Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ for any nonprime dimension $d=d_1\times d_2$. Moreover, based on the structure of the orthogonal product basis in $\mathbb{C}^2\otimes \mathbb{C}^n$, we obtain another construction of absolutely entangled set with $2n+1$ elements in $\mathbb{C}^2\otimes \mathbb{C}^n$.
Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on Cd1⊗Cd2 for any nonprime dimension d=d1×d2. Moreover, based on the structure of the orthogonal product basis in C2⊗Cn, we obtain another construction of absolutely entangled set with 2n+1 elements in C2⊗Cn.
A set of quantum states is said to be absolutely entangled when at least one state in the set remains entangled for any definition of subsystems, i.e., for any choice of the global reference frame. In this work we investigate the properties of absolutely entangled sets (AESs) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of N states such that $N\ensuremath{-}3$ (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition d=d1d2, we prove that sets of $N>\ensuremath{\lfloor}({d}_{1}+1)({d}_{2}+1)/2\ensuremath{\rfloor}$ states are AESs with Haar measure 1. Then, we define AESs for multipartitions. We derive a general lower bound on the number of states in an AES for a given multipartition, and also construct explicit examples. In particular, we exhibit an AES with respect to any possible multipartitioning of the total system.
The notion of entanglement of quantum states is usually defined with respect to a fixed bipartition. Indeed, a global basis change can always map an entangled state to a separable one. The situation is, however, different when considering a set of states. In this work we define the notion of an ``absolutely entangled set'' of quantum states: for any possible choice of global basis, at least one of the states in the set is entangled. Hence, for all bipartitions, i.e., any possible definition of the subsystems, the set features entanglement. We present examples of such sets, including sets with minimal size. Moreover, we propose a quantitative measure for absolute set entanglement. To lower bound this quantity, we develop a method based on polynomial optimization to perform convex optimization over unitaries, which is of independent interest.
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.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.
A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving measurement can be performed to this set. In general, it is difficult to show such a orthogonality-preserving measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully construct orthogonal product sets (OPSs) in any possible three, four and five-partite systems that are locally irreducible in every bipartition. Moreover, we also show the existence of unextendible product bases (UPBs) with such properties in d⊗d⊗d for any d≥3. Therefore, these OPSs and UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Both results solve the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. Our results shed a new light on the strong quantum nonlocality without entanglement.
Quantum nonlocality, rooted in the quantum state discrimination task, contributes to the development of the theory of local distinguishability of quantum states. Recently, Zhu et al. [Phys. A (Amsterdam, Neth.) 624, 128956 (2023)] found that a nonlocal set of orthogonal product states (OPSs) can be perfectly identified by local operations and classical communication after removing a specific quantum state in a bipartite quantum system. They called this phenomenon minimal nonlocality. It is obvious that minimal nonlocality effectively reflects the lower boundary of the elements contained in a nonlocal set of OPSs. In this paper, we generalize the methods of constructing nonlocal sets with minimal nonlocality from bipartite systems to multipartite systems and from orthogonal product states to entangled states. First, we give a method to construct a set of OPSs with minimal nonlocality in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and then we extend it to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system for $d\ensuremath{\ge}4$ and $n\ensuremath{\ge}3$. In addition, we give a method to construct a nonlocal set of orthogonal states with entanglement relations in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and we prove that this set has minimal nonlocality, which can also be extended to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system.