We present an experimental method to produce 4-qubit phased Dicke states,
based on a source of 2-photon hyperentangled states. By introducing quantum
noise in the multipartite system in a controlled way, we have tested the
robustness of these states. To this purpose the entanglement of the resulting
multipartite entangled mixed states has been verified by using a new kind of
structural witness.Comment: 6 pages, 3 figure, supplementary information include

Bruno, N.

Bruß, D.

Chiuri, A.

Macchiavello, C.

Mataloni, P.

Vallone, G.

English

arXiv

A recently introduced family of multipartite entangled states, the 4-qubit phased Dicke states, has been created by 2-photon hyperentanglement. Our experimental method allows high state fidelity and generation rate. By introducing quantum noise in the multipartite system in a controlled way, we have tested the robustness of these states. To this purpose the entanglement of the resulting multipartite entangled mixed states has been verified by using a new kind of structural witness

CHIURI, ANDREA

VALLONE, GIUSEPPE

BRUNO, Nicola

MATALONI, Paolo

C. Macchiavello

D. Bruss

Archivio della ricerca- Università di Roma La Sapienza

Hyperentangled Mixed Phased Dicke States: Optical Design and Detection

A. Chiuri

N. Bruno

D. Bruß

P. Mataloni

Archivio istituzionale della ricerca - Università di Padova

Hyperentangled Mixed Phased Dicke States: Optical Design and DetectionA. Chiuri,1 G. Vallone,2,1 N. Bruno,1 C. Macchiavello,3 D. Bruß,4 and P. Mataloni1,51Dipartimento di Fisica, Sapienza Universitá di Roma, I-00185 Roma, Italy2Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi,Via Panisperna 89/A, Compendio del Viminale, I-00184 Roma, Italy3Dipartimento di Fisica ‘‘A. Volta’’ and INFN-Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy4Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany5Istituto Nazionale di Ottica Applicata (INO-CNR), L.go E. Fermi 6, I-50125 Firenze, Italy(Received 28 July 2010; revised manuscript received 25 October 2010; published 13 December 2010)A recently introduced family of multipartite entangled states, the 4-qubit phased Dicke states, has beencreated by 2-photon hyperentanglement. Our experimental method allows high state fidelity andgeneration rate. By introducing quantum noise in the multipartite system in a controlled way, we havetested the robustness of these states. To this purpose the entanglement of the resulting multipartiteentangled mixed states has been verified by using a new kind of structural witness.DOI: 10.1103/PhysRevLett.105.250501 PACS numbers: 03.67.Mn, 03.65.YzThe generation and detection of multipartite entangledstates is a remarkable challenge that needs to be accom-plished in order to fully explore and exploit the genuinequantum features of quantum information and many-bodyphysics. So far only a limited number of families of puremultipartite entangled states has been experimentally pro-duced. In view of future applications, it is particularlyimportant to test the robustness of the generated states inthe presence of unavoidable noise coming from the envi-ronment. Here, we produce a new family of multipartiteentangled states, we experimentally introduce certain typesof noise in a controlled way and test the robustness prop-erties of the states.The experimental generation of multipartite entangledstates that we propose is based on hyperentangled photons[1,2], which allows us to produce symmetric and phasedDicke states. Dicke states have recently attracted muchinterest, due to their resistance against photon loss [3]and projection measurements [4], and have been producedin experiments with photons [4–6]. Phased Dicke states areachieved by introducing phase changes starting from ordi-nary Dicke states [7]. Thus, they do not belong to thesymmetric subspace and offer new possibilities for multi-partite communication protocols, in particular, becausetheir degree of entanglement (in terms of the geometricmeasure of entanglement [8]) is higher or equal withrespect to the symmetric ones [9].In order to test the presence of multipartite entanglementwe may adopt different kinds of entanglement witnesses.Their