The main subject of this thesis is the investigation of the transformations of pure multipartite entangled states having Schmidt rank 2 by using only local operations assisted with classical communications (LOCC). A new parameterization is used for describing the entangled state of p particles distributed to p distant, spatially separated persons. Product, bipartite and truly multipartite states are identified in this new parametrization. Moreover, alternative parameterizations of local operations carried out by each party are provided. For the case of a deterministic transformation to a truly multipartite final state, one can find an analytic expression that determines whether such a transformation is possible. In this case, a chain of measurements by each party for carrying out the transformation is found. It can also be seen that, under deterministic LOCC transformations, there are some quantities that remain invariant. For the purpose of applying the results of this thesis in the context of the quantum information and computation, brief reviews of the entanglement purification, measurement based quantum computation and quantum codes are given.
The main subject of this thesis is the investigation of the transformations of pure multipartite entangled states having Schmidt rank 2 by using only local operations assisted with classical communications (LOCC). A new parameterization is used for describing the entangled state of p particles distributed to p distant, spatially separated persons. Product, bipartite and truly multipartite states are identified in this new parametrization. Moreover, alternative parameterizations of local operations carried out by each party are provided. For the case of a deterministic transformation to a truly multipartite final state, one can find an analytic expression that determines whether such a transformation is possible. In this case, a chain of measurements by each party for carrying out the transformation is found. It can also be seen that, under deterministic LOCC transformations, there are some quantities that remain invariant. For the purpose of applying the results of this thesis in the context of the quantum information and computation, brief reviews of the entanglement purification, measurement based quantum computation and quantum codes are given.
Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n\ensuremath{\ge}5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.
Transformations involving only local operations assisted with classical communication are investigated for multipartite entangled pure states having tensor rank 2. All necessary and sufficient conditions for the possibility of deterministically converting truly multipartite, rank-2 states into each other are given. Furthermore, a chain of local operations that successfully achieves the transformation has been identified for all allowed transformations. The identified chains have two nice features: (1) each party needs to carry out at most one local operation and (2) all of these local operations are also deterministic transformations by themselves. Finally, it is found that there are disjoint classes of states, all of which can be identified by a single real parameter, which remain invariant under deterministic transformations.
Entanglement is one of the fundamental features of quantum information science.\nThough bipartite entanglement has been analyzed thoroughly in theory and shown to\nbe an important resource in quantum computation and communication protocols, the\ntheory of entanglement shared between more than two parties, which is called multipartite\nentanglement, is still not complete. Specifically, the classification of multipartite\nentanglement and the transformation property between different multipartite states by\nlocal operators and classical communications (LOCC) are two fundamental questions in\nthe theory of multipartite entanglement.\nIn this thesis, we present results related to the LOCC transformation between multipartite\nentangled states. Firstly, we investigate the bounds on the LOCC transformation\nprobability between multipartite states, especially the GHZ class states. By analyzing\nthe involvement of 3-tangle and other entanglement measures under weak two-outcome\nmeasurement, we derive explicit upper and lower bound on the transformation probability\nbetween GHZ class states. After that, we also analyze the transformation between\nN-party W type states, which is a special class of multipartite entangled states that\nhas an explicit unique expression and a set of analytical entanglement monotones. We\npresent a necessary and sufficient condition for a known upper bound of transformation\nprobability between two N-party W type states to be achieved.\nWe also further investigate a novel entanglement transformation protocol, the random\ndistillation, which transforms multipartite entanglement into bipartite entanglement\nii\nshared by a non-deterministic pair of parties. We find upper bounds for the random distillation\nprotocol for general N-party W type states and find the condition for the upper\nbounds to be achieved. What is surprising is that the upper bounds correspond to entanglement\nmonotones that can be increased by Separable Operators (SEP), which gives\nthe first set of analytical entanglement monotones that can be increased by SEP.\nFinally, we investigate the idea of a new class of multipartite entangled states, the\nAbsolutely Maximal Entangled (AME) states, which is characterized by the fact that\nany bipartition of the states would give a maximal entangled state between the two sets.\nThe relationship between AME states and Quantum secret sharing (QSS) protocols is\nexhibited and the application of AME states in novel quantum communication protocols\nis also explored.
