Closed quantum systems evolve unitarily. Real quantum systems interact with environments, lose coherence, exchange energy and undergo noise. Quantum channels are the mathematical language for those more general transformations.
Guide 5 replaced pure state vectors with density matrices so that mixed states and subsystems could be described correctly. This guide now asks what kinds of maps can act on those density matrices while still producing valid quantum states.
Closed system: ρ → UρU†. Open system: ρ → 𝓔(ρ), where 𝓔 is a physically valid quantum channel.
1. Why unitary evolution is not enough
A unitary operator preserves the eigenvalues of a density matrix. Therefore it preserves purity and entropy. But real processes can reduce purity, erase phase coherence or relax an excited state toward a lower-energy state. None of those effects can be produced on an isolated system by unitary conjugation alone.
The resolution is not to abandon unitary quantum mechanics. Instead, enlarge the system to include an environment, evolve the total system unitarily, and then ignore the environment by taking a partial trace. The effective transformation on the subsystem is generally non-unitary.
2. Quantum channels as linear maps on density matrices
A quantum channel is a linear map 𝓔 that takes density operators to density operators while remaining physically consistent even when the system is entangled with an external reference.
The standard finite-dimensional requirement is that 𝓔 be completely positive and trace preserving, abbreviated CPTP.
- Linear: statistical mixtures transform consistently.
- Positive: positive operators remain positive.
- Completely positive: positivity is preserved even after adjoining an arbitrary untouched reference system.
- Trace preserving: unit trace remains unit trace.
Complete positivity is stronger than ordinary positivity. The extra strength is necessary because a local operation must remain valid when its input is part of an entangled joint state.
3. Kraus representation
Every finite-dimensional quantum channel can be written in operator-sum form
𝓔(ρ)=Σ_k K_k ρ K_k†,
where the Kraus operators satisfy
Σ_k K_k†K_k = I.
The completeness relation guarantees trace preservation. Indeed, using cyclicity of trace,
Tr(𝓔(ρ))=Σ_k Tr(K_kρK_k†)=Tr(ρΣ_kK_k†K_k)=Tr(ρ)=1.
4. Kraus representations are not unique
A physical channel is the map 𝓔, not one particular list of Kraus operators. Different Kraus sets can represent the same channel. A unitary mixing of one Kraus list can produce another list with identical action on every density matrix.
This resembles the non-uniqueness of ensemble decompositions for mixed states. The operational object is the channel itself.
5. Unitary evolution is a one-Kraus channel
Take one Kraus operator K₁=U, where U is unitary. Then
𝓔(ρ)=UρU†
and U†U=I. Thus ordinary closed-system evolution is a special case of the channel framework.
6. The bit-flip channel
A simple qubit noise model flips the state with probability p:
𝓔(ρ)=(1-p)ρ+pXρX.
A Kraus representation is
K₀=√(1-p) IK₁=√p X
The completeness relation is immediate:
K₀†K₀+K₁†K₁=(1-p)I+pI=I.
Worked example: bit flip on |0⟩
For ρ=|0⟩⟨0|,
𝓔(ρ)=(1-p)|0⟩⟨0|+p|1⟩⟨1|.
The output is mixed unless p is 0 or 1. A pure input has become a probabilistic mixture.
7. The phase-flip channel
The phase-flip channel applies Z with probability p:
𝓔(ρ)=(1-p)ρ+pZρZ.
Computational-basis populations are unchanged because Z does not exchange |0⟩ and |1⟩. But relative phase can be disturbed, which changes off-diagonal density-matrix elements.
Worked example: phase noise on |+⟩
Because Z|+⟩=|−⟩,
𝓔(|+⟩⟨+|)=(1-p)|+⟩⟨+|+p|−⟩⟨−|.
At p=1/2, the X-basis coherence is completely erased and the output becomes I/2.
8. Dephasing as suppression of off-diagonal terms
A common dephasing model leaves populations unchanged while multiplying the off-diagonal terms by a factor λ with |λ|≤1:
[[a,c],[c*,b]] → [[a,λc],[λ*c*,b]].
As |λ| decreases, interference capability is reduced. When λ=0, all coherence in that chosen basis has disappeared.
This provides a precise mathematical meaning for one common use of the word decoherence: environmentally induced loss of off-diagonal coherence in a physically relevant basis.
9. The depolarising channel
A standard qubit depolarising model moves a state toward the maximally mixed state:
𝓔(ρ)=(1-p)ρ+p I/2.
In Bloch-vector form, if ρ=(I+r·σ)/2, then
r → (1-p)r.
