Discriminating multi-partite entangled states
Christian Schmid, Nikolai Kiesel, Wiesław Laskowski, Witlef Wieczorek, Marek Żukowski, Harald Weinfurter
aa r X i v : . [ qu a n t - ph ] A p r Discriminating multi-partite entangled states
Christian Schmid , , Nikolai Kiesel , , Wies law Laskowski ,Witlef Wieczorek , , Marek ˙Zukowski and Harald Weinfurter , Department f¨ur Physik, Ludwig-Maximilians-Universit¨at, D-80797 M¨unchen, Germany Max-Planck-Institut f¨ur Quantenoptik, D-85748 Garching, Germany Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gda´nski, PL-80-952 Gda´nsk, Poland (Dated: November 2, 2018)The variety of multi-partite entangled states enables numerous applications in novel quantuminformation tasks. In order to compare the suitability of different states from a theoretical pointof view classifications have been introduced. Accordingly, here we derive criteria and demonstratehow to experimentally discriminate an observed state against the ones of certain other classes ofmulti-partite entangled states. Our method, originating in Bell inequalities, adds an important toolfor the characterization of multi-party entanglement.
PACS numbers: 03.65.Ud, 03.67.Mn, 03.67.-a.
Entanglement is the crucial resource for quantum in-formation processing and as such the ”currency” to paywith in almost all applications. For two-partite quantumstates measures have been developed that uniquely spec-ify the value of this resource. In contrast, for n-partitestates the picture changes significantly. First, one has todistinguish not only between fully separable or entangled,but also between genuine n-partite, bi-, and tri- separa-ble entangled states, etc. Second, even states with thesame level of separability are different in the sense thatthey have, for example, different Schmidt rank [1] or thatthey cannot be transformed into each other, e.g., by, lo-cal unitary (LU) or, more generally, by stochastic localoperations and classical communication (SLOCC) [2, 3].From an experimental point of view, classifying statesaccording to the latter property is reasonable, as statesfrom one SLOCC-class are suited for the same multi-party quantum communication applications. Thus, forthe usage of multi-partite states it is of importance toknow not only the amount but also the type of entangle-ment contained in a particular state. In other words, thevalue and the type of the ”currency” is what matters.Tools to detect the entanglement of a state exist, mostprominently entanglement witnesses [4]. An alternativemethod, relying on the correlations between results ob-tained by local measurements, are Bell inequalities. Be-ing originally devised to test fundamental issues of quan-tum physics they allow to distinguish entangled fromseparable two-qubit quantum systems [5, 6]. Bell in-equalities, meanwhile extended to three- and more par-tite quantum states [7, 8, 9], can thus serve as witnessfor both entanglement and the violation of local realism.Recently it was observed that for each graph state allnon-vanishing correlations (or even a restricted numberthereof) form a Bell-inequality, which is maximally vio-lated only by the respective quantum state [10, 11]. Inparticular, the Bell inequality for the four-qubit clusterstate is not violated at all by GHZ states [10]. Naturallyseveral questions arise: Whether one can in general ap-ply such Bell inequalities to discriminate particular statesfrom other classes of multi-partite entangled states, if so, whether they can also be constructed and applied fornon-graph states, and finally, whether there are otheroperators that allow to experimentally discriminate en-tanglement classes.In this article we address these problems starting fromBell inequalities. We present a way to construct Bell op-erators [12] that are characteristic for a particular quan-tum state, i.e., operators that have maximal expectationvalue for this multi-partite state, only. With respect toexperimental applications we further aim that the ex-pectation value can be obtained by a minimal numberof measurement settings. Under certain conditions, wecan relax the initial requirement that characteristic op-erators have to be also Bell operators, which allows fur-ther reduction of the number of settings. Comparison ofthe experimentally obtained expectation values with themaximal expectation values for states from other entan-glement classes enables us to clearly distinguish observedstates from other multi-party entangled states.In order to construct a Bell operator, we exploit thefact that certain correlations between measurement re-sults on