C_{i} = L_{i} L_{i}^{*}

and

C_{j} = L_{j} L_{j}^{*}

R

d_{P} (C_{i}, C_{j}) = in f_{R} ∥ L_{i} - L_{j} R ∥_{HS} .

Unitary invariance of the Hilbert-Schmidt norm gives

∥ L_{i} - L_{j} R ∥_{HS}^{2} = ∥ L_{i} ∥_{HS}^{2} + ∥ L_{j} ∥_{HS}^{2} - 2 Re tr (L_{i}^{*} L_{j} R) .

sup_{R} Re tr (L_{i}^{*} L_{j} R) = ∥ L_{j}^{*} L_{i} ∥_{1} = \sum_{k = 1}^{\infty} σ_{k},

where the last equality expresses the trace norm as the sum of the singular values. Taking the infimum proves

d_{P} (C_{i}, C_{j})^{2} = ∥ L_{i} ∥_{HS}^{2} + ∥ L_{j} ∥_{HS}^{2} - 2 \sum_{k = 1}^{\infty} σ_{k} .

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