Entanglement verification via nonlinear witnesses
Megan Agnew, Jeff Z. Salvail, Jonathan Leach, Robert W. Boyd
EEntanglement verification via nonlinear witnesses
Megan Agnew, Jeff Z. Salvail, Jonathan Leach, Robert W. Boyd , Dept. of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, K1N 6N5 Canada and Institute of Optics, University of Rochester, Rochester, USA (Dated: October 26, 2018)The controlled generation of entangled states and their subsequent detection are integral aspectsof quantum information science. In this work, we analyse the application of nonlinear witnessesto the verification of entanglement, and we demonstrate experimentally that nonlinear witnessesperform significantly better than linear witnesses. Specifically, we demonstrate that a single non-linear entanglement witness is able to determine to a high degree of certainty that a mixed statecontaining orbital angular momentum (OAM) entanglement of the form ( | j, j (cid:105) + e iϕ | k, k (cid:105) ) / √ | j, k (cid:105) + e iϕ | k, j (cid:105) ) / √
2) is entangled for any relative phase ϕ and sufficient fidelity; j and k are OAMazimuthal quantum numbers. This is a significant improvement over linear witnesses, which cannotprovide the same level of performance. We envisage that nonlinear witnesses and our method ofstate preparation will have further uses in areas of quantum science such as superdense coding andquantum key distribution. Introduction : Entanglement, which can produce nonlo-cal correlations that are stronger than those predicted byclassical physics, is an essential part of quantum mechan-ics [1]. As a result, it has been studied extensively as ameans to test quantum mechanics [2–4]. Entanglement isa vital resource in many quantum information protocols.For example, entangled photonic qubits are importantfor communication protocols such as quantum key distri-bution [5, 6], superdense coding [7, 8], and quantum tele-portation [9–11], and entangled qubits are required forimplementations of quantum computing. Consequently,the efficient detection of entangled states [12] plays a vitalrole in many quantum information science applications.Determining the full quantum state of a system can beaccomplished through tomography [13–15], which con-sists of taking many measurements on identical copies ofa quantum state. The resulting real-valued probabilitiesare then used to estimate the complex-valued state thatbest fits the measurements. Tomography can determinethe complete density matrix that describes the system;however, it is inefficient for determining entanglementas it requires a large number of measurements. This isparticularly relevant when considering high-dimensionalor multipartite systems. As the number of elements re-quired to describe the state increases, so too does thetotal number of measurements required for reconstruc-tion [16] [35]. Compressive sensing is one approach toreducing the required number of measurements, but re-constructing the state from less information leads to aless accurate estimation of the final state [17].Entanglement witnesses provide an alternative to to-mography in the case where complete knowledge ofthe state is not required and detecting entanglement isthe goal. An entanglement witness establishes directlywhether a quantum state belonging to a certain class isentangled [18–22]. The use of entanglement witnessescan be more efficient than tomography as witnesses re-quire fewer measurements and no reconstruction. More generally, the application of witnesses in quantum scienceplays a vital role in establishing particular properties ofsystems [23].Linear entanglement witnesses, which are witnessesthat depend linearly on expectation value, have beenused to detect entanglement in bipartite polarisationstates [24] and orbital angular momentum (OAM) states[25, 26]. Multipartite entanglement has also been de-tected using a linear entanglement witness [27]. Linearentanglement witnesses are efficient as they require thefewest possible number of measurements that will givesufficient information about the state; however, in orderto function optimally, they require prior knowledge of theform of the entangled state. For example, different lin-ear witness are required for each of the four Bell states | Φ + (cid:105) , | Φ − (cid:105) , | Ψ + (cid:105) and | Ψ − (cid:105) ; using a linear witness that isnot appropriate to the form of the state may produce aninconclusive result when used to detect entanglement.Alternatively, nonlinear entanglement witnesses im-prove upon an existing linear entanglement witness witha term that relies nonlinearly on expectation value [28–31]. The improvement is that a nonlinear witness is ableto verify entanglement over a significantly larger set ofstates compared to its linear counterpart. Returningto the example of the Bell states, one can construct asingle nonlinear witness that will detect both correlatedBell states | Φ + (cid:105) and | Φ − (cid:105) and a single nonlinear witnessthat will detect both anti-correlated Bell states | Ψ + (cid:105) and | Ψ − (cid:105) . Specifically, one nonlinear witness works for almostall correlated states regardless of relative phase betweenthe modes, and one nonlinear witness works for almost allanti-correlated states. Importantly, these nonlinear wit-nesses are accessible when the nonlinear extension can beachieved using the same measurements as for the linearwitness [32].In this work, we demonstrate the controlled generationof a wide range of spatially entangled states and the sub-sequent experimental realisation of a class of nonlinear a r X i v : . [ qu a n t - ph ] O c t Separable Entangled | Φ + w Φ + ∞ > W Φ + L W Φ + ∞ W Φ - L c ba EntangledstatesSeparablestates w Φ + ∞ < w Φ + ∞ > ρ Φ + ρ Φ − ρ Φ − ρ Φ − ρ Φ + ρ Φ + FIG. 1:
Visual representation of nonlinear and linear en-tanglement witnesses.
