Chapter 6. Dynamics of Open Quantum Systems: Quantum Channels and Representations

An ideal quantum system is perfectly isolated from the outside and evolves unitarily, but all real systems cannot be free from interaction with the surrounding environment (Environment). The time evolution of such open quantum systems (Open Quantum System) is no longer a simple unitary transformation and includes irreversible processes such as information loss and decoherence (Decoherence). The quantum channel (Quantum Channel) is a powerful framework for mathematically describing these general quantum processes, and there are three key perspectives—Kraus representation, Stinespring extension, Choi representation—to view it.

1. Fundamental Concepts

  • Quantum Channel (Quantum Channel / CPTP Map): This generalizes the unitary transformation of closed systems to open systems. All physically possible changes of quantum states are described by a linear map \(\mathcal{E}\) that sends a density matrix to another density matrix. This map must satisfy two key conditions.

    1. Trace-Preserving (Trace-Preserving, TP): \(\mathrm{Tr}(\mathcal{E}(\rho)) = \mathrm{Tr}(\rho)=1\). The total probability must always be preserved as 1.
    2. Completely Positive (Completely Positive, CP): The map \(\mathcal{E}\) must not only send all positive operators to positive operators but also maintain positivity even for states entangled with arbitrary auxiliary systems.

    Detailed Explanation: Why must it be “completely positive” rather than just “positive”? 🤔

    A transformation being positive means that it sends valid states (those with positive eigenvalues, \(\rho\)) to valid states. However, this is not sufficient.

    Imagine our system \(S\) is entangled with another system \(R\) outside. Although the state of the total system \(SR\) is valid, applying a local physical transformation \(\mathcal{E}\) only to \(S\) must not result in an unphysical state with negative probabilities for the entire \(SR\).

    Complete positivity guarantees this condition. That is, it is a strong condition that “no matter how entangled we are with unknown external systems, local operations never violate the physical validity of the entire system.” A typical example of an operation that is positive but not completely positive is the matrix transpose operation.

  • Kraus Representation (Kraus Representation): This theorem states that all quantum channels (CPTP maps) can be decomposed into the sum of several Kraus operators (Kraus Operator) \(\{K_k\}\). This provides an “operational” perspective by representing the channel’s action as the sum of easily understandable operations. \(\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger\) The trace-preserving condition is expressed as \(\sum_k K_k^\dagger K_k = \mathbf{1}\).

  • Stinespring Dilation: This is a ‘physical’ perspective that shows that every quantum channel can be viewed as a unitary evolution of a larger ‘system+environment’ whole. That is, it provides a profound insight that any complex irreversible process can be understood as following simple quantum mechanical laws (unitary evolution) if we introduce an environment (Environment) that we do not observe. \(\mathcal{E}(\rho) = \mathrm{Tr}_E \left[ U(\rho \otimes |0\rangle\langle 0|_E) U^\dagger \right]\) Here, \(U\) is the unitary operator acting on the entire system, and \(\mathrm{Tr}_E\) is the partial trace operation that discards information about the environment.

  • Choi Representation: This is a ‘mathematical’ perspective that maps the quantum channel, which is a ‘process (map)’, one-to-one to a ‘state (state)’. The Choi Matrix \(J(\mathcal{E})\) obtained by passing one qubit of the maximally entangled state \(|\Phi^+\rangle\) through the channel \(\mathcal{E}\) is like a ‘fingerprint’ that contains all information about the channel. Simply checking whether this matrix is a positive semi-definite matrix allows us to determine whether the channel is completely positive or not.

  • Decoherence: This is a key phenomenon in open quantum systems, where the phase information (coherence) of quantum superposition states disappears as the system becomes entangled with the environment. In the density matrix, this is represented by the off-diagonal elements (off-diagonal elements) gradually decaying to zero, making it appear as if quantum superposition changes into a classical probabilistic mixture.

