Quantum Wasserstein isometries of the $n$-qubit state space: a Wigner-type result

We determine the isometry group of the $n$-qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all $n \in \mathbb{N}.$ It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.

💡 Research Summary

The paper investigates the group of isometries of the n‑qubit state space when the distance is given by a quantum Wasserstein metric induced by the so‑called symmetric transport cost. The symmetric cost is generated by the full set of tensor products of Pauli operators (including the identity), i.e. the observable family
A = { ⊗{i=1}^n σ{m(i)} | m(i)∈{0,1,2,3} }.
For a generic collection of observables A = {A_k}, the authors recall the De Palma‑Trevizan framework: a quantum coupling between two states ρ and ω is realized by a quantum channel Φ sending ρ to ω, and the associated coupling Π_Φ is built from the canonical purification of ρ. The quadratic transport cost of a coupling is defined as the expectation of the cost operator
C_A = Σ_k (A_k⊗I – I⊗A_k^T)^2.
The quantum Wasserstein distance D_A(ρ,ω) is the square root of the minimal cost over all couplings.

Specialising to the symmetric cost, the cost operator becomes
C_sym,n = Σ_{m∈{0,1,2,3}^n} (⊗{i=1}^n σ{m(i)}⊗I – I⊗⊗{i=1}^n σ{m(i)}^T)^2.
Proposition 2 shows that C_sym,n simplifies dramatically: it is a scalar multiple of the projector onto the orthogonal complement of the maximally mixed “identity” vector |I⟩⟨I| in the Hilbert–Schmidt space. Explicitly,
C_sym,n = 2^{2n+1} I – 2^n |I⟩⟨I|,
so its spectrum consists of two eigenvalues only (0 and a positive constant), and the eigenvectors are precisely the tensor products of Pauli matrices. This common eigenbasis is crucial for the subsequent analysis.

The authors then examine any map Φ: S(ℂ^{2^n})→S(ℂ^{2^n}) that preserves the distance D_sym,n. By expanding a state ρ in the Pauli basis, one obtains a real vector v(ρ) of expectation values v_m = tr