Quantum Foundations

Density Matrices

A density matrix is a positive operator of trace one that describes any quantum state, pure or mixed, including classical uncertainty.

Beyond the state vector

A state vector describes a perfectly known, isolated system. When there is classical uncertainty about which state was prepared, or when a subsystem is entangled with an inaccessible environment, the vector is not enough. The density matrix rho handles both cases uniformly.

Definition

Kronos motion — classical vs quantum

For a pure state |psi>, the density matrix is the outer product rho = |psi> occurs with probability p_i, it is rho = sum_i p_i |psi_i>

Defining properties

Hermitian: rho = rho-dagger
Positive semidefinite: all eigenvalues >= 0
Unit trace: Tr(rho) = 1 (probabilities sum to one)

Any matrix with these three properties is a valid quantum state. The diagonal entries are outcome probabilities in that basis; the off-diagonal entries, called coherences, encode superposition and vanish as a state decoheres.

Pure versus mixed

A state is pure exactly when Tr(rho^2) = 1, equivalently when rho has a single nonzero eigenvalue equal to one. If Tr(rho^2) < 1 the state is mixed. The quantity Tr(rho^2), called purity, ranges from 1 (pure) down to 1/d for the maximally mixed state of dimension d.

Dynamics and measurement

Under a unitary U the density matrix evolves as rho -> U rho U-dagger. The Born rule becomes prob(i) = Tr(P_i rho). Expectation of an observable A is Tr(A rho). These formulas reduce to the state-vector versions for pure states but also cover noise, making the density matrix the standard language for real hardware where noise channels act.

python

import numpy as np
plus = np.array([1,1])/np.sqrt(2)
rho = np.outer(plus, plus.conj())
print(np.trace(rho@rho).real)   # 1.0 -> pure