Two-step procedure to discriminate discordant from classical correlated or factorized states
Simone Cialdi, Andrea Smirne, Matteo G. A. Paris, Stefano Olivares, Bassano Vacchini
TTwo-step procedure to discriminate discordant from classical correlated or factorized states
Simone Cialdi , , Andrea Smirne , , Matteo G.A. Paris , Stefano Olivares , Bassano Vacchini , Dipartimento di Fisica, Universit`a degli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy Department of Physics, University of Trieste, Strada Costiera 11, I-34151 Trieste, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy (Dated: October 11, 2018)We devise and experimentally realize a procedure capable to detect and distinguish quantum discord andclassical correlations, as well the presence of factorized states in a joint system-environment setting. Our schemebuilds on recent theoretical results showing how the distinguishability between two reduced states of a quantumsystem in a bipartite setting can convey important information about the correlations present in the bipartitestate and the interaction between the subsystems. The two addressed subsystems are the polarization and spatialdegrees of freedom of the signal beam generated by parametric downconversion, which are suitably prepared bythe idler beam. Different global and local operations allow for the detection of different correlations by studyingvia state tomography the trace distance behavior between suitable polarization subsystem states.
PACS numbers: 03.65.Yz,42.50.Dv,03.67.-a
The study of a bipartite system is an ever present themewhich has led to important advancements in the understand-ing of quantum mechanics, especially when the two partiescannot be put on an equal footing. The prototypical situationis a measurement interaction, in which the interest is all onthe side of the system, calling for tools and ideas allowing foran ever improving description of such interactions [1]. Thetheory of open quantum systems has provided a natural ex-tension of these efforts, in which the quantum features of themeasurement apparatus are put in major evidence [2], whilecorrelations between system and environment have receivedimportant attention only more recently, thanks to the consol-idation of quantum information theory [3]. The latest theo-retical developments as well as the refinement in the experi-mental techniques has led to a change of paradigm in facingthe system-environment ( SE ) dynamics. The possibility hasbeen envisaged of actually exploiting the open quantum sys-tem, supposed to be liable to a relatively easy and accurate ex-perimental observation, as a quantum probe of features of theenvironment, typically to be considered as a complex system.Properties of the environment which might be unveiled by anobservation on the system up to now include the detection ofquantum phase transitions [4], as well as the assessment ofcorrelations within the state of the environment [5]. Theseadvancements have been based on the study in time of the dis-tinguishability of different initial system states [6], which hasproven to be a fruitful strategy in order to exploit a quantumsystem as probe of features of a bipartite dynamics [7–13].In this paper we improve this approach to devise a novelmethod for the determination of quantum correlations, whichplay a crucial role both in quantum information and in the de-velopment of quantum technologies. The approach is basedon a two-step procedure, which relying on measurements onthe system only allows to determine whether a given initial SE state actually contains quantum correlations, as quantifiedby quantum discord or, if this is not the case, decide whetherit contains classical correlations or it is factorized. The rel-evance of this characterization lies in the fact that quantum discord has proven useful for different tasks in quantum infor-mation processing (see e.g. [14]). Our scheme goes beyondprevious studies on the detection of initial correlations [7] andof quantum discord [10], takes as figure of merit for the distin-guishability the trace distance among statistical operators [15]and is experimentally realized in an all-optical setup based onparametric downconversion (PDC) for the generation of cor-relations [16]. At variance with other approaches, relying ona measurement on multiple copies of the total system [17], wehere only perform tomographic measurements on one of thesubsystems. Detection of correlations
