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arXiv:1801.00405 · Physical Review A

Family of bound entangled states on the boundary of the Peres set

Saronath Halder, Manik Banik, Sibasish Ghosh A

Published Jun 24, 2019• 42 citations
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Abstract

Bound entangled (BE) states are strange in nature: a nonzero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though the usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize them for an arbitrary composite quantum system. Here we present a $(d\ensuremath{-}3)/2$-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ with $d$ odd and $d\ensuremath{\ge}5$. The range of each such PPT-BE state is contained in a $2(d\ensuremath{-}1)$-dimensional entangled subspace, whereas the associated UPB subspace is of dimension ${(d\ensuremath{-}1)}^{2}+1$. We further show that each of these PPT-BE states can be written as a convex combination of $(d\ensuremath{-}1)/2$ rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set $\mathcal{P}$ of all PPT states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, namely, the Peres set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points---the $(d\ensuremath{-}3)/2$ simplex---is sitting on the boundary between the set $\mathcal{P}$ and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.

DOI
10.1103/physreva.99.062329

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New family of bound entangled states residing on the boundary of Peres set

Saronath Halder, Manik Banik, Sibasish Ghosh A

Bound entangled (BE) states are strange in nature: non-zero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize all these states for an arbitrary composite quantum system. Here we present a $(d-3)/2$-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in $\mathbb{C}^d\otimes\mathbb{C}^d$ with $d$ odd and $d\ge 5$. We also provide the `tile structure' corresponding to these UPBs. The range of each such PPT-BE state is contained in a $2(d-1)$ dimensional entangled subspace whereas the associated UPB-subspace is of dimension $(d-1)^2+1$. We further show that each of these PPT-BE states can be written as a convex combination of $(d-1)/2$ number of rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set $\mathcal{P}$ of all PPT states in $\mathbb{C}^d\otimes\mathbb{C}^d$, namely the {\it Peres} set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points -- the $(d-3)/2$-simplex -- is sitting on the boundary between the set $\mathcal{P}$ and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.

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Abstract In this work, we concentrate on identifying bound entangled states with positive partial transposition in tripartite system. Since directly extending PPT-BE states from bipartite to tripartite systems is nontrivial, we adopt unextendible product bases (UPBs) as a tool to generate tripartite PPT-BE states. First, a set of UPB with cardinalities d 3 − 4 d + 3 is presented to characterize the extremal states of PPT set in the C d ⨂ C d ⨂ C d system. Using this basis, the missing states are introduced and employed to identify a 2 d − 1 -parameter family of PPT entangled edge states. Additionally, a generation circuit is provided which transforms the states in Z-basis to the missing states in three-qudit composite system. This circuit intuitively reflects the structure of the missing states and prepares for the generation of PPT entangled states. Our work demonstrates a different construction method of PPT entangled edge states in higher-dimensional multipartite system, providing a general insight into the relationship between nonlocality and entanglement in such system.

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Bound-entangled states with positive partial transpose (PPT) can exhibit quantum steering and Bell nonlocality, disproving the long-standing Peres conjecture and its stronger steering counterpart. These counterexamples, obtained through explicit constructions or numerical searches, raise the question of whether they share a common structural origin. We address this question in the low-rank regime by asking whether known counterexamples can be obtained, up to invertible local filtering, from the complement of an unextendible product basis (UPB). We find that the answer depends crucially on the local dimension. In $3\otimes3$, the rank-four structure of PPT entangled states implies that both the steerable PPT state of Moroder \textit{et al.} and the Bell-nonlocal PPT state of Vértesi and Brunner can be obtained from UPB-complement states under invertible local filtering. For three qubits, in contrast, rank-four states that are PPT across every bipartition are separable across each cut. Moreover, comparison of the product-vector structure of the relevant kernels shows that the known three-qubit Bell-nonlocal PPT states cannot be obtained by invertible local filtering from the complement of any three-qubit UPB. Going beyond rank four, we further show that a known rank-eight three-qutrit state, supported on the complement of a UPB and PPT across every bipartition, is genuinely multipartite entangled. These results show that UPBs provide a unifying structural origin for important low-rank PPT entangled states in $3\otimes3$, but that nonlocal bound entanglement is not, in general, a UPB phenomenon.

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Construction of multipartite unextendible product bases and geometric measure of entanglement of positive-partial-transpose entangled states

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In quantum information theory, it is a fundamental problem to construct multipartite unextendible product bases (UPBs). We show that there exist two families UPBs in Hilbert space $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4$ by merging two different systems of an existing $7$-qubit UPB of size $11$. Moreover, a new family of $7$-qubit positive-partial-transpose (PPT) entangled states of rank $2^7-11$ is constructed. We analytically derive a geometric measure of entanglement of a special PPT entangled states. Also an upper bound are given by two methods.

