In this paper, we investigate the characterization of multipartite entanglement based on the unified
(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure
Eq,sA∣B(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures
Aq,sk(ρ) and
Gq,sk(ρ) for quantifying
k-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters
q and
s are in certain ranges,
Aq,sk(ρ) and
Gq,sk(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures
Aq,sk(ρ) and
Gq,sk(ρ) with respect to the parameters
q and
s for fixed quantum states. In addition, for fixed
q and
s, we also give their order relations for any two pure states.
We propose to use a set of averaged entropies, the multiple entropy measures (MEMS), to partially quantify quantum entanglement of multipartite quantum state. The MEMS is vector-like with m = [N/2] components: [S 1 , S 2 , ..., S m ], and the i-th component S i is the geometric mean of i-qubits partial entropy of the system. The S i measures how strong an arbitrary i qubits from the system are correlated with the rest of the system. It satisfies the conditions for a good entanglement measure. We have analyzed the entanglement properties of the GHZ-state, the W-states, and cluster-states under MEMS.
We propose a new approach to the problem of defining the degree of entanglement between two particles in a pure state with Hilbert spaces of arbitrary finite dimensions. The central idea is that entanglement gives rise to correlations between the particles that do not occur in separable states. We individuate the contributions of these correlations to the joint and the conditional probabilities of local measurements outcomes. We use these probabilities to define the measure of entanglement. Our measure turns out to be proportional to the so-called 2-entropy and therefore satisfies the properties required for any measure of entanglement. We conclude with an outlook on the problem of extending our approach to the case of multipartite systems and mixed states.
A natural way of quantifying the degree of entanglement for a pure quantum state is to compare how far this state is from the set of all unentangled pure states. This geometric measure of entanglement is explored for bipartite and multipartite pure and mixed states. It is determined analytically for arbitrary two‐qubit mixed states and for generalized Werner and isotropic states. It is also applied to certain multipartite mixed states, including two multipartite bound entangled states discovered by Smolin and Dür. Moreover, the geometric measure of entanglement is applied to the ground state of the Ising model in a transverse magnetic field. From this model the entanglement is shown to exhibit singular behavior at the quantum critical point.
The paper is an elaborate comparative assessment of the most relevant entropy-based quantities to be applied in quantifying quantum entanglement in pure, mixed, bipartite, and multipartite quantum systems. The study fills the existing gap in the literature in which most of the previous literature examines individual measures, but a lot of knowledge is lacking on what these measures are regarding their strengths, weaknesses, and the operational usefulness against each other. The study can prove the behaviour of each measure under noise, decoherence, dimensional scaling, and variation of state-type by analysing von Neumann entropy, the Tsallis entropy, entropy of entanglement of formation, concurrence and logarithmic negativity. Numerical studies with pure two-qubit states, Werner states, GHZ states, W states and mixed random states demonstrate specific sensitivity properties and computational limitations which ensure that no single measure can be a universal quantifier. The paper goes on to say that entropy-only measures do not work in mixed regimes because of a dominance in classical correlations. The results have several important implications to quantum communication, quantum simulation, and quantum computing in the NISQ-era, where the choice of measurement has a direct impact on benchmarking, error-correction scheme, and protocol performance based on entanglement. The paper ends with a recommendation of contextual, purpose-based scheme of selecting entanglement measures and suggests a further study on scalable benchmarking, hybrid quantifiers, and operational performance-based entanglement measurements. These observations reinforce the theoretical premise of entanglement studies and enable the advancement of more effective quantum technologies in the field of practice.