Chapter 27: Quantum Channels and Operations

Quantum states evolve, interact with environments, and undergo noise. The mathematical framework for describing all such processes is the theory of quantum channels - completely positive, trace-preserving (CPTP) linear maps that transform density operators into density operators. Just as density operators generalize state vectors, quantum channels generalize unitary evolution. This chapter develops the three equivalent representations of quantum channels - Kraus, Choi, and Stinespring - and explores the most important channels in quantum information theory.

27.1 Kraus Representation

A quantum channel is a linear map $\mathcal{E}$ that takes density operators to density operators. It must satisfy two properties: it must be completely positive (mapping positive operators to positive operators, even when acting on part of a larger system) and trace-preserving ($\text{Tr}(\mathcal{E}(\rho)) = \text{Tr}(\rho)$ for all $\rho$). Together, these make $\mathcal{E}$ a CPTP map.

The most computationally convenient way to specify a quantum channel is through its Kraus representation (also called the operator-sum representation):

$$\mathcal{E}(\rho) = \sum_{k} K_k \rho K_k^\dagger$$

where the Kraus operators $\{K_k\}$ satisfy the completeness relation:

$$\sum_k K_k^\dagger K_k = I$$

This condition ensures trace preservation: $\text{Tr}(\mathcal{E}(\rho)) = \text{Tr}\!\left(\sum_k K_k \rho K_k^\dagger\right) = \text{Tr}\!\left(\rho \sum_k K_k^\dagger K_k\right) = \text{Tr}(\rho) = 1$.

Key Concept.

The Kraus representation theorem states that a linear map $\mathcal{E}$ is completely positive and trace-preserving if and only if it can be written as $\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger$ with $\sum_k K_k^\dagger K_k = I$. This is the fundamental characterization of physically realizable quantum operations.

Non-Uniqueness of Kraus Operators

The Kraus operators of a channel are not unique. If $\{K_k\}$ and $\{L_j\}$ are two sets of Kraus operators for the same channel, they are related by a unitary matrix $u_{jk}$:

$$L_j = \sum_k u_{jk} K_k$$

(padding with zero operators if the two sets have different sizes). This non-uniqueness parallels the non-uniqueness of ensemble decompositions of density operators and reflects the same physical principle: different microscopic models can produce the same observable channel.

Examples of Single-Kraus-Operator Channels

When there is exactly one Kraus operator, the completeness relation gives $K^\dagger K = I$, so $K$ must be unitary. Unitary evolution $\rho \mapsto U\rho U^\dagger$ is therefore the special case of a quantum channel with a single Kraus operator. All other channels have multiple Kraus operators and represent irreversible (noisy) processes.

Interpretation: Channels as Averaged Interactions

Each Kraus operator $K_k$ can be interpreted as the evolution conditioned on a particular (unobserved) outcome of an environment. The sum over $k$ represents averaging over all possible environmental outcomes. This connects to the physical picture of decoherence: information leaks into the environment, and because we do not observe the environment, we must average over its possible states.

Worked Example: The Bit-Flip Channel

The bit-flip channel flips a qubit with probability $p$ and leaves it unchanged with probability $1 - p$. The Kraus operators are:

$$K_0 = \sqrt{1 - p}\, I = \sqrt{1-p}\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}, \quad K_1 = \sqrt{p}\, \sigma_x = \sqrt{p}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$$

Verification: $K_0^\dagger K_0 + K_1^\dagger K_1 = (1-p)I + p I = I$. Applying the channel to a general qubit state:

$$\mathcal{E}(\rho) = (1-p)\rho + p \sigma_x \rho \sigma_x = (1-p)\begin{pmatrix}\rho_{00}&\rho_{01}\\\rho_{10}&\rho_{11}\end{pmatrix} + p\begin{pmatrix}\rho_{11}&\rho_{10}\\\rho_{01}&\rho_{00}\end{pmatrix}$$

The populations get mixed ($\rho_{00} \to (1-p)\rho_{00} + p\rho_{11}$) and the coherences are multiplied by $(1 - 2p)$. At $p = 1/2$, the off-diagonal elements vanish and the output is diagonal in the computational basis. The bit-flip channel is the quantum analog of the binary symmetric channel from classical information theory.

