Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1409.5093 · Physical Review A

Entanglement properties of positive operators with ranges in completely entangled subspaces

Ritabrata Sengupta, Arvind, Ajit Iqbal Singh

Published Dec 15, 2014• 11 citations
arXivHTMLPDF

Abstract

We prove that the projection on a completely entangled subspace $\mathcal{S}$ of maximum dimension obtained by Parthasarathy [K. R. Parthasarathy, Proc. Indian Acad. Sci. Math. Sci. 114, 365 (2004)] in a multipartite quantum system is not positive under partial transpose. We next show that a large number of positive operators with a range in $\mathcal{S}$ also have the same property. In this process we construct an orthonormal basis for $\mathcal{S}$ and provide a theorem to link the constructions of completely entangled subspaces due to Parthasarathy (as cited above), Bhat [B. V. R. Bhat, Int. J. Quantum Inf. 4, 325 (2006)], and Johnston [N. Johnston, Phys. Rev. A 87, 064302 (2013)].

DOI
10.1103/physreva.90.062323

Related papers

Non-positive-partial-transpose subspaces can be as large as any entangled subspace

Nathaniel Johnston

It is known that, in an $(m\ensuremath{\bigotimes}n)$-dimensional quantum system, the maximum dimension of a subspace that contains only entangled states is $(m\ensuremath{-}1)(n\ensuremath{-}1)$. We show that the exact same bound is tight if we require the stronger condition that every state with range in the subspace has non-positive partial transpose. As an immediate corollary of our result, we solve an open question that asks for the maximum number of negative eigenvalues of the partial transpose of a quantum state. In particular, we give an explicit method of construction of a bipartite state whose partial transpose has $(m\ensuremath{-}1)(n\ensuremath{-}1)$ negative eigenvalues, which is necessarily maximal, despite recent numerical evidence that suggested such states may not exist for large $m$ and $n$.

arXiv:1305.0257•Physical Review A•Jun 28, 2013•44 citations

Asymptotic Indistinguishability of Bipartite Quantum States by Operations Completely Preserving Positivity of Partial Transpose

Yinan Li, Xin Wang, Runyao Duan

A bipartite subspace $S$ is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer $k$, there is no PPT state supporting on the orthogonal complement of $S^{\otimes k}$. We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Applying this result, we demonstrate a simple criterion for verifying whether a set of bipartite quantum states is asymptotically indistinguishable by operations completely preserving positivity of partial transpose. This criterion is based on the observation that if the support of some bipartite state is strongly PPT-unextendible, then any set of bipartite quantum states which contains this state cannot be asymptotically distinguished by PPT operations. Utilizing this criterion, we further show that any entangled pure state and its orthogonal complement cannot be distinguished asymptotically by PPT operations. Meanwhile, we also investigate that the minimum dimension of strongly PPT-unextendible subspaces in a $m\otimes n$ system is $m+n-1$, which involves an extension of the result that non-positive-partial-transpose (NPT) subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87: 064302 (2013)].

•Feb 1, 2017•1 citation

Fully non-positive-partial-transpose genuinely entangled subspaces

Owidiusz Makuta, Błażej Kuzaka, Remigiusz Augusiak

Genuinely entangled subspaces are a class of subspaces in the multipartite Hilbert spaces that are composed of only genuinely entangled states. They are thus an interesting object of study in the context of multipartite entanglement. Here we provide a construction of multipartite subspaces that are not only genuinely entangled but also fully non-positive-partial-transpose (NPT) in the sense that any mixed state supported on them has non-positive partial transpose across any bipartition. Our construction originates from the stabilizer formalism known for its use in quantum error correction. To this end, we first introduce a couple of criteria allowing to assess whether any state from a given non-trivial stabilizer subspace is genuinely multipartite entangled. We then use these criteria to construct genuinely entangled stabilizer subspaces for any number of parties and arbitrary local dimension and conjecture them to be of maximal dimension achievable within the stabilizer formalism. At the same time, we prove that every genuinely entangled subspace is fully NPT in the above sense, which implies a quite surprising fact that no genuinely entangled stabilizer subspace can support PPT entangled states.

arXiv:2203.16902•Quantum•Feb 9, 2023•9 citations

Indistinguishability of bipartite states by positive-partial-transpose operations in the many-copy scenario

Yinan Li, Xin Wang, Runyao Duan

A bipartite subspace $S$ is called strongly positive-partial-transpose (PPT) unextendible if for every positive integer $k$, there is no PPT operator supporting on the orthogonal complement of ${S}^{\ensuremath{\bigotimes}k}$. We show that a subspace is strongly PPT unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Based on these, we are able to propose a simple criterion for verifying whether a set of bipartite orthogonal quantum states is indistinguishable by PPT operations in the many-copy scenario. Utilizing this criterion, we further point out that any entangled pure state and its orthogonal complement cannot be distinguished by PPT operations in the many-copy scenario. On the other hand, we investigate that the minimum dimension of strongly PPT-unextendible subspaces in an $m\ensuremath{\bigotimes}n$ system is $m+n\ensuremath{-}1$, which involves a generalization of the result that non-positive-partial-transpose subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87, 064302 (2013)].