We start considering a SE state,which might contain correlations of some kind. For the ex-perimental realization at hand we encode the system in thepolarization degrees of freedom of one of the beams in thePDC (referred to as the signal). The environment correspondsto momentum (spatial) degrees of freedom of the signal, whilethe other beam (usually referred to as the idler) is exploited toprepare the initial state ρ SE (0) . In the first stage, the eigen-states of the reduced system state ρ S (0) = Tr E [ ρ SE (0)] areobtained by performing state tomography. This allows us todefine the two orthogonal projections on the system eigen-states, Π and − Π , and to introduce a dephasing operation Φ d such that ρ SE (0) → ρ d SE (0) ≡ Φ d [ ρ SE (0)] , where ρ d SE (0) = Π ρ SE (0)Π + ( − Π) ρ SE (0)( − Π) . (1)The dephased state has the same marginals of the initial onebut, according to its expression, has zero quantum discord[18]. As suggested in [19], the difference between ρ d SE (0) and ρ SE (0) , as given by the trace distance, provides a quantifier ofthe quantum discord in the original state, namely: T = 12 (cid:107) ρ d SE (0) − ρ SE (0) (cid:107) = (cid:107) Π ρ SE (0)Π − { Π , ρ SE (0) }(cid:107) . (2)Now, one can prove the presence of non-classical correlationsin ρ SE (0) by just measuring the system. In fact, if quantumcorrelations are present the marginals of the system states after a r X i v : . [ qu a n t - ph ] N ov k ⇢ S ( t ) ⇢ dS ( t ) k E U t x?? x?? Tr E U t ⇢ SE ( ) d ! ⇢ dSE ( ) k ⇢ S ( t ) ⇢ uS ( t ) k E U t x?? x?? Tr E U t ⇢ SE ( ) V u ! ⇢ uSE ( ) Tr E ??y ??y Tr E k ⇢ S ( ) ⇢ uS ( ) k T u (t) - T u (0)T d (t) =0>0>0 CC ⇢ SE = ⇢ S ⌦ ⇢ E F =0 QC Figure 1. (Color online) Logical scheme of the cascading, two-step procedure exploited to discriminate among quantum correlated (QC),classically correlated (CC) or simply factorized (F) SE states. In the first stage (left box) a dephasing operation Φ d which leaves invariantthe marginals is applied, allowing to detect quantum correlations using as witness the trace distance T d ( t ) between the reduced states evolvedfrom original and dephased state. If no quantum correlations are detected, the second stage (right box) is entered, in which the growth in timeof the trace distance of initial states differing by a local unitary operation on the system V u , i.e. T u ( t ) − T u (0) , allows to detect classicalcorrelations or to conclude that the initial state is factorized. See the text for details. a time evolution U t will generally differ, even if coinciding atthe initial time [10]. This implies that the quantity T d ( t ) = 12 (cid:107) ρ d S ( t ) − ρ S ( t ) (cid:107) = (cid:13)(cid:13) Tr E ◦ U t [ ρ d SE (0) − ρ SE (0)] (cid:13)(cid:13) (3)acts as a local witness of quantum discord in the initial state.This witness is probabilistic in nature, since not every timeevolution is bound to reveal the existing quantum correlations.However, as argued in [11], the efficiency of the method isvery high, and in the case considered a fixed time evolution al-lows for the detection of quantum discord in the whole rangeof states which can be prepared, apart from a set of measurezero. In the general case it has been shown that the averageover the set of unitaries not only detects the quantum discord,but also allows to quantify it. This first stage of the detectionscheme is described in the first section of the logical schemein Fig. 1. If the witness provided by the expression Eq. (3) ispositive, then the state ρ SE (0) does contain quantum correla-tions corresponding to non zero discord. On the other hand, if T d ( t ) = 0 , then the second stage of the cascading procedureis entered (second section in Fig. 1). At this level we havealready checked the absence of quantum correlations, there-fore we should perform a measurement involving only the sys-tem to check whether ρ SE (0) is a factorized state or containsclassical correlations. Also in this case the presence of initialcorrelations can be unveiled by a growth of the trace distancebetween different initial states above the initial value as a con-sequence of the SE time evolution [7]: while the consideredcondition is in principle only sufficient, the considered timeevolution allows to detect with unit efficiency the consideredclass of states. In order to generate another initial SE statewithout introducing correlations we perform a local unitarytransformation denoted by V u , which only affects the systemdegrees of freedom. Given the fact that the marginal statesof the environment are left unchanged by V u , the growth ofthe trace distance indeed witnesses the presence of initial cor-relations, rather than of different initial environmental states.We are then led to consider the behavior of the trace distancebetween the reduced system state ρ S and its transformed coun- Figure 2. (Color online) Scheme of the apparatus. L is the pumplaser, HWP1 and HWP2 are half-wave plates, BBOx2 two BBO crys-tals, Π a polarizer and U is the spatial light modulator. T is the to-mographic scheme (a quarter-wave plate, a HWP and a polarizer),H and V denote two polarizers aligned along the horizontal and ver-tical axes, respectively, DS is the double slit, F1 a high pass filter(780 nm), F2 an interference filter (bandwidth of 10 nm and centralwavelength of 810 nm); D1, D2 are detectors and C the coincidencescounter. The components drawn with a dashed line have to be modi-fied or moved according to the different stages of the procedure. terpart ρ u S at the initial and at a later time. If the difference T u ( t ) − T u (0) = 12 (cid:107) ρ u S ( t ) − ρ S ( t ) (cid:107) − (cid:107) ρ u S (0) − ρ S (0) (cid:107) (4)which acts as correlation witness is greater than zero, then ρ SE (0) has classical correlations, otherwise the state is actu-ally factorized, i.e. ρ SE (0) = ρ S (0) ⊗ ρ E (0) (see Fig. 1). Experimental realization