The Number of Kraus Operators

For a channel on a $d$-dimensional system, at most $d^2$ Kraus operators are needed. This follows from the rank of the Choi matrix (Section 27.2), which is at most $d^2$. In practice, many important channels have far fewer Kraus operators: the amplitude damping channel needs only 2, and the depolarizing channel on a qubit can be written with 4 (corresponding to the four Pauli matrices including the identity).

Note.

If we do learn which Kraus operator was applied (e.g., by measuring the environment), the map $\rho \mapsto K_k \rho K_k^\dagger / \text{Tr}(K_k \rho K_k^\dagger)$ describes the conditional state update. The probability of outcome $k$ is $p_k = \text{Tr}(K_k \rho K_k^\dagger)$. This is exactly the quantum instrument formalism from Chapter 26.

27.2 Complete Positivity and the Choi-Jamiolkowski Isomorphism

Why Complete Positivity?

A map that is merely positive (sends positive operators to positive operators) is not necessarily physical. The reason: quantum systems may be entangled with other systems. If Alice applies a positive but not completely positive map to her half of an entangled state, the result can be a non-positive operator - not a valid density matrix.

Complete positivity demands that $\mathcal{E} \otimes \mathcal{I}_n$ is positive for all $n$, where $\mathcal{I}_n$ is the identity map on an $n$-dimensional ancilla. This ensures that the channel produces valid states even when applied to part of an entangled system. The transpose map $\rho \mapsto \rho^T$ is the textbook example of a positive but not completely positive map: applying it to one half of a maximally entangled state produces an operator with a negative eigenvalue.

The Choi Matrix

The Choi-Jamiolkowski isomorphism provides a one-to-one correspondence between linear maps and operators on a tensor product space. Given a channel $\mathcal{E}: \mathcal{L}(\mathcal{H}_A) \to \mathcal{L}(\mathcal{H}_B)$, the Choi matrix (or Choi state) is:

$$J(\mathcal{E}) = \sum_{i,j} |i\rangle\langle j| \otimes \mathcal{E}(|i\rangle\langle j|) = (\mathcal{I} \otimes \mathcal{E})(|\Omega\rangle\langle\Omega|)$$

where $|\Omega\rangle = \sum_i |i\rangle|i\rangle$ is the (unnormalized) maximally entangled state. The Choi matrix encodes everything about the channel.

Key Concept.

A linear map $\mathcal{E}$ is completely positive if and only if its Choi matrix $J(\mathcal{E})$ is positive semi-definite. The map is trace-preserving if and only if $\text{Tr}_B(J(\mathcal{E})) = I_A$. This gives an elegant, checkable criterion: a map is a valid quantum channel if and only if its Choi matrix is positive semi-definite with the correct partial trace. This is known as the Choi theorem on completely positive maps.

Channel-State Duality

When we normalize the Choi matrix as $\rho_{\mathcal{E}} = J(\mathcal{E})/d$, we get a valid density operator on the bipartite system $A'B$. This is the channel-state duality: every channel corresponds to a bipartite state, and every bipartite state with the correct marginal corresponds to a channel. This duality is extremely powerful:

Channel properties become state properties: entanglement-breaking channels correspond to separable Choi states, unitary channels correspond to pure maximally entangled Choi states.
Optimization over channels becomes optimization over states: many capacity formulas and channel comparison problems reduce to semidefinite programs on the Choi matrix.
Recovering Kraus operators: the eigenvectors of the Choi matrix (reshaped into matrices) give a set of Kraus operators. If $J(\mathcal{E}) = \sum_k |v_k\rangle\langle v_k|$, then the Kraus operators are obtained by reshaping each $|v_k\rangle$ into a matrix $K_k$.

