Kraus Operators

Kraus operators give a concrete, computable representation of any quantum channel as a sum of one-sided products.

The operator-sum representation

Any quantum channel can be written E(rho) = sum_k K_k rho K_k-dagger, where the Kraus operators K_k satisfy sum_k K_k-dagger K_k = I. This operator-sum form makes channels easy to apply, compose, and simulate. The completeness relation encodes trace preservation, and the one-sided sandwich form guarantees complete positivity automatically.

Interpretation

Each K_k can be read as one possible thing the environment could learn or do. The term K_k rho K_k-dagger is the unnormalized state given that branch k occurred, and Tr(K_k rho K_k-dagger) is its probability. Summing over k discards knowledge of which branch happened, producing the average channel output. This connects Kraus operators directly to generalized measurements whose outcomes are forgotten.

import numpy as np
def apply(Ks, rho):
    return sum(K@rho@K.conj().T for K in Ks)
g = 0.2
K0 = np.array([[1,0],[0,np.sqrt(1-g)]])
K1 = np.array([[0,np.sqrt(g)],[0,0]])
rho = np.array([[0.5,0.5],[0.5,0.5]])
print(np.round(apply([K0,K1], rho),3))  # amplitude damping

Non-uniqueness

A channel does not have a unique Kraus set. Two sets {K_k} and {L_j} describe the same channel if and only if they are related by an isometry, L_j = sum_k V_{jk} K_k with V-dagger V = I. This freedom mirrors the freedom in purifications and lets one choose a minimal set, whose size equals the Kraus rank of the channel, at most the square of the Hilbert-space dimension.

Uses

Kraus operators are the standard input to density-matrix simulators, where noise is applied gate by gate. They express canonical noise models compactly: amplitude damping needs two, depolarizing can use four Pauli terms, dephasing two. From the Kraus set one can build the Choi matrix, extract error rates, and reason about how errors accumulate, making them the working representation of noise in quantum computing.