The study of state transformations by spatially separated parties with local operations assisted by classical communication (LOCC) plays a crucial role in entanglement theory and its applications in quantum information processing. Transformations of this type among pure bipartite states were characterized long ago and have a revealing theoretical structure. However, it turns out that generic fully entangled pure multipartite states cannot be obtained from nor transformed to any inequivalent fully entangled state under LOCC. States with this property are referred to as isolated. Nevertheless, multipartite states are classified into families, the so-called SLOCC classes, which possess very different properties. Thus, the above result does not forbid the existence of particular SLOCC classes that are free of isolation, and therefore, display a rich structure regarding LOCC convertibility. In fact, it is known that the celebrated n -qubit GHZ and W states give particular examples of such classes and in this work, we investigate this question in general. One of our main results is to show that the SLOCC class of the 3-qutrit totally antisymmetric state is isolation-free as well. Actually, all states in this class can be converted to inequivalent states by LOCC protocols with just one round of classical communication (as in the GHZ and W cases). Thus, we consider next whether there are other classes with this property and we find a large set of negative answers. Indeed, we prove weak isolation (i.e., states that cannot be obtained with finite-round LOCC nor transformed by one-round LOCC) for very general classes, including all SLOCC families with compact stabilizers and many with non-compact stabilizers, such as the classes corresponding to the n -qunit totally antisymmetric states for n≥4 . Finally, given the pleasant feature found in the family corresponding to the 3-qutrit totally antisymmetric state, we explore in more detail the structure induced by LOCC and the entanglement properties within this class.
This thesis studies the manipulation of entanglement in three-qubit quantum systems. I consider the operational setting in which the qubits are distributed to three spatially separated parties. The parties act locally on their quantum systems and share classical communication with one another, a scenario commonly called local operations and classical communication (LOCC). In the LOCC setting, there are two different classes of entanglement in multipartite systems, called the GHZ and W classes, which are inequivalent in the sense that states from one class cannot be transformed into the other without the consumption of additional entanglement. In this thesis, I first show that the LOCC conversion of certain GHZ and W-class states becomes possible by using only one additional ebit (“entangled bit”) of shared entanglement. In many cases, this can be proven as the minimal amount of needed entanglement. I then consider the problem of one-way communication transformations from general three-qubit states into two-qubit maximally entangled states, known as EPR states. An optimal protocol in terms of success probability is provided for W-class states. The success probability of this protocol coincides with the optimal success probability if two of the parties are allowed to act jointly within the same laboratory. In other words, forcing the locality constraint on all three parties does not weaken their capabilities for obtaining bipartite entanglement when starting from a W-class state. I also present that this property holds for certain types of GHZ-class states as well.
Entanglement plays an important role in quantum computation and information. We can only manipulate an entangled quantum state shared among several distant parties by local operations with classical communication (LOCC). Thus, it is of great importance to characterize the conditions by which we can determine whether one pure quantum state can be transformed to another pure state by LOCC. It is well known that separable operations and LOCC are not equivalent. Dozens of results have illustrated this difference. Little do we know about this difference despite these recent results. It is shown in this paper that if the initial state $|\ensuremath{\psi}\ensuremath{\rangle}$ and final state $|\ensuremath{\phi}\ensuremath{\rangle}$ are genuine tripartite pure states in the Greenberger-Horne-Zeilinger (GHZ) class then $|\ensuremath{\psi}\ensuremath{\rangle}$ can be transformed to $|\ensuremath{\phi}\ensuremath{\rangle}$ by separable operations if and only if $|\ensuremath{\psi}\ensuremath{\rangle}$ can be transformed to $|\ensuremath{\phi}\ensuremath{\rangle}$ by deterministic LOCC. That is, SEP equals LOCC on tripartite GHZ states.