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Physical Review A

Deterministic entanglement extraction

Luis Roa, Ariana S. Muñoz, Carlos A Muñoz, A. B. Klimov

Published May 28, 2019• 3 citations
Publisher

Abstract

We propose a scheme for a deterministic extraction of entanglement by means of a reduction process involving local von Neumann measurements. In an example of a tripartite system, we show that by choosing appropriate measurement bases for a given qubit, one can map an initial three-qubit state into outcome pure bipartite states with the same amount of entanglement of formation. In addition, by optimizing local projective measurements, one can efficiently convert higher-order correlations, contained in a multipartite state, into their lower-order counterparts, and thus significantly enhance the initial (reduced mixed-state) entanglement. We find the analytical expressions for the bounds of the deterministically extracted entanglement and relate them with the initial three-tangle and bipartite (mixed-state) concurrences.

DOI
10.1103/physreva.99.052344

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We propose a scheme for a deterministic extraction of entanglement by means of a reduction process involving local von Neumann measurements. In an example of a tripartite system, we show that by choosing appropriate measurement bases for a given qubit, one can map an initial three-qubit state into outcome pure bipartite states with the same amount of entanglement of formation. In addition, by optimizing local projective measurements, one can efficiently convert higher-order correlations, contained in a multipartite state, into their lower-order counterparts, and thus significantly enhance the initial (reduced mixed-state) entanglement. We find the analytical expressions for the bounds of the deterministically extracted entanglement and relate them with the initial three-tangle and bipartite (mixed-state) concurrences.
The basic entanglement swapping protocol allows to achieve the main aim of deterministically projecting two qubits, which have never interacted, onto a maximally entangled state. For deterministic swapping the key ingredient is the maximal entanglement initially contained in two pairs of qubits and the ability to measure onto a Bell basis. In other words, the basic and deterministic entanglement swapping scheme involves three maximal level of entanglement. In this work we address probabilistic entanglement swapping processes performed with different amounts of initial entanglement. Besides, we suggest a non Bell measuring-basis, thus we introduce a third entanglement level in the process. Additionally, we propose the \emph{unambiguous state extraction} scheme as the local mechanism for probabilistically achieving the main aim. Amalgamating these three elements allows us to design four strategies for performing probabilistic entanglement swapping. Surprisingly, we encountered an twofold entanglement matching effect related to the concurrence of the measuring-basis. Specifically, the maximal probability of accomplishing the main aim becomes a constant for concurrences higher than or equal to the matching entanglement values. Thus, we show that maximal entanglement in the measuring-basis is not required for attaining the main aim with optimal probability.
The Von Neumann entropy of reduced states is a measure of bipartite entanglement. Despite its name, the entanglement entropy cannot by itself be used as a resource for creating thermodynamic heat flows. In order to extract heat from an entangled pure state, it first needs to be converted into a stochastically mixed state by a process of quantum state reduction. Here we show that even in a system with only two degrees of freedom, for which bipartite entanglement is the sole form of entanglement available, the entanglement entropy cannot be converted into thermodynamic entropy in a one-to-one fashion. Moreover, we show that the stochastic dynamics which is necessarily present in any realistic model of quantum state reduction, allows for multiple definitions of entropy. We indicate why quantum state reduction does not allow construction of a perpetuum mobile, despite some measures of entropy evolving non-monotonically during its dynamics. Finally, we relate the different measures of entropy to the information they contain about quantum entanglement and extractable heat, and show that models of quantum state reduction based on physical, correlated stochastic driving forces give rise to observable thermodynamic signatures of quantum state reduction that can be unambiguously distinguished from environment-induced dephasing.
The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the WWW type. A state belonging to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having standard GHZ and standard WWW states, under the local measurement operations carried out by a single party and ppp (p=2,3p=2,3p=2,3) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which all its bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.
Quantum entanglement is the fundamental hallmark of quantum mechanics and a core resource for realizing long-distance quantum communication and scalable linear quantum computing. Accordingly, the precise detection and quantitative quantification of entanglement constitute a foundational and critical problem in quantum information theory. To date, researchers have proposed numerous sufficient conditions for entanglement detection as well as a variety of entanglement measures to characterize the entanglement strength of quantum states; nevertheless, efficient and direct measurement schemes for core entanglement parameters remain underdeveloped. Based on unitary transformations and auxiliary measurements, this paper proposes a set of quantum circuit schemes capable of directly measuring the bipartite concurrence and the tripartite 3-tangle entanglement measure. By introducing auxiliary qubits and constructing specific controlled unitary operations, the proposed scheme maps the analytical expressions of the two entanglement measures onto the measurement probabilities of output states from quantum circuits. It enables efficient and direct quantitative measurement of bipartite and tripartite entanglement without performing full quantum state tomography. This work provides a feasible technical route for the experimental characterization of entanglement properties and lays a groundwork for the practical deployment of multipartite entanglement resources in quantum information processing.
Entanglement is known to be a promising resource which enables us to execute various quantum tasks. Its quantification has been successful for bipartite pure states. However, the quantification of entanglement for multipartite states has been in a difficult situation. The difficulty is due to the difference of structures between bipartite and multipartite states. In the present thesis, we introduce a new approach for this problem. We give a necessary and sufficient condition of the possibility of a deterministic LOCC transformation of three-qubit pure states. This condition is expressed as a transformation law of six entanglement parameters jAB, jAC , jBC , jABC , J5 and Qe, where jAB, jAC and jBC are bipartite entanglements, jABC is a tripartite entanglement, J5 is a tripartite parameter which means a kind of phase, and Qe is a new tripartite parameter which means a kind of charge. This fact shows that three-qubit pure states are a partially ordered set parametrized by the six entanglement parameters. The order of the partially ordered set is defined by the possibility of a deterministic LOCC transformation from a state to another state. In this sense, the present condition is an extension of Nielsen’s work [14] to three-qubit pure states. We also clarify the rules of transfer and dissipation of entanglement. These rules guarantee that the tripartite entanglement can be transformed into bipartite entanglements, but that the bipartite entanglements cannot be transformed into the tripartite entanglement. This implies that the tripartite entanglement is a higher entity than the bipartite entanglements. With a new combination of the six entanglement parameters, the present condition can be simplified enough to determine easily whether a deterministic LOCC transformation from an arbitrary state |ψ〉 to another arbitrary state |ψ′〉 is possible or not. Moreover, the minimum number of times of measurements to reproduce an arbitrary deterministic LOCC transformation is given. This is an extension of Horodecki et al.’s work [24] to three-qubit pure states.
Local Operations enhancing the entanglement of bipartite quantum states are of great interest in quantum information processing. Subject of this paper are local selective operations acting on single copies of states. Such operations can lead to larger entanglement with respect to a certain measure as studies before by the Horodeckis and A. Kent et al. (PRA 60,PRL 81 and 83). We present a complete characterisation of all local operations yielding optimal entanglement for pairs of qubits, extending former results of A. Kent et al. We introduce a new technique for the classification of states according to their behaviour under entanglement optimizing operations, using the entanglement properties of the support of density matrices.