individual qubits are specific for multi-partitequantum states [9]. All correlations for a state | X i aresummarized by the correlation tensor T . If we focus onthe case of four qubits, then T ijkl = h X | ( σ i ⊗ σ j ⊗ σ k ⊗ σ l ) | X i , with i, j, k, l ∈ { , x, y, z } , where σ = σ x,y,z are the Pauli spin operators. To obtain a Bell op-erator ˆ B X which is characteristic for a state | X i , werequire that | X i is the eigenstate of ˆ B X with the highesteigenvalue λ max . If the eigenstate is not degenerate, thisimplies that ˆ B X , acting on another state cannot lead toan expectation value greater or equal λ max .An operator, which is in general not a Bell operator,but trivially fulfills the condition to have | X i as the onlyeigenstate with λ max = 1, is the projector or fidelity op-erator ˆ F X = | X ih X | andˆ F X = 116 X i,j,k,l T ijkl ( σ i ⊗ σ j ⊗ σ k ⊗ σ l ) . (1)For most of the relevant quantum states the major partof the 256 coefficients T ijkl is zero. Therefore, the num-ber of measurement settings necessary for the evaluationof ˆ F X is much smaller than for a complete state tomog-raphy. We consider the non-vanishing terms as relevantcorrelations for characterizing the state and take them asa starting point for the construction of ˆ B X . As we will seein the following two examples, there are quantum statesfor which a small subset of the relevant correlations isenough to construct ˆ B X . Once this is accomplished onecan calculate the upper bound, v ∗ Y , on the expectationvalues v Y = h Y | ˆ B X | Y i = h ˆ B X i Y for states | Y i whichbelong to other classes than | X i . Consequently, a stateunder investigation with h ˆ B X i Z = v Z cannot be an ele-ment of any class of states with v ∗ Y < v Z .Note, h ˆ B X i induces a particular ordering of stateswhich is neither absolute nor related to some entangle-ment of the states and, similarly to the entanglement wit-ness, depends on the operator ˆ B X . Yet, now we do notonly detect higher or lower degree of entanglement: wedistinguish different types of entanglement. One mightsay that a state with a higher h ˆ B X i is more ” | X i -type”entangled. The same is true for a mixed state ρ with ex-pectation value v ρ = Tr[ ˆ B X ρ ] = h ˆ B X i ρ , in the sense thatit cannot solely be expressed as a mixture of pure states | Y i i with v ∗ Y i < v ρ , but it has to contain contributionswith a higher ”X-type” entanglement.Summarizing, we point at the fact that one can ob-tain a witness of ” | X i -type” entanglement by construct-ing a discrimination operator, which has | X i as non-degenerate eigenvector with the highest eigenvalue. Af-ter all, such an operator is not unique, neither does itnecessarily have to be a Bell operator. However, a Belloperator unconditionally detects the entanglement of theinvestigated state, even if the state space is not fullyknown. For example, witness operators might detect astate to be entangled though a description of measure-ment results based on local realistic models, or for thatpurpose, based on separable states in higher dimensionalHilbert spaces, is possible [13]. If one trusts in the repre-sentation of the state, as shown below, even more efficientoperators for state discrimination can be devised.Let us now apply our method to the state | Ψ i [14]: | Ψ i = √ ( | i + | i − ( | i + | i + | i + | i )) . (2)This state was observed in multi-photon experiments [15]and can be used, for example, for decoherence free quan-tum communication [16], quantum telecloning [17], andmulti-party secret sharing [18].The fidelity operator for that state ˆ F Ψ contains 40relevant correlation operators ( σ i ⊗ σ j ⊗ σ k ⊗ σ l ), outof which 21 describe four-qubit correlations (i.e. do notcontain σ ). Already 10 are enough to construct a char-acteristic Bell operator that has | Ψ i as non-degenerate TABLE I: Maximal expectation values h ˆ B Ψ i State under LU under SLOCC | Ψ i | D (2)4 i | GHZ i | C i | W i | bi-sep i | sep i h ˆ B D (2)4 i State under LU under SLOCC | D (2)4 i | Ψ i | GHZ i | C i | bi-sep i | W i | sep i eigenstate with maximum eigenvalue λ max = 1:6 ˆ B Ψ = σ x ⊗ σ y ⊗ σ y ⊗ σ x + σ y ⊗ σ x ⊗ σ y ⊗ σ x − σ y ⊗ σ y ⊗ σ x ⊗ σ x + σ x ⊗ σ z ⊗ σ x ⊗ σ z + σ z ⊗ σ x ⊗ σ x ⊗ σ z − σ z ⊗ σ z ⊗ σ x ⊗ σ x + σ z ⊗ σ z ⊗ σ z ⊗ σ z − σ y ⊗ σ y ⊗ σ z ⊗ σ z + σ y ⊗ σ z ⊗ σ y ⊗ σ z + σ z ⊗ σ y ⊗ σ y ⊗ σ z . (3)ˆ B Ψ can be used to discriminate an experimentally ob-served state with respect to other four-qubit states. Withthe chosen normalization we obtain the limit for any lo-cal realistic theory by replacing σ i by some locally pre-determined values I i = ±