2. Symbols and Key Relations

  • Conditions for CPTP Map \(\mathcal{E}\):
    • Linearity: \(\mathcal{E}(a\rho_1 + b\rho_2) = a\mathcal{E}(\rho_1) + b\mathcal{E}(\rho_2)\)
    • Trace Preservation (TP): \(\mathrm{Tr}(\mathcal{E}(\rho)) = \mathrm{Tr}(\rho)\) for all \(\rho\).
    • Complete Positivity (CP): \(\mathcal{E} \otimes \mathsf{id}_k\) must be a positive map for all auxiliary systems of dimension \(k\). (\(\mathsf{id}_k\) is the identity channel on a \(k\)-dimensional system)
  • Kraus Representation (Operator-Sum Representation):
    • Action of the channel: \(\mathcal{E}(\rho) = \sum_{k=0}^{L-1} K_k \rho K_k^\dagger\)
    • Trace Preservation Condition: \(\sum_{k=0}^{L-1} K_k^\dagger K_k = \mathbf{1}_S\) (identity operator)

    💡 Why does \(\sum K_k^\dagger K_k = \mathbf{1}\) preserve the trace?

    It can be easily shown by using the cyclic property of the trace operator (\(\mathrm{Tr}(ABC) = \mathrm{Tr}(CAB)\)).

    \(\mathrm{Tr}(\mathcal{E}(\rho)) = \mathrm{Tr}\left(\sum_k K_k \rho K_k^\dagger\right) = \sum_k \mathrm{Tr}(K_k \rho K_k^\dagger)\)

    \(= \sum_k \mathrm{Tr}(K_k^\dagger K_k \rho) = \mathrm{Tr}\left(\left(\sum_k K_k^\dagger K_k\right) \rho\right) = \mathrm{Tr}(\mathbf{1} \cdot \rho) = \mathrm{Tr}(\rho)\).

  • Stinespring Dilation:
    • System \(H_S\), environment \(H_E\). Total space \(H_S \otimes H_E\).
    • Action of the channel: \(\mathcal{E}(\rho) = \mathrm{Tr}_E \left[ U(\rho \otimes |e_0\rangle\langle e_0|) U^\dagger \right]\)
    • Here, \(|e_0\rangle\) is the environment’s initial state, \(U\) is the unitary operator acting on the total space.
  • Choi Representation and Criterion:
    • Maximal entangled state: \(|\Phi^+\rangle = \frac{1}{\sqrt{d}} \sum_{i=0}^{d-1} |i\rangle_S \otimes |i\rangle_R\)
    • Definition of Choi matrix: \(J(\mathcal{E}) = (\mathcal{E} \otimes \mathsf{id}_R)(|\Phi^+\rangle\langle\Phi^+|)\)
    • CPTP Necessary and Sufficient Condition:
      1. \(\mathcal{E}\) is CP \(\iff\) \(J(\mathcal{E}) \ge 0\) (Choi matrix is positive semidefinite)
      2. \(\mathcal{E}\) is TP \(\iff\) \(\mathrm{Tr}_S(J(\mathcal{E})) = \frac{1}{d}\mathbf{1}_R\) (partial trace over the system gives the identity matrix)

