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arXiv:1508.02524 · Physical Review A

Source and accessible entanglement of few-body systems

David Sauerwein, Katharina Schwaiger, Martí Cuquet, Julio I. de Vicente, Barbara Kraus

Published Dec 28, 2015• 12 citations
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Abstract

Entanglement is the resource to overcome the natural limitations of spatially separated parties restricted to local operations assisted by classical communication (LOCC). Recently, two new classes of operational entanglement measures, the source and the accessible entanglement, for arbitrary multipartite states have been introduced. Whereas the source entanglement measures from how many states the state of interest can be obtained via LOCC, the accessible entanglement measures how many states can be reached via LOCC from the state at hand. We consider here pure bipartite as well as multipartite states and derive explicit formulas for the source entanglement. Moreover, we obtain explicit formulas for a whole class of source entanglement measures that characterize the simplicity of generating a given bipartite pure state via LOCC. Furthermore, we show how the accessible entanglement can be computed numerically. For generic four-qubit states we first derive the necessary and sufficient conditions for the existence of LOCC transformations among these states and then derive explicit formulas for their accessible and source entanglement.

DOI
10.1103/physreva.92.062340

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Entanglement is the resource to overcome the natural limitations of spatially separated parties restricted to local operations assisted by classical communication (LOCC). Recently, two new classes of operational entanglement measures, the source and the accessible entanglement, for arbitrary multipartite states have been introduced. Whereas the source entanglement measures from how many states the state of interest can be obtained via LOCC, the accessible entanglement measures how many states can be reached via LOCC from the state at hand. We consider here pure bipartite as well as multipartite states and derive explicit formulas for the source entanglement. Moreover, we obtain explicit formulas for a whole class of source entanglement measures that characterize the simplicity of generating a given bipartite pure state via LOCC. Furthermore, we show how the accessible entanglement can be computed numerically. For generic four-qubit states we first derive the necessary and sufficient conditions for the existence of LOCC transformations among these states and then derive explicit formulas for their accessible and source entanglement.
We investigate the entanglement of bipartite systems from an operational point of view. Main emphasis is put on bipartite pure states in the single copy regime. First, we present an operational characterization of bipartite pure state entanglement, viewing the state as a multipartite state. Then, we investigate the properties and relations of two classes of operational bipartite and multipartite entanglement measures, the so-called source and the accessible entanglement. The former measures how easy it is to generate a given state via local operations and classical communication (LOCC) from some other state, whereas the latter measures the potentiality of a state to be convertible to other states via LOCC. We investigate which parameter regime is physically available, i.e. for which values of these measures does there exist a bipartite pure state. Moreover, we determine, given some state, which parameter regime can be accessed by it and from which parameter regime it can be accessed. We show that this regime can be determined analytically using the Positivstellensatz. We compute the boundaries of these sets and the boundaries of the corresponding source and accessible sets. Furthermore, we relate these results to other entanglement measures and compare their behaviors.
We introduce two operational entanglement measures that are applicable for arbitrary multipartite (pure or mixed) states. One of them characterizes the potentiality of a state to generate other states via local operations assisted by classical communication and the other characterizes the simplicity of generating the state at hand. We show how these measures can be generalized to two classes of entanglement measures. Moreover, we compute the new measures for pure few-partite systems and use them to characterize the entanglement contained in a three-qubit state. We identify the Greenberger-Horne-Zeilinger and W state as the most powerful pure three-qubit states regarding state manipulation.
The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit states cannot be obtained from nor transformed to any state (i.e., they are isolated), which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that generic nnn-qubit symmetric states are also isolated, we consider particular families for which we can determine to be, on the contrary, endowed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. In the course of the study of LOCC transformations, we also characterize the local symmetries of symmetric states.
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.
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 nnn-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 (LOCCNLOCC_{\mathbb{N}}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 LOCCNLOCC_{\mathbb{N}}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.