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arXiv · 2204.14134

Quantum Wasserstein isometries on the qubit state space

György Pál Gehér·József Pitrik·Tamás Titkos·Dániel Virosztek
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Abstract

We describe Wasserstein isometries of the quantum bit state space with respect to distinguished cost operators. We derive a Wigner-type result for the cost operator involving all the Pauli matrices: in this case, the isometry group consists of unitary or anti-unitary conjugations. In the Bloch sphere model, this means that the isometry group coincides with the classical symmetry group $\mathbf{O}(3).$ On the other hand, for the cost generated by the qubit "clock" and "shift" operators, we discovered non-surjective and non-injective isometries as well, beyond the regular ones. This phenomenon mirrors certain surprising properties of the quantum Wasserstein distance.

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György Pál Gehér, József Pitrik, Tamás Titkos, Dániel Virosztek. 2022-10-28. Quantum Wasserstein isometries on the qubit state space. https://doi.org/10.1016/j.jmaa.2022.126955

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