The state ρ Φ + is entangled and is sep-arated from the set of separable states by the linear entanglementwitness W Φ + L ( a ), and the entangled state ρ Φ − is separated fromthe set of separable states by the linear entanglement witness W Φ − L ( b ). However, both states are separated from the set of separablestates by the nonlinear witness W Φ + ∞ ( c ). entanglement witnesses. We compare the expectationvalues of the nonlinear witnesses to those of the stan-dard linear witnesses and establish that the nonlinearwitnesses are capable of detecting entanglement over awide range of states. The particular degree of freedomwe choose to investigate is orbital angular momentum;however, our results are general in that the procedurescan be applied to other degrees of freedom such as polar-isation or spin. Theory : The characteristic feature of entanglement isthe observation of non-local correlations between twoqubits that belong to spatially separated systems. Wedenote the eigenstates of the qubits by | j (cid:105) and | k (cid:105) andthe two spatially separated systems by A and B . Thenthe general entangled state that contains two qubits inthe same state, i.e. correlated qubits, can be written as | Φ (cid:105) = 1 √ ε (cid:16) | j, j (cid:105) + εe iϕ | k, k (cid:105) (cid:17) . (1)Here, ε defines the degree of entanglement, ϕ is the phasebetween the modes and defines the nature of the corre-lations, and we use | j, j (cid:105) to be equivalent to | j (cid:105) A ⊗ | j (cid:105) B .Two of the four Bell states | Φ ± (cid:105) are particular cases ofequation (1) where ε is equal to unity such that the stateis maximally entangled, and the phase ϕ is equal to ei-ther 0 ( | Φ + (cid:105) ) or π ( | Φ − (cid:105) ). The general entangled state | Ψ (cid:105) that contains two qubits in opposite states, i.e. anti-correlated qubits, can be denoted by replacing | j, j (cid:105) with | j, k (cid:105) and | k, k (cid:105) with | k, j (cid:105) . The remaining two Bell states | Ψ ± (cid:105) are particular cases of the anti-correlated entangledstate.However, in practice, the incident state need not bepure; that is, the photons are described by ρ ψ = | ψ (cid:105)(cid:104) ψ | p + (1 − p ) / , (2)where p is the probability of obtaining the entangled state | ψ (cid:105) and is the identity matrix, which represents un-coloured noise. We use the superscript ψ to indicatethat the convex combination ρ is partially composed ofthe entangled state | ψ (cid:105) . Whether or not the state ρ ψ isentangled is determined by the probability p : states with p > / w of a witness W on a quantum state ρ provides the relevant information: a negative expectationvalue indicates entanglement, whereas a positive expec-tation value gives an inconclusive result. If a positiveexpectation value is obtained, the information gained isthat either the state is separable, or the witness chosenwas not appropriate for the form of the entangled state.The simplicity of such a result has led to the widespreaduse of linear entanglement witnesses.Linear entanglement witnesses are the simplest formof entanglement witness. However, linear witnesses func-tion only over restricted sets of states: entanglement ofthe set of states ρ Φ (or ρ Ψ ) cannot be verified with a sin-gle linear witness. As an example, the entanglement ofthe Bell state | Φ − (cid:105) cannot be confirmed using the linearwitness W Φ + L (constructed for the state | Φ + (cid:105) ) becausethe expectation value w Φ + L is positive.Recently, it was shown that it is possible to improvea linear witness with a term that relies nonlinearly onexpectation value [31–33]. One improvement is that en-tanglement of a significantly larger fraction of the set ofstates ρ Φ (or ρ Ψ ) can be verified with a single nonlinearwitness that contains the same observables as the linearwitness. For any value of p , there exists a nonlinear im-provement of a linear witness that always verifies the en-tanglement of a larger set of states compared to its linearcounterpart. As an example, the entanglement of boththe Bell states | Φ − (cid:105) and | Φ + (cid:105) can be confirmed using asingle nonlinear witness. A