The monogamy of entanglement is one of the basic quantum mechanical features, which says that when two partners Alice and Bob are more entangled then either of them has to be less entangled with the third party. Here we qualitatively present the converse monogamy of entanglement: given a tripartite pure system and when Alice and Bob are weakly entangled, then either of them is generally strongly entangled with the third party. Our result leads to the classification of tripartite pure states based on bipartite reduced density operators, which is an effective way to this longstanding problem compared to the means by stochastic local operations and classical communications. We also systematically indicate the structure of the classified states and generate them.The monogamy of entanglement is one of the basic quantum mechanical features, which says that when two partners Alice and Bob are more entangled then either of them has to be less entangled with the third party. Here we qualitatively present the converse monogamy of entanglement: given a tripartite pure system and when Alice and Bob are weakly entangled, then either of them is generally strongly entangled with the third party. Our result leads to the classification of tripartite pure states based on bipartite reduced density operators, which is an effective way to this longstanding problem compared to the means by stochastic local operations and classical communications. We also systematically indicate the structure of the classified states and generate them.

arXiv:1102.0439Physical Review AJul 21, 201135 citations

Maximal extension of converse monogamy of entanglement for tripartite pure states

Jiaxing An, Soojoon Lee

Unlike classical correlations, entanglement cannot be freely shared among multiple parties. This unique feature of quantum systems is known as the monogamy of entanglement. While it holds for all multipartite pure states, its converse -- weak entanglement between two parties enforces strong entanglement with a third party -- occurs only under specific conditions. In particular, Hayashi and Chen [Phys. Rev. A \textbf{84}, 012325 (2011)] demonstrated a qualitative version of the converse monogamy of entanglement (CMoE) for tripartite pure states by employing a hierarchy of bipartite entanglement defined through the relations among various separability criteria, and Singh and Datta [IEEE Trans. Inf. Theory \textbf{69}, 6564 (2023)] later extended this notion of the CMoE from the viewpoint of distillability under one-way or two-way classical communication. In this work, we extend their results to the CMoE with broader conditions, and furthermore show that our extensions are maximal with respect to the hierarchies they considered.Unlike classical correlations, entanglement cannot be freely shared among multiple parties. This unique feature of quantum systems is known as the monogamy of entanglement. While it holds for all multipartite pure states, its converse -- weak entanglement between two parties enforces strong entanglement with a third party -- occurs only under specific conditions. In particular, Hayashi and Chen [Phys. Rev. A \textbf{84}, 012325 (2011)] demonstrated a qualitative version of the converse monogamy of entanglement (CMoE) for tripartite pure states by employing a hierarchy of bipartite entanglement defined through the relations among various separability criteria, and Singh and Datta [IEEE Trans. Inf. Theory \textbf{69}, 6564 (2023)] later extended this notion of the CMoE from the viewpoint of distillability under one-way or two-way classical communication. In this work, we extend their results to the CMoE with broader conditions, and furthermore show that our extensions are maximal with respect to the hierarchies they considered.

arXiv:2510.27264Physical Review AApr 20, 20260 citations

Fully Undistillable Quantum States Are Separable

Satvik Singh, Nilanjana Datta

Assume that Alice, Bob, and Charlie share a tripartite pure state$ \vert \psi _{ABC}\rangle $. We prove that if Alice cannot distill entanglement with either Bob or Charlie using$ \vert \psi _{ABC}\rangle $and local operations with any one of the following configurations for classical communication:$ (A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C) $, and$ (A\leftrightarrow B, A\leftrightarrow C) $, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems$ AB $and$ AC$are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. In proving these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.Assume that Alice, Bob, and Charlie share a tripartite pure state$ \vert \psi _{ABC}\rangle $. We prove that if Alice cannot distill entanglement with either Bob or Charlie using$ \vert \psi _{ABC}\rangle $and local operations with any one of the following configurations for classical communication:$(A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C)$, and$(A\leftrightarrow B, A\leftrightarrow C)$, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems$AB$and$AC$are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. In proving these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.

arXiv:2207.05193IEEE Transactions on Information TheoryJun 12, 20233 citations

Quantification of Unextendible Entanglement and Its Applications in Entanglement Distillation