Worked Example: Choi Matrix of the Dephasing Channel

For the single-qubit dephasing channel $\mathcal{E}(\rho) = (1-p)\rho + p Z\rho Z$ with $d = 2$, the maximally entangled state is $|\Omega\rangle = |00\rangle + |11\rangle$ (unnormalized). The Choi matrix is:

$$J(\mathcal{E}) = (\mathcal{I} \otimes \mathcal{E})(|\Omega\rangle\langle\Omega|)$$

We compute: $|\Omega\rangle\langle\Omega| = |00\rangle\langle 00| + |00\rangle\langle 11| + |11\rangle\langle 00| + |11\rangle\langle 11|$. Applying $\mathcal{E}$ to the second qubit of each term:

$$J = |0\rangle\langle 0| \otimes \mathcal{E}(|0\rangle\langle 0|) + |0\rangle\langle 1| \otimes \mathcal{E}(|0\rangle\langle 1|) + |1\rangle\langle 0| \otimes \mathcal{E}(|1\rangle\langle 0|) + |1\rangle\langle 1| \otimes \mathcal{E}(|1\rangle\langle 1|)$$

Since $\mathcal{E}(|0\rangle\langle 0|) = |0\rangle\langle 0|$, $\mathcal{E}(|1\rangle\langle 1|) = |1\rangle\langle 1|$, and $\mathcal{E}(|0\rangle\langle 1|) = (1-2p)|0\rangle\langle 1|$, the Choi matrix in the $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$ basis is:

$$J = \begin{pmatrix} 1 & 0 & 0 & (1-2p) \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ (1-2p) & 0 & 0 & 1\end{pmatrix}$$

The eigenvalues are $1 + (1-2p)$, $1 - (1-2p)$, $0$, and $0$, i.e., $2(1-p)$ and $2p$. Both are non-negative for $p \in [0, 1]$, confirming complete positivity. The rank is 2 (for $0 < p < 1$), which means the channel needs exactly 2 Kraus operators.

Useful Properties of the Choi Matrix

Several important channel properties have clean Choi-matrix characterizations:

Channel property	Choi matrix condition
Completely positive	$J(\mathcal{E}) \geq 0$
Trace-preserving	$\text{Tr}_B(J(\mathcal{E})) = I_A$
Unital ($\mathcal{E}(I) = I$)	$\text{Tr}_A(J(\mathcal{E})) = I_B$
Unitary channel	$J(\mathcal{E})$ is rank 1
Entanglement-breaking	$J(\mathcal{E})$ is separable

27.3 Common Quantum Channels

Several quantum channels appear repeatedly throughout quantum information theory. Each models a different type of noise or interaction with the environment. Understanding these channels builds intuition for the general theory and provides concrete examples for capacity calculations in Chapter 29.

The Depolarizing Channel

The depolarizing channel with parameter $p \in [0, 1]$ replaces the input state with the maximally mixed state with probability $p$:

$$\mathcal{E}_{\text{depol}}(\rho) = (1 - p)\rho + p \frac{I}{d}$$

For a single qubit ($d = 2$), this can be written with Kraus operators:

$$\mathcal{E}_{\text{depol}}(\rho) = \left(1 - \frac{3p}{4}\right)\rho + \frac{p}{4}(\sigma_x \rho \sigma_x + \sigma_y \rho \sigma_y + \sigma_z \rho \sigma_z)$$

Geometrically, the depolarizing channel uniformly shrinks the Bloch vector: $\vec{r} \mapsto (1 - p)\vec{r}$. It is the "worst-case" noise model in the sense that it treats all directions equally. The channel is unital ($\mathcal{E}(I) = I$) since the maximally mixed state is a fixed point.

