Quantum Operations - Lecture Notes

Recap from Last Week

Discussed the most general quantum operations on quantum states.
Conclusion: Quantum operations must be completely positive trace-preserving (CPTP) maps.
Derived mathematically and physically as the most general way a quantum state can transform.

Mathematical Statement: If and only if E is a CPTP map, there exists a set of operators ( {E_i} ) such that:
- ( E(\rho) = \sum_i E_i \rho E_i^{\dagger} )
- This is known as the Choi-Kraus-Sudarshan representation theorem for CPTP maps.
Physical Interpretation: Derived the operator-sum representation via a system-environment (or bath) picture where:
- The system interacts with an environment (bath) through a unitary transformation.
- Initial state of system and bath is a product state (\rho_s \otimes \sigma_b).
- Environment typically starts in a pure state, (\sigma_b = |e_0\rangle \langle e_0|).
- The final state of the system alone is obtained by tracing out the environment: (\text{Tr}_B(U(\rho_s \otimes \sigma_b)U^{\dagger})).

Example: If the system is a superconducting qubit on a chip in vacuum, the chip's surroundings are the environment.
The environment is a large system with many degrees of freedom, out of the experimenter’s control.
System’s state is controlled, environment often assumed pure due to purification.
Tracing out the environment involves discarding environmental degrees of freedom, effectively taking the marginal state.
Operator Sum Representation: Based on tracing out the environment:
- Defined by operators ({E_i}) such that (E_i) are derived from the system-bath unitary interaction.
- The operator sum representation helps in characterizing noise and decoherence effects in quantum operations.
Kraus Operators: Derived from tracing out the environment, related to the number of degrees of freedom in the environment.

Consider a system-bath scenario with system initially in state ( \rho_s ) and environment in state ( |0\rangle ).
A CNOT gate acting on the composite system (system and bath) as a joint unitary operation.
The environment is then traced out.
The final state of the system, ( \rho_s'), results from an operator-sum representation with projectors on 0 and 1.
- ( \rho_s' = P_0 \rho_s P_0 + P_1 \rho_s P_1 )
- ( P_0 = |0\rangle \langle 0|, P_1 = |1\rangle \langle 1| )
Action similar to a projective measurement in the Z basis (leading to phase damping).

Decoherence: Loss of off-diagonal elements in the density matrix due to partial trace over environment.
Measurement Operator: When applied, results in information loss similar to decoherence induced by system-environment interaction.
CPTP Maps as Models of Noise: Showing how interactions with a bath and tracing out the bath leads to noise and decoherence.

Choi Isomorphism: Establishes a one-to-one correspondence between CPTP maps and states on a larger system.
- Define a joint state of the reference system and the system.
- Show that projecting onto a state in the reference system results in the map acting on the system.
- Demonstrates that the action of the map on the system can be captured by projecting onto a corresponding state on a larger reference-system composite system.
**Constructing Kraus Operators: (To be Continued) **
- Defined operators related to pure state decomposition.
- Next steps include describing the formal construction of these operators.

Continue reviewing the lecture material to reinforce understanding.
Practice similar examples to deepen comprehension of the concepts discussed.
Prepare for further discussion on Kraus operator construction in the next lecture.

Clarifications needed on any of the discussed concepts should be brought up in the next class session.