Entanglement of purification: from spin chains to holography

TL;DR

This work investigates the entanglement of purification $E_p$—the minimal entanglement in purifications of a mixed state—across three frameworks: holographic CFTs, Ising spin chains, and random stabilizer tensor networks. It introduces and tests a holographic proposal $E_{ph}$ that equates $E_p$ with the minimal cross-section of the entanglement wedge, and validates this with analytic AdS$_3$/BTZ results, numerical MPS studies of spin chains, and exact results in random stabilizer networks. The findings show substantial reductions in purification entanglement relative to the thermofield double, align with holographic expectations in many regimes, and reveal a nuanced relationship between $E_p$ and the mutual information in different models. The results point to practical gains for tensor-network simulations, illuminate the geometry-information connection in AdS/CFT, and motivate further exploration of time dependence, differential entropy, and bit-thread formalisms in the study of quantum entanglement and emergent geometry.

Abstract

Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.

Figures (20)

• Figure 1:
• Figure 2: Left: For sufficiently large $A$ and $B$, the entanglement wedge is connected (the RT surface $\Sigma$ is shown in red) and the $E_{p}$ is computed by the length of the green geodesic $X$. Center: For small $A$ and $B$, the entanglement wedge is disconnected and the $E_{p}$ is zero. Right: Tensors under a causal cut in MERA (red line) can be gotten rid of by a unitary transformation. In each case, the region to be cut out is shaded in gray.
• Figure 3: Left: we vary the position of $a$ and keep $b$, $c$, $d$ fixed. The values chosen here are $b=0.6\pi$, $c=1.4\pi$ and $d = 1.7\pi$. The green geodesics are the shortest curves connecting $(ab)$ to $(cd)$. Here we use the Beltrami-Klein coordinate system (explained in Appendix \ref{['App:Distance']}), in which geodesics are straight lines. Right: Plot of the $E_{ph}$ as a function of $a$, over the range $a \in [0,b]$. The $E_{ph}$ diverges when $a=b$, and undergoes a phase transition near $a \approx 1.256$ (where the RT surface changes topology). We set $4G_{N}=1$.
• Figure 4: Plot of $E_{ph}$ for 2 adjacent intervals as a function of $\alpha_{2}$, at fixed $\alpha_{1}$. The values of $\alpha_{1}$ are: $\pi/6$ (red), $\pi/4$ (green) and $\pi/3$ (black). We set the cutoff $\epsilon$ to $0.1$ and $4G_{N}=1$.
• Figure 5: Left: The $E_{ph}$ geodesic is in green, and the RT surface (including the horizon) is in red. Right: When the Araki-Lieb inequality is saturated, the $E_{ph}$ coincides with $S(B)$.