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Tripartite Genuinely Entangled Subspaces inArbitrary Dimensions using UPBs

Sristy Agrawal, Supervisor Prof. Prasanta K. Panigrahi, Guruprasad Kar

While we possess a basic understanding of bipartite entangled systems, multipartite entanglement remains mostly unexplored. Specifically, it is quite unclear how several concepts developed for the bipartite case can be generalized in a meaningful way to study multipartite scenarios. At the heart of this research on multiparty quantum states lies the problem of testing the entanglement present in a given state. It has been proved, that in the most general scenario, the question of determining the entanglement or separability of a state is NP-hard. From this perspective, it is desirable to have a construction of states for which some prior knowledge about entanglement properties already exist. One particular approach to this problem relies on the construction of Completely Entangled Subspaces (CESes), i.e., subspaces void of product states. It follows that states in such subspaces are necessarily entangled and form a convex set. This means we possess prior knowledge of entanglement of all states in that subspace. In our study of multiparty systems, we have constructed a specific subclass of the CESes such that all the states residing in the newly constructed subspace is not only entangled but also Genuinely Entangled. A given state is Genuinely Entangled if it is not just entangled for any pair among the multiparty but is entangled with all the parties present and at the same time. The notion of a CES is intimately connected with the idea of Unextendible Product Bases (UPBs). UPBs are sets of twisted orthogonal product states/vectors spanning a proper subset of a given Hilbert space. They have the property that no other product vector exists in the ortho-complement of their span. While UPBs have been relatively well studied for bipartite systems, no known multipartite orthogonal UPBs give rise to Genuinely Entangled Subspace (GES). In our study, we provide schemes for the construction of GES for a tripartite scenario. The construction proposed is linked to a Symmetric Block structure such that it can be extended to any arbitrary dimension.One of the primary motivations for construction of UPBs is to create entangled states which remain Positive under Partial Transposition (PPT). These classes of Entangled States are known to be un-distillable and hence called Bound Entangled, i.e., states in which the resource of Entanglement is not free to be used. These Bound Entangled states are essential to our understanding the basics of Entanglement Theory such as distillability, quantification of entanglement, and the relationship between nonlocality and entanglement (apart from their applications in other quantum information tasks). The construction itself guarantees the PPTness of the state residing in the complementary subspace i.e. formed by the equal mixture of the basis spanning the complementary space. In our study, we propose a symmetric and an asymmetric construction of the Biseperable and Genuinely Entangled states. Via the symmetric construction, we have derived two subspaces, an eight-dimensional CES and a five-dimensional GES within the CES mentioned above. The state formed by the equal mixtures of the CES is a Genuinely Bound Entangled PPT state i.e., it entangled as well as PPT in each cut. To the best of author’s knowledge, the proposed state is the only know Genuinely Entangled PPT state derived from a UPB construction. Additionally, the state lying in the ortho-complement of UPB (i.e., states formed by an equal mixture of the basis of CES) is known to be a PPT Bound Entangled state. However, unlike in case of UPBs, the state in the ortho-complement of Unextendible Biseparable Product basis (i.e. states formed by an equal mixture of the basis of GES) is found to be NPT states across all bipartitions. This is striking and prompts us to believe that the state in the ortho-complement of CES is the only examples of bound entangled state and distinct from other states in the space. We further expect that the state we obtain must exhibit some distinctive features. We verify that this state is one copy undistillable. However, the n-copy undistillablity needs to be probed further, and no conclusive statement regarding the same can be made with our current analysis. The existence of NPT-bound entangled state would prove that the distillable entanglement is nonadditive and not convex. The constructed subspace of Genuinely Entangled States in-turn reveals some peculiar features. As mentioned above, unlike the Bennett et al., 1999 case, the state in the ortho-complement is an NPT state across all bipartition. Furthermore, it was numerically verified by taking one hundred thousand random probability distributions that the entire five-dimensional genuine subspace obtained happens to be NPT. This leads us to conjecture that the subspace obtained is a Genuinly Entangled NPT subspace. To the best of the author’s knowledge, no known genuine NPT subspace exists for the multipartite scenario. The exists only one other NPT subspace for Biseparable States and was suggested by Johnston, 2013. Here, the construction is not related to UPBs. We further argue that if our conjecture on the NPTness of the entire subspace is correct, any four-dimensional subspace of the constructed tripartite NPT GES is also distillable. This would then be the only known distillable subspace in a multipartite scenario. We believe that our study brings us closer to a systematic analysis of multipartite entanglement and provides us with tools to probe further into the current mess of an entangled (read communicating) world of multiple actors.