3. Easy Examples (Examples with Deeper Insight)

  • Example 1: Dephasing Channel
    • Physical Meaning: The system’s energy does not change (\(|0\rangle \leftrightarrow |0\rangle, |1\rangle \leftrightarrow |1\rangle\)), but the relative phase information between \(|0\rangle\) and \(|1\rangle\) becomes randomly altered due to interaction with the environment. It is the most basic channel modeling “decoherence” in quantum computers. 🕰️
    • Kraus operators: \(K_0 = \sqrt{1-p}\mathbf{1}, K_1 = \sqrt{p}\sigma_z\).
    • Action result: \(\rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \quad \xrightarrow{\mathcal{E}} \quad \begin{pmatrix} \rho_{00} & (1-2p)\rho_{01} \\ (1-2p)\rho_{10} & \rho_{11} \end{pmatrix}\)
    • Interpretation: Diagonal components (classical probabilities) remain unchanged, while the off-diagonal components (quantum coherence) decay at a rate of \(1-2p\).
  • Example 2: Amplitude Damping Channel
    • Physical Meaning: Models energy loss processes such as an excited state atom spontaneously emitting a photon and falling into the ground state. \(|1\rangle\) state collapses to \(|0\rangle\) state with probability \(\gamma\). ⚛️
    • Kraus Operators: \(K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix}, K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix}\).
    • Resulting Action: \(\rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \quad \xrightarrow{\mathcal{E}} \quad \begin{pmatrix} \rho_{00} + \gamma\rho_{11} & \sqrt{1-\gamma}\rho_{01} \\ \sqrt{1-\gamma}\rho_{10} & (1-\gamma)\rho_{11} \end{pmatrix}\)
    • Interpretation: The probability of being in \(|1\rangle\) (\(\rho_{11}\)) decreases by a factor of \((1-\gamma)\), and correspondingly, the probability of being in \(|0\rangle\) (\(\rho_{00}\)) increases. The coherence (\(\rho_{01}\)) also damps accordingly.
  • Example 3: Stinespring Extension of the Amplitude Damping Channel
    • Story: There is a system qubit (\(S\)) and an environment qubit (\(E\), initial state \(|0\rangle_E\)). The unitary operation \(U\) causes the environment to transition to \(|1\rangle_E\) with probability \(\sqrt{\gamma}\) only when the system is in the \(|1\rangle_S\) state, and the system falls to \(|0\rangle_S\).
    • Unitary Definition: \(U (|0\rangle_S |0\rangle_E) = |0\rangle_S |0\rangle_E\) \(U (|1\rangle_S |0\rangle_E) = \sqrt{1-\gamma}|1\rangle_S |0\rangle_E + \sqrt{\gamma}|0\rangle_S |1\rangle_E\)
    • Result: After applying this \(U\) to the initial state \(\rho_S \otimes |0\rangle\langle 0|_E\) and taking the partial trace over the environment \(E\), the amplitude damping channel from Example 2 is exactly reproduced. This demonstrates that complex energy loss processes are actually part of the energy conservation law (unitary evolution) of a larger system.

4. Practice Problems

  1. Trace Preservation Check: Prove how the condition \(\sum_k K_k^\dagger K_k = \mathbf{1}\) in the Kraus representation \(\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger\) ensures trace preservation \(\mathrm{Tr}(\mathcal{E}(\rho)) = \mathrm{Tr}(\rho)\).
  2. Non-uniqueness of Kraus Representation: Show that two Kraus operator sets \(\{K_k\}\) and \(\{L_j\}\), related by a unitary matrix \(U\) as \(L_j = \sum_k U_{jk} K_k\), describe the same quantum channel.
  3. Phase Damping and the Bloch Sphere: Calculate how the phase damping channel (\(K_0 = \sqrt{1-p}\mathbf{1}, K_1 = \sqrt{p}\sigma_z\)) transforms the Bloch vector \(\vec{r} = (r_x, r_y, r_z)\).
  4. Depolarizing Channel: Determine the Kraus operators for the depolarizing channel \(\mathcal{E}(\rho) = (1-p)\rho + p \frac{\mathbf{1}}{2}\), which transforms a qubit into a completely random state (\(\mathbf{1}/2\)) with probability \(p\) and leaves it unchanged with probability \(1-p\).
  5. Stinespring Construction: Describe the procedure for constructing the unitary operator \(U\) for the Stinespring extension from a given Kraus operator set \(\{K_k\}_{k=0}^{L-1}\). (Hint: \(U|\psi\rangle_S |0\rangle_E = \sum_k K_k |\psi\rangle_S |k\rangle_E\))
  6. Choi Matrix Calculation: Directly calculate the Choi matrix \(J(\mathcal{E})\) for the bit flip channel (\(K_0 = \sqrt{1-p}\mathbf{1}, K_1 = \sqrt{p}\sigma_x\)).

