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arXiv:quant-ph/0206192

Local vs. Joint Measurements for the Entanglement of Assistance

T. Laustsen, Frank Verstraete, S. J. van Enk

Published Jun 27, 2002• 7 citations
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Abstract

We consider a variant of the entanglement of assistance, as independently introduced by D.P. DiVincenzo {\em et al.} ({\tt quant-ph/9803033}) and O. Cohen (Phys. Rev. Lett. {\bf 80}, 2493 (1998)). Instead of considering three-party states in which one of the parties, the assistant, performs a measurement such that the remaining two parties are left with on average as much entanglement as possible, we consider four-party states where two parties play the role of assistants. We answer several questions that arise naturally in this scenario, such as (i) how much more entanglement can be produced when the assistants are allowed to perform joint measurements, (ii) for what type of states are local measurements sufficient, (iii) is it necessary for the second assistant to know the measurement outcome of the first, and (iv) are projective measurements sufficient or are more general POVMs needed?

DOI
10.48550/arxiv.quant-ph/0206192

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We consider a variant of the entanglement of assistance, as independently introduced by D.P. DiVincenzo {\em et al.} ({\tt quant-ph/9803033}) and O. Cohen (Phys. Rev. Lett. {\bf 80}, 2493 (1998)). Instead of considering three-party states in which one of the parties, the assistant, performs a measurement such that the remaining two parties are left with on average as much entanglement as possible, we consider four-party states where two parties play the role of assistants. We answer several questions that arise naturally in this scenario, such as (i) how much more entanglement can be produced when the assistants are allowed to perform joint measurements, (ii) for what type of states are local measurements sufficient, (iii) is it necessary for the second assistant to know the measurement outcome of the first, and (iv) are projective measurements sufficient or are more general POVMs needed?
We consider a variant of the entanglement of assistance, as independently introduced by D.P. DiVincenzo et al. (quant-ph/9803033) and O. Cohen (Phys. Rev. Lett. 80, 2493 (1998)). Instead of considering three-party states in which one of the parties, the assistant, performs a measurement such that the remaining two parties are left with on average as much entanglement as possible, we consider four-party states where two parties play the role of assistants. We answer several questions that arise naturally in this scenario, such as (i) how much more entanglement can be produced when the assistants are allowed to perform joint measurements, (ii) for what type of states are local measurements sufficient, (iii) is it necessary for the second assistant to know the measurement outcome of the first, and (iv) are projective measurements sufficient or are more general POVMs needed?
We consider a variant of the entanglement of assistance, as independently introduced by D.P. DiVincenzo et al. (quart-ph/9803033) and O. Cohen (Phys. Rev. Lett. 80, 2493 (1998)). Instead of considering three-party states in which one of the parties, the assistant, performs a measurement such that the remaining two parties are left with on average as much entanglement as possible, we consider four-party states where two parties play the role of assistants. We answer several questions that arise naturally in this scenario, such as (i) how much more entanglement can be produced when the assistants are allowed to perform joint measurements, (ii) for what type of states are local measurements sufficient, (iii) is it necessary for the second assistant to know the measurement outcome of the first, and (iv) are projective measurements sufficient or are more general POVMs needed?.
The entanglement of assistance quantifies the amount of entanglement that can be concentrated among a group of spatially separated parties using the assistance of some auxiliary system. In this paper we study the entanglement of assistance in the simplest scenario of three qubits. We consider how much entanglement is lost when the helper party becomes uncoupled by local measurement and classical communication. For three-qubit pure states, it is found that lossless decoupling is always possible when entanglement is quantified by the maximum probability of locally generating a Bell state. However, with respect to most other entanglement measures, such as concurrence and entanglement entropy, a lossless decoupling of the helper is possible only for a special family of states that we fully characterize.
For certain joint measurements on a pair of spatially separated particles, we ask how much entanglement is needed to carry out the measurement exactly. For a class of orthogonal measurements on two qubits with partially entangled eigenstates, we present upper and lower bounds on the entanglement cost. The upper bound is based on a recent result by Berry [Phys. Rev. A 75, 032349 (2007)]. The lower bound, based on the entanglement production capacity of the measurement, implies that for almost all measurements in the class we consider, the entanglement required to perform the measurement is strictly greater than the average entanglement of its eigenstates. On the other hand, we show that for any complete measurement in $d\ifmmode\times\else\texttimes\fi{}d$ dimensions that is invariant under all local generalized Pauli operations, the cost of the measurement is exactly equal to the average entanglement of the states associated with the outcomes.
The Unruh response thermalizes accelerated detectors, but the entanglement remaining in a register does not by itself specify what distributed measurements can recover. We study this distinction for a four-qubit cluster state whose two assisting nodes independently undergo generalized amplitude damping, while the target nodes remain isolated and the environments are inaccessible. We derive the exact joint-assisted concurrence and the exact one-way value for arbitrary local measurements. The joint optimum equals the bipartition negativity, and computational-basis measurements attain the one-way optimum. A polar-decomposition argument, together with compactness, establishes a necessary and sufficient condition for separable measurements to attain the joint optimum and proves a strict gap whenever this condition fails. For assisting detectors at equal Unruh temperature with relaxation-rate ratio r>1r>1r>1, joint advantage exists precisely when the thermal excitation factor satisfies q<1/rq<1/rq<1/r. At finite positive temperature it ends at a unique finite time, while the remaining entanglement is nonzero. For r=2r=2r=2, both the transition time and remaining resource have closed expressions. These results identify when joint assistance is necessary within the independent weak-coupling thermalization model.
We study under which conditions it is possible to assert that a joint measurement cannot be simulated by local operations and classical communication. More concretely, we consider a scenario where two parties, Alice and Bob, send each a state selected from a labeled set of unknown states to a third party, Charlie, who in turn interacts with the states in some undisclosed way and then announces an outcome. We show that, under the assumption that Alice and Bob know the dimensionality of their systems, there exist situations where the statistics of the outcomes reveals the nature of Charlie's measurement.