•May 1, 2018•0 citations

Constructing $2\times2\times4$ and $4\times4$ unextendible product bases and positive-partial-transpose entangled states

Lin Chen, Wang, Kai, Yi Shen +2 more

The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size $6$ and six UPBs of size $9$ in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size $9$ in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size $7$ in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a special $\r$.

arXiv:1810.08932•Oct 21, 2018•1 citation
Bound entangled (BE) states are strange in nature: a nonzero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though the usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize them for an arbitrary composite quantum system. Here we present a (d\ensuremath−3)/2(d\ensuremath{-}3)/2(d\ensuremath−3)/2-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd with ddd odd and d\ensuremath≥5d\ensuremath{\ge}5d\ensuremath≥5. The range of each such PPT-BE state is contained in a 2(d\ensuremath−1)2(d\ensuremath{-}1)2(d\ensuremath−1)-dimensional entangled subspace, whereas the associated UPB subspace is of dimension (d\ensuremath−1)2+1{(d\ensuremath{-}1)}^{2}+1(d\ensuremath−1)2+1. We further show that each of these PPT-BE states can be written as a convex combination of (d\ensuremath−1)/2(d\ensuremath{-}1)/2(d\ensuremath−1)/2 rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set P\mathcal{P}P of all PPT states in Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd, namely, the Peres set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points---the (d\ensuremath−3)/2(d\ensuremath{-}3)/2(d\ensuremath−3)/2 simplex---is sitting on the boundary between the set P\mathcal{P}P and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.
Bound entangled (BE) states are strange in nature: non-zero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize all these states for an arbitrary composite quantum system. Here we present a (d−3)/2(d-3)/2(d−3)/2-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd with ddd odd and d≥5d\ge 5d≥5. We also provide the `tile structure' corresponding to these UPBs. The range of each such PPT-BE state is contained in a 2(d−1)2(d-1)2(d−1) dimensional entangled subspace whereas the associated UPB-subspace is of dimension (d−1)2+1(d-1)^2+1(d−1)2+1. We further show that each of these PPT-BE states can be written as a convex combination of (d−1)/2(d-1)/2(d−1)/2 number of rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set P\mathcal{P}P of all PPT states in Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd, namely the {\it Peres} set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points -- the (d−3)/2(d-3)/2(d−3)/2-simplex -- is sitting on the boundary between the set P\mathcal{P}P and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.
Abstract In this work, we concentrate on identifying bound entangled states with positive partial transposition in tripartite system. Since directly extending PPT-BE states from bipartite to tripartite systems is nontrivial, we adopt unextendible product bases (UPBs) as a tool to generate tripartite PPT-BE states. First, a set of UPB with cardinalities d 3 − 4 d + 3 is presented to characterize the extremal states of PPT set in the C d ⨂ C d ⨂ C d system. Using this basis, the missing states are introduced and employed to identify a 2 d − 1 -parameter family of PPT entangled edge states. Additionally, a generation circuit is provided which transforms the states in Z-basis to the missing states in three-qudit composite system. This circuit intuitively reflects the structure of the missing states and prepares for the generation of PPT entangled states. Our work demonstrates a different construction method of PPT entangled edge states in higher-dimensional multipartite system, providing a general insight into the relationship between nonlocality and entanglement in such system.
Bound-entangled states with positive partial transpose (PPT) can exhibit quantum steering and Bell nonlocality, disproving the long-standing Peres conjecture and its stronger steering counterpart. These counterexamples, obtained through explicit constructions or numerical searches, raise the question of whether they share a common structural origin. We address this question in the low-rank regime by asking whether known counterexamples can be obtained, up to invertible local filtering, from the complement of an unextendible product basis (UPB). We find that the answer depends crucially on the local dimension. In 3⊗33\otimes33⊗3, the rank-four structure of PPT entangled states implies that both the steerable PPT state of Moroder \textit{et al.} and the Bell-nonlocal PPT state of Vértesi and Brunner can be obtained from UPB-complement states under invertible local filtering. For three qubits, in contrast, rank-four states that are PPT across every bipartition are separable across each cut. Moreover, comparison of the product-vector structure of the relevant kernels shows that the known three-qubit Bell-nonlocal PPT states cannot be obtained by invertible local filtering from the complement of any three-qubit UPB. Going beyond rank four, we further show that a known rank-eight three-qutrit state, supported on the complement of a UPB and PPT across every bipartition, is genuinely multipartite entangled. These results show that UPBs provide a unifying structural origin for important low-rank PPT entangled states in 3⊗33\otimes33⊗3, but that nonlocal bound entanglement is not, in general, a UPB phenomenon.
In quantum information theory, it is a fundamental problem to construct multipartite unextendible product bases (UPBs). We show that there exist two families UPBs in Hilbert space C2⊗C2⊗C2⊗C2⊗C2⊗C4\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4C2⊗C2⊗C2⊗C2⊗C2⊗C4 by merging two different systems of an existing 777-qubit UPB of size 111111. Moreover, a new family of 777-qubit positive-partial-transpose (PPT) entangled states of rank 27−112^7-1127−11 is constructed. We analytically derive a geometric measure of entanglement of a special PPT entangled states. Also an upper bound are given by two methods.