arXiv:1702.00231•Physical Review A•May 30, 2017•20 citations

High-Dimensional Entanglement in States with Positive Partial Transposition

Marcus Huber, Ludovico Lami, Cécilia Lancien +1 more

Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.

arXiv:1802.04975•Physical Review Letters•Nov 15, 2018•46 citations

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
We prove that the projection on a completely entangled subspace S\mathcal{S}S of maximum dimension obtained by Parthasarathy [K. R. Parthasarathy, Proc. Indian Acad. Sci. Math. Sci. 114, 365 (2004)] in a multipartite quantum system is not positive under partial transpose. We next show that a large number of positive operators with a range in S\mathcal{S}S also have the same property. In this process we construct an orthonormal basis for S\mathcal{S}S and provide a theorem to link the constructions of completely entangled subspaces due to Parthasarathy (as cited above), Bhat [B. V. R. Bhat, Int. J. Quantum Inf. 4, 325 (2006)], and Johnston [N. Johnston, Phys. Rev. A 87, 064302 (2013)].
It is known that, in an $(m\ensuremath{\bigotimes}n)$-dimensional quantum system, the maximum dimension of a subspace that contains only entangled states is $(m\ensuremath{-}1)(n\ensuremath{-}1)$. We show that the exact same bound is tight if we require the stronger condition that every state with range in the subspace has non-positive partial transpose. As an immediate corollary of our result, we solve an open question that asks for the maximum number of negative eigenvalues of the partial transpose of a quantum state. In particular, we give an explicit method of construction of a bipartite state whose partial transpose has $(m\ensuremath{-}1)(n\ensuremath{-}1)$ negative eigenvalues, which is necessarily maximal, despite recent numerical evidence that suggested such states may not exist for large mmm and nnn.
A bipartite subspace SSS is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer kkk, there is no PPT state supporting on the orthogonal complement of S⊗kS^{\otimes k}S⊗k. We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Applying this result, we demonstrate a simple criterion for verifying whether a set of bipartite quantum states is asymptotically indistinguishable by operations completely preserving positivity of partial transpose. This criterion is based on the observation that if the support of some bipartite state is strongly PPT-unextendible, then any set of bipartite quantum states which contains this state cannot be asymptotically distinguished by PPT operations. Utilizing this criterion, we further show that any entangled pure state and its orthogonal complement cannot be distinguished asymptotically by PPT operations. Meanwhile, we also investigate that the minimum dimension of strongly PPT-unextendible subspaces in a m⊗nm\otimes nm⊗n system is m+n−1m+n-1m+n−1, which involves an extension of the result that non-positive-partial-transpose (NPT) subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87: 064302 (2013)].
Genuinely entangled subspaces are a class of subspaces in the multipartite Hilbert spaces that are composed of only genuinely entangled states. They are thus an interesting object of study in the context of multipartite entanglement. Here we provide a construction of multipartite subspaces that are not only genuinely entangled but also fully non-positive-partial-transpose (NPT) in the sense that any mixed state supported on them has non-positive partial transpose across any bipartition. Our construction originates from the stabilizer formalism known for its use in quantum error correction. To this end, we first introduce a couple of criteria allowing to assess whether any state from a given non-trivial stabilizer subspace is genuinely multipartite entangled. We then use these criteria to construct genuinely entangled stabilizer subspaces for any number of parties and arbitrary local dimension and conjecture them to be of maximal dimension achievable within the stabilizer formalism. At the same time, we prove that every genuinely entangled subspace is fully NPT in the above sense, which implies a quite surprising fact that no genuinely entangled stabilizer subspace can support PPT entangled states.
A bipartite subspace SSS is called strongly positive-partial-transpose (PPT) unextendible if for every positive integer kkk, there is no PPT operator supporting on the orthogonal complement of ${S}^{\ensuremath{\bigotimes}k}$. We show that a subspace is strongly PPT unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Based on these, we are able to propose a simple criterion for verifying whether a set of bipartite orthogonal quantum states is indistinguishable by PPT operations in the many-copy scenario. Utilizing this criterion, we further point out that any entangled pure state and its orthogonal complement cannot be distinguished by PPT operations in the many-copy scenario. On the other hand, we investigate that the minimum dimension of strongly PPT-unextendible subspaces in an $m\ensuremath{\bigotimes}n$ system is $m+n\ensuremath{-}1$, which involves a generalization of the result that non-positive-partial-transpose subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87, 064302 (2013)].
Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.
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.