Density Matrices

The density operator generalizes the state vector to describe statistical mixtures, subsystems, and open-system dynamics.

Beyond the state vector

A pure state |psi> captures maximal knowledge, but we often face a probabilistic ensemble of states or a subsystem of a larger entangled whole. The density matrix rho handles both. For an ensemble that yields |psi_i> with probability p_i, rho = sum_i p_i |psi_i>

Defining properties

• rho = rho-dagger (Hermitian, real eigenvalues)
• rho >= 0 (all eigenvalues nonnegative, so probabilities are valid)
• Tr(rho) = 1 (normalization)
• Tr(rho^2) <= 1, with equality iff the state is pure

Everything observable is a trace

The Born rule takes the uniform form p(a) = Tr(P_a rho) and expectation values are = Tr(A rho). Time evolution of a closed system is rho -> U rho U-dagger. Because these are all trace expressions, the density matrix is the natural language whenever classical uncertainty and quantum superposition coexist.

import numpy as np
plus = np.array([1,1])/np.sqrt(2)
rho_pure = np.outer(plus, plus.conj())
rho_mix  = 0.5*np.eye(2)            # maximally mixed
print(np.trace(rho_pure@rho_pure))  # 1.0 pure
print(np.trace(rho_mix@rho_mix))    # 0.5 mixed

There are two routes to a mixed rho: classical ignorance about which pure state was prepared, and tracing out an entangled partner. Remarkably, these are operationally indistinguishable from the local point of view. A subsystem of an entangled pure state is described by a proper mixed state even though the global state is pure, a fact formalized by purification and the partial trace.