The study of transformations among pure states via Local Operations assisted by Classical Communication (LOCC) plays a central role in entanglement theory. The main emphasis of these investigations is on the deterministic, or probabilistic transformations between two states and mainly tools from linear algebra are employed. Here, we go one step beyond that and analyze all optimal protocols. We show that for all bipartite and almost all multipartite (of arbitrarily many d-level systems) pairs of states, there exist infinitely many optimal intermediate states to which the initial state can first be transformed (locally) before it is transformed to the final state. The success probability of this transformation is nevertheless optimal. We provide a simple characterization of all intermediate states. We generalize this concept to differentiable paths in Hilbert space that connect two states of interest. With the help of survival analysis we determine the success probability for the continuous transformation along such a path and derive necessary and sufficient conditions on the optimality. We show, in strong contrast to previous results on state transformations, that optimal paths are characterized as solutions of a differential equation. Whereas, for almost all pairs of states, there exist infinitely many optimal paths, we present examples of pairs of states for which not even a single optimal intermediate state exists. Furthermore, we introduce a physically motivated distance measure, the interconversion metric, and show that (generically) any minimal geodesic with respect to the interconversion metric is an optimal path. Moreover, we identify infinitely many easily computable entanglement monotones for generic multipartite pure states. We show that a given finite set of these entanglement monotones can be used to completely characterize the entanglement contained in a generic state.
The study of transformations among pure states via Local Operations assisted by Classical Communication (LOCC) plays a central role in entanglement theory. The main emphasis of these investigations is on the deterministic, or probabilistic transformations between two states and mainly tools from linear algebra are employed. Here, we go one step beyond that and analyze all optimal protocols. We show that for all bipartite and almost all multipartite (of arbitrarily many d-level systems) pairs of states, there exist infinitely many optimal intermediate states to which the initial state can first be transformed (locally) before it is transformed to the final state. The success probability of this transformation is nevertheless optimal. We provide a simple characterization of all intermediate states. We generalize this concept to differentiable paths in Hilbert space that connect two states of interest. With the help of survival analysis we determine the success probability for the continuous transformation along such a path and derive necessary and sufficient conditions on the optimality. We show, in strong contrast to previous results on state transformations, that optimal paths are characterized as solutions of a differential equation. Whereas, for almost all pairs of states, there exist infinitely many optimal paths, we present examples of pairs of states for which not even a single optimal intermediate state exists. Furthermore, we introduce a physically motivated distance measure, the interconversion metric, and show that (generically) any minimal geodesic with respect to the interconversion metric is an optimal path. Moreover, we identify infinitely many easily computable entanglement monotones for generic multipartite pure states. We show that a given finite set of these entanglement monotones can be used to completely characterize the entanglement contained in a generic state.
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.
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.
Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled, pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local-unitary-inequivalent (LU-inequivalent) states via local operations, nor can they be deterministically transformed to pure, fully entangled LU-inequivalent states. In order to derive this result, we prove a more general statement, namely, that, generically, a state possesses no nontrivial local symmetry. We discuss further consequences of this result for the characterization of optimal, probabilistic single copy and probabilistic multi-copy LOCC transformations and the characterization of LU-equivalence classes of multipartite pure states.
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.
We consider generic pure n-qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems. We characterize those states which can be reached from some other state via local operations assisted by finitely many rounds of classical communication (LOCC_{N}). For n qubits with n>3, we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of LOCC_{N}. Moreover, we identify a class of states for which any LOCC_{N} protocol can be realized via a concatenation of deterministic steps. We show, however, that in general there exist state transformations which require a probabilistic step within the protocol, which highlights the difference between bipartite and multipartite LOCC.
We consider generic pure n-qubit states and a general class of pure states of arbitrary dimensions and arbitrary many subsystems. We characterize those states which can be reached from some other state via Local Operations assisted by finitely many rounds of Classical Communication (LOCCN). For qubits we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of finite-round LOCC. Moreover, we identify a class of states for which any LOCCN protocol can be realized via a concatenation of deterministic steps. We show, however, that in general there exist state transformations which require a probabilistic step within the protocol, which highlights the difference between bipartite and multipartite LOCC.