1, leading to the inequality |h ˆ B Ψ i avg | ≤ . Table I shows the bounds on the ex-pectation value of ˆ B Ψ acting on some classes of promi-nent four-qubit states (including a fully separable state | sep i , any bi-separable state | bi-sep i , as well as the four-partite entangled Dicke state D (2)4 [19], the GHZ [20],W [2] and Cluster ( C ) [21] state). These bounds wereobtained by numerical optimization over either LU- orSLOCC-transformations, respectively. In particular withthe bound for an arbitrary bi-separable state ˆ B Ψ pro-vides also a sufficient condition for genuine four-partiteentanglement.We now employ these results for the analysis of experi-mental data. To observe the state | Ψ i we used photonsgenerated by type II non-collinear spontaneous paramet-ric down conversion (SPDC) and a variable linear op-tics setup. Essentially, a four photon emission into twomodes is overlapped on a polarizing beam splitter (PBS)and subsequently split into four modes. Depending onthe setting of a half-wave plate (in our case oriented at45 ◦ ) preceding the PBS and conditioned on detecting aphoton in each of the four outputs, a variety of statescan be observed [22]. The fidelity of the experimentalstate ρ Ψ , determined from 21 four-qubit correlations,was F Ψ = Tr[ ˆ F Ψ ρ Ψ ] = 0 . ± .
01. The analysis ofthe experimental state using the Bell operator ˆ B Ψ re-quired less than half of the measurement settings andleads to v ρ Ψ4 = 0 . ± .
02 (see Fig. 1a). This value is,according to Table I, sufficient to prove that the experi-mental state is genuine four-qubit entangled and cannotbe of W-, Cluster-, or GHZ-type in the sense describedabove.The class of states that can experimentally not be ex-cluded as it has the second largest expectation value inTable I is represented by the so-called symmetric fourqubit Dicke state [19, 23] | D (2)4 i = √ ( | i + | i + | i + | i + | i + | i ) . (4)In turn, for the Dicke state a separate, characteristic Belloperator ˆ B D (2)4 can be constructed. Again, | D (2)4 i has 40correlation operators with non zero expectation value,out of which 21 describe original four-qubit correlations.Naturally, the exact values of the correlations T ijkl dif-fer compared to | Ψ i . In the case of | D (2)4 i they aresuch that eight of the correlation operators are alreadysufficient for the construction of ˆ B D (2)4 :6 ˆ B D (2)4 = − σ x ⊗ σ z ⊗ σ z ⊗ σ x − σ x ⊗ σ z ⊗ σ x ⊗ σ z − σ x ⊗ σ x ⊗ σ z ⊗ σ z + σ x ⊗ σ x ⊗ σ x ⊗ σ x − σ y ⊗ σ z ⊗ σ z ⊗ σ y − σ y ⊗ σ z ⊗ σ y ⊗ σ z − σ y ⊗ σ y ⊗ σ z ⊗ σ z + σ y ⊗ σ y ⊗ σ y ⊗ σ y , (5)with λ max = 1 for | D (2)4 i . This operator has a remark-able structure: It is of the form σ x ⊗ M + σ y ⊗ M ′ , where M and M ′ are three-qubit Mermin inequality operators[7, 24]. Thus, by applying a kind of GHZ-argument [20],the bound for any local realistic theory can be determinedto be |h ˆ B D (2)4 i avg | ≤ .Table II shows the maximal expectation values of ˆ B D (2)4 by the same set of four-qubit states as before. Con-sidering the structure of ˆ B D (2)4 , further omitting corre-lation operators, for example one whole block σ x ⊗ M (or σ y ⊗ M ′ ), leaves us with a four-qubit Mermin-typeBell operator. The corresponding Bell inequality is stillviolated by | D (2)4 i . However, it is not characteristicanymore for | D (2)4 i as it is maximally violated by thestate | GHZ i y = √ ( | RRRR i ± |
LLLL i ) and the bi-separable state | BS i = √ ( | + i ( | RRR i ± i | LLL i ))(where | ± i = √ ( | i±| i ) and | R, L i = √ ( | i± i | i )are the eigenstates of σ x and σ y , respectively). It is a par-ticular property of the Dicke state to have correlationsin two planes (x-z- and y-z-plane) of the Bloch sphere,whereas a GHZ state, for instance, is correlated only in FIG. 1: Histogramms of the four-photon coincidence statisticsfor the different measurement settings. Slots at the ordinateindicate different events for a particular basis setting: e.g.0011 for basis zzzz means detection of photons in the state | HHV V i . a) Statistics of the ten correlation measurements,required for the evaluation of the operator ˆ B Ψ . b) Statis-tics of the eight correlation measurements, required for theevaluation of the operator ˆ B D (2)4 . one plane (here the x-z-plane). This quite characteristicfeature is reflected in the construction of ˆ B D (2)4 . Recently,an experiment has been performed to observe the state | D (2)4 i [23]. In order to increase the state fidelity F bya higher degree of indistinguishability, here we reducedthe filter bandwidth from 3 nm to 2 nm, resulting in F = 0 . ± .
