We put forward a simple construction of genuinely entangled subspaces – subspaces supporting only genuinely multipartite entangled states – of any permissible dimensionality for any number of parties and local dimensions. The method uses nonorthogonal product bases, which are built from totally nonsingular matrices with a certain structure. We give an explicit basis for the constructed subspaces. An immediate consequence of our result is the possibility of constructing in the general multiparty scenario genuinely multiparty entangled mixed states with ranks up to the maximal dimension of a genuinely entangled subspace.
We put forward a simple construction of genuinely entangled subspaces – subspaces supporting only genuinely multipartite entangled states – of any permissible dimensionality for any number of parties and local dimensions. The method uses nonorthogonal product bases, which are built from totally nonsingular matrices with a certain structure. We give an explicit basis for the constructed subspaces. An immediate consequence of our result is the possibility of constructing in the general multiparty scenario genuinely multiparty entangled mixed states with ranks up to the maximal dimension of a genuinely entangled subspace.
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.
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.
Abstract Genuinely entangled subspaces (GESs) are the class of completely entangled subspaces that contain only genuinely multiparty entangled states. They constitute a particularly useful notion in the theory of entanglement but also have found an application, for instance, in quantum error correction and cryptography. In a recent study (Demianowicz and Augusiak in Phys Rev A 98:012313, 2018), we have shown how GESs can be efficiently constructed in any multiparty scenario from the so-called unextendible product bases. The provided subspaces, however, are not of maximal allowable dimensions, and our aim here is to put forward an approach to building such. The method is illustrated with few examples in small systems. Connections with other mathematical problems, such as spaces of matrices of equal rank and the numerical range, are discussed.
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.
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.
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.