A variational quantum algorithm for entanglement quantification

TL;DR

The paper tackles the problem of quantifying entanglement for general mixed states by defining a Bures-entanglement measure and proposing a variational quantum algorithm (EvQA) that leverages Uhlmann's theorem to recast fidelity as an overlap between purifications. By employing the SWAP test, EvQA estimates $F(\rho,\sigma)$ and minimizes the Bures distance $\mathcal{D}^2_B(\rho,\sigma)=2\left(1-\sqrt{F(\rho,\sigma)}\right)$ over purifications of separable (free) states, with a controllable cardinality $N$ for the separable decomposition. The method is demonstrated on two-, three-, and four-qubit states, reproducing known analytic benchmarks (including bound-entangled regimes) and illustrating how the expressiveness of the ansatz affects convergence and trainability. Beyond entanglement, the framework extends naturally to other quantum resources and enables reconstruction of the closest free states, offering a scalable approach for resource quantification on near-term quantum devices and a pathway to broader quantum-information tasks.

Abstract

Quantum entanglement is a foundational resource in quantum information science, underpinning applications across physics. However, detecting and quantifying entanglement remains a significant challenge. In this article, we introduce a variational quantum algorithm inspired by Uhlmann's theorem to quantify the Bures entanglement of general quantum states, a method that naturally extends to other quantum resources, including genuine multipartite entanglement, quantum discord, quantum coherence, and total correlations, while also enabling reconstruction of the closest free states. The algorithm requires a polynomial number of ancillary qubits and circuit depth relative to the system size, dimensionality, and free state cardinality, making it scalable for practical implementations. Thus, it provides a versatile framework for quantifying quantum resources, demonstrated here through several applications.

Figures (16)

• Figure 1: Illustration of the quantum circuit used to calculate the fidelity between $|\Psi'\rangle := |\Psi'(\rho)\rangle$ and $|\Phi'\rangle := |\Phi'(\sigma)\rangle$.
• Figure 2: Comparison between the analytical expression of the Bures entanglement and EvQA results for the Werner state \ref{['eq:wenerState']}. The hyperparameters are given by: $N=2$ (number of qubits in subsystem $C$); $l_1=1$ (number of layers employed in the parametrization of $V_C$); $l_2=16$ (number of layers in the parametrization of $U_C$); $\eta=0.01$ (learning rate); and $1000$ Epochs.
• Figure 3: EvQA results for the tripartite noisy graph-state \ref{['eq:lineargraphstate']}. The hyperparameters are given by: $N=3$ (number of qubits in subsystem $C$); $l_1=1$ (number of layers employed in the parametrization of $V_C$); $l_2=24$ (number of layers in the parametrization of $U_C$); $\eta=0.01$ (learning rate); and $1500$ Epochs.
• Figure 4: EvQA results for the fourpartite bound entangled Smolin state \ref{['eq:noiseSmolinState']}. The hyperparameters are given by: $N=5$ (number of qubits in subsystem $C$); $l_1=2$ (number of layers employed in the parametrization of $V_C$); $l_2=36$ (number of layers in the parametrization of $U_C$); $\eta=0.01$ (learning rate); and $3000$ Epochs.
• Figure 5: Illustration of the circuit used for the preparation of the $|\Psi(\rho_{sep}^{AB})\rangle$ state from Eq. \ref{['eq:rho_sep_AB_purification']}.