The Bloch vector contracts toward the origin. Pure states move into the interior of the Bloch sphere.
10. Amplitude damping
Amplitude damping models energy relaxation such as an excited two-level system decaying toward a ground state. A standard Kraus pair is
K₀=[[1,0],[0,√(1-γ)]]K₁=[[0,√γ],[0,0]]
where 0≤γ≤1.
Check completeness:
K₀†K₀+K₁†K₁=diag(1,1-γ)+diag(0,γ)=I.
Worked example: excited-state decay
For input |1⟩⟨1|, the output is
(1-γ)|1⟩⟨1|+γ|0⟩⟨0|.
At γ=0, no decay occurs. At γ=1, the system has fully relaxed to |0⟩.
11. Why amplitude damping is not unital
A channel is unital if 𝓔(I)=I, equivalently if it preserves the maximally mixed state. Bit-flip, phase-flip and depolarising channels are unital in their standard forms. Amplitude damping is not.
This makes sense physically: energy relaxation has a preferred direction toward the ground state. It does not merely blur all directions symmetrically.
12. Stinespring dilation: noise from a larger unitary world
A foundational theorem says that every finite-dimensional quantum channel can be realised by adjoining an environment, applying a unitary transformation to system plus environment, and tracing out some environmental degrees of freedom.
Schematically,
𝓔(ρ_S)=Tr_E[U(ρ_S⊗|0⟩⟨0|_E)U†].
This explains why subsystem evolution can look irreversible even though the larger closed system evolves unitarily. Information can move into correlations with the environment.
Local non-unitarity can arise from global unitarity plus discarded information.
13. Deriving Kraus operators from an environment
Choose an orthonormal basis {|k⟩_E} for the environment and initial environmental state |0⟩_E. Define
K_k = ⟨k|_E U |0⟩_E.
Then tracing out E produces
𝓔(ρ)=Σ_k K_kρK_k†.
The Kraus representation is therefore not an arbitrary algebraic trick. It is what remains after environmental degrees of freedom are resolved in a basis and then ignored.
14. Channel composition
If channel 𝓔 is followed by channel 𝓕, the combined channel is 𝓕∘𝓔. If {K_i} are Kraus operators for 𝓔 and {L_j} for 𝓕, then one Kraus set for the composition is {L_jK_i}.
This matters in quantum circuits and error models because several individually simple noise processes can combine into a more complicated effective channel.
15. Channels acting on part of an entangled state
If a channel acts only on subsystem A of a joint state ρ_AB, the correct transformation is
(𝓔_A⊗I_B)(ρ_AB).
This is exactly why complete positivity is required. A map that looks positive on isolated A states can still fail when extended to an entangled A-B state. Physical local operations must remain positive in every such extension.
16. Worked example: local dephasing of a Bell state
Start from |Φ+⟩=(|00⟩+|11⟩)/√2. Its density matrix contains coherence terms |00⟩⟨11| and |11⟩⟨00|.
Apply a complete computational-basis dephasing channel to the first qubit. The off-diagonal joint terms vanish, leaving
ρ'=(1/2)|00⟩⟨00|+(1/2)|11⟩⟨11|.
The computational-basis correlations remain, but the coherent entangled superposition has become a classically correlated mixture.
17. Bloch-sphere picture of qubit channels
A general qubit density matrix can be represented by a Bloch vector r. Any qubit CPTP map acts affinely:
r → Mr+t
for a real 3×3 matrix M and real vector t satisfying physicality constraints. Unital channels have t=0. Amplitude damping typically has non-zero t because the Bloch ball is shifted toward the ground-state pole.
This geometric picture is powerful: noise can shrink, rotate, shear within physical constraints, or translate the Bloch ball.
18. Fixed points of a channel
A fixed state satisfies 𝓔(ρ*)=ρ*. Repeated application of a noisy channel may drive many initial states toward one or more fixed points.
For amplitude damping with non-zero relaxation, |0⟩⟨0| is a fixed point. For the fully depolarising channel, I/2 is the unique output and therefore a fixed point.
Fixed-point analysis turns long-time noise behaviour into an eigenvalue problem for a linear map on operators.
19. Superoperators and vectorisation
Because channels are linear maps acting on matrices, they can themselves be represented as matrices once operators are vectorised. A d×d density matrix has d² entries, so a channel can be represented by a d²×d² matrix in a chosen operator basis.
This higher-level matrix is often called a superoperator representation. It allows channel composition, eigenvalue analysis and numerical propagation to be handled with ordinary linear algebra on an enlarged space.
