Reduced Density Matrices

A reduced density matrix is the complete local description of a subsystem, obtained by tracing out the rest.

The local state

When a system is divided into parts, each part has its own reduced density matrix found by partial trace: rho_A = Tr_B(rho_AB). This operator contains everything measurable within A alone. If the global state is a product |a>|b>, then rho_A = |a>

Purity as an entanglement gauge

For a bipartite pure state, both reduced density matrices share the same nonzero eigenvalues (the squared Schmidt coefficients). Their common von Neumann entropy S(rho_A) = S(rho_B) is the entanglement entropy. A product state gives pure reduced states with zero entropy; a maximally entangled state gives maximally mixed reduced states with maximal entropy. Thus a purely local measurement of mixedness reports a global property, correlation.

import numpy as np
def reduced(psi, dimA, dimB, keep='A'):
    m = psi.reshape(dimA, dimB)
    return m @ m.conj().T if keep=='A' else m.conj().T @ m
psi = np.array([1,0,0,1])/np.sqrt(2)
print(np.round(reduced(psi,2,2),3))   # I/2

What reduced states forget

Reduced density matrices discard correlations between subsystems. From rho_A and rho_B alone you cannot reconstruct rho_AB, because the same marginals are consistent with many different joint states, including product, classically correlated, and entangled ones. Recovering the joint state requires measuring correlations, i.e. tomography of the whole.

Uses in computation and simulation

Reduced density matrices are central to quantum simulation of materials and molecules, where local observables and few-body correlation functions are the quantities of interest. In tensor-network methods the spectrum of a reduced density matrix (the entanglement spectrum) controls how much bond dimension is needed to represent a state faithfully. Rapid growth of reduced-state entropy under evolution is what makes classical simulation of some quantum dynamics intractable.