Abstract We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. https://doi.org/10.1103/PhysRevA.95.032308 ), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces.
Abstract We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. https://doi.org/10.1103/PhysRevA.95.032308 ), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces.
Abstract Genuine entanglement is the strongest form of multipartite entanglement. Genuinely entangled pure states contain entanglement in every bipartition and as such can be regarded as a valuable resource in the protocols of quantum information processing. A recent direction of research is the construction of genuinely entangled subspaces (GESs)—the class of subspaces consisting entirely of genuinely entangled pure states. In this paper we present methods of construction of such subspaces including those of maximal possible dimension. The approach is based on the composition of bipartite entangled subspaces and quantum channels of certain types. The examples include maximal subspaces for systems of three qubits, four qubits, three qutrits. We also provide lower bounds on two entanglement measures for mixed states, the concurrence and the convex-roof extended negativity, which are directly connected with the projection on GESs.
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.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.
Entanglement is an important concept in quantum information, quantum communication, and quantum computing. We provide a geometrical analysis of entanglement and separability for all the rank 2 quantum mixed states: complete analysis for the bipartite states and partial analysis for the multipartite states. For each rank 2 mixed state, we define its unique Bloch sphere, that is spanned by the eigenstates of its density matrix. We characterize those Bloch spheres into exactly five classes of entanglement and separability, give examples for each class, and prove that those are the only classes.
A geometric understanding of entanglement is proposed based on local measurements. Taking recourse to the general structure of density matrices in the framework of Euclidean geometry, we first illustrate our approach for bipartite Werner states. It is demonstrated that separable states satisfy certain geometric constraints that entangled states do not. A separability criterion for multiparty Werner states of arbitrary dimension is derived. This approach can be used to determine separability across any bipartition of a general density matrix and leads naturally to a computable measure of entanglement for multiparty pure states. It is known that all density matrices within a certain distance of the normalized identity are separable. This distance is determined for a general setting of n qudits, each of dimension d.