Scalable experimental estimation of multipartite entanglement

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Scalable experimental estimation of multipartite entanglement

Leandro Aolita,1,2Andreas Buchleitner,2,3and Florian Mintert 2,3,4

1

Instituto de F´ısica,Universidade Federal do Rio de Janeiro,Caixa Postal 68528,21941-972Rio de Janeiro,RJ,Brasil

2

Max-Planck-Institut f¨u r Physik komplexer Systeme,N¨o thnitzerstraße 38,D-01187,Dresden,Germany 3

Albert-Ludwigs-Universit¨a t Freiburg,Physikalisches Institut,Hermann-Herder-Str.3,D-79104Freiburg,Germany

4

Department of Physics,Harvard University,17Oxford Street,Cambridge Massachusetts,USA

(Dated:February 4,2008)We present an efficient experimental estimation of the multipartite entanglement of mixed quantum states in terms of simple parity measurements.

PACS numbers:03.67.-a,03.67.Mn,42.50.-p

Introduction.–Entanglement has been identified as a key resource for quantum information processing tasks.Further-more,it is clear that the dramatic advantage in using quan-tum mechanical systems instead of classical ones to process information emerges only in the limit of a very large num-ber of system components.Taking into account all the ex-tra qubits necessary for error correction,a quantum computer has to run on quantum registers composed of at least several thousand qubits to outperform its present-day classical coun-terpart.This explains the tremendous effort dedicated during the last few years to the experimental production and coher-ent manipulation of multiparticle entangled states of photons [1,2,3,4,5,6],ions [8,9,10,11],and in cavity QED devices [7].

Also the experimental quantification of multipartite entan-glement has thus become a major issue of interest.In prin-ciple,such quantification can be carried out through quantum state tomography [9,12,13],i.e.,the complete reconstruction of the state’s density matrix via the measurement of a com-plete set of observables,followed by the subsequent evalua-tion of a valid entanglement measure.In practice,however,to-mography rapidly saturates the available resources and is thus no viable strategy under the perspective of scalability.Clear evidence of this is given by the experimental characterization of genuine multiparticle entangled states of up to eight ions [11]:Ten hours of data aquisition –implementing measure-ments in 38=6561detection bases,each corresponding to a different experimental setting –were followed by computa-tionally expensive data processing,to reconstruct the eight-ion density matrix of the experimentally prepared state.There-fore,full tomography of entangled ion chains composed of more than only eight ions appears largely impracticable.Quantum non-locality tests [1,2,4,5,6,7]and entangle-ment witnesses [3,8,9,10,11,14,15,16]provide alterna-tive means to assess the degree of entanglement of a quantum state,and were used in several experiments.Both these tech-niques require the measurement of only a few observables,but allow to detect the entanglement of only a small class of states.This implies that some a priori knowledge on the state to be analyzed is necessary.A simple entanglement measurement scheme for arbitrary mixed states is therefore highly desir-able.

First steps in this direction were taken in [17,18],where

multipartite concurrence [19]was shown to be directly acces-sible through projective measurements on two identically pre-pared quantum states.This original approach [17,18]–exper-imentally demonstrated for twin photon entanglement [20,21]–was restricted to the ideal case of pure states,and a first gen-eralization for mixed states was given in [22],yet applicable only for bipartite systems.Here we come up with the ultimate formulation of this approach to direct experimental entangle-ment estimation,for mixed states of quantum systems with an arbitrary number of constituents.Our procedure,based on lo-cal parity measurements,features excellent scaling properties:the number of required observables –which can all be probed in one single experimental setting –is equal to the number of subsystems.

The observable lower bound.–Consider two copies of an arbitrary mixed state of an N -partite quantum system with Hilbert space H .We introduce an observable V such that

C 2( )≥Tr ⊗ V

,(1)i.e.,that allows us to experimentally bound the concurrence

of the state from below.The Hermitian operator V acts on the composite Hilbert space associated with the two-fold copy of the system H⊗H ,and has the two following remarkable prop-erties:(i)it can be detected through projective measurements of only N two-particle observables ,and (ii)a single experi-mental setting is required throughout the detection process.The required two-particle measurements are simultaneous parity measurements on each particle and its copy.In each run of the experiment,measuring the local parity state of all N pairs defines an event in which each pair is projected onto either a symmetric or an antisymmetric state.From all possi-ble events we distinguish three types:

(i)the entire system together with its copy is projected onto a globally symmetric state –symmetric with re-spect only to the exchange of both copies of the entire system;(ii)the entire system and copy are projected onto a globally

antisymmetric state –antisymmetric with respect only to the exchange of both copies of the entire system;or,(iii)system and copy are projected onto a full locally sym-metric state in which all N particle-copy pairs are si-

a r X i v :0710.3529v 2 [q u a n t -p h ] 19 O c t 2007

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