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arXiv:2412.07032 · Physical Review A

Entanglement of assistance as a measure of multiparty entanglement

Indranil Biswas, Atanu Bhunia, Subrata Bera, Indrani Chattopadhyay, Debasis Sarkar

Published Mar 21, 2025• 1 citation
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Abstract

Quantifying multipartite entanglement poses a significant challenge in quantum information theory, prompting recent advancements in methodologies to assess it. We introduce the notion of ``volume of assistance'' (VoA), which computes the geometric mean of entanglement of assistance across all potential parties. We demonstrate the feasibility of VoA for three-qubit pure states and certain classes of pure tripartite qudit states. We have extended this measure to four-qubit states and general multipartite scenarios. We have done a comparative analysis to illustrate VoA's distinctiveness from established entanglement measures, notably showing it serves as an upper bound for the much celebrated generalized geometric measure. Remarkably, VoA excels in distinguishing a broad class of states that elude differentiation by the recently proposed Minimum Pairwise Concurrence measure. Finally, VoA is applied to quantify genuine entanglement in the ground states of a three-qubit Heisenberg $XY$ model, which highlights its practical utility in quantum information processing tasks.

DOI
10.1103/physreva.111.032423

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Quantifying multipartite entanglement poses a significant challenge in quantum information theory, prompting recent advancements in methodologies to assess it. We introduce the notion of ``volume of assistance'' (VoA), which computes the geometric mean of entanglement of assistance across all potential parties. We demonstrate the feasibility of VoA for three-qubit pure states and certain classes of pure tripartite qudit states. We have extended this measure to four-qubit states and general multipartite scenarios. We have done a comparative analysis to illustrate VoA's distinctiveness from established entanglement measures, notably showing it serves as an upper bound for the much celebrated generalized geometric measure. Remarkably, VoA excels in distinguishing a broad class of states that elude differentiation by the recently proposed Minimum Pairwise Concurrence measure. Finally, VoA is applied to quantify genuine entanglement in the ground states of a three-qubit Heisenberg XYXYXY model, which highlights its practical utility in quantum information processing tasks.
We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite mixed states. We show that this assisted entanglement, when measured in terms of the generalized concurrence (called the GGG concurrence), is (tightly) bounded by an entanglement monotone, which we call the GGG concurrence of assistance. The GGG concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the geometric mean of the Schmidt numbers. For a large (nontrivial) class of $(d\ifmmode\times\else\texttimes\fi{}d)$-dimensional mixed states, we are able to generalize Wootters' formula for the concurrence into lower and upper bounds on the GGG concurrence. Moreover, we have found an explicit formula for the GGG concurrence of assistance that generalizes the expression for the concurrence of assistance for a large class of $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite pure states.
In this paper, we propose the geometric mean of bipartite concurrences as a genuine multipartite entanglement measure. This measure achieves the maximum value for absolutely maximally entangled states and has desirable properties for quantifying potential quantum resources. The simplicity and symmetry in the definition facilitates its computation for various multipartite entangled states including the GHZ states and the WWW states. With explicit examples, we show that our measure results in distinct entanglement orderings from other measures and can detect differences in certain types of genuine multipartite entanglement while other measures cannot. These results justify the potential application of our measure for tasks involving genuine multipartite entanglement.
We propose an entanglement tensor to quantitatively compute the entanglement of a general pure multipartite quantum state. We compare the ensuing tensor with the concurrence for bipartite state and apply the tensor measure to some interesting examples of entangled three-qubit and four-qubit states. It is shown that in defining the degree of entanglement of a multi-partite state, one needs to make assumptions about the willingness of the parties to cooperate. For such states our tensor becomes a measure of generalized entanglement of assistance. We also discuss the degree of entanglement and the concurrence of assistance of two generic multi-qubit states.
The entanglement of assistance quantifies the entanglement that can be generated between two parties, Alice and Bob, given assistance from a third party, Charlie, when the three share a tripartite state and where the assistance consists of Charlie initially performing a measurement on his share and communicating the result to Alice and Bob through a one-way classical channel. We argue that if this quantity is to be considered an operational measure of entanglement, then it must be understood to be a tripartite rather than a bipartite measure. We compare it with a distinct tripartite measure that quantifies the entanglement that can be generated between Alice and Bob when they are allowed to make use of a two-way classical channel with Charlie. We show that the latter quantity, which we call the entanglement of collaboration, can be greater than the entanglement of assistance. This demonstrates that the entanglement of assistance (considered as a tripartite measure of entanglement), and its multipartite generalizations such as the localizable entanglement, are not entanglement monotones, thereby undermining their operational significance.
Determining whether a subspace spanned by certain quantum states is entangled and its entanglement dimensionality remains a fundamental challenge in quantum information science. This paper introduces a geometric measure of rrr-bounded rank, Er(S){E}_{r}(\mathcal{S})Er​(S), for a given subspace S\mathcal{S}S. Derived from the established geometric measure of entanglement, this measure is specifically designed to assess the entanglement within S\mathcal{S}S. It not only serves as a tool for determining the entanglement dimensionality but also illuminates the subspace's capacity to preserve such entanglement. By employing developed nonconvex optimization techniques utilized in machine learning area, we can accurately calculate Er(S){E}_{r}(\mathcal{S})Er​(S) within the manifold optimization framework. Our approach demonstrates notable advantages over existing hierarchical methods, PPT relaxation techniques, and the seesaw strategy, particularly by combining computational efficiency with broad applicability. More importantly, it paves the way for high-dimensional entanglement certification, which is crucial for numerous quantum information tasks. We showcase its effectiveness in validating high-dimensional entangled subspaces in bipartite systems, determining the border rank of multipartite pure states, and identifying genuinely or completely entangled subspaces.
Abstract Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of genuine multipartite entanglement for four-qubit systems. Based on the concurrence of nine different classes of four-qubit states, with each class being closed under stochastic local operation and classical communication, we construct a concurrence tetrahedron. Proper genuine four-qubit entanglement measure is presented by using the volume of the concurrence tetrahedron. For non genuine entangled pure states, the four-qubit entanglement measure classifies the bi-separable entanglement. We show that the concurrence tetrahedron based measure of genuine four-qubit entanglement is not equivalent to the genuine four-partite entanglement concurrence. We illustrate the advantages of the concurrence tetrahedron by detailed examples.