This paper introduces a novel approach to quantifying quantum entanglement using entanglement entropy. Traditional entanglement measures, such as concurrence and entanglement of formation, often struggle to fully capture the intricate correlations present in entangled states. We propose a new definition of entanglement entropy based on the density matrix of the entangled system, explicitly accounting for the correlations between subsystems. This approach leverages information-theoretic entropy calculations to provide a more precise and comprehensive measure of entanglement strength and the degree of correlation. We demonstrate that this entropy-based metric offers advantages over existing measures, particularly in characterizing complex entangled states and distinguishing between different levels of entanglement. The core claim is that entanglement entropy can serve as a more accurate indicator of entanglement than existing metrics. The underlying mechanism centers around a density matrix-based entropy calculation, recognizing the interconnectedness within entangled states. The novelty lies in moving beyond purely state-based descriptions to incorporate a measure of subsystem correlations.
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.
In this paper, we consider the problem of how to quantify entanglement for any multipartite quantum states. For bipartite pure states partial entropy is a good entanglement measure. By using partial entropy, we firstly introduce the Combinatorial Entropy of Fully entangled states (CEF) which can be used to quantify entanglement for any fully entangled pure states. In order to quantify entanglement for any multipartite states we also need another concept the Entanglement Combination (EC) which can be used to completely describe the entanglement between any parties of the given quantum states. Combining CEF with EC, we define the Combinatorial Entropy (CE) for any multipartite pure states and present some nice properties which indicate CE is a good entanglement measure. Finally, we point out the feasibility of extending these three concepts to mixed quantum states.
A new entanglement measure, the multiple entropy measures (MEMS), is proposed to quantify quantum entanglement of multi-partite quantum state. The MEMS is vector-like with
m=[N/2], the integer part of
N/2, components:
[S1,S2,...,Sm], and the
i-th component
Si is the geometric mean of
i-body partial entropy of the system. The
Si measures how strong an arbitrary
i bodies from the system are entangled with the rest of the system. The MEMS is not only transparent in physical picture, but also simple to calculate. 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. The cluster-state is more entangled than the GHZ-state and W-state under MEMS.
k-entanglement provides a hierarchical characterization of multipartite correlations according to their entanglement structure, ranging from multipartite entanglement (
k=n) to genuine multipartite entanglement (
k=2). Its quantitative treatment for mixed states faces three related obstacles: the standard axioms for such quantifications does not incorporate the requirements from many-body resource theory, as it
does not regulate how to quantify the entanglement assigned to part systems of the same multipartite state and how to describe the rule that the entanglement possessed by a part system never exceeds the entanglement possed by whole system;
no explicit measures are known that satisfy the above requirements from many-body resource theory; and the extreme difficulty in numerical evaluation at mixed states of known quantifications. In this work, we first formulate an axiomatic framework for \emph{true}
k-entanglement measures that obeys the many-body resource theory, by supplementing usual faithfulness and LOCC monotonicity with two new ones: unification condition and hierarchy condition. Then, for every
2≤k≤n, we construct a witness-based true
k-entanglement measure
Ew(k,n) without the convex-roof extension, which is valid for both finite- and infinite-dimensional
n-partite systems. Finally, we derive a method of constructing finite-dimensional
k-entanglement witnesses from positive operator and
k-separable-state optimizations, design an evaluation scheme of
Ew(k,n) based on pre-established database of
k-entanglement witnesses sampled by positive operators. To check the feasibility of this computing scheme, we develop a software tool, which implements the computation of the approximation of
Ew(k,n) for systems up to four qubits. Numerical tests show that the software
is sufficiently accurate and reliable.