon entropy to quantum systems:

where are eigenvalues of . Key properties:

• with equality iff is pure
• for a -dimensional system
• is invariant under unitary transformations:

For a bipartite pure state , the entanglement entropy is .

Quantum Relative Entropy

The quantum relative entropy measures distinguishability between states:

Properties:

• (Gibbs inequality)
• iff
• Monotonicity under quantum channels:

The relative entropy is central to quantum hypothesis testing and quantum Stein's lemma.

Holevo Bound

The Holevo bound limits the classical information extractable from a quantum state:

where is the Holevo information. Even though a -dimensional quantum system can carry qubits, at most classical bits can be extracted by measurement.

This bound is achievable in the asymptotic limit using many copies of the state.

Quantum Channel Capacity

The quantum capacity of a channel is:

where is the coherent information:

A remarkable result: the quantum capacity can be superadditive.

Resource Theory of Entanglement

In the resource theory of entanglement:

• Free states: separable states
• Free operations: Local Operations and Classical Communication (LOCC)

Entanglement measures quantify the resource:

• Entanglement of formation:
• Squashed entanglement:
• Negativity:

The monogamy of entanglement constrains how entanglement can be distributed.

Quantum Data Compression

Schumacher compression is the quantum analogue of Shannon compression. For a source producing states :

The source can be compressed to qubits per symbol, and this is optimal.

The quantum Shannon theorem states that for a source with entropy rate , there exist encoders and decoders that achieve arbitrarily low error rate using qubits per source symbol.

Python: Entropy Calculations

import numpy as np

def von_neumann_entropy(rho):
    # Compute von Neumann entropy of density matrix.
    eigenvalues = np.linalg.eigvalsh(rho)
    eigenvalues = eigenvalues[eigenvalues > 1e-12]
    return -np.sum(eigenvalues * np.log2(eigenvalues))

def quantum_relative_entropy(rho, sigma):
    # Compute D(rho || sigma).
    evals_rho, evecs_rho = np.linalg.eigh(rho)
    evals_rho = evals_rho[evals_rho > 1e-12]
    evecs_rho = evecs_rho[:, :len(evals_rho)]
    sigma_eig = evecs_rho.conj().T @ sigma @ evecs_rho
    sigma_diag = np.diag(sigma_eig).real
    sigma_diag = sigma_diag[sigma_diag > 1e-12]
    D = np.sum(evals_rho * (np.log2(evals_rho) - np.log2(sigma_diag[:len(evals_rho)])))
    return D

bell = np.array([1,0,0,1], dtype=complex) / np.sqrt(2)
rho_bell = np.outer(bell, bell.conj())
print(f"Bell state entropy: {von_neumann_entropy(rho_bell):.4f}")

rho_A = np.array([[0.5, 0], [0, 0.5]], dtype=complex)
print(f"Reduced state entropy: {von_neumann_entropy(rho_A):.4f}")

This computes von Neumann entropy and quantum relative entropy for key quantum states.

Entropy Inequalities

Key inequalities for von Neumann entropy:

1. Subadditivity:
2. Araki-Lieb:
3. Strong subadditivity:
4. Monotonicity: for pure states