experimental implementations are presented, e.g., inRefs. [10] for bipartite qubits, and [4–6,11] for pure sym-metric multipartite states. In this work we implement arecently proposed new class of entanglement witnesses, so-called structural witnesses [7], and further extend such aclass in order to achieve higher efficiency in entanglementdetection. Moreover, we test the robustness of the phasedDicke states by introducing dephasing noise in a controlledfashion and provide a measurement of the lower bound onthe robustness of entanglement. In this way we provide anew experimental tool to investigate the entanglementproperties of multipartite mixed states. The methodadopted to create 4-qubit phased Dicke states is based on2-photon hyperentanglement. This technique makes pos-sible the realization of such multipartite states, with rele-vant advantages in terms of generation rate and statefidelity compared to 4-photon states. The measurementswere performed by a closed-loop Sagnac scheme thatallows high stability. Moreover, we were able to controlthe noise in a photonic 4-qubit experiment (so far, to ourknowledge controlled noise on 4 qubits has only beenachieved with ions, see [12]).An entanglement witness is defined as a Hermitianoperator W that detects the entanglement of a state  if ithas a negative expectation value for this state, hWi ¼TrðWÞ< 0 while at the same time TrðWÞ  0 for allseparable states  [13,14]. For a composite system of Nparticles, the structural witnesses [7] have the formWðkÞ :¼ 1N  ðkÞ ¼ 1N  12½ ðkÞ þ ðkÞ; (1)where k is a real parameter (the wave-vector transfer in ascattering scenario), 1N is the identity operator, andðkÞ ¼ 1BðN; 2Þ ½cxŜxxðkÞ þ cyŜyyðkÞ þ czŜzzðkÞ; (2)with ci 2 R, jcij  1. Here BðN; 2Þ is the binomial coeffi-cient and the structure factor operators ŜðkÞ are defined asŜ ðkÞ :¼ Xi<jeikðrirjÞSi Sj ; (3)where i, j denote the ith and jth spins, ri, rj their positionsin a one-dimensional scenario, and Si are the spin operatorsPRL 105, 250501 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending17 DECEMBER 20100031-9007=10=105(25)=250501(4) 250501-1  2010 The American Physical Societywith ,  ¼ x, y, z. In the following we normalize thedistances with the labels of the qubits as ri  rj ¼ i j. Inthe present work we focus on the case of 4-qubits phasedDicke states defined as [7]:jDph4 i ¼1ffiffiffi6p ðj0011i þ j1100i þ j0110i þ j1001i j0101i  j1010iÞ: (4)A suitable structural witnessW for the above phasedDickestate is given by the operator (1) with k ¼ , cx ¼ cy ¼cz ¼ 1 and Si being the Pauli operators [7]. This witnessleads to TrðjDph4 ihDph4 jW Þ ¼  49 .A wider class of structural witness can be obtained bygeneralizing the operator given in (2) to linear superposi-tions of structure factor operators ŜðkÞ evaluated fordifferent values of k:ðkx; ky; kzÞ ¼ 1BðN; 2Þ ½cxŜxxðkxÞ þ cyŜyyðkyÞþ czŜzzðkzÞ; (5)with ci 2 R, jcij  1. Following the same argument as in[7], it can be shown that any operator of the form (5)combined as in (1) has non-negative expectation valuesfor separable states and is therefore an entanglement wit-ness. Using this more general construction for the presentexperiment we consider a witness operator with kx ¼ky ¼  and kz ¼ 0:W ¼ 1N  16 ½ŜxxðÞ þ ŜyyðÞ  Ŝzzð0Þ: (6)The expectation value of the above witness for the phasedDicke state (4) is given by TrðjDph4 ihDph4 j W Þ ¼  23 , that issmaller than the one achieved by the structural witnessWbased on (2), thus leading to a more robust entanglementdetection in the presence of noise.State generation.