ENTCALC: Toolkit for calculating geometric entanglement in multipartite quantum systems

TL;DR

entcalc provides a robust numerical toolkit for estimating multipartite geometric entanglement by bracketing the true value with complementary lower- and upper-bound bounds. It combines SDP-based PPT-relaxation bounds and purity-based relaxations with a gradient-descent upper-bound procedure, enabling high-precision estimates for both pure and mixed states. The paper demonstrates the method on PPT entangled 3×3 states, GHZ/W mixtures, thermal states of spin chains, and noisy multipartite states, revealing phase-transition signatures and long-range entanglement activation under fields. The results show very small gaps between bounds across tested cases, offering a practical framework for studying entanglement in realistic quantum systems and informing experimental and theoretical analyses.

Abstract

We present entcalc, a Python and MATLAB package for estimating the geometric entanglement of multipartite quantum states. The package operates as follows: given a multipartite quantum state as input, it outputs an estimate of its geometric entanglement. For pure states, it computes the geometric entanglement together with an estimation error. For mixed states, it provides both lower and upper bounds on the geometric entanglement, thereby identifying an interval in which the true value lies. We provide several methods to compute the lower bound, enabling users to balance accuracy against computational cost. We apply entcalc to several representative examples, including for $3\otimes3$ PPT entangled states, mixtures of GHZ and W states, thermal states of selected three-qubit spin chains, and noisy GHZ and W states. We observe signatures of quantum phase transitions by quantifying entanglement in spin chains. We also demonstrate that entanglement between non-neighbouring sites can be activated by tuning the external magnetic field. In all tested cases, the gap between the lower and upper bounds is found to be very small, indicating that entcalc provides highly accurate estimates of the geometric entanglement for these states.

Figures (10)

• Figure 1: The lower bound on the geometric entanglement for the PPT entangled state defined in Eq. \ref{['eq::ppt33']}. Although the state remains PPT entangled for all $a\in(0,1),$ our method successfully quantifies its entanglement. The maximal difference between the lower and upper bounds is $5.15\times10^{-6}.$
• Figure 2: Plot of the geometric entanglement of the mixture from Eq. \ref{['eq::ghzmix']} for a system consisting of 3, 4 and 5-parties, each having one qubit. The maximal difference between lower and upper bound was smaller than $5.05\times 10^{-6}$ for 3 parties, $1.5\times 10^{-5}$ for 4 parties and $3.5\times 10^{-3}$ for 5 parties.
• Figure 3: Thermal entanglement of three qubit Heisenberg spin chain with Hamiltonian \ref{['eq::hxx']}, with $J=1$ in blue and $J=-1$ in orange. The difference between the calculated lower and upper bound difference is at most $2.42\times10^{-6}.$ State with $J=-1$ is multipartite entangled even though it is pairwise separable.
• Figure 4: The geometric entanglement of a XX model. The plot has the same setting as Fig. \ref{['fig::hxx']}, but we plot only those values of $\beta$ which gives PPT entangled state. For $J=-1$ states with higher $\beta$ are bound entangled. The difference between the calculated lower and upper bound is atmost $4.5\times 10^{-6}$.
• Figure 5: The plot shows the geometric entanglement of a spin chain described by the XXX model with a magnetic field, as a function of the inverse temperature. Blue dots correspond to a weak magnetic field ($h = 1$), orange dots to a medium field ($h = 1.5$), and green dots to a strong field ($h = 2$). Note the distinct values of geometric entanglement in the ground state ($\beta \rightarrow \infty$), which suggests the possible presence of a quantum phase transition in this system. Each curve consists of 155 sample points, and the maximum estimation error is $3.81 \times 10^{-6}$.

Theorems & Definitions (6)

• Theorem 1
• proof
• Definition B.1
• Definition B.2
• Theorem 2
• proof