Abstract An experimental verification of the maximally entangled state ensures that the constructed state is close to the maximally entangled state, but it does not guarantee that the state is exactly the same as the maximally entangled state. Further, the entanglement structure is not uniquely determined in general probabilistic theories even if we impose that the local subsystems are fully equal to quantum systems. Therefore, the existence of the maximally entangled state depends on whether the standard entanglement structure (SES) is valid. To examine this issue, we introduce pseudo SES as a structure of quantum composite system under natural assumptions based on the existence of projective measurements and the existence of approximations of all maximally entangled standard states. Surprisingly, there exist infinitely many pseudo SESs different from the SES. In our setting, any maximally entangled state can be arbitrarily approximated by an entangled state that belongs to our obtained pseudo standard entanglement structure. That is, experimental verification does not exclude the possibility of our obtained pseudo standard entanglement structure that is different from the standard entanglement structure. On the other hand, such pseudo structures never possess global unitary symmetry, i.e. global unitary symmetry is essential condition for the SES.
We study the entanglement structure, i.e., the structure of quantum composite system from operational aspects. The structure is not uniquely determined in General Probabilistic Theories (GPTs) even if we impose reasonable postulate about local systems. In this paper, we investigate the possibility that the standard entanglement structure can be determined uniquely by repeatability of measurement processing and its saturated situation called self-duality. Surprisingly, self-duality cannot determine the standard entanglement structure even if we additionally impose local unitary symmetry assumption. In this paper, we show the existence of infinite structures of quantum composite system such that it is self-dual with local unitary symmetry. Besides, we also show the existence of a structure of quantum composite system such that non-orthogonal states in the structure are perfectly distinguishable. In addition, as a byproduct, we derive an sufficient condition to achieve the detection of the entanglement property with a finite number of parameterized minimizations.
This thesis deals with General Probabilistic Theories (GPTs) and Entanglement Structures (ESs). An ES is a possible structure of a quantum composite system in GPTs, which is not uniquely determined as the Standard Entanglement Structure (SES). It is an important problem to find reasonable postulates that determine various ESs as the SES. In order to solve this problem, this thesis explores the diversity of ESs. The topics of this thesis are roughly divided into two parts. First, this thesis considers state discrimination in ESs. As a main result, this thesis gives equivalent conditions for a given measurement in ESs to have a performance superior to standard quantum theory. Second, this thesis focuses on symmetry and self-duality. As a main result, this thesis gives derivations of the SES by symmetric conditions. On the other hand, this thesis clarifies that there are infinitely many self-dual ESs, even if they cannot be distinguished from the SES by a certain physical experiment with small errors.
We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for
n-qubit states, we express the condition that all
k-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all
k vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group
Pn that is isomorphic to
Z2n. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed
k for fixed
n. In particular, the ratio
k/n can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed
k-qubit reductions with
k>1. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though
k>1 is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.
Within the quantum networks scenario we introduce a single scheme allowing one to certify three different types of composite projective measurements acting on a three-qubit Hilbert space: one constructed from genuinely entangled GHZ-like states, one constructed from fully product vectors that exhibit the phenomenon of nonlocality without entanglement, and a hybrid measurement obtained from an unextendible product basis. Noticeably, we certify a basis exhibiting nonlocality without entanglement in the smallest dimension capable of supporting this phenomenon. On the other hand, the possibility of certification of a measurement obtained from an unextendible product basis has an interesting implication that one can also self-test a bound entangled state in the considered quantum network. Such a possibility does not seem to exist in the standard Bell scenario. Furthermore, we also analyze the robustness of our scheme toward experimental errors.
We find that the mixed maximally entangled states exist and prove that the form of the mixed maximally entangled states is unique in terms of the entanglement of formation. Moreover, even if the entanglement is quantified by other entanglement measures, this conclusion is still proven right. This result is a supplementary to the generally accepted fact that all maximally entangled states are pure. These states possess important properties of the pure maximally entangled states, for example, these states can be used as a resource for faithful teleportation and they can be distinguished perfectly by local operations and classical communication.
We find that the mixed maximally entangled states exist and prove that the form of the mixed maximally entangled states is unique in terms of the entanglement of formation. Moreover, even if the entanglement is quantified by other entanglement measures, this conclusion is still proven right. This result is a supplementary to the generally accepted fact that all maximally entangled states are pure. These states possess important properties of the pure maximally entangled states, for example, these states can be used as a resource for faithful teleportation and they can be distinguished perfectly by local operations and classical communication.