The Dephasing Channel

The dephasing (or phase damping) channel destroys coherence without affecting populations. With parameter $p$:

$$\mathcal{E}_{\text{deph}}(\rho) = (1 - p)\rho + p Z\rho Z$$

where $Z = \sigma_z$. In the computational basis, this gives:

$$\begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \mapsto \begin{pmatrix} \rho_{00} & (1 - 2p)\rho_{01} \\ (1 - 2p)\rho_{10} & \rho_{11} \end{pmatrix}$$

The diagonal elements (populations) are untouched, but the off-diagonal elements (coherences) are multiplied by $(1 - 2p)$. At $p = 1/2$, all coherence is destroyed and the output is diagonal in the computational basis. Geometrically, the Bloch sphere is compressed into an ellipsoid: the $z$-component is preserved while $x$ and $y$ are shrunk.

The Amplitude Damping Channel

The amplitude damping channel models energy dissipation - the process by which an excited state $|1\rangle$ decays to the ground state $|0\rangle$ with probability $\gamma$. It is the quantum analog of a lossy channel. The Kraus operators are:

$$K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1 - \gamma} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix}$$

One can verify $K_0^\dagger K_0 + K_1^\dagger K_1 = I$. The effect on a general qubit state is:

$$\begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \mapsto \begin{pmatrix} \rho_{00} + \gamma \rho_{11} & \sqrt{1-\gamma}\,\rho_{01} \\ \sqrt{1-\gamma}\,\rho_{10} & (1-\gamma)\rho_{11} \end{pmatrix}$$

Population flows from $|1\rangle$ to $|0\rangle$: the $\rho_{11}$ entry decreases while $\rho_{00}$ increases. Unlike the depolarizing and dephasing channels, amplitude damping is not unital - the fixed point is $|0\rangle\langle 0|$, not $I/2$. At $\gamma = 1$, every input state is mapped to the ground state.

Note.

The amplitude damping channel models spontaneous emission of a photon from a two-level atom. $K_0$ represents the "no emission" branch (the atom stays in its current state but with reduced $|1\rangle$ amplitude), while $K_1$ represents the "emission" branch (the atom transitions from $|1\rangle$ to $|0\rangle$). The parameter $\gamma$ is related to the decay time $T_1$ in the language of experimental quantum computing.

The Erasure Channel

The quantum erasure channel with parameter $\epsilon$ either transmits the input perfectly (with probability $1 - \epsilon$) or replaces it with a fixed "erasure flag" state $|e\rangle$ (with probability $\epsilon$):

$$\mathcal{E}_{\text{erase}}(\rho) = (1 - \epsilon)\rho + \epsilon |e\rangle\langle e|$$

where $|e\rangle$ is orthogonal to the input Hilbert space (a third level for a qubit). Unlike other noise channels, the erasure channel is "honest" - the receiver knows when an error occurred. This makes it particularly important for understanding quantum capacity, as its capacity formulas are exactly computable.

The Phase-Flip Channel

The phase-flip channel applies a $Z$ gate with probability $p$:

$$\mathcal{E}_{\text{pf}}(\rho) = (1-p)\rho + p Z\rho Z$$

This is mathematically identical to the dephasing channel above - they are two names for the same map. The term "phase-flip" emphasizes its role as the $Z$-error analog of the bit-flip ($X$-error) channel. The bit-flip errors in the $Z$ basis while the phase-flip errors in the $X$ basis. Combining all three Pauli errors ($X$, $Z$, and $Y = iXZ$) with equal probability gives the depolarizing channel. Understanding these three error types separately is the starting point for quantum error correction theory.

Summary Table

Channel	Effect on Bloch vector	Unital?	Key property
Depolarizing	$\vec{r} \mapsto (1-p)\vec{r}$	Yes	Isotropic noise
Dephasing	Shrinks $x,y$; preserves $z$	Yes	Destroys coherence only
Amplitude damping	Shrinks and shifts toward $\|0\rangle$	No	Models energy decay
Erasure	State or flag	No	Receiver knows if error occurred

27.4 The Stinespring Dilation Theorem

The Kraus representation shows that any channel can be decomposed into a sum of "sandwich" terms $K_k \rho K_k^\dagger$. The Stinespring dilation theorem provides a different, more physically transparent picture: every quantum channel arises from a unitary interaction on a larger system followed by discarding (tracing out) part of that system.

