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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 323

Each rank-one <density operator> is
$$
|z\rangle\langle z|
=\frac1d\sum_{j,k}z_jz_k^*|j\rangle\langle k|,
\qquad
|z^*\rangle\langle z^*|
=\frac1d\sum_{j',k'}z_{j'}^*z_{k'}|j'\rangle\langle k'|.
$$
Taking their <tensor product> and averaging over the $4^d$ independent choices of the fourth roots of unity gives
$$
\boxed{
\sigma
=\frac1{4^dd^2}
\sum_{j,k,j',k'}
\left(\sum_z z_jz_k^*z_{j'}^*z_{k'}\right)
|j\rangle\langle k|\otimes|j'\rangle\langle k'|}.
$$
This is the claimed expansion in <matrix elements>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 2 / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 2 / iii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 2 / iii / Solution

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