02 (compared to F = 0 . ± .
01 in [23]).For the state’s experimental analysis with the Bell op-erator (5) we find v ρ D (2)4 = 0 . ± .
04 (see Fig. 1b),from which we can conclude that it is genuine four-qubitentangled and cannot be, e.g., of W-, Cluster- or GHZ-type. Yet, this value is again just at the limit to separateagainst | Ψ i .If one is sure about the structure of the state space,that means that in our case it is spanned by four qubits,we can equally well use other operators instead of theBell operators. Let us first drop some of the correla-tions from ˆ B D (2)4 , e.g., the terms ( σ x ⊗ σ x ⊗ σ x ⊗ σ x )and ( σ y ⊗ σ y ⊗ σ y ⊗ σ y ). The resulting discriminationoperator ˆ D D (2)4 is not a Bell operator anymore, but stillhas | D (2)4 i as the only eigenstate with maximal eigen-value λ max = 1 (after proper normalization). Interest-ingly, as seen in Table III, it introduces a new orderingof states with a bigger separation between | D (2)4 i and | Ψ i . With v D ρ D (2)4 = 0 . ± .
05 we can discriminateagainst this state with a better significance. Note, thereordering, which results in the GHZ state having nowthe second highest eigenvalue, indicates that this opera-tor analyzes the various states from a different point ofview. This is quite plausible as it uses different correla-tions for the analysis. An even more radical change in
TABLE III: Alternative characteristic operators for D (2)4 State |h ˆ D D (2)4 i| (SLOCC) |h ˆ D ′ D (2)4 i| (SLOCC) | D (2)4 i | GHZ i | C i | W i | Ψ i | bi-sep i | sep i the point of view is possible with the data we droppedabove, i.e., ( σ x ⊗ σ x ⊗ σ x ⊗ σ x ) and ( σ y ⊗ σ y ⊗ σ y ⊗ σ y ).Relying on the particular symmetries of the Dicke state,from these measurements we can evaluate the discrim-ination operator ˆ D ′ D (2)4 = (( P k σ kx ) + ( P k σ ky ) ),where e.g. σ x/y = ⊗ ⊗ σ x/y ⊗ v D ′ ρ D (2)4 = 0 . ± .