Key Concept.

Stinespring Dilation Theorem. For every quantum channel $\mathcal{E}: \mathcal{L}(\mathcal{H}_A) \to \mathcal{L}(\mathcal{H}_B)$, there exists an environment (ancilla) system $E$ with initial pure state $|0\rangle_E$ and a unitary $U$ on $A \otimes E$ such that: $$\mathcal{E}(\rho) = \text{Tr}_E\!\left(U(\rho \otimes |0\rangle\langle 0|_E)U^\dagger\right)$$ This is called the Stinespring dilation or unitary dilation of the channel.

The theorem says that all noise and irreversibility in quantum mechanics is ultimately a consequence of entanglement with an unobserved environment. If we could access the environment, the global evolution would be perfectly reversible (unitary). The channel is "open system" evolution - unitary evolution of a closed system seen from the perspective of a subsystem. This perspective is fundamental to understanding decoherence: a qubit does not "lose" coherence spontaneously. Rather, it becomes entangled with its environment, and because we trace out the environment, the coherence appears to vanish from the qubit's reduced state.

Connection to Kraus Operators

The Kraus operators are extracted from the Stinespring dilation by expanding the environment trace. If $\{|k\rangle_E\}$ is an orthonormal basis for the environment, then:

$$K_k = \langle k|_E U |0\rangle_E$$

and the channel becomes $\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger$, recovering the Kraus form. Different choices of environment basis give different (but unitarily equivalent) sets of Kraus operators.

The Complementary Channel

The Stinespring dilation naturally defines the complementary channel $\mathcal{E}^c$, which describes what the environment receives:

$$\mathcal{E}^c(\rho) = \text{Tr}_B\!\left(U(\rho \otimes |0\rangle\langle 0|_E)U^\dagger\right)$$

While the channel $\mathcal{E}$ traces out the environment, the complementary channel traces out the output system. The complementary channel plays a crucial role in quantum capacity theory (Chapter 29): roughly, information that the environment captures is information that the receiver loses.

Note.

The relationship between a channel and its complement generalizes the classical wiretap channel. The quantum capacity of a channel is determined by the balance between what the receiver gets and what the environment captures. For degradable channels (where the complementary channel can be obtained by processing the output), the quantum capacity has a simple single-letter formula. The amplitude damping channel is degradable; the depolarizing channel generally is not.

Three Equivalent Representations

We now have three equivalent ways to describe a quantum channel, each with its own advantages:

Kraus representation $\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger$: most convenient for computation and simulation. Directly gives the channel's action on any input.
Choi matrix $J(\mathcal{E}) = (\mathcal{I} \otimes \mathcal{E})(|\Omega\rangle\langle\Omega|)$: best for checking complete positivity (just check $J \geq 0$), optimization via semidefinite programming, and reading off channel properties.
Stinespring dilation $\mathcal{E}(\rho) = \text{Tr}_E(U(\rho \otimes |0\rangle\langle 0|_E)U^\dagger)$: most physically transparent. Shows that every channel comes from a unitary on a larger space. Essential for defining the complementary channel.

Any one representation can be converted to any other. The choice of which to use depends on the problem at hand.

Worked Example: Stinespring Dilation of Amplitude Damping

The amplitude damping channel has Kraus operators $K_0 = \begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix}$ and $K_1 = \begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix}$. The Stinespring dilation uses a single environment qubit initialized to $|0\rangle_E$. The unitary $U$ on the system-environment space acts as:

$$U|0\rangle_S|0\rangle_E = |0\rangle_S|0\rangle_E$$ $$U|1\rangle_S|0\rangle_E = \sqrt{1-\gamma}|1\rangle_S|0\rangle_E + \sqrt{\gamma}|0\rangle_S|1\rangle_E$$

This describes the physical process: if the atom is in the ground state $|0\rangle$, nothing happens. If it is in the excited state $|1\rangle$, it has probability $\gamma$ of emitting a photon (the environment transitions from $|0\rangle_E$ to $|1\rangle_E$) and decaying to $|0\rangle$. The Kraus operators are recovered as $K_k = \langle k|_E U |0\rangle_E$.