Kun Wang, Xin Wang, Mark M. Wilde

The unextendibility or monogamy of entangled states is a key property of quantum entanglement. Unlike conventional ways of expressing entanglement monogamy via entanglement measure inequalities, we develop a state-dependent resource theory to quantify the unextendibility of bipartite entangled states. First, we introduce a family of entanglement measures called unextendible entanglement. Given a bipartite state ρAB, the key idea behind these measures is to minimize a divergence between ρABand any possibly reduced state ρAB' of an extension ρABB' of ρAB. These measures are intuitively motivated by the fact that the more a bipartite state is entangled, the less each of its individual systems can be entangled with a third party. Second, we show that the unextendible entanglement is an entanglement monotone under two-extendible operations, which include local operations and one-way classical communication as a special case. Unextendible entanglement has several other desirable properties, including normalization and faithfulness. As applications, we show that the unextendible entanglement provides efficiently computable benchmarks for the rate of perfect entanglement distillation, as well as for the overhead of entanglement distillation.The unextendibility or monogamy of entangled states is a key property of quantum entanglement. Unlike conventional ways of expressing entanglement monogamy via entanglement measure inequalities, we develop a state-dependent resource theory to quantify the unextendibility of bipartite entangled states. First, we introduce a family of entanglement measures called unextendible entanglement. Given a bipartite state ρAB, the key idea behind these measures is to minimize a divergence between ρABand any possibly reduced state ρAB' of an extension ρABB' of ρAB. These measures are intuitively motivated by the fact that the more a bipartite state is entangled, the less each of its individual systems can be entangled with a third party. Second, we show that the unextendible entanglement is an entanglement monotone under two-extendible operations, which include local operations and one-way classical communication as a special case. Unextendible entanglement has several other desirable properties, including normalization and faithfulness. As applications, we show that the unextendible entanglement provides efficiently computable benchmarks for the rate of perfect entanglement distillation, as well as for the overhead of entanglement distillation.

IEEE International Symposium on Information Theory (ISIT)Jun 1, 20201 citation

The distillability of entanglement of bipartite reduced density matrices of a tripartite state

Yize Sun, Lin Chen

Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.

Journal of Physics A Mathematical and TheoreticalJun 1, 20204 citations

Tripartite genuinely entangled states from entanglement-breaking subspaces

Yize Sun, Lin Chen, Li-Jun Zhao

Abstract The determination of genuine entanglement is a central problem in quantum information processing. We investigate the tripartite state as the tensor product of two bipartite entangled states by merging two systems. We show that the tripartite state is a genuinely entangled (GE) state when the range of both bipartite states are entanglement-breaking (EB) subspaces. We further investigate the tripartite state when one of the two bipartite states has rank two. Our results provide the latest progress on a conjecture proposed in the paper [Yi Shen et al 2020 J. Phys. A 53 125302]. We apply our results to construct multipartite states whose bipartite reduced density operators have additive entanglement of formation (EOF). Further, such states are distillable across every bipartition under local operations and classical communications.Abstract The determination of genuine entanglement is a central problem in quantum information processing. We investigate the tripartite state as the tensor product of two bipartite entangled states by merging two systems. We show that the tripartite state is a genuinely entangled (GE) state when the range of both bipartite states are entanglement-breaking (EB) subspaces. We further investigate the tripartite state when one of the two bipartite states has rank two. Our results provide the latest progress on a conjecture proposed in the paper [Yi Shen et al 2020 J. Phys. A 53 125302]. We apply our results to construct multipartite states whose bipartite reduced density operators have additive entanglement of formation (EOF). Further, such states are distillable across every bipartition under local operations and classical communications.

arXiv:2006.05128Journal of Physics A Mathematical and TheoreticalNov 26, 20205 citations

Assume that Alice, Bob, and Charlie share a tripartite pure state$ \vert \psi _{ABC}\rangle

. We prove that if Alice cannot distill entanglement with either Bob or Charlie using

\vert \psi _{ABC}\rangle

and local operations with any one of the following configurations for classical communication:

(A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C)

, and

(A\leftrightarrow B, A\leftrightarrow C)

, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems

and

AC$are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. In proving these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.