20. The Choi matrix
A channel can also be represented by its Choi matrix. Let |Φ⟩ be a maximally entangled state on a reference and input space. Applying the channel to one half gives an operator proportional, depending on normalisation convention, to the Choi representation of the channel.
The importance is structural: complete positivity becomes ordinary positive semidefiniteness of the Choi matrix, while trace-preservation becomes a partial-trace constraint. A difficult property of a map becomes a concrete matrix condition.
21. Decoherence versus dissipation
The words are related but should not be merged.
- Decoherence: loss of phase coherence, often represented by shrinking off-diagonal terms in a preferred basis.
- Dissipation: exchange of energy with an environment, as in amplitude damping.
A process can contain both. Different noise mechanisms therefore require different channel models.
22. Repeated channels and Markovian intuition
If the same channel is applied repeatedly,
ρ_n=𝓔^n(ρ_0).
This resembles a discrete-time Markov process, except the state space consists of density operators and the map must satisfy quantum physicality constraints. In continuous time, certain memoryless open-system models form quantum dynamical semigroups generated by Lindblad equations.
The important caution is that not every real environment is memoryless. Non-Markovian dynamics can return information or coherence to the system, so a simple semigroup model may be inadequate.
23. Common misconception: noise means random unitary gates only
Some channels can be written as classical mixtures of unitaries, but not every channel can. Amplitude damping is a standard counterexample. Quantum noise includes processes that change the spectrum of ρ and move the state toward a preferred physical state.
24. Common misconception: decoherence destroys total probability
A trace-preserving channel keeps Tr(ρ)=1. What is lost can be coherence, purity or locally accessible information—not probability normalisation itself.
25. Common misconception: irreversible local dynamics contradict unitary quantum theory
They do not. A subsystem can undergo an effectively irreversible channel because information has become encoded in an environment that we do not control or observe. The enlarged system can still evolve unitarily.
26. Worked synthesis problem
Let a qubit start in |+⟩ and undergo phase-flip noise with probability p.
Step 1: Initial density matrix.
ρ=(1/2)[[1,1],[1,1]].
Step 2: Apply the channel.
ρ'=(1-p)ρ+pZρZ.
Since ZρZ=(1/2)[[1,-1],[-1,1]],
ρ'=(1/2)[[1,1-2p],[1-2p,1]].
Step 3: Interpret coherence. The off-diagonal term is multiplied by 1-2p. At p=1/2, it vanishes.
Step 4: Purity. The Bloch vector lies along X with length |1-2p|, so the purity is [1+(1-2p)²]/2.
Step 5: Check endpoints. At p=0 or p=1 the output is pure; at p=1/2 the output is maximally mixed. Noise strength and mixedness need not vary monotonically under this particular “apply Z with probability p” parameterisation because p=1 is a deterministic unitary Z operation.
27. Practice set
- State the Kraus form of a quantum channel.
- State the completeness condition for trace preservation.
- Write Kraus operators for a bit-flip channel with probability p.
- Apply that channel to
|0⟩⟨0|. - What happens to the off-diagonal terms under complete dephasing?
- What state does a fully depolarising qubit channel produce?
- Apply amplitude damping with parameter γ to
|1⟩⟨1|. - Explain why complete positivity is stronger than positivity.
- Write the Stinespring form of a channel using an environment and partial trace.
- Give one difference between decoherence and dissipation.
Answers
𝓔(ρ)=Σ_kK_kρK_k†.Σ_kK_k†K_k=I.√(1-p)Iand√pX.(1-p)|0⟩⟨0|+p|1⟩⟨1|.- They vanish in the dephasing basis.
I/2.(1-γ)|1⟩⟨1|+γ|0⟩⟨0|.- A local map must remain positive even when tensored with the identity on any reference system.
Tr_E[U(ρ⊗|0⟩⟨0|)U†].- Decoherence primarily removes phase coherence; dissipation involves energy exchange or relaxation.
28. Series navigation
- Guide 5: Density Matrices, Mixed States, Trace and Partial Trace
- Guide 6: Quantum Channels, Kraus Operators, Noise and Decoherence
- Guide 7: Generalised Measurements, POVMs, State Discrimination and Fidelity
- Guide 8: Quantum Entropy, Purity, Mutual Information and Correlations
- Return to the BTT Mathematics Learning Hub
Educational note: channel models are abstractions. Real hardware noise can be time-dependent, correlated, non-Markovian, state-dependent and calibration-sensitive. A useful model must be validated against the physical regime it is intended to represent.