9.3 Quantum operations and completely positive maps
4 min read•Last Updated on July 23, 2024
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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Circuits of space and time quantum channels – Quantum View original
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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†U=UU†=I, where U† 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 E as a sum of Kraus operators Ei
The action of E on a state ρ is given by E(ρ)=∑iEiρEi†, where Ei are the Kraus operators
Kraus operators must satisfy the completeness relation ∑iEi†Ei=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 E is positive if it maps positive operators to positive operators, i.e., if ρ≥0, then E(ρ)≥0
A map E is completely positive if (I⊗E)(ρ) is positive for any positive operator ρ, where 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 E is given by J(E)=(I⊗E)(∣Φ+⟩⟨Φ+∣), where ∣Φ+⟩=d1∑i=1d∣i⟩⊗∣i⟩ is a maximally entangled state
A map E is CP if and only if its Choi matrix J(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, E(ρ)=TrE[U(ρ⊗∣0⟩⟨0∣)U†], where U is a unitary operator acting on the system and an ancillary environment, and ∣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 ∣ψ⟩, the transformed state is U∣ψ⟩
For a density matrix ρ, the transformed state is UρU†
Non-unitary operations, such as measurements or dissipative processes, transform quantum states according to their Kraus representation
Applying Kraus operators Ei to a state ρ yields a new state E(ρ)=∑iEiρEi†
Projective measurements transform a state ρ into a post-measurement state ρi=PiρPi/Tr(Piρ) with probability pi=Tr(Piρ), where Pi are the measurement projectors
Effects on quantum systems
State fidelity quantifies the similarity between two quantum states ρ and σ
Fidelity is defined as F(ρ,σ)=Trρσρ and ranges from 0 (orthogonal states) to 1 (identical states)
For pure states ∣ψ⟩ and ∣ϕ⟩, fidelity reduces to the squared overlap F(∣ψ⟩,∣ϕ⟩)=∣⟨ψ∣ϕ⟩∣2
Quantum channel capacity characterizes the maximum rate at which information can be reliably transmitted through a quantum channel E
Classical capacity C(E) quantifies the maximum rate of classical information transmission, given by C(E)=maxpi,ρiχ({pi,E(ρi)}), where χ is the Holevo information
Quantum capacity Q(E) quantifies the maximum rate of quantum information transmission, given by Q(E)=limn→∞n1maxρIc(ρ,E⊗n), where Ic 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