visual comparison betweenlinear and nonlinear witnesses is shown in figure 1.In our experiment, we compare the accessible nonlin-ear witnesses W Φ + ∞ and W Ψ + ∞ to their corresponding lin-ear witnesses. In order to construct the nonlinear im-provement to the linear witnesses, we use the methodoutlined in Ref. [32]. One starts with the original lin-ear witness W Φ + L , which can be considered a first-orderwitness W Φ + . The n th -order witness W Φ + n can be found A n t i - c o rr e l a t ed s t a t e s C o rr e l a t ed s t a t e s Q uan t u m s t a t e p r epa r a t i on P ho t on - pa i r gene r a t i on P ha s e c on t r o l N on li nea r w i t ne ss m ea s u r e m en t s C o i n c i den c e de t e c t i on P r o j e c t i on m ea s u r e m en t s BBO crystal Far-field ofcrystal R e l a y op t i cs SLM SMFDove prism θ /2 a b FIG. 2:
Schematic of the setup . The experiment has two main stages: ( a ) quantum state preparation and ( b ) nonlinear witnessmeasurements. In the quantum state preparation stage, entangled photon pairs are generated by parametric downconversion. The numberof Dove prisms then allows us to prepare either a correlated (one prism) or anti-correlated state (two prisms), and the angle of the first prismgives control of the phase between the constituent modes. For the nonlinear witness measurements, projective measurements are madewith spatial light modulators (SLMs) and single-mode fibres (SMFs) used in combination with single-photon detectors and coincidencedetection electronics. by iteration [32, 33]. Taking the limit as n → ∞ , oneobtains W Φ + ∞ .The expectation value w Φ + ∞ of this witness can be ex-pressed as a combination of expectation values of mea-surable operators. Contained within the measurementsfor the nonlinear witness is a unitary operator U , whichprovides some freedom in choosing the exact form of thewitness. By choosing U to be equal to − σ z ⊗ σ z , weshow in the supplementary information that the expec-tation value of the particular nonlinear witness that weconsider in this experiment is given by w Φ + ∞ ( ρ ) = Tr( ρW Φ + L ) − | Tr( ρW Φ + L ) | (3) − | Tr( ρW Φ + L ) − Tr( ρW Φ + L )Tr( ρ ( − σ z ⊗ σ z )) | − | Tr( ρ ( − σ z ⊗ σ z )) | . We see that Tr( ρW Φ + L ) and Tr( ρ ( − σ z ⊗ σ z )) are the onlymeasurements that are required for the nonlinear witness.A similar method may be used to generate the nonlinearimprovement of the linear witness W Ψ + L , but in this casewe require Tr( ρ ( σ z ⊗ σ z )) to achieve the same result.The general form of the nonlinear witness is given in thesupplementary information.For the correlated Bell states | Φ ± (cid:105) , the decompositionof the operator W Φ ± L with the fewest local measurements is given by [19] W Φ ± L = 12 (cid:16) | j, k (cid:105)(cid:104) j, k | + | k, j (cid:105)(cid:104) k, j | (cid:17) ± (cid:16) | x + , x − (cid:105)(cid:104) x + , x − | + | x − , x + (cid:105)(cid:104) x − , x + |− | y + , y − (cid:105)(cid:104) y + , y − | − | y − , y + (cid:105)(cid:104) y − , y + | (cid:17) , (4)and the unitary operator is − σ z ⊗ σ z = | j, j (cid:105)(cid:104) j, j | + | k, k (cid:105)(cid:104) k, k | − ( | j, k (cid:105)(cid:104) j, k | + | k, j (cid:105)(cid:104) k, j | ) (cid:124) (cid:123)(cid:122) (cid:125) Contained within W Φ ± L . (5)The linear witnesses W Ψ ± L have similar decompositions,and in all cases, | x ± (cid:105) = 1 √ (cid:16) | j (cid:105) ± | k (cid:105) (cid:17) and | y ± (cid:105) = 1 √ (cid:16) | j (cid:105) ± i | k (cid:105) (cid:17) . It follows that the nonlinear witness that we choose toinvestigate has a total of eight projective measurements,which is a nearly twofold improvement over the numberrequired for complete tomography of the state [16]. Thisis because there are always two measurements that oc-cur in both W Φ ± L and − σ z ⊗ σ z or in both W Ψ ± L and σ z ⊗ σ z . For the correlated case these are | j, k (cid:105)(cid:104) j, k | and | k, j (cid:105)(cid:104) k, j | , and for the anti-correlated case theseare | j, j (cid:105)(cid:104) j, j | and | k, k (cid:105)(cid:104) k, k | . Additionally, we note thatTr( ρ ( − σ z ⊗ σ z )) is the measure of the strength of