Notes on Wasserstein distance and wormholes

Abstract

We develop the Boltzmann-Wasserstein (BW) distance, a temperature-dependent metric on the space of quantum theories, defined as the optimal $W_2$ distance between Boltzmann-weighted energy spectra. Computing it is an optimisation over wormholes: each unitary identification of the two energy bases defines a coupling of the two boundaries in the doubled Hilbert space, and the optimum - the comonotone partition function $C_{\max}$, which pairs states by rank - is the dominant wormhole connecting the two theories. For semiclassical theories differing by a small entropy shift, the normalised BW distance collapses to a squared horizon-area comparator, $\tilde{\mathcal{W}}^2 \approx (\delta A/4G)^2/8$, with the two areas evaluated at equal energy. When the Hamiltonians differ by an operator $V$, the BW distance equals a long-time average of the real-time thermal two-point function of $V$; when the thermal one-point function of $V$ vanishes - for instance for $V$ odd under an unbroken discrete global symmetry - a four-point representation appears at the next order. On the gravity side we construct the classical saddle that computes $C_{\max}$: a Schwinger-Keldysh wormhole built from two Euclidean caps sharing a single horizon, joined by Lorentzian segments that adiabatically interpolate between the two theories. Its on-shell action reproduces the spectral saddle of $C_{\max}$ - both the saddle-point conditions and the on-shell value - and the Lorentzian segments are essential: a purely Euclidean interpolation is exponentially suppressed. The saddle captures only the rearrangement of the spectrum; the perturbative representations retain in addition the variance of the matrix elements of $V$, invisible to the classical geometry. We work out two examples - two BTZ black holes with different cosmological constants and a $T\bar{T}$ deformation of BTZ.