In our experiment SE stateswith different correlations have been generated, and the two-step procedure described above for the discrimination of cor-relations has been tested, providing in particular an experi-mental verification of the scheme for the detection of quantumdiscord proposed in [10].Our experimental apparatus, sketched in Fig. 2, is based onPDC generated by two 1 mm adjacent type-I Beta-Barium Bo-rate (BBO) crystals, oriented with their optical axes alignedin perpendicular planes and pumped by a 10 mW, 405 nmcw diode laser (Newport LQC405-40P). The two BBO crys-tals generate the signal and idler states with perpendicularpolarization and the interference filter (F2) ensures a goodspatial correlation between signal and idler [5, 20, 21]. Wegenerate two channels 0 and 1 (corresponding to the mo-mentum states | (cid:105) and | (cid:105) , respectively) with a double slit(DS) positioned along the idler path. This scheme allowsus to act on the idler beam to prepare the signal beam inthe three cases of interest and to easily control and changethe amplitude of the polarizations. The arrangement of thetwo BBO crystals produces a factorized state between po-larization and momentum, namely ρ p ⊗ ρ m . Both compo-nents are generally described by a mixture of the form [8] ρ k = P k ρ ent k + (1 − P k ) ρ mix k , where k = p, m , the statisticaloperator ρ ent k = | ψ k (cid:105)(cid:104) ψ k | denotes a pure entangled state and ρ mix k the corresponding mixed counterpart. The weight P k isnaturally interpreted as purity of the state, but does not play arole in the present treatment which studies the correlations be-tween polarization and momentum. The states for polarizationand momentum read | ψ p (cid:105) = √ λ | HH (cid:105) + √ − λ | V V (cid:105) and | ψ m (cid:105) = √ ( | (cid:105) + | (cid:105) ) respectively, where H and V denotehorizontal and vertical polarizations. The relative weight ofthe two polarization states parametrized by ≤ λ ≤ can beadjusted at will by means of a half-wave plate (HWP) locatedin the path of the pump laser, while the balance in the momen-tum degrees of freedom is obtained by a careful alignment ofthe preparation apparatus and optimizing the phase-matchingbetween the crystals. Controlled correlations between sys-tem (polarization) and environment (momentum) can be in-troduced inserting in the idler beam a horizontal and a verticalpolarizer in the paths corresponding to the momenta denotedby and respectively. If no further operation is performed,the obtained state is of the form ρ CCSE = λ | H (cid:105)(cid:104) H | ⊗ | (cid:105)(cid:104) | + (1 − λ ) | V (cid:105)(cid:104) V | ⊗ | (cid:105)(cid:104) | , (5)which clearly exhibits only classical correlations, while stateswith non zero quantum discord are generated by inserting ahalf-wave plate (HPW2) in the momentum channel of thesignal beam, thus obtaining ρ QCSE = λ | H (cid:105)(cid:104) H | ⊗ | (cid:105)(cid:104) | + (1 − λ ) | θ (cid:105)(cid:104) θ | ⊗ | (cid:105)(cid:104) | , (6)where | θ (cid:105) = cos( θ ) | H (cid:105) +sin( θ ) | V (cid:105) . In the left panel of Fig. 3we plot the quantum discord in such a state as quantified byEq. (2). The absence of polarizers in the idler path leads totake the trace over the idler degrees of freedom and thereforeto the factorized state ρ FSE = ( λ | H (cid:105)(cid:104) H | +(1 − λ ) | V (cid:105)(cid:104) V | ) ⊗
12 ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) . (7)The eigenstates of the reduced system states, whose knowl-edge is necessary to determine the dephasing operation de-scribed in Eq. (1), that is the projections Π and − Π , areobtained through the full tomography (T) of the polarizationstates [22], as depicted in Fig. 2. The projections are im-plemented by means of polarizers according to the measured Figure 3. (Color online) (Left) The amount of quantum discord T defined in Eq. (2) for a state given in Eq. (6), as a function of λ and θ .