The complementary channel sends $\rho$ to the environment's state: $\mathcal{E}^c(\rho) = \text{Tr}_S(U(\rho \otimes |0\rangle\langle 0|)U^\dagger)$. For this channel, the environment receives a state that encodes information about whether a decay event occurred. Because amplitude damping is degradable, the complementary channel can be obtained by further processing the system output, which simplifies the quantum capacity calculation.

Channels as Resources

A channel $\mathcal{E}$ can itself be viewed as a resource to be consumed. How much classical information can it transmit? How much quantum information? Can it generate entanglement? These questions define the various capacities of the channel, the central topic of Chapter 29. The Stinespring picture makes clear why capacity is fundamentally about the information split between receiver and environment: whatever the environment learns, the receiver cannot.

Composition of Channels

Channels can be composed in series and parallel. If $\mathcal{E}_1$ has Kraus operators $\{A_i\}$ and $\mathcal{E}_2$ has Kraus operators $\{B_j\}$:

Sequential composition (first $\mathcal{E}_1$, then $\mathcal{E}_2$): The composite channel $\mathcal{E}_2 \circ \mathcal{E}_1$ has Kraus operators $\{B_j A_i\}_{i,j}$.
Parallel composition ($\mathcal{E}_1$ on system $A$, $\mathcal{E}_2$ on system $B$): The channel $\mathcal{E}_1 \otimes \mathcal{E}_2$ has Kraus operators $\{A_i \otimes B_j\}_{i,j}$.

Sequential composition models a signal passing through multiple noisy stages. Parallel composition models independent noise on separate qubits. The set of all quantum channels on a fixed system forms a monoid under sequential composition: it is closed under composition, associative, and has the identity channel as its unit.

Interactive: Channel Bloch Deformation

Each quantum channel deforms the Bloch sphere in a characteristic way. The depolarizing channel shrinks it uniformly, the dephasing channel flattens it into a disc, and amplitude damping both shrinks and shifts it toward $|0\rangle$. Select a channel and adjust the noise parameter to see the deformation.

Channel: Strength $p$: 0.20

OPENQASM 2.0; include "qelib1.inc"; qreg q[1]; creg c[1]; h q[0]; c[0] = measure q[0];

Interactive: Channel Representations

Explore different mathematical representations of quantum noise channels. Adjust the noise parameter to see how the Kraus operators, Choi matrix, and Pauli transfer matrix change. All representations describe the same physical process.

Things to explore:

Depolarizing at p = 0: The single Kraus operator is the identity, the Choi matrix is the maximally entangled state projector, and the PTM is the 4 x 4 identity. This is the "do nothing" channel.
Amplitude damping: Notice that the PTM is not diagonal - the channel is not unital, so the identity Pauli component is not preserved. The Bloch vector shifts toward the north pole.
Phase damping vs phase flip: Compare these two channels. They have identical Kraus structure (both use I and Z), differing only in the relationship between the parameter and the Kraus coefficients.
Any channel at p = 0: All channels reduce to the identity. At p = 1, observe how each channel reaches its maximum effect - the depolarizing channel replaces everything with the maximally mixed state, while amplitude damping sends everything to the ground state.

Interactive: Stinespring Dilation Walkthrough

Every quantum channel arises from a unitary on a larger system followed by tracing out the environment. This walkthrough shows the Stinespring dilation of the amplitude damping channel: the system qubit interacts unitarily with an environment qubit, then the environment is discarded.