thecorrelations. Perfect correlations (or anti-correlations)correspond to | Tr( ρ ( − σ z ⊗ σ z )) | equal to unity, and nocorrelations correspond to | Tr( ρ ( − σ z ⊗ σ z )) | equal to zero. Quantum state preparation : To experimentally realisenonlinear entanglement witnesses, we require precise con-trol of the form of the entangled state; see figure 2. Tofully test the nonlinear witnesses, we need to prepare arange of correlated ρ Φ and anti-correlated ρ Ψ states. Asthe benefit of the particular nonlinear witnesses that weare investigating is that they detect entangled states re-gardless of the phase between the modes, we require theability to adjust this parameter.Our investigation concerns entanglement of the spa-tial degree of freedom. More specifically, we look for en-tanglement between orbital angular momentum states oflight in two-dimensional state spaces. Consequently, wecan achieve exact quantum state preparation through theuse of unitary transformations applied to the signal andidler photons of the downconverted light. We accomplishsuch transformations using Dove prisms; the number ofprisms allows us to choose between a correlated and anti-correlated entangled state, and the angle of the prismsallows us to manipulate the phase between the entangledmodes.A Dove prism placed at an angle θ/ (cid:96) → − (cid:96) ; secondly, an (cid:96) -dependent phase shift is in-troduced such that the modes within the beam acquirethe additional phase (cid:96)θ . It follows that two Dove prismscan be used to introduce an (cid:96) -dependent phase shift be-tween different OAM modes whilst leaving the sign of (cid:96) unchanged.Placing a Dove prism oriented at an angle θ B / | (cid:96) = 0 (cid:105) modeas these are correlated entangled states of the form ofequation (1) | Φ (cid:96) (cid:105) = 1 (cid:112) ε (cid:96) (cid:16) | , (cid:105) + ε (cid:96) e iϕ | (cid:96), (cid:96) (cid:105) (cid:17) , (6)where ϕ = (cid:96)θ B . Placing a second Dove prism oriented atan angle θ A / | Φ (cid:96) (cid:105) to an anti-correlated state | Ψ (cid:96) (cid:105) = 1 (cid:112) ε (cid:96) (cid:16) | , (cid:105) + ε (cid:96) e iφ | − (cid:96), (cid:96) (cid:105) (cid:17) , (7)where φ = (cid:96) ( θ B − θ A ). Using the states | Φ (cid:96) (cid:105) and | Ψ (cid:96) (cid:105) ,we can test the ability of the nonlinear witness for de-tecting entanglement for a large range of different states.Although we do not use them, we also note that all fourBell states can be produced by this method of quantumstate preparation. Experiment results : Before we calculate any expecta-tion value, we perform quantum state tomography oneach input state to ensure that it is indeed entangled.We find that our method of quantum state preparationis able to produce the desired quantum states of the formgiven in equations (6) and (7) to a high degree of confi-dence. Thus, we then proceed to calculate the relevantexpectation values so that we can assess the performanceof nonlinear and linear witnesses for the verification ofentanglement.In figure 3 we see the main result of our work: a singlenonlinear witness is able to verify the entanglement ofa large range of input states, that is states of the formeither ρ Φ or ρ Ψ . In contrast, no single linear witnessis able to verify entanglement over the same range; theexpectation value of each linear witness is above zero forhalf of the states we measure. Since we use quantum statetomography to confirm that our states are entangled, thismeans that each linear witness delivers an inconclusiveresult and thus cannot detect entanglement in a largerange of states that are entangled. These results are forthe two-dimensional subspaces described in equations (6)and (7) where (cid:96) = 2.In the anti-correlated case, all expectation values ofthe nonlinear witness are negative, indicating entangledstates for all phases observed. In the correlated case, thenonlinear witness is negative for the majority of the ob-served states. However, near π/ π/
2, there arethree states that have slightly positive expectation val-ues. This is attributable to noise introduced during mea-surement, resulting in lower purity of the state.