(Right) Measured values of T d for different values of the parameter φ , which describes the interaction between system and environment.The two points for φ = π/ and φ = π/ have been obtained for λ = 0 . , whereas for φ = π we have set λ = 0 . . The solid anddashed lines correspond to λ = 0 . and λ = 0 . , respectively. eigenstates. The interaction between system and environmentis obtained by a spatial light modulator (U), which can in-sert a position and polarization sensitive phase in the sig-nal. In particular we have realized an evolution correspond-ing to a phase-gate, acting on the momentum correspondingto channel only by applying a phase to the polarization de-grees of freedom according to the operator Diag(e iφ , in the {| H (cid:105) , | V (cid:105)} basis. As shown in the right panel of Fig. 3, theoptimal performance in the correlation detection is obtainedfor φ = π , which has thus been taken as reference value. Inthe following, the time specification “ ” will denote the stateright after the preparation, while the time “ t ”’ will identify thestate after the interaction. The unitary transformation V u onthe system degrees of freedom only, used to prepare the otherreference state for the second stage of the two-step procedureof Fig. 1, is obtained by inserting a half-wave plate intercept-ing both momenta in the idler beam.The experimental results are summarized in Fig. 4, that re-ports the data of the tomographic analysis. In the first rowexamples of the system-environment states corresponding toEqs. (6), (5) and (7), respectively, are considered for specificvalues of λ and θ . From the tomographic data we retrieve theexpression for the dephasing operation Φ d to be implemented.In the second row the reduced system states after the timeevolution corresponding to a phase gate are given, to be com-pared via trace distance with the reduced states plotted in thethird row and obtained by applying Φ d to the overall state be-fore the evolution. The experimentally measured value of thetrace distance growth corresponding to Eq. (3) is given in thefourth row. When this value is zero (within the experimentalerrors), thus pointing to the absence of quantum discord, a fur-ther analysis corresponding to the second stage of the schemein Fig. 1 is performed. Therefore, we first apply a local uni-tary operation V u to the system and then measure the quantityEq. (4), whose positivity reveals the presence of correlationsin the initial state, as detected by a growth of the distinguisha-bility in time between different initial reduced system states.The experimental values for the quantity in Eq. (4) are givenin the last row of Fig. 4, showing that indeed a factorized state Figure 4. (Color online) Tomographic measurements of the statesinvolved in the experiment. In the left column the case of a state ofthe form Eq. (6) with λ = 0 . and θ = π/ has been considered.From top to bottom we have plotted the observed values for ρ SE (0) , ρ S ( t ) and ρ d S ( t ) respectively. In the central column we provide thecorresponding measurements for the state Eq. (5) with λ = 0 . .The value T d ( t ) of the trace distance Eq. (3) is here compatible withzero according to the experimental error, testifying the absence ofquantum discord, while the positivity of T u ( t ) − T u (0) given byEq. (4) shows the detection of classical correlations. In the rightcolumn the considered state corresponds to Eq. (7) with λ = 0 . ,and the factorized structure of the state is unveiled by the value of T u ( t ) − T u (0) , which is zero within the experimental value. Thetime specification 0 and t denote the states right after the preparationand the interaction stages respectively. can be detected within the experimental accuracy. In fact theindistinguishability of two statistical operators correspondingto zero trace distance can be consistently assessed within atomographic approach since quantum tomography is a statis-tically reliable procedure, meaning that for any finite numberof repeated preparations one obtains an estimate with a pre-dictable standard deviation, thus leading to error bars follow-ing the standard statistical scaling for any quantity evaluatedusing the reconstructed density matrix [23].The reliability of the method has been further tested bymeasuring the growth of the trace distance between the de-phased states after the interaction as quantified by Eq. (3) fordifferent values of λ and θ and comparing it with the theoret-ical prediction. The result is plotted in Fig. 5, where differentexperimental points are measured along lines with fixed rela-tive weight λ and varying angle θ , as well as vice-versa. Thetheoretical expression is given by the smooth surface. As itappears the trace distance Eq. (3) lies above zero, thus detect- Figure 5. (Color online) Experimental results for the trace distanceEq. (3) corresponding to different values of the parameters λ and θ ,as compared to the theoretical prediction given by the smooth sur-face. The red curve corresponds to λ = 0 . , while the blue curveis fixed by θ = π/ . The experimental points are plotted on theprojected curves to improve their visibility. ing the quantum discord of the state plotted in there left panelof Fig. 3, for all possible values of the parameters λ and θ ,apart from a set of measure zero corresponding to the pointson the line λ [cos(2 θ ) −