Step 1: System + Environment

Begin with the system qubit in state $\rho$ and an environment qubit initialized to $|0\rangle_E$. The joint state is $\rho \otimes |0\rangle\langle 0|_E$ - a product state with no entanglement.

System

$\rho$

Environment

$|0\rangle\langle 0|$

Step 2: Joint Unitary

Apply a global unitary $U$ that couples system and environment. For amplitude damping: $U|1\rangle_S|0\rangle_E = \sqrt{1-\gamma}|1\rangle_S|0\rangle_E + \sqrt{\gamma}|0\rangle_S|1\rangle_E$. The system and environment become entangled.

$U_{SE}$

$U(\rho \otimes |0\rangle\langle 0|)U^\dagger$

System and environment are now entangled

Step 3: Trace Out Environment

Discard the environment by taking the partial trace over $E$. The resulting system state is $\mathcal{E}(\rho) = \text{Tr}_E(U(\rho \otimes |0\rangle\langle 0|)U^\dagger)$. This produces the noisy channel output.

System output

$\mathcal{E}(\rho)$

Environment

(traced out)

Result: Kraus Operators Emerge

The Kraus operators $K_k = \langle k|_E U |0\rangle_E$ are extracted from the unitary by projecting onto environment basis states. For amplitude damping: $K_0 = \begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix}$ and $K_1 = \begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix}$.

$K_0 = \langle 0|U|0\rangle$
No decay branch

$K_1 = \langle 1|U|0\rangle$
Decay (emission) branch

$\mathcal{E}(\rho) = K_0\rho K_0^\dagger + K_1\rho K_1^\dagger$

Interactive: Channel Composition

Compose two channels in sequence and observe the cumulative effect. Notice that channel composition is generally non-commutative: applying dephasing then amplitude damping gives a different result from amplitude damping then dephasing.

Sandbox: Quantum Channel Effects

The sandbox below demonstrates the effect of noise channels on a qubit. We start with a qubit in the $|+\rangle$ state (equal superposition) and apply different noise models. Run the circuit as-is to see the pure $|+\rangle$ state, then uncomment different noise sections to observe their effects.

Experiments to try:

No noise: Leave only h q[0]; before the final Hadamard. The output should be deterministically 0 since $H|+\rangle = |0\rangle$.
Simulated dephasing: Uncomment the rz(1.0) q[0]; line. This rotates the phase, partially destroying the coherence measured by the final Hadamard. You should see a mixture of 0 and 1 outcomes.
Entanglement with environment: Uncomment the cx q[0], q[1]; line. This entangles the qubit with an "environment" qubit. Since we only measure q[0], the entanglement causes decoherence - the output becomes mixed.

Interactive: Quantum Channel Explorer

Apply different quantum noise channels to a single qubit in the $|+\rangle$ state and observe how each channel affects the output. The purity measures how mixed the state has become (1.0 for a pure state, 0.5 for maximally mixed), while fidelity tracks overlap with the ideal noise-free state. Watch how different channels degrade the state in qualitatively different ways as you increase the noise strength.

Compare the channels at the same noise strength:

Depolarizing: shrinks the Bloch vector uniformly in all directions. The histogram spreads symmetrically toward 50/50.
Amplitude damping: pushes the state toward $|0\rangle$. The histogram becomes biased, and the StateCity plot shows population flowing to $\rho_{00}$.
Phase damping: destroys off-diagonal coherence without changing populations. Since $|+\rangle$ has equal populations, the histogram shifts toward 50/50 in the computational basis as the coherent superposition decoheres.
Bit flip: randomly applies $X$. For $|+\rangle$ (an eigenstate of $X$), the bit-flip channel has no effect on measurement statistics - try switching the initial state by modifying the circuit.
Phase flip: randomly applies $Z$. Since $Z|+\rangle = |-\rangle$, this channel mixes $|+\rangle$ and $|-\rangle$, destroying coherence in the $X$ basis.