Critical Scaling of the Quantum Wasserstein Distance

Abstract

Distinguishing quantum states with minimal sampling overhead is of fundamental importance to teach quantum data to an algorithm. Recently, the quantum Wasserstein distance emerged from the theory of quantum optimal transport as a promising tool in this context. Here we show on general grounds that the quantum Wasserstein distance between two ground states of a quantum critical system exhibits critical scaling. We demonstrate this explicitly using known closed analytical expressions for the magnetic correlations in the transverse field Ising model, to numerically extract the critical exponents for the distance close to the quantum critical point, confirming our analytical derivation. Our results have implications for learning of ground states of quantum critical phases of matter.

Figures (3)

• Figure 1: The Wasserstein distance in Eq. \ref{['eq:wd_def']} scaled by $L^2$, as a function of $g_\sigma$ for different values of $g_\rho$, for a system size $L=500$. Regions of magnetic order and disorder are identified by the non-analytic behavior of the distance close to the quantum critical point.
• Figure 2: (a) The QFI (self-distance) scaled by $L^{7/4}$, for different system sizes $L$. Close to the quantum critical point $g=1$, the QFI develops a narrow peak separating the two phases of the model when approaching the thermodynamic limit $L\to\infty$, in accordance with the results of Ref. Hauke2016. (b) The critical exponent for the quantum Wasserstein distance in the TFIM, when the two quantum states $\rho,\sigma$ are close to the critical point $g=1$, as a function of $\tilde{g}_\sigma=|g_\sigma - 1|$. In the limit $g_\rho,g_\sigma\to 1$, we obtain the scaling exponent from Ref. Hauke2016. We note that in regions away from the critical point, the scaling with $L$ might be ill-defined under Assumption 1. The different system sizes used to extract the exponents are $L=20,40,80,120,150,200,250,300,350,400,450,500,600,700$
• Figure 3: (a) The power-law behavior for the sub-leading contribution in Eq. \ref{['eq:main_eq']}, for $g_\rho=0$ and the state $\sigma$ being close to the critical point, for different system sizes $L$. Note that since the contribution is subleading, we have scaled by a factor of $L$ after subtracting the analytic value of $A_\rho=\frac{1}{2}$. Note the logarithmic scale on both axes. The critical exponent is extracted numerically for the largest $L$, showing good agreement with the predicted analytic result $\nu(\eta-1)=-3/4$. (b) Power-law behavior of the leading contribution in Eq. \ref{['eq:main_eq']} when $g_\sigma=10$. The critical exponent is obtained numerically, in close agreement with the predicted analytic value for the TFIM $2\beta=\frac{1}{4}$.