Entanglement of Formation of Mixed Many-Body Quantum States via Tree Tensor Operators

We present a numerical strategy to efficiently estimate bipartite entanglement measures, and in particular the entanglement of formation, for many-body quantum systems on a lattice. Our approach exploits the tree tensor operator tensor network Ansatz, a positive loopless representation for density matrices which, as we demonstrate, efficiently encodes information on bipartite entanglement, enabling the upscaling of entanglement estimation. Employing this technique, we observe a finite-size scaling law for the entanglement of formation in 1D critical lattice models at finite temperature for up to 128 spins, extending to mixed states the scaling law for the entanglement entropy.

Figure 1

(a) The tree tensor operator (TTO) representing a density matrix $\rho =X{X}^{†}$. $K_0$ is the number of pure states in the representation used, while $M$ is the maximal dimension for all bonds. The gray dashed square highlights the root tensor $R$, containing all the information about entanglement between the red and green bipartitions of the physical space. (b) Change of representation for the EOF minimization using $R$, after having compressed the state with some maximal bond dimension $M$. (c) Same as (b), but without compression, so that $M=d^{N/2}$. Optimizations are possible for any system size and state that can be efficiently represented as TTOs.

Figure 2

Computational times of EOF estimation “from scratch” as a function of system size $N$, for critical ${\widehat{H}}_{\text{Ising}}$ ($h=1$) at temperature $k_{B}T=0.5\mathrm{\Delta }$, where $\mathrm{\Delta }\sim N^{-z}$ ($z=1$) is the finite-size gap. Two data sets show EOF estimation without TTO compression: either using the full exact density matrix (green diamonds), or from the exact $X$ matrix at convergence in columns $K_0$ (orange squares). The last data set shows EOF estimation with TTO method, where the Hamiltonian eigenstates were calculated via tree-tensor network eigensolver algorithm. Inset: bond dimensions $M$ required to achieve convergence of the EOF estimator (approximation less than 1% from its exact value). Red pentagons and purple diamonds refer, respectively, to the critical Ising model at $k_{B}T=0.1J$ and to the $XXZ$ model with $\xi =0.5$ (critical) at $k_{B}T=0.5J$.

Figure 3

Scale invariance of the EOF at temperatures $T$ (in units of $J/k_B$) in the range $k_{B}T\le 0.5\mathrm{\Delta }$, where $\mathrm{\Delta }\propto N^{-z}$, for the critical Ising model in Eq. (2) (top panel) and the $XXZ$ model in Eq. (3) in the critical phase at $\xi =0.5$ (bottom panel). Data for $N=16$, 32, 64, 128 (top, approximate method for TTO computation) and $N=8$, 12, 16, 20 (bottom, exact TTO from ED), from the flattest to the steepest curve. Inset: curves in the main figures after rescaling according to Eq. (4). The agreement is stunning, using $c=1/2$ and $z=1.00\pm 0.01$ (top) and $c=1$ and $z=0.98\pm 0.02$ (bottom). The gray area highlights the temperature range $T\le 0.2\mathrm{\Delta }(N)$.