Quantum Wasserstein isometries on the qubit state space

Gehér, György and Pitrik, József and Titkos, Tamás and Virosztek, Dániel (2023) Quantum Wasserstein isometries on the qubit state space. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 522 (2). ISSN 0022-247X

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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 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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