Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 1 / ii

Table of contents
- Solution ii

Solution

For a fixed input, the Holevo–Helstrom theorem gives the optimal error for the two output states. Optimizing the input and quantum ancilla therefore gives

P_{err}^{anc} = \frac{1}{2} (1 - ∥ p T_{1} - (1 - p) T_{2} ∥_{⋄}) .

The stabilization in the diamond norm is exactly the optimization over ancillary systems. Without an ancilla the same argument instead gives

P_{err}^{no anc} = \frac{1}{2} (1 - ∥ p T_{1} - (1 - p) T_{2} ∥_{1}),

where

∥ \cdot ∥_{1}

is the induced trace norm.

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