Discussion : Our results clearly show that our nonlin-ear witnesses are able to establish the entanglement of therelevant class of states. Nonetheless, it is interesting thatthe numerical values of the expectation value of the wit-ness are quite different in the two cases, as the witnessesare defined in such a manner that we expect the valuesto be comparable. Consequently, the difference betweenthe correlated and anti-correlated cases highlights an in-teresting aspect of nonlinear entanglement witnesses: theextreme sensitivity of the expectation value with regardsto the outcome of a single projective measurement. Ascan be seen from equation (3), the last term that issubtracted in the calculation is inversely proportional to1 − |
Tr( ρ ( − σ z ⊗ σ z )) | (one minus the square of the con-trast of the OAM correlations), and consequently, theprecise expectation value that is measured is highly sen-sitive to Tr( ρ ( − σ z ⊗ σ z )). As the strength of the OAMcorrelations depends critically on a few measurements, sotoo does the obtained value of w ∞ .The difference in the range of measured expectationvalues originates in the strength of the OAM correla-tions for each case. For the anti-correlated states of theform ρ Ψ , the average measured value of | Tr( ρ ( − σ z ⊗ σ z )) | a Correlated states b Anti-correlated states w Φ+ ∞ w Ψ+ ∞ theoreticalpredictions E n t ang l e m en t v e r i f i ed E n t ang l e m en t v e r i f i ed or oror E x pe c t a t i on v a l ue E x pe c t a t i on v a l ue -5.0-4.0-3.0-2.0-1.00.0 φ = ( θ B − θ A ) ϕ = B Inconclusive result + w Φ L w Ψ L + w Ψ L − w Φ L − FIG. 3:
Expectation values for nonlinear witnesses andlinear witnesses.
Experimentally recorded expectation values ofthe nonlinear entanglement witness and the two linear witnesses forcorrelated ( a ) and anti-correlated ( b ) entangled states for (cid:96) = 2.A negative expectation value indicates that the state is entangled.Note that the nonlinear witness correctly detects entanglement un-der almost all cases, whereas each of the linear witnesses often failsto detect entanglement, even though it is present. For the cor-related states, we measured w Φ + ∞ , w Φ + L , and w Φ − L , and for theanti-correlated states, we measured w Ψ + ∞ , w Ψ + L , and w Ψ − L . Thecircles give the experimental data points and the lines are theoreti-cal prediction obtained using average parameters obtained from thedata. The vertical error bars were obtained by applying √ N fluctu-ations to the measured coincidence counts, then averaging over 100iterations to obtain the standard deviation. The horizontal errorbars are estimated to be π/ was equal to 0.92, whereas for the correlated states of theform ρ Φ , the average measured value of | Tr( ρ ( − σ z ⊗ σ z )) | was equal to 0.69. We attribute the reduced contrast inthe correlated case to the asymmetry introduced by plac-ing only one Dove prism in the system. The theoretical fits to the data are adjusted to reflect the appropriatemeasured contrasts.In certain situations it would be desirable to increasefurther the range of states accessible to nonlinear wit-nesses. There are two possible avenues for doing so.The first method involves adjusting the exact form ofthe witness. The only degree of flexibility in the con-struction of the accessible nonlinear witnesses describedin Ref. [32] is in the choice of unitary operator U .For our particular choice of U , a nonlinear improve-ment on a correlated linear witness will not be able todetect entanglement in anti-correlated states, and viceversa. Other choices of U are possible, for example U = ( − σ x ⊗ σ x − σ y ⊗ σ y + σ z ⊗ σ z ) /
