Norm Estimates of the Partial Transpose Map on the Tensor Products of Matrices

Norm Estimates of the Partial Transpose Map on the Tensor Products of Matrices We present a norm estimate for the partial transpose map Θ on the tensor product $$M_m \otimes M_n$$ $$\Theta \left( \sum_k A_k \otimes B_k \right) := \sum_k A_k \otimes B_k^T$$ with respect to a unitarily invariant norm. This is related to the norm estimates of the following maps on M m,n in terms of the spectral norm of $$\displaystyle \sum_k A_k \otimes B_k$$ : $$X \longmapsto \sum_k A_k X B_k \quad \textrm{and} \quad X \longmapsto \sum_k A_k X B_k^T.$$ We show further that in the special case of $$A \otimes I_n + I_m \otimes B$$ as well as AX + XB and AX + XB T those estimates are much improved and that $$\| A \otimes I_n + I_m \otimes B^T \|_p = \| A \otimes I_n + I_m \otimes B \|_p$$ for certain Schatten p-norms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Norm Estimates of the Partial Transpose Map on the Tensor Products of Matrices

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Publisher
Birkhäuser-Verlag
Copyright
Copyright © 2007 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2121-7
Publisher site
See Article on Publisher Site

Abstract

We present a norm estimate for the partial transpose map Θ on the tensor product $$M_m \otimes M_n$$ $$\Theta \left( \sum_k A_k \otimes B_k \right) := \sum_k A_k \otimes B_k^T$$ with respect to a unitarily invariant norm. This is related to the norm estimates of the following maps on M m,n in terms of the spectral norm of $$\displaystyle \sum_k A_k \otimes B_k$$ : $$X \longmapsto \sum_k A_k X B_k \quad \textrm{and} \quad X \longmapsto \sum_k A_k X B_k^T.$$ We show further that in the special case of $$A \otimes I_n + I_m \otimes B$$ as well as AX + XB and AX + XB T those estimates are much improved and that $$\| A \otimes I_n + I_m \otimes B^T \|_p = \| A \otimes I_n + I_m \otimes B \|_p$$ for certain Schatten p-norms.

Journal

PositivitySpringer Journals

Published: Oct 29, 2007

References

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