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Some Decompositions of Matrix Variances

Léka Z., Petz D.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

191--199

When D is a density matrix and A1, A2 are self-adjoint operators, then the standard variance is a 2 × 2 matrix: VarD(A1, A2)i,j := TrDAiAj − (TrDAi)(TrDAj) (1 ≤ i, j ≤ 2). The main result in this work is that there are projections Pk such that D = Σk λk Pk with 0 < λk and Σk λk = 1 and VarD (A1, A2) = Σk λk VarPk (A1, A2). In a previous paper only the A1 = A2 case was included and the relevance is motivated by the paper [8].