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arXiv:quant-ph/0209166 · Physical Review A

Protocols for entanglement transformations of bipartite pure states

Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi

Published Apr 17, 2003• 3 citations
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Abstract

We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication. This framework unifies and greatly simplifies previous works. A necessary condition for ``pure contraction'' transformations is given. Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.

DOI
10.1103/physreva.67.042312

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We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication. This framework unifies and greatly simplifies previous works. A necessary condition for ``pure contraction'' transformations is given. Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.
It is well known that the majorization condition is the necessary and sufficient condition for the deterministic transformations of both pure bipartite entangled states by local operations and coherent states under incoherent operations. In this paper, we present two explicit protocols for these transformations. We first present a permutation-based protocol which provides a method for the single-step transformation of ddd-dimensional coherent states. We also obtain generalized solutions of this protocol for some special cases of ddd-level systems. Then, we present an alternative protocol where we use ${d}^{\ensuremath{'}}$-level (${d}^{\ensuremath{'}}<d$) subspace solutions of the permutation-based protocol to achieve the complete transformation as a sequence of coherent-state transformations. We show that these two protocols also provide solutions for deterministic transformations of pure bipartite entangled states.
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 studied pure state transformations using local operations assisted by finitely many rounds of classical communication (LOCCIN{\text{LOCC}}_{I\phantom{\rule{-0.30em}{0ex}}N}LOCCIN​) [C. Spee, J. I. de Vicente, D. Sauerwein, and B. Kraus [Phys. Rev. Lett. (to be published)], arXiv:1606.04418]. Here, we present the details of some of the proofs and generalize the construction of examples of state transformations via LOCCIN{\text{LOCC}}_{I\phantom{\rule{-0.30em}{0ex}}N}LOCCIN​ which require a probabilistic step. However, we also present explicit examples of SLOCC classes where any separable transformation can be realized by a protocol in which each step is deterministic (all-det-LOCCIN{\text{LOCC}}_{I\phantom{\rule{-0.30em}{0ex}}N}LOCCIN​). Such transformations can be considered as natural generalizations of bipartite transformations. Furthermore, we provide examples of pure state transformations which are possible via separable transformations, but not via LOCCIN{\text{LOCC}}_{I\phantom{\rule{-0.30em}{0ex}}N}LOCCIN​. We also analyze an interesting genuinely multipartite effect which we call locking or unlocking the power of other parties. This means that one party can prevent or enable the implementation of LOCC transformations by other parties. Moreover, we investigate the maximally entangled set restricted to LOCCIN{\text{LOCC}}_{I\phantom{\rule{-0.30em}{0ex}}N}LOCCIN​ and show how easily computable bounds on some entanglement measures can be derived by restricting to LOCCIN{\text{LOCC}}_{I\phantom{\rule{-0.30em}{0ex}}N}LOCCIN​.
We study the problem of transforming a set of pure bipartite states into another using deterministic LOCC (local operations and classical communication). Necessary conditions for the existence of such a transformation are obtained using LOCC constraints on state transformation, entanglement, and distinguishability. These conditions are shown to be independent but not sufficient. We discuss their satisfiability and classify all possible input-output pairs of sets accordingly. We also prove that strict inclusions hold between LOCC, separable, and positive partial transpose operations for set transformation problems.