9.3 Quantum operations and completely positive maps

Quantum operations are the building blocks of quantum information processing. They transform quantum states, enabling computation and communication. These operations come in two flavors: unitary (reversible) and non-unitary (irreversible), each playing a crucial role in quantum systems.

Completely positive maps provide a mathematical framework for describing quantum operations. They ensure that physical transformations preserve the positivity of density matrices. This concept is fundamental to understanding how quantum information is processed and transmitted in real-world systems.

Quantum Operations

Types of quantum operations

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• Quantum operations transform quantum states and can be classified into two main categories
  • Unitary operations are reversible transformations that preserve the inner product between states (orthogonality)
    • Represented mathematically by unitary matrices $U$ satisfying $U^\dagger U = UU^\dagger = I$, where $U^\dagger$ is the conjugate transpose of $U$ and $I$ is the identity matrix
  • Non-unitary operations are irreversible transformations that do not preserve the inner product between states
    • Include processes such as measurement (projective or POVM), dissipation (interaction with the environment), and decoherence (loss of quantum coherence)
• Kraus representation provides a general framework to describe any quantum operation $\mathcal{E}$ as a sum of Kraus operators $E_i$
  • The action of $\mathcal{E}$ on a state $\rho$ is given by $\mathcal{E}(\rho) = \sum_i E_i \rho E_i^\dagger$, where $E_i$ are the Kraus operators
  • Kraus operators must satisfy the completeness relation $\sum_i E_i^\dagger E_i = I$ to ensure the operation is trace-preserving

Concept of completely positive maps

• Completely positive (CP) maps are a special class of quantum operations that preserve positivity of density matrices
  • A map $\mathcal{E}$ is positive if it maps positive operators to positive operators, i.e., if $\rho \geq 0$, then $\mathcal{E}(\rho) \geq 0$
  • A map $\mathcal{E}$ is completely positive if $(\mathcal{I} \otimes \mathcal{E})(\rho)$ is positive for any positive operator $\rho$, where $\mathcal{I}$ is the identity map on an ancillary system
• Choi-Jamiołkowski isomorphism establishes a one-to-one correspondence between CP maps and positive semidefinite matrices
  • The Choi matrix of a CP map $\mathcal{E}$ is given by $J(\mathcal{E}) = (\mathcal{I} \otimes \mathcal{E})(\ket{\Phi^+}\bra{\Phi^+})$, where $\ket{\Phi^+} = \frac{1}{\sqrt{d}} \sum_{i=1}^d \ket{i} \otimes \ket{i}$ is a maximally entangled state
  • A map $\mathcal{E}$ is CP if and only if its Choi matrix $J(\mathcal{E})$ is positive semidefinite
• Stinespring dilation theorem states that any CP map can be realized as a unitary operation on a larger Hilbert space followed by a partial trace
  • Mathematically, $\mathcal{E}(\rho) = \text{Tr}_E[U(\rho \otimes \ket{0}\bra{0})U^\dagger]$, where $U$ is a unitary operator acting on the system and an ancillary environment, and $\ket{0}$ is a fixed ancillary state
  • This representation provides an operational interpretation of CP maps and their connection to open quantum systems

Transformation of quantum states

• Unitary operations transform quantum states by applying a unitary matrix $U$ to the state vector or density matrix
  • For a pure state $\ket{\psi}$, the transformed state is $U\ket{\psi}$
  • For a density matrix $\rho$, the transformed state is $U\rho U^\dagger$
• Non-unitary operations, such as measurements or dissipative processes, transform quantum states according to their Kraus representation
  • Applying Kraus operators $E_i$ to a state $\rho$ yields a new state $\mathcal{E}(\rho) = \sum_i E_i \rho E_i^\dagger$
  • Projective measurements transform a state $\rho$ into a post-measurement state $\rho_i = P_i \rho P_i / \text{Tr}(P_i \rho)$ with probability $p_i = \text{Tr}(P_i \rho)$, where $P_i$ are the measurement projectors

Effects on quantum systems

• State fidelity quantifies the similarity between two quantum states $\rho$ and $\sigma$
  • Fidelity is defined as $F(\rho, \sigma) = \text{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$ and ranges from 0 (orthogonal states) to 1 (identical states)
  • For pure states $\ket{\psi}$ and $\ket{\phi}$, fidelity reduces to the squared overlap $F(\ket{\psi}, \ket{\phi}) = |\braket{\psi|\phi}|^2$
• Quantum channel capacity characterizes the maximum rate at which information can be reliably transmitted through a quantum channel $\mathcal{E}$
  • Classical capacity $C(\mathcal{E})$ quantifies the maximum rate of classical information transmission, given by $C(\mathcal{E}) = \max_{p_i, \rho_i} \chi(\{p_i, \mathcal{E}(\rho_i)\})$, where $\chi$ is the Holevo information
  • Quantum capacity $Q(\mathcal{E})$ quantifies the maximum rate of quantum information transmission, given by $Q(\mathcal{E}) = \lim_{n \to \infty} \frac{1}{n} \max_{\rho} I_c(\rho, \mathcal{E}^{\otimes n})$, where $I_c$ is the coherent information
• Quantum error correction aims to protect quantum information from errors induced by quantum operations
  • Quantum information is encoded into a larger Hilbert space using error-correcting codes (stabilizer codes, topological codes)
  • Syndrome measurements are performed to detect and correct errors without disturbing the encoded information
  • Examples of quantum error-correcting codes include the Shor code, the Steane code, and the surface code