1] = cos(2 θ ) . Conclusions and outlook
We have suggested and demon-strated a simple all-optical setup to detect and discriminatedifferent kind of SE correlations by performing measurementson the system only. The scheme consists of a two-step proce-dure. At each step information about the presence and the na-ture of correlations is extracted by tomographically estimatingthe distinguishability between system states after the action ofsuitable global or local quantum operations. In particular, wefirst assess the presence of quantum discord as quantified bythe measurement induced disturbance [10, 19], and then, inthe absence of quantum discord, we further determine the fac-torizability of the state versus the presence of classical corre-lations, exploiting the connection between initial correlationsand growth of trace distance [7]. The successful realization ofour procedure is based on the implementation of a dephasingmap on the SE state and on the reliable detection of quan-tum discord. Our procedure can be easily adapted to differentexperimental settings, the basic requirement being the realiza-tion of the dephasing map and the capability to perform statetomography on the sole system. Our results pave the way forreliable detection and discrimination of environments or SE features in systems of interest for quantum technology. Acknowledgments
This work has been supported byMIUR (FIRB RBFR10YQ3H “LiCHIS”). BV and AS grate-fully acknowledge financial support by the EU projects COSTAction MP 1006 and NANOQUESTFIT respectively. [1] A. S. Holevo,
Probabilistic and statistical aspects of quantumtheory (North-Holland, Amsterdam, 1982)[2] H.-P. Breuer and F. Petruccione,
The Theory of Open QuantumSystems (Oxford University Press, Oxford, 2002)[3] M. Nielsen and I. Chuang,
Quantum Computation and Quan-tum Information (Cambridge University Press, Cambridge,2000)[4] P. Haikka, S. McEndoo, and S. Maniscalco, Phys. Rev. A ,012127 (2013); M. Borrelli, P. Haikka, G. De Chiara, andS. Maniscalco, Phys. Rev. A , 010101 (2013); M. Gessner,M. Ramm, H. H¨affner, A. Buchleitner, H.-P. Breuer, EPL ,40005 (2014)[5] A. Smirne, S. Cialdi, G. Anelli, M. G. A. Paris, and B. Vacchini,Phys. Rev. A , 012108 (2013)[6] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. ,210401 (2009)[7] E.-M. Laine, J. Piilo, and H.-P. Breuer, EPL , 60010 (2010)[8] A. Smirne, D. Brivio, S. Cialdi, B. Vacchini, and M. G. A. Paris,Phys. Rev. A , 032112 (2011)[9] S. Cialdi, D. Brivio, E. Tesio, and M. G. A. Paris, Phys. Rev. A , 042308 (2011)[10] M. Gessner and H.-P. Breuer, Phys. Rev. Lett. , 180402(2011)[11] M. Gessner and H.-P. Breuer, Phys. Rev. A , 042107 (2013)[12] B. P. Lanyon, P. Jurcevic, C. Hempel, M. Gessner, V. Vedral,R. Blatt, and C. F. Roos, Phys. Rev. Lett. , 100504 (2013)[13] M. Gessner, M. Ramm, T. Pruttivarasin, A. Buchleitner, H.-P.Breuer, and H. H¨affner, Nature Physics , 105 (2014)[14] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral Rev.Mod. Phys. , 1655 (2012); A. Brodutch, Phys. Rev. A ,022307 (2013); D. Girolami & al., Phys. Rev. Lett. , 210401 (2014)[15] C. A. Fuchs and J. van de Graaf, IEEE Trans. Inf. Th. , 1216(1999)[16] S. Cialdi, D. Brivio, and M. G. A. Paris, Phys. Rev. A ,042322 (2010)[17] C. Zhang, S. Yu, Q. Chen, and C. H. Oh, Phys. Rev. A ,032122 (2011); G. H. Aguilar, O. Jim´enez Farias, J. Maziero,R. M. Serra, P. H. Souto Ribeiro, and S. P. Walborn, Phys. Rev.Lett. , 063601 (2012); D. Girolami and G. Adesso, Phys.Rev. Lett. , 150403 (2012)[18] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. , 017901(2001); L. Henderson and V. Vedral, J. Phys. A: Math. Gen. , 6899 (2001);[19] S. Luo, Phys. Rev. A , 022301 (2008)[20] S. Cialdi, D. Brivio, E. Tesio, and M. G. A. Paris, Phys. Rev. A , 043817 (2011)[21] S. Cialdi, D. Brivio, A. Tabacchini, A. M. Kadhim, andM. G. A. Paris, Opt. Lett. , 3951 (2012)[22] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White,Phys. Rev. A , 052312 (2001); K. Banaszek, G. M. D’Ariano,M. G. A. Paris, and M. F. Sacchi, Phys. Rev. A , 010304(1999)[23] For each tomographic reconstruction we need 4 acquisitions,each consisting of 30 counts of 1 second. We thus ob-tain 4 mean counts with the relative standard deviations.Errors on the trace distances are then evaluated by MonteCarlo sampling starting from the experimental results, ac-cording to BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IU-PAP and OIML 2008 Evaluation of Measurement Data—Supplement 1 to the Guide to the Expression of Uncer-tainty in Measurement—Propagation of distributions using aMonte Carlo method