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Nielsen & Chuang · Chapter 8, Quantum noise and quantum operations · p. 373

Exercise 8.10 - The unitary that diagonalizes the Gram matrix of the operation elements gives the same operation with at most d2d^2 nonzero elements

Mixing operation elements into orthogonal ones.

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The Exercise

A quantum operation ℰ\mathcal{E} on a system whose state space has dimension dd acts on a density operator as ℰ(ρ)=∑jEjρEj†\mathcal{E}(\rho) = \sum_j E_j \rho E_j^\dagger, with operation elements E1,…,EmE_1, \dots, E_m whose number mm can be as large as one likes. The exercise mixes the EjE_j by a unitary matrix into at most d2d^2 operation elements of the same operation, orthogonal to one another. The unitary is the one that diagonalizes the matrix of inner products of the EjE_j.

Operators and Their Gram Matrix

Let HH and GG be complex Hilbert spaces, HH of finite dimension dd, and let E1,…,EmE_1, \dots, E_m be linear operators from HH to GG. The Gram matrix WW of the family is the m×mm \times m complex matrix of their trace inner products, Wjk=tr⁡(Ej†Ek).W_{jk} = \operatorname{tr}\bigl(E_j^\dagger E_k\bigr).(1)✦

Let λ1,…,λm\lambda_1, \dots, \lambda_m be the eigenvalues of WW, and uu a unitary m×mm \times m matrix whose aath column is an eigenvector of WW for λa\lambda_a. The diagonalized operation elements F1,…,FmF_1, \dots, F_m of the family are the operators from HH to GG obtained by mixing the EjE_j with uu, Fa=∑j=1mujaEj.F_a = \sum_{j=1}^m u_{ja} E_j.(2)✦

The Claim

Theorem 1. Let WW be the Gram matrix of E1,…,EmE_1, \dots, E_m, with eigenvalues λ1,…,λm\lambda_1, \dots, \lambda_m, and let F1,…,FmF_1, \dots, F_m be their diagonalized operation elements. Then the FaF_a define the same operation as the EjE_j: for every operator ρ\rho on HH, ∑a=1mFaρFa†=∑j=1mEjρEj†.\sum_{a=1}^m F_a \rho F_a^\dagger = \sum_{j=1}^m E_j \rho E_j^\dagger.(3)✦ They are orthogonal in the trace inner product: tr⁡(Fa†Fb)=δabλa.\operatorname{tr}\bigl(F_a^\dagger F_b\bigr) = \delta_{ab} \lambda_a.(4)✦ When G=HG = H, the rank of WW is at most d2d^2, rank⁡W≤d2,\operatorname{rank} W \le d^2,(5)✦ and at most d2d^2 of the FaF_a are nonzero: #{a:Fa≠0}≤d2.\#\{a : F_a \ne 0\} \le d^2.(6)✦

These nonzero FaF_a are the operation elements of ℰ\mathcal{E} that the exercise asks for.

What Is Checked

The claim is stated over the Gram matrix WW of (1) and the diagonalized operation elements FaF_a of (2), for any finite family of operators EjE_j from HH to GG. Of these constructions the library proves that WW is Hermitian.✦