—Let us consider the following stateji  1ffiffi6p ðj0010i  j1000i þ 2j0111iÞ. It is easy to showthat the phased Dicke state can be obtained by applying aunitary transformation U to the state ji:jDðphÞ4 i ¼ Z4CZ12CZ34CX12CX34H1H3ji  Uji (7)where Hj and Zj stands for the Hadamard and the Pauli ztransformations on qubit j, CXij ¼ j0iih0j1j þ j1iih1jXj isthe controlled-NOT gate, and CZij ¼ j1iih1j1j þ j0iih0jZjthe controlled Z. An equivalent and more simple trans-formation is given by U ¼ Z1CX12CX34H1H3. We usedthe transformation given in (7) in order to compensate theoptical delay introduced by theCX gates in the Sagnac loopof Fig. 1(b). We realized the Dicke state by using4-qubits encoded into polarization and path of two para-metric photons [A and B in Fig. 1(a)]. The j0i and j1i statesare encoded into horizontal jHi and vertical jVi polariza-tion or into right jri and left j‘i path. Explicitly, we used thefollowing correspondence between physical states andlogical qubits: fj0i1; j1i1g ! fjriA; j‘iAg, fj0i2; j1i2g !fjHiA; jViAg, fj0i3; j1i3g ! fjriB; j‘iBg, and fj0i4; j1i4g !fjHiB; jViBg. According to these relations the state ji readsji ¼ 1ffiffiffi6p ½jHHiðjr‘i  j‘riÞ þ 2jVVijr‘i (8)and may be obtained by suitably modifying the source usedto realize polarization-momentum hyperentangled states[1,15]. In each ‘‘ket’’ of (8) the first (second) term refersto particle A (B). A vertically polarized UV laser beam(P ¼ 40 mW) impinges on a type I -barium borate(BBO) nonlinear crystal in two opposite directions, backand forth, and determines the generation of the polarizationentangled state corresponding to the superposition of thespontaneous parametric down conversion (SPDC) emissionat degenerate wavelength [see Fig. 1(a)]. A 4-hole maskselects four optical modes (two for each photon), namelyjriA, j‘iA, jriB, and j‘iB, within the emission cone of thecrystal. The SPDC contribution, due to the pump beamFIG. 1 (color online). Generation of the 4-qubit phased Dicke state. (a) Scheme of the entangled 2-photon 4-qubit parametric sourcethat generates the state ji. Mirror M reflects both the UV pump beam and the parametric radiation. The lens L is used to obtainparallel modes at the output of the 4-hole screen. HWP (QWP) is an half- (quarter-) wave plate. (b) Optical setup used to transform jiinto the state jDðphÞ4 i and to measure Pauli operators. The phase ’ is used to properly generate jiwhile ’A and ’B are used to measurePauli path operators for the A and B photon, respectively.PRL 105, 250501 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending17 DECEMBER 2010250501-2incoming after reflection on mirror M, corresponds to theterm jHHi  ðjr‘i  j‘riÞ, whose weight is determined bya half wave plate intercepting the UV beam (see [16] formore details on the generation of the nonmaximally polar-ization entangled state). The other SPDC contribution2jVVijr‘i is determined by the first excitation of thepump beam: here the j‘ri modes are intercepted by twobeam stops and a quarter wave plate QWP transforms thejHHi SPDC emission into jVVi after reflection on mirrorM. The relative phase between the jVVi and jHHi is variedby translation of the spherical mirror.The transformation (7) ji ! jDðphÞ4 i is realized by usingwave plates and one beam splitter (BS): the twoHadamards H1 and H3 in (7), acting on both path qubits,are implemented by a single BS for both A and B modes.For each controlled-NOT (or controlled-Z) gate appearing in(7) the control and target qubit are, respectively, repre-sented by the path and the polarization of a single photon: ahalf wave plate (HWP) with axis oriented at 45 (0) withrespect to the vertical direction and located into the left j‘i(right jri) mode implements a CX (CZ) gate.After these transformations, the optical modes are spa-tially matched for a second time on the BS, closing in thisway a ‘‘displaced Sagnac loop’’ interferometer that allowshigh stability in the path Pauli operator measurements [seeFig. 1(b)]. Polarization Pauli operators are measured bystandard polarization analysis setup in front of detectorsDA and DB (not shown in the figure). The overall detectionrate is500 Hz. Note that the j0i (j1i) states are identifiedby the counterclockwise (clockwise) modes in the Sagnacloop. It is worth stressing the high stability allowed in pathanalysis by the Sagnac interferometric scheme. This par-ticular configuration, operating on both the up anddown photon of the state, has made it possible to performa detailed investigation of the robustness of a multipartiteentangled state.Decoherence.