L6-02 · Channels in the Choi representation
Use an explicit Choi convention to recover channel outputs.
Level 6 · Graduate
Objective. Use an explicit Choi convention to recover channel outputs.
Before this lesson. L5-06, L3-04
Hands-on lab. Qubits and the Bloch sphere
Learn¶
A Choi operator represents a linear map as an operator on input and output spaces. This representation makes channel constraints accessible through matrix tests. Several normalization and transpose conventions exist. State the convention before importing a formula from another paper.
Use the unnormalized maximally entangled vector sum_i |i>|i>. Apply the identity map to its first half and the channel to its second half. A channel has a positive Choi operator. Trace preservation requires that tracing out the output gives the input identity.
The channel action then contains a transpose on the input density operator. Omitting that transpose changes predictions for complex states. In causal networks, the same issue appears when connecting operators. A correct diagram must also specify the algebra used to connect its ports.
Practice¶
Solution
Its Choi operator is I_in tensor sigma. Tracing out the output gives I_in times Tr(sigma), hence I_in. The recovery formula gives Tr(rho) sigma, which equals sigma for a state.
Check your understanding¶
What is the trace of a trace-preserving qubit Choi operator in this convention?
A. One half
B. One
C. Two
Show the answer
C. Two
The output partial trace is the two-dimensional input identity. Taking the remaining trace therefore gives two in this unnormalized convention.
Sources¶
This lesson draws on Watrous, 2018Allen et al., 2017.
Next: L6-03 · Quantum DAGs and Markov assumptions
- Watrous, J. (2018). The Theory of Quantum Information. https://cs.uwaterloo.ca/~watrous/TQI/
- Allen, J.-M. A., Barrett, J., Horsman, D. C., Lee, C. M., & Spekkens, R. W. (2017). Quantum common causes and quantum causal models. Physical Review X, 7(3). 10.1103/PhysRevX.7.031021