This is Marc Rieffel's "improvement" on classical Gromov-Hausdorff distance (GHD). It may or may not be of use to some physicists.

The QGHD between two compact quantum metric spaces {A, L(A)} and {B, L(B)} is defined to be the infinum of the GHD between the state spaces of the order unit spaces A and B, where the metric used to calculate the GHD is the metric obtained from a seminorm allowed to vary over all seminorms on the direct sum of A and B which yield L(A) and L(B) when restricted to A and B respectively.

Rieffel has shown that the QGHD satisfies the triangle inequality, and that if the QGHD between two compact quantum metric spaces is 0 then the two spaces are isometrically isomorphic. Symmetry is obvious, so QGHD has the three standard properties of distance.

Source: Rieffel, Marc. Gromov-Hausdorff Distance for Quantum Metric Spaces. 2001, draft manuscript/in press.

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