—We will now describe how we intro-duced a controlled decoherence into the system. Considera single photon in aMach-Zehnder interferometer with twoarms (left and right). Varying the relative delay x¼‘rbetween the right and left arm corresponds to a single qubitpath decoherence channel given by  ! ð1 pÞþpZZ. The parameter p is related to x: when x > c,where  represents the photon coherence time, then p ¼ 12 ,while when x¼0 we have p¼0. This can be understoodby observing that there are two time bins (one for eachpath). By varying the optical delay, we entangle the pathwith the time bin degree of freedom (DOF). Hence, bytracing over time we obtain decoherence in the path DOFdepending on the overlap between the two time bins. In oursetup, this can be obtained by changing the relative delayx¼‘r between the right and the left modes of thephotons in the first interferometer shown in Fig. 1. Sincethe translation stage acts simultaneously on both photons,this operation corresponds to two path decoherencechannels: ! ð1 q2Þ2þ q2ð1 q2Þ½Z1Z1 þ Z3Z3þ q22Z1Z3Z1Z3 (9)where the parameter q2 is related to x in the followingway. Let us consider the path terms in the jHHi contribu-tion in ji, namely jci ¼ 1ffiffi2p ðjr‘i  j‘riÞ. The decoher-ence acts by (partially) spoiling the coherence between thejr‘i and j‘ri term giving the state 12 ðj‘rih‘rj þ jr‘ihr‘jÞ 12 ð1 2q2Þ2ðj‘rihr‘j þ jr‘ih‘rjÞ. By assuming that forjci the decoherence (9) is the main source of imperfec-tions, the measured visibility ~VexpðxÞ of first interferenceon BS may be compared with the calculated value ~V ¼ð1 2q2Þ2: then, the relation between x and q2, shown inFig. 2, is obtained. The measured visibility is defined as~VexpðxÞ ¼ BCB where B are the coincidences measuredout of interference (i.e., measured forxmuch longer thanthe single photon coherence length) andC the coincidencesmeasured in a given position of x. It is worth noting thatat x ¼ 0 we have q2 ¼ 0:0175	 0:0001 which corre-sponds to a maximum visibility V0 ¼ 0:9313	 0:0005 atx ¼ 0.The decoherence channel (9) acts on the state ji.However, it can be interpreted as a decoherence actingon the phased Dicke state jDðphÞ4 i. Using Eq. (7) and therelations UZ1Uy ¼ Y1Y2 and UZ3Uy ¼ Y3Y4, thechannel (9) may be interpreted as a collective decoherencechannel on jDðphÞ4 i:jDðphÞ4 ihDðphÞ4 j !X4j¼1BjjDðphÞ4 ihDðphÞ4 jByj (10)with B1 ¼ ð1 q2Þ1, B2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2ð1 q2ÞpY1Y2, B3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2ð1 q2ÞpY3Y4, and B4 ¼ q2Y1Y2Y3Y4. A collective de-coherence is a decoherence process that cannot be seen asthe action of several channels acting separately on two (ormore) qubits. A different type of collective noise, intro-duced in [17], was experimentally demonstrated in [18] fortwo polarization qubits in optical fibers.Two other main sources of imperfections must be con-sidered in our setup (see supplementary information for adetailed discussion [19]): the first one is due to a non-perfect superposition between forward and backwardSPDC emission, i.e., between the jHHi and jVVi contri-butions. This imperfection can be modeled as a phasepolarization decoherence channel acting on qubit 2:FIG. 2 (color online). Values of q2 corresponding to differentvalues of the path delay x.PRL 105, 250501 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending17 DECEMBER 2010250501-3 ! ð1 q1Þþ q1Z2Z2. By selecting in ji the corre-lated modes jrli and by suitably setting the HWP on thepump beam we obtain the following state: 1ffiffi2p ðjHHiAB þeijVViABÞjrli. Even in this case the value of the measuredpolarization visibility (V ’ 0:90) can be related to thepolarization decoherence channel as q1 ¼ 1V2 ’ 0:05.The second interference on the BS (i.e., after the Sagnacloop) has been also investigated. In the measurement con-dition we obtained an average visibility of Vk2 ’ 0:80corresponding to a decoherence channel  ! ð1 q3Þ2 þ q3ð1  q3Þ½Z1Z1 þ Z3Z3 þ q23Z1Z3Z1Z3with q3 ¼ 0:05.Measurements.