David Jekel: Quantum Wasserstein metrics for the quantum permutation group

Title: Quantum Wasserstein metrics for the quantum permutation group

Abstract: We describe several quantum analogs of the Hamming distance on the permutation group S_n, using blend of ideas from quantum optimal transport theory, non-commutative geometry, and free probability.  The quantum permutation group S_n⁺ is a quantum group that describes the quantum symmetries of an n-point space.  The C*-algebra C(S_n⁺) is the universal C*-algebra generated by projections (u_ij)_i,j=1ⁿ where the row and column sums are 1.  In particular, any collection of n² projection matrices in M_k(C) satisfying these conditions defines a representation of C(S_n⁺), and by pulling back the trace on M_k(C) one obtains a tracial state on C(S_n⁺).  We define three distances on the space of tracial states on C(S_n⁺) that serve as quantum analogs of the L¹-Wasserstein distance on P(S_n) where the underlying metric is the Hamming distance.  We prove inequalities that compare these metrics to each other, as well as describe how they naturally interact with the quantum group (co)multiplication.

This is based on joint work with Anshu and Therese Landry, arXiv:2505.19269.