bibkey: audenaert2009variance authors: Koenraad M. R. Audenaert year: 2009 title: “Variance bounds, with an application to norm bounds for commutators” doi: null url: https://arxiv.org/abs/0907.3913 claim: “Cartesian variance is the density expectation of (A* A + A A*)/2 minus the squared modulus of the mean; numerical-range convexity and the maximal variance control matrix commutators.” strata_touched:

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Variance bounds, with an application to norm bounds for commutators

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Source: https://arxiv.org/abs/0907.3913

Section 7, page 17, defines the Cartesian modulus and variance:

For that reason we need a name for the expression ((X∗X + XX∗)/2)1/2, and we have chosen to call it the Cartesian modulus.

Each modulus builds a different variance, which we’ll distinguish by the corresponding subscript too.

Equation (27) reads ${\mathrm{Var}}_{\ast }(X)=\mathrm{Tr}[\rho |X{|}_{\ast }^2]-|\mathrm{Tr}[\rho X]{|}^2$, where the subscript is L, R or C. For C, $|X{|}_C^2=(X^{\ast }X+X{X}^{\ast })/2$.

The numerical range on pages 17–18 is $W(X)=\{\psi X{\psi }^{\ast }:\psi \in {\mathbb{C}}^d,\Vert \psi \Vert =1\}$.

Theorem 9 and its proof relate the maximum Cartesian variance to the Cartesian radius; the proof uses the Toeplitz-Hausdorff convexity of the complex numerical range. The commutator application uses the density matrix $(B^{\ast }B+B{B}^{\ast })/(2\Vert B{\Vert }_F^2)$ for nonzero $B$. The pure maximizer is represented by a normalized vector, whose rank-one matrix is $v{v}^{\ast }$.