  Quantum entanglement is a key resource in quantum technology, and its
quantification is a vital task in the current Noisy Intermediate-Scale Quantum
(NISQ) era. This paper combines hybrid quantum-classical computation and
quasi-probability decomposition to propose two variational quantum algorithms,
called Variational Entanglement Detection (VED) and Variational Logarithmic
Negativity Estimation (VLNE), for detecting and quantifying entanglement on
near-term quantum devices, respectively. VED makes use of the positive map
criterion and works as follows. Firstly, it decomposes a positive map into a
combination of quantum operations implementable on near-term quantum devices.
It then variationally estimates the minimal eigenvalue of the final state,
obtained by executing these implementable operations on the target state and
averaging the output states. Deterministic and probabilistic methods are
proposed to compute the average. At last, it asserts that the target state is
entangled if the optimized minimal eigenvalue is negative. VLNE builds upon a
linear decomposition of the transpose map into Pauli terms and the recently
proposed trace distance estimation algorithm. It variationally estimates the
well-known logarithmic negativity entanglement measure and could be applied to
quantify entanglement on near-term quantum devices. Experimental and numerical
results on the Bell state, isotropic states, and Breuer states show the
validity of the proposed entanglement detection and quantification methods.
