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Tripartite Genuinely Entangled Subspaces inArbitrary Dimensions using UPBs

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

Published May 1, 2018• 0 citations
OpenAlex

Abstract

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.

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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.
Unextendible product bases (UPBs) provide a versatile tool with various applications across different areas of quantum information theory. Their comprehensive characterization is thus of great importance and has been a subject of vital interest for over two decades now. An open question asks about the existence of UPBs, which are genuinely unextendible, i.e., they are not extendible even with biproduct vectors. In other words, the problem is to verify whether there exist genuinely entangled subspaces (GESs), subspaces composed solely of genuinely multiparty entangled states, complementary to UPBs. We solve this problem in the negative for many sizes of UPBs in different multipartite scenarios. In particular, in the all-important case of equal local dimensions, we show that there are always forbidden cardinalities for such UPBs, including the minimal ones corresponding to GESs of the maximal dimensions.
Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled but genuinely entangled. Here we explore the connection between UPBs and such genuinely entangled subspaces (GESs) and provide classes of nonorthogonal UPBs that lead to GESs for any number of parties and local dimensions. We then show how these subspaces can be immediately utilized for a simple general construction of genuinely entangled states in any such multipartite scenario.
In a multipartite scenario quantum entanglement manifests its most dramatic form when the state is genuinely entangled. Such a state is more beneficial for information theoretic applications if it contains distillable entanglement in every bipartition. It is, therefore, of significant operational interest to identify subspaces of multipartite quantum systems that contain such properties a priori. In this article we introduce the notion of unextendible biseparable bases (UBB) that provides an adequate method to construct genuinely entangled subspaces (GES). We provide explicit construction of two types of UBBs---party symmetric and party asymmetric---for every 3-qudit quantum system, with local dimension $d\ensuremath{\ge}3$. Furthermore, we show that the GES resulting from the symmetric construction is indeed a bidistillable subspace, i.e., all the states supported on it contain distillable entanglement across every bipartition.
We introduce a class of entangled subspaces: completely entangled subspaces of entanglement depth kkk $(k\text{\ensuremath{-}}\text{CESs})$. These are subspaces of multipartite Hilbert spaces containing only pure states with an entanglement depth of at least kkk. We present an efficient construction of $k\text{\ensuremath{-}}\text{CESs}$ of any achievable dimensionality in any multipartite scenario. Further, we discuss the relation between these subspaces and unextendible product bases (UPBs). In particular, we establish that there is a nontrivial bound on the cardinality of a UPB whose orthocomplement is a $k\text{\ensuremath{-}}\text{CES}$. Further, we discuss the existence of such UPBs for qubit systems.
We introduce a class of entangled subspaces: completely entangled subspaces of entanglement depth kkk (kkk-CESs). These are subspaces of multipartite Hilbert spaces containing only pure states with an entanglement depth of at least kkk. We present an efficient construction of kkk-CESs of any achievable dimensionality in any multipartite scenario. Further, we discuss the relation between these subspaces and unextendible product bases (UPBs). In particular, we establish that there is a non-trivial bound on the cardinality of a UPB whose orthocomplement is a kkk-CES. Further, we discuss the existence of such UPBs for qubit systems.
A set of orthogonal pure states is an unextendible biseparable basis (UBB), which means that its complementary subspace contains only genuinely entangled states. UBBs thus serve as an effective tool for constructing genuinely entangled subspaces. If every state within such a subspace exhibits distillable entanglement across all bipartitions, it becomes particularly advantageous for applications in quantum information. In this paper, we mainly conduct research on the 4-qudit quantum systems, where the local dimension ddd is not less than 3. We present an approach for constructing UBB and prove that the UBB established in this way is strongly nonlocal. We build several genuinely entangled subspaces and demonstrate the distillability of the genuinely entangled subspaces across all bipartitions. In addition, we also describe the specific orthonormal basis for some genuinely entangled subspaces. These results will not only contribute to the development of quantum nonlocality theory, but also provide a crucial theoretical foundation for practical quantum information processing tasks.