Reduced Density Matrix and Partial Trace

The reduced density matrix describes a subsystem alone, obtained by partial-tracing out the rest; it reveals entanglement as local mixedness.

Describing part of a system

When you can access only one part of a larger quantum system, its state is given by the reduced density matrix. You obtain it from the full density matrix by the partial trace, which sums over the degrees of freedom you cannot see.

The partial trace

For a two-part system with joint state rho_AB, the reduced state of A is rho_A = Tr_B(rho_AB), tracing out B. Concretely, you take the matrix elements of B and sum the diagonal ones, collapsing B's indices while keeping A's. The result is a valid density matrix on A alone.

Entanglement becomes mixedness

Here is the key insight. If A and B are entangled, then rho_A is a mixed state even though the whole rho_AB is pure. For the Bell state (|00>+|11>)/sqrt(2), tracing out one qubit leaves the other in the maximally mixed state I/2 — no local information at all. Local mixedness is the fingerprint of entanglement.

python

import numpy as np
bell=np.array([1,0,0,1])/np.sqrt(2)
rho=np.outer(bell,bell).reshape(2,2,2,2)
rhoA=np.trace(rho,axis1=1,axis2=3)  # trace out qubit B
print(np.round(rhoA,3))  # [[0.5 0],[0 0.5]] maximally mixed

Why it is the right tool

The reduced density matrix gives correct predictions for any measurement confined to the subsystem, and it is the only object that does — no state vector for A alone exists when A is entangled. This makes it essential for open systems, decoherence (the environment is traced out), and quantifying entanglement.

Measuring entanglement

The mixedness of rho_A, quantified by its von Neumann entropy, measures how entangled A is with B. Zero entropy means a product state; maximal entropy means a maximally entangled pair like a Bell state.