2; in this case, U = 2 W Ψ + L . With this choice of U , the nonlinear witnesscan access different states. More specifically, startingwith a linear witness ( W Ψ + L ) that detects anti-correlatedstates, the nonlinear improvement using U = 2 W Ψ + L candetect both anti-correlated and correlated states.The second method involves using two carefully cho-sen nonlinear witnesses. In fact, using two nonlinear wit-nesses, it is possible to extend the range sufficiently todetect entanglement in all qubit entangled states. Forexample, this can be achieved using the two witnessesshown in this paper: one witness that detects the cor-related states (e.g. W Φ + ∞ ) and one witness that detectsthe anti-correlated states (e.g. W Ψ + ∞ ). Using these twowitnesses enables the verification of entanglement of thefull range of pure quantum states, which includes all fourBell states, with only ten measurements. Conclusions : In this work we demonstrate experimen-tally that a single nonlinear witness is able to verify theentanglement of states of the form either ρ Φ or ρ Ψ ; thisis the largest range of entangled states detected with thefewest number of measurements. This is a significant im-provement over linear witnesses, which are not able toprovide the same level of performance. Such nonlinearwitnesses require only a few measurements and no statereconstruction, thus drastically reducing processing timeas compared to tomography. We also demonstrate labo-ratory procedures that allow us to vary the precise formof entangled quantum states, which provides an addi-tional resource for quantum information protocols. Thus,the combination of state preparation and nonlinear wit-nesses provides a clear indication of the significance ofour approach to such applications as superdense codingand quantum teleportation. Moreover, we envisage thecontinued application of nonlinear witnesses to other ar-eas of quantum information science, where it is advanta-geous to extract maximal information with the minimumnumber of measurements. Acknowledgements : We thank J. M. Arrazola,O. Gittsovich, and N. L¨utkenhaus for valuable discus-sions regarding this work. This work was supported bythe Canada Excellence Research Chairs (CERC) Pro-gram and the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC).
Author Contributions : The experiment was designedby J. L. and performed by J. L., J. Z. S., and M. A. Thetheoretical calculations and data analysis were performedby M. A. All authors contributed to the writing of themanuscript.
Methods : We generate photon pairs entangled in theorbital angular momentum basis by means of parametricdownconversion. A Nd:YAG laser at 355 nm with anaverage power of 150 mW is used to pump a 3-mm-longtype I BBO crystal. We use a spatial light modulator(SLM) coupled with a single mode fibre in each arm ofthe system in order to select a particular mode of light.The SLMs display computer-generated holograms thatmodify the phase profile of the incoming light so that it isconverted into the fundamental mode. The light in eacharm then propagates to a single mode fibre, which createsan effective means of mode selection by only allowing thefundamental mode of light. The single mode fibres areconnected to avalanche photodetectors and a coincidencecounting card with a timing resolution of 25 ns. Theplane of the crystal is imaged onto the plane of the SLMsusing a 4-f imaging system with a magnification of ∼− .
33; the focal lengths of the lenses are f = 150 mmand f = 500 mm. The planes of the SLMs are thenimaged onto the fibre facets using a second 4-f imagingsystem with a magnification of ∼ − . × − ; the focallengths of the lenses are f = 400 mm and f = 1 .
45 mm.Each iteration of the experiment involves first prepar-ing the state by setting the angle of each dove prism.Quantum state tomography is then performed on thephoton pair to ensure that it is entangled and has therequired phase. We use an overcomplete set of measure-ments in order to accurately determine the state, andwe reconstruct the density matrix using the method inRef. [15]. We measure one nonlinear witness and two lin-ear witnesses for each state. For the correlated case (oneDove prism), we measure the nonlinear witness W Φ + ∞ and the linear witnesses W Φ + L and W Φ − L . For the anti-correlated case (two Dove prisms), we measure the non-linear witness W Ψ + ∞ and the linear witnesses W Ψ + L and W Ψ − L . We repeat this process for varying values of ϕ (correlated case) and φ (anti-correlated case) in order toproduce the two plots in figure 3. [1] M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cav-alcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, andG. Leuchs, Rev. Mod. Phys. , 1727 (2009). [2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[3] J. Bell, Physics , 195 (1964).[4] L. Hardy, Phys. Rev. Lett. , 1665 (1993).