Some Decompositions of Matrix Variances

• Autorzy: Léka Z. , Petz D.
• Języki publikacji: EN

Abstrakty

EN

When D is a density matrix and A₁, A₂ are self-adjoint operators, then the standard variance is a 2 × 2 matrix: Var_D(A₁, A₂)_i,j := TrDA_iA_j − (TrDA_i)(TrDA_j) (1 ≤ i, j ≤ 2). The main result in this work is that there are projections P_k such that D = Σ_k λ_k P_k with 0 < λ_k and Σ_k λ_k = 1 and Var_D (A₁, A₂) = Σ_k λ_k Var_Pk (A₁, A₂). In a previous paper only the A₁ = A₂ case was included and the relevance is motivated by the paper [8].

Słowa kluczowe

EN

density matrix; variance; covariance; decomposition; projections

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 191--199
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Léka Z.

• Alfréd Rényi Institute of Mathematics, H-1364 Budapest, POB 127, Hungary

autor

Petz D.

• Alfréd Rényi Institute of Mathematics, H-1364 Budapest, POB 127, Hungary

Bibliografia

• [1] R. Bhatia, Positive Definite Matrices, Princeton University Press, Princeton 2007.
• [2] O. Gühne, P. Hyllus, O. Gittsovich, and J. Eisert, Covariance matrices and the separability problem, Phys. Rev. Lett. 99 (2007), 130504.
• [3] F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, to appear in Hindustan Book Agency.
• [4] P. D. Lax, Functional Analysis, Wiley, 2002.
• [5] D. Petz, Quantum Information Theory and Quantum Statistics, Springer, Heidelberg 2008.
• [6] D. Petz and G. Tóth, Matrix variances with projections, Acta Sci. Math. (Szeged) 78 (2012), pp. 683-688.
• [7] J. M. Steele, The Cauchy-Schwarz Master Class, Cambridge University Press, Cambridge 2004.
• [8] G. Tóth and D. Petz, Extremal properties of the variance and the quantum Fisher information, Phys. Rev. A 87 (2013), 032324.
• [9] A. Uhlmann, Roofs and convexity, Entropy 12 (2010), pp. 1799-1832.
• [10] S. Yu, Quantum Fisher information as the convex roof of variance, http://arxiv.org/abs/1302.5311.

Uwagi

This paper is dedicated to Rajendra Bhatia on the occasion of his 60th birthday

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).