—We measured the witness operator (6)for different values of q2. The results are shown in Fig. 3.The dark curve corresponds to the theoretical curve ob-tained by considering all the three decoherence channelsdescribed above and setting q1 ¼ 0:05 and q3 ¼ 0:05 (seethe supplementary information about the details on thetheoretical curve [19]). Notice that the noise parameterfor which the witness expectation value vanishes gives alower bound on the robustness of the entanglement of theproduced state with respect to the implemented noise. Thewitness W measured for the phased Dicke state ish W iexp ¼ 0:382	 0:012: (11)We also measured a witness Wmult introduced in [20] todemonstrate that the generated state jDðphÞ4 i is a genuinemultipartite state and to obtain a bound on the fidelity F. Itsexpression is given in the supplementary information [19].We obtainedhWmulti ¼ 0:341	 0:015 ! F  0:780	 0:005: (12)Varying the noise parameter q2, we obtained a negativeexpectation value ofWmult for q2  0:114, thus proving theexistence of genuine multipartite entanglement up to thisnoise level.Following the approach of quantitative entanglementwitnesses [21], we can also use the experimental resulton the expectation value of the witness to provide a lowerbound on the random robustness of entanglement Er. Thisis defined in [22] to be the maximum amount of white noisethat one can add to a given state  before it becomesseparable. When h W i is negative, a lower bound onErðÞ is given byErðÞ  DjTrðW ÞjTrð W Þ; (13)where D is the dimension of the Hilbert space on which acts. In our experiment the witness from Eq. (6) and itsexpectation value given in Eq. (11) lead toErðÞ  jh W iexpj ¼ 0:382	 0:012: (14)Other bounds for different values of q2 are shown in Fig. 3.Equation (13) thus relates the value of W in the presenceof the collective noise (10) with the resilience of entangle-ment under the presence of a general white noise.In summary, by two hyperentangled photons we studiedthe structural properties and decoherence resilience of thephased 4-qubit Dicke states. The multipartite entanglementwas detected up to considerable noise level and the robust-ness of entanglement was tested by using an intrinsicallyhigh phase stability setup. The realized phase Dicke stateshave a high fidelity and, compared with other Dicke statesbased on 4-photon entanglement, are produced at higherrepetition rate.This work was supported in part by the EU projectsCORNER, by a DFG project, and by FinanziamentoAteneo 2009 of Sapienza Università di Roma.[1] M. Barbieri et al., Phys. Rev. A 72, 052110 (2005).[2] J. T. Barreiro et al., Phys. Rev. Lett. 95, 260501 (2005).[3] M. Koashi V. Bužek, N. Imoto, Phys. Rev. A 62, 050302(2000).[4] N. Kiesel et al., Phys. Rev. Lett. 98, 063604 (2007).[5] R. Prevedel et al., Phys. Rev. Lett. 103, 020503 (2009).[6] W. Wieczorek et al., Phys. Rev. Lett. 103, 020504 (2009).[7] P. Krammer et al., Phys. Rev. Lett. 103, 100502 (2009).[8] T.-C. Wei and P.M. Goldbart, Phys. Rev. A 68, 042307(2003).[9] M. Aulbach (private communication).[10] M. Barbieri et al., Phys. Rev. Lett. 91, 227901 (2003).[11] M. Bourennane et al., Phys. Rev. Lett. 92, 087902 (2004).[12] J. T. Barreiro et al., Nature Phys. 6, 943 (2010).[13] M. Horodecki et al., Phys. Lett. A 223, 1 (1996).[14] B.M. Terhal, Phys. Lett. A 271, 319 (2000).[15] R. Ceccarelli et al., Phys. Rev. Lett. 103, 160401 (2009).[16] G. Vallone et al., Phys. Rev. A 76, 012319 (2007).[17] C. Macchiavello and G.M. Palma, Phys. Rev. A 65,050301 (2002).[18] K. Banaszek et al., Phys. Rev. Lett. 92, 257901 (2004).[19] See supplementary material at http://link.aps.org/supplemental/10.1103/PhysRevLett.105.250501 for a de-tailed analysis of the decoherence channels and the ex-pression of Wmult.[20] G. Tóth et al., New J. Phys. 11, 083002 (2009).[21] J. Eisert et al., New J. Phys. 9, 46 (2007).[22] G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999).FIG. 3 (color online). Experimental values of the witness Wand the bound on Er as a function of q2. The curves correspondto the theoretical behavior obtained by setting q1 ¼ 0:05 andq3 ¼ 0:05.PRL 105, 250501 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending17 DECEMBER 2010250501-4

Hyperentangled mixed phased Dicke states: optical design and detection

Abstract

Similar works

Full text

Available Versions

Archivio della ricerca- Università di Roma La Sapienza

Archivio istituzionale della ricerca - Università di Padova

Archivio Istituzionale della Ricerca - Università degli Studi di Pavia

Crossref