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Characterization of Bistochastic Kadison–Schwarz Operators on \(M_2(\mathbb C)\)

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Abstract

We characterize bistochastic Kadison–Schwarz operators acting on \(M_2(\mathbb C)\). The obtained characterization allows us to find positive operators that are not Kadison–Schwarz ones. Moreover, we provide several examples of Kadison–Schwarz operators which are not completely positive.

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Acknowledgments

We are grateful to an anonymous referee, whose useful suggestions and comments allowed us to improve the presentation of this paper.

Funding

The work is supported by the UAEU UPAR grant (grant code G00003447).

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Correspondence to Farrukh Mukhamedov.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 313, pp. 178–191 https://doi.org/10.4213/tm4179.

Appendix. Diagonalizability

Let \(\Phi\) be a bistochastic KS operator on \(M_2({\mathbb C})\). Then it can be represented by (3.2). Following [19], let us decompose the matrix \(T\) as \(T=RS\), where \(R\) is a rotation and \(S\) is a self-adjoint matrix (see [19]). Define a mapping \(\Phi_S\) as follows:

$$ \Phi_S(w_0 {\mathbb I} + {\mathbf w} \cdot {\boldsymbol\sigma} )=w_0 {\mathbb I} +(S {\mathbf w} )\cdot {\boldsymbol\sigma} .$$
(A.1)
Every rotation is implemented by a unitary matrix in \(M_2({\mathbb C})\); therefore, there is a unitary \(U\in M_2({\mathbb C})\) such that
$$ \Phi(x)=U\Phi_S(x)U^*, \qquad x\in M_2({\mathbb C}).$$
(A.2)

On the other hand, every self-adjoint operator \(S\) can be diagonalized by some unitary operator; i.e., there is a unitary \(V\in M_2({\mathbb C})\) such that \(S=VD_{\lambda_1,\lambda_2,\lambda_3}V^*\), where

$$ D_{\lambda_1,\lambda_2,\lambda_3}=\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{pmatrix},\qquad \lambda_1,\lambda_2,\lambda_3\in{\mathbb R}.$$
(A.3)

Consequently, the mapping \(\Phi\) can be represented as

$$ \Phi(x)= \widetilde{U}{} \Phi_{D_{\lambda_1,\lambda_2,\lambda_3}}(x) \widetilde{U}{} ^*, \qquad x\in M_2({\mathbb C}),$$
(A.4)
for some unitary \( \widetilde{U}{} \). Due to Theorem 2.1 the mapping \(\Phi_{D_{\lambda_1,\lambda_2,\lambda_3}}\) is also a KS operator. Hence, all bistochastic KS operators can be characterized by \(\Phi_{D_{\lambda_1,\lambda_2,\lambda_3}}\) and unitaries. In what follows, for the sake of shortness we denote the mapping \(\Phi_{D_{\lambda_1,\lambda_2,\lambda_3}}\) by \(\Phi_{(\lambda_1,\lambda_2,\lambda_3)}\). It is easy to observe from (3.3) that \(|\lambda_k|\leq1\), \(k=1,2,3\).

Using Theorem 2.1, one can characterize KS operators of the form \(\Phi_{(\lambda_1,\lambda_2,\lambda_3)}\).

Theorem A.1 [8].

Let \(|\lambda_k|\leq 1,\) \(k=1,2,3\). If one has

$$ \lambda_1^2+\lambda_2^2+\lambda_3^2\leq 1+2\lambda_1\lambda_2\lambda_3,$$
(A.5)
then \(\Phi_{(\lambda_1,\lambda_2,\lambda_3)}\) is a KS operator.

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Mukhamedov, F., Akın, H. Characterization of Bistochastic Kadison–Schwarz Operators on \(M_2(\mathbb C)\). Proc. Steklov Inst. Math. 313, 165–177 (2021). https://doi.org/10.1134/S0081543821020164

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