Quantum Cookbook

Part II · Chapter 11

Kraus Operators

Read ~20 min Prerequisites Ch. 4, Ch. 8–10 Tools channel applicator, code cell Live
Basic

Ch. 4's evolution postulate said a closed system evolves by a single unitary U. But no real qubit is ever truly closed — it's always coupled, at least a little, to an environment you don't control. Kraus operators are the general tool for describing what evolution looks like once you stop pretending the environment isn't there.

The general quantum operation

A quantum channel (or quantum operation) maps an input ρ to an output ρ′ via a set of matrices {Kᵢ}, called Kraus operators:

ρ′ = Σᵢ Kᵢ ρ Kᵢ†

For this to be a legitimate, probability-preserving operation, the Kᵢ must satisfy the completeness relation:

Σᵢ Kᵢ†Kᵢ = I

Fig. 11.1 — where Kraus operators come fromsystem + environment, then forget the environment
qubit ρ environment U trace out environment (Ch. 10)
This is the real origin story: let the qubit and its environment interact via one big unitary U (Ch. 4 still applies — to the whole system), then use Ch. 10's partial trace to throw away the environment. What's left over, from the qubit's point of view, is exactly the Kraus-operator form above — the Kᵢ are literally pieces of U.
Intermediate
Fig. 11.2 — pick a channel, watch it act on |+⟩checks completeness live

Completeness: —

Input ρ (always |+⟩)

Output ρ

Bit-flip has 2 Kraus operators; depolarizing has 4. All four presets pass completeness — try the math in Fig. 11.3's code cell with your own (possibly broken) Kraus set to see what failing the check actually looks like.

Special cases you already know

  • One Kraus operator, K₀=U: the completeness relation becomes U†U=I — exactly Ch. 4's unitary evolution. Closed-system evolution is the special case of a quantum channel with no actual noise.
  • Projective measurement (Ch. 7): the Kraus operators are the projectors |φᵢ⟩⟨φᵢ|, and ΣKᵢ†Kᵢ=Σ|φᵢ⟩⟨φᵢ|=I is exactly the completeness of an orthonormal basis from Ch. 2.
Advanced

The same channel, different Kraus operators

Just like Ch. 8's "different ensemble, same ρ" surprise, a single channel can be written with more than one valid set of Kraus operators — any set related by a unitary "mixing" of the original Kᵢ describes the identical physical channel. The completeness relation is the only hard requirement; everything else is a choice of description. This is why you'll sometimes see the same depolarizing channel written with 4 operators in one paper and a different-looking 4 in another — both are correct, and both pass the same completeness check Fig. 11.2 runs.

Fig. 11.3 — code, runs in your browser
The "broken" example is deliberate — it's worth seeing what a failed completeness check actually looks like numerically, not just being told the rule exists.
Check your understanding

What's the completeness relation Kraus operators must satisfy, and what does it guarantee?

ΣᵢKᵢ†Kᵢ = I. It guarantees the channel preserves trace — i.e. it takes a valid (Tr=1) density matrix to another valid density matrix, so probabilities still add up to 1 after the channel acts.

Check your understanding

What does a Kraus-operator channel with exactly one operator, K₀=U, reduce to?

Ordinary unitary evolution from Ch. 4 — ρ′ = UρU†, with the completeness relation becoming U†U=I, exactly the unitarity condition.

Check your understanding

Conceptually, where do Kraus operators actually come from?

Let the system interact with its environment via one big unitary (Ch. 4 applies to the combined system), then use the partial trace (Ch. 10) to discard the environment. The Kraus operators are the pieces of that unitary left behind from the system's point of view.

Check your understanding

In Fig. 11.2, why does the bit-flip channel need exactly the factors √(1-p) and √p, rather than just 1-p and p?

Because completeness requires K₀†K₀ + K₁†K₁ = I, and K†K involves squaring the scalar factor. (1-p)I + pX†X = (1-p)I+pI = I works out, but only if the scalars in front of K₀ and K₁ are the square roots √(1-p) and √p in the first place.

Check your understanding

How do projective measurements (Ch. 7) fit into the Kraus-operator framework?

They're a special case where the Kraus operators are the projectors |φᵢ⟩⟨φᵢ| onto each possible outcome — completeness ΣKᵢ†Kᵢ=I is then exactly the orthonormal-basis completeness relation from Ch. 2.

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