013 with the boundsfor other states (Table III) we see that we can discrimi-nate our state against all states of the respective classeswith only two settings. Analogous considerations can beapplied for the construction of characteristic operatorsfor other states [26], where the number of settings scales polynomially with the number of qubits compared to theexponentially increasing effort for state tomography.In conclusion, here we showed that characteristic(Bell-)operators, i.e., operators for which a particularstate only has maximal expectation value, allow to dis-tinguish this state from the ones out of other classes ofmulti-partite entangled states. A simple, though not yetconstructive, method to design discrimination operatorsis based on the correlations between local measurementsettings that are typical for the respective quantum state.The low number of measurement settings significantlydiminishes the effort compared with standard analysis.Employing characteristic symmetries and properties ofthe state under investigation can even further reduce theeffort to a number of settings which scales polynomiallywith the number of qubits, thereby rendering the newmethod a truly efficient tool for the characterization ofmulti-partite entanglement.We thank D. Bruß, M. Horodecki, and M. Wolffor stimulating discussions. We acknowledge the sup-port by the DFG-Cluster of Excellence MAP, theDAAD/MNiSW exchange program, the EU ProjectsQAP and SECOQC. W.W. is supported by QCCC ofthe ENB and the Studienstiftung des dt. Volkes, W.L.by FNP. [1] B. M. Terhal and P. Horodecki, Phys. Rev. A ,040301(R) (2000); A. Sanpera, D. Bruß, and M. Lewen-stein, Phys. Rev. A , 050301(R) (2001); Y. Tokunaga,T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. A , 020301(R) (2006).[2] W. D¨ur, G. Vidal, and J. I. Cirac, Phys. Rev. A ,062314 (2000).[3] F. Verstraete, J. Dehaene, B. DeMoor, and H. Verschelde,Phys. Rev. A , 052112 (2002).[4] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.Lett. A , 1 (1996):[5] N. Gisin, Phys. Lett. A , 201 (1991).[6] B.M. Terhal, Phys. Lett. A , 319 (2000).[7] N. D. Mermin, Phys. Rev. Lett. , 1838 (1990).[8] A. V. Belinski˘i and D. N. Klyshko, Phys. Usp. , 653(1993); W. Laskowski, T. Paterek, M. Zukowski, andC. Brukner, Phys. Rev. Lett. , 200401 (2004); K. Na-gata, W. Laskowski, M. Wie´sniak, and M. ˙Zukowski,Phys. Rev. Lett. , 230403 (2004).[9] R. F. Werner and M. M. Wolf, Phys. Rev. A , 032112(2001); M. ˙Zukowski and ˇC. Brukner, Phys. Rev. Lett. , 210401 (2002).[10] V. Scarani, A. Acin, E. Schenck, and M. Aspelmeyer,Phys. Rev. A , 042325 (2005).[11] O. G¨uhne, G. T´oth, P. Hyllus, and H. J. Briegel, Phys.Rev. Lett. , 120405 (2005); G. T´oth, O. G¨uhne, andH. J. Briegel, Phys. Rev. A , 022303 (2006).[12] S. L. Braunstein, A. Mann, and M. Revzen, Phys. Rev.Lett. , 3259 (1992); R. F. Werner, and M. M. Wolf,Phys. Rev. A , 062102 (2000).[13] A. Acin, N. Gisin, and L. Masanes, Phys. Rev. Lett. , 120405 (2006).[14] H. Weinfurter and M. ˙Zukowski, Phys. Rev. A ,010102(R) (2001).[15] M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer,M. ˙Zukowski, and H. Weinfurter, Phys. Rev. Lett. ,200403 (2003); S. Gaertner, M. Bourennane, M. Eibl,C. Kurtsiefer, and H. Weinfurter, Appl. Phys. B , 803(2003); J.-S. Xu, C.-F. Li, and G.-C. Guo, Phys. Rev. A , 052311 (2006).[16] M. Bourennane, M. Eibl, S. Gaertner, C. Kurtsiefer,A. Cabello, and H. Weinfurter, Phys. Rev. Lett. ,107901 (2004).[17] M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral,Phys. Rev. A , 156 (1999).[18] S. Gaertner, C. Kurtsiefer, M. Bourennane and H. We-infurter, Phys. Rev. Lett. , 020503 (2007).[19] R. H. Dicke, Phys. Rev. , 99 (1954).[20] D. Greenberger, M. A. Horne, and A. Zeilinger, Goingbeyond Bell’s Theorem (Kluwer Academic, Dordrecht,1989); D. M. Greenberger, M. A. Horne, and A. Zeilinger,Am. J. Phys. , 1131 (1990).[21] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. ,5188 (2001).[22] W. Wieczorek et al., in preparation.[23] N. Kiesel, C. Schmid, G. T´oth, E. Solano, and H. Wein-furter, Phys. Rev. Lett. , 063604 (2007).[24] The Bell inequality found by our method for the symmet-ric six-qubit Dicke state with three excitations, | D (3)6 i , isof the same structure: σ x ⊗ M + σ y ⊗ M ′ . The bound forlocal realistic theories in this case is 0.4 and the expecta-tion value for the Dicke state is 1 compared to e.g. 0.85 for any six-qubit GHZ state.[25] G. T´oth and O. G¨uhne, Phys. Rev. A , 022340 (2005);G. T´oth, J. Opt. Soc. Am. B , 275 (2007).[26] G. T´oth and O. G¨uhne, Phys. Rev. Lett. , 060501 (2005); O. G¨uhne, C.-Y. Lu, W.-B. Gao, and J.-W. Pan,Phys. Rev. A , 030305(R) (2007). c (cid:13)(cid:13)