While we possess a basic understanding of bipartite entangled systems, multipartite entanglement remains mostly unexplored. Specifically, it is quite unclear how several concepts developed for the bipartite case can be generalized in a meaningful way to study multipartite scenarios. At the heart of this research on multiparty quantum states lies the problem of testing the entanglement present in a given state. It has been proved, that in the most general scenario, the question of determining the entanglement or separability of a state is NP-hard. From this perspective, it is desirable to have a construction of states for which some prior knowledge about entanglement properties already exist. One particular approach to this problem relies on the construction of Completely Entangled Subspaces (CESes), i.e., subspaces void of product states. It follows that states in such subspaces are necessarily entangled and form a convex set. This means we possess prior knowledge of entanglement of all states in that subspace. In our study of multiparty systems, we have constructed a specific subclass of the CESes such that all the states residing in the newly constructed subspace is not only entangled but also Genuinely Entangled. A given state is Genuinely Entangled if it is not just entangled for any pair among the multiparty but is entangled with all the parties present and at the same time. The notion of a CES is intimately connected with the idea of Unextendible Product Bases (UPBs). UPBs are sets of twisted orthogonal product states/vectors spanning a proper subset of a given Hilbert space. They have the property that no other product vector exists in the ortho-complement of their span. While UPBs have been relatively well studied for bipartite systems, no known multipartite orthogonal UPBs give rise to Genuinely Entangled Subspace (GES). In our study, we provide schemes for the construction of GES for a tripartite scenario. The construction proposed is linked to a Symmetric Block structure such that it can be extended to any arbitrary dimension.One of the primary motivations for construction of UPBs is to create entangled states which remain Positive under Partial Transposition (PPT). These classes of Entangled States are known to be un-distillable and hence called Bound Entangled, i.e., states in which the resource of Entanglement is not free to be used. These Bound Entangled states are essential to our understanding the basics of Entanglement Theory such as distillability, quantification of entanglement, and the relationship between nonlocality and entanglement (apart from their applications in other quantum information tasks). The construction itself guarantees the PPTness of the state residing in the complementary subspace i.e. formed by the equal mixture of the basis spanning the complementary space. In our study, we propose a symmetric and an asymmetric construction of the Biseperable and Genuinely Entangled states. Via the symmetric construction, we have derived two subspaces, an eight-dimensional CES and a five-dimensional GES within the CES mentioned above. The state formed by the equal mixtures of the CES is a Genuinely Bound Entangled PPT state i.e., it entangled as well as PPT in each cut. To the best of author’s knowledge, the proposed state is the only know Genuinely Entangled PPT state derived from a UPB construction. Additionally, the state lying in the ortho-complement of UPB (i.e., states formed by an equal mixture of the basis of CES) is known to be a PPT Bound Entangled state. However, unlike in case of UPBs, the state in the ortho-complement of Unextendible Biseparable Product basis (i.e. states formed by an equal mixture of the basis of GES) is found to be NPT states across all bipartitions. This is striking and prompts us to believe that the state in the ortho-complement of CES is the only examples of bound entangled state and distinct from other states in the space. We further expect that the state we obtain must exhibit some distinctive features. We verify that this state is one copy undistillable. However, the n-copy undistillablity needs to be probed further, and no conclusive statement regarding the same can be made with our current analysis. The existence of NPT-bound entangled state would prove that the distillable entanglement is nonadditive and not convex. The constructed subspace of Genuinely Entangled States in-turn reveals some peculiar features. As mentioned above, unlike the Bennett et al., 1999 case, the state in the ortho-complement is an NPT state across all bipartition. Furthermore, it was numerically verified by taking one hundred thousand random probability distributions that the entire five-dimensional genuine subspace obtained happens to be NPT. This leads us to conjecture that the subspace obtained is a Genuinly Entangled NPT subspace. To the best of the author’s knowledge, no known genuine NPT subspace exists for the multipartite scenario. The exists only one other NPT subspace for Biseparable States and was suggested by Johnston, 2013. Here, the construction is not related to UPBs. We further argue that if our conjecture on the NPTness of the entire subspace is correct, any four-dimensional subspace of the constructed tripartite NPT GES is also distillable. This would then be the only known distillable subspace in a multipartite scenario. We believe that our study brings us closer to a systematic analysis of multipartite entanglement and provides us with tools to probe further into the current mess of an entangled (read communicating) world of multiple actors.
Constructing 2×2×42\times2\times42×2×4 and 4×44\times44×4 unextendible product bases and positive-partial-transpose entangled states
The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size 666 and six UPBs of size 999 in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size 999 in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size 777 in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a special $\r$.