Quantum Mutual Information

Quantum mutual information measures the total correlation, classical and quantum, between two subsystems.

Total correlation in one number

The quantum mutual information of a bipartite state is I(A:B) = S(rho_A) + S(rho_B) - S(rho_AB), where S is the von Neumann entropy. It generalizes the classical mutual information and quantifies how much knowing one subsystem tells you about the other. It is nonnegative and vanishes exactly for product states rho_AB = rho_A tensor rho_B, which carry no correlation at all.

Classical and quantum together

Unlike entanglement entropy, which measures only quantum correlation and only for pure states, mutual information captures all correlations in any state: classical correlation, discord, and entanglement combined. For a pure global state it equals twice the entanglement entropy. For mixed states it exceeds the entanglement, since it also counts classical correlations, and the split between the two is precisely what discord tries to isolate.

import numpy as np
def vN(rho):
    ev=np.linalg.eigvalsh(rho); ev=ev[ev>1e-12]
    return -np.sum(ev*np.log2(ev))
def mutual_info(rhoAB,rhoA,rhoB):
    return vN(rhoA)+vN(rhoB)-vN(rhoAB)
bell=np.array([1,0,0,1])/np.sqrt(2)
rho=np.outer(bell,bell)
print(mutual_info(rho, 0.5*np.eye(2), 0.5*np.eye(2)))  # 2 bits

Bounds and monotonicity

Mutual information is bounded by 2 log d for a d-dimensional subsystem, twice the classical maximum, and this quantum enhancement is achieved by maximally entangled states. It obeys the data-processing inequality: local operations on either side cannot increase it. Strong subadditivity of the von Neumann entropy, one of the deepest results in the field, guarantees mutual information behaves consistently when subsystems are grouped or partitioned.

Uses

Quantum mutual information sets channel capacities: the quantum channel capacity and the entanglement-assisted capacity are expressed through it. In many-body physics it detects correlations and phase transitions and, unlike entanglement entropy, remains meaningful at finite temperature where states are mixed. It is also the starting point for defining conditional entropy and discord, making it a central quantity in quantum information theory.