Quantum Wasserstein isometries on the qubit state space

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.

18 pages, 3 figures. v2: new references added, v3: minor changes

References in corpus (3)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• Isometric Embeddings into Heisenberg Groups
• Isometric rigidity of Wasserstein tori and spheres

Cited by in corpus (6)

• Quantum Wasserstein distance based on an optimization over separable states
• Quantum Optimal Transport: Quantum Channels and Qubits
• On the metric property of quantum Wasserstein divergences
• Extending quantum detailed balance through optimal transport
• Isometries of the qubit state space with respect to quantum Wasserstein distances
• Critical Scaling of the Quantum Wasserstein Distance