5. Explanation

  1. \(\mathrm{Tr}(\mathcal{E}(\rho)) = \mathrm{Tr}(\sum_k K_k \rho K_k^\dagger) = \sum_k \mathrm{Tr}(K_k \rho K_k^\dagger) = \sum_k \mathrm{Tr}(K_k^\dagger K_k \rho) = \mathrm{Tr}((\sum_k K_k^\dagger K_k)\rho) = \mathrm{Tr}(\mathbf{1}\cdot\rho) = \mathrm{Tr}(\rho)\).
  2. \(\sum_j L_j \rho L_j^\dagger = \sum_j (\sum_k U_{jk} K_k) \rho (\sum_l U_{jl}^* K_l^\dagger) = \sum_{k,l} (\sum_j U_{jk} U_{jl}^*) K_k \rho K_l^\dagger\). Since \(U\) is unitary, \(\sum_j U_{jk} U_{jl}^* = (UU^\dagger)_{kl} = \delta_{kl}\). Therefore, it becomes \(\sum_k K_k \rho K_k^\dagger\), resulting in the same channel.
  3. \(\rho = \frac{1}{2}(\mathbf{1}+\vec{r}\cdot\vec{\sigma})\). \(\mathcal{E}(\rho) = (1-p)\rho + p\sigma_z\rho\sigma_z = \dots = \frac{1}{2}(\mathbf{1} + (1-2p)r_x\sigma_x + (1-2p)r_y\sigma_y + r_z\sigma_z)\). Thus, \(\vec{r} \to ((1-2p)r_x, (1-2p)r_y, r_z)\), which corresponds to a contraction in the \(xy\) plane.
  4. \(\mathcal{E}(\rho)\) can be expressed using \(K_0=\sqrt{1-p/2}\mathbf{1}, K_1=\sqrt{p/6}\sigma_x, K_2=\sqrt{p/6}\sigma_y, K_3=\sqrt{p/6}\sigma_z\). \(\sum K_i^\dagger K_i = (1-p/2+3p/6)\mathbf{1}=\mathbf{1}\) is satisfied. (Other forms are also possible)
  5. Prepare the system Hilbert space \(H_S\), and the environment Hilbert space \(H_E=\mathbb{C}^L\). Denote the environment basis as \(\{|k\rangle_E\}_{k=0}^{L-1}\), and define the action of the unitary operator \(U\) on the subspace \(H_S \otimes \{|0\rangle_E\}\) as \(U(|\psi\rangle_S |0\rangle_E) = \sum_{k=0}^{L-1} (K_k |\psi\rangle_S) \otimes |k\rangle_E\). This definition preserves inner products, so \(U\) can be extended to the orthogonal complement of the subspace, resulting in a unitary operator over the entire space.
  6. \(|\Phi^+\rangle\langle\Phi^+| = \frac{1}{2}(|00\rangle\langle 00| + |00\rangle\langle 11| + |11\rangle\langle 00| + |11\rangle\langle 11|)\). Applying \(\mathcal{E}\otimes\mathsf{id}\) to this, \(\mathcal{E}\) acts only on the first qubit. \(\mathcal{E}(|0\rangle\langle 0|) = (1-p)|0\rangle\langle 0|+p|1\rangle\langle 1|\), \(\mathcal{E}(|1\rangle\langle 1|) = (1-p)|1\rangle\langle 1|+p|0\rangle\langle 0|\), \(\mathcal{E}(|0\rangle\langle 1|) = \sqrt{1-p}^2|0\rangle\langle 1|+\sqrt{p}^2|1\rangle\langle 0| = (1-2p)|0\rangle\langle 1|\) (not needed in this example). The final \(J(\mathcal{E})\) is obtained by combining these results.