[5] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[6] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev.Mod. Phys. , 145 (2002).[7] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. ,2881 (1992).[8] J. Barreiro, T. Wei, and P. Kwiat, Nature physics , 282(2008).[9] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa,A. Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895(1993).[10] D. Bouwmeester, J. Pan, K. Mattle, M. Eibl, H. We-infurter, and A. Zeilinger, Nature (London) , 575(1997).[11] I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden,and N. Gisin, Nature , 509 (2003).[12] O. G¨uhne and G. T´oth, Physics Reports , 1 (2009).[13] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G.White, Phys. Rev. A , 052312 (2001).[14] K. J. Resch, P. Walther, and A. Zeilinger, Phys. Rev.Lett. , 070402 (2005).[15] M. Agnew, J. Leach, M. McLaren, F. S. Roux, and R. W.Boyd, Phys. Rev. A , 062101 (2011).[16] R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro,Phys. Rev. A , 012303 (2002).[17] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eis-ert, Phys. Rev. Lett. , 150401 (2010).[18] M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki,Phys. Rev. A , 052310 (2000).[19] O. G¨uhne, P. Hyllus, D. Bruß, A. Ekert, M. Lewen-stein, C. Macchiavello, and A. Sanpera, Phys. Rev. A , 062305 (2002).[20] G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris,Phys. Rev. A , 042310 (2003).[21] A. O. Pittenger and M. H. Rubin, Phys. Rev. A ,012327 (2003).[22] R. A. Bertlmann, K. Durstberger, B. C. Hiesmayr, andP. Krammer, Phys. Rev. A , 052331 (2005).[23] M. Hendrych, R. Gallego, M. Miˇcuda, N. Brunner,A. Ac´ın, and J. P. Torres, Nat Phys , 2012 (2012).[24] M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni,G. M. D’Ariano, and C. Macchiavello, Phys. Rev. Lett. , 227901 (2003).[25] J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett,Science , 662 (2010).[26] M. Agnew, J. Leach, and R. W. Boyd, The EuropeanPhysical Journal D , 6 (2012).[27] M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner,H. Weinfurter, O. G¨uhne, P. Hyllus, D. Bruß, M. Lewen-stein, and A. Sanpera, Phys. Rev. Lett. , 087902(2004).[28] J. Uffink, Phys. Rev. Lett. , 230406 (2002).[29] V. Giovannetti, S. Mancini, D. Vitali, and P. Tombesi,Phys. Rev. A , 022320 (2003).[30] P. Hyllus and J. Eisert, New Journal of Physics , 51(2006).[31] O. G¨uhne and N. L¨utkenhaus, Phys. Rev. Lett. ,170502 (2006).[32] J. M. Arrazola, O. Gittsovich, and N. L¨utkenhaus, Phys.Rev. A , 062327 (2012). [33] T. Moroder, O. G¨uhne, and N. L¨utkenhaus, Phys. Rev.A , 032326 (2008).[34] D. Bruß, J. Math. Phys. , 4237 (2002).[35] For n identical particles in d dimensions, the number ofmeasurements for a tomographically complete set scalesas d n − SUPPLEMENTARY INFORMATION
Linear witnesses : For an anti-correlated state | Ψ ± (cid:105) ,the operator W Ψ ± L can be decomposed as follows: W Ψ ± L = 12 (cid:16) | j, j (cid:105)(cid:104) j, j | + | k, k (cid:105)(cid:104) k, k | (cid:17) ∓ (cid:16) | x + , x + (cid:105)(cid:104) x + , x + | + | x − , x − (cid:105)(cid:104) x − , x − |− | y + , y − (cid:105)(cid:104) y + , y − | − | y − , y + (cid:105)(cid:104) y − , y + | (cid:17) . (8)In this case, since our OAM anti-correlations are of theform | Ψ (cid:105) = 1 √ ε (cid:16) | j, k (cid:105) + εe iϕ | k, j (cid:105) (cid:17) = 1 √ ε (cid:16) | , (cid:105) + εe iϕ | − (cid:96), (cid:96) (cid:105) (cid:17) , (9)we note that | j (cid:105) A = | (cid:105) , | j (cid:105) B = | (cid:96) (cid:105) , | k (cid:105) A = | − (cid:96) (cid:105) and | k (cid:105) B = | (cid:105) . How to generate a nonlinear witness : A linear witnessof a state ρ with nonpositive partial transpose can beconstructed using [18, 34] W L = ( | η (cid:105)(cid:104) η | ) T B , (10)where ρ T B denotes the partial transpose of ρ on theHilbert space of photon B and | η (cid:105) denotes the eigenvectorof ρ T B corresponding to the minimum eigenvalue. Thenan entangled state ρ ent gives w = Tr( ρ ent W L ) < . (11)In the following, we demonstrate how to generate the lin-ear and nonlinear witnesses for the correlated Bell state | Φ + (cid:105) = 1 √ (cid:16) | j, j (cid:105) + | k, k (cid:105) (cid:17) . (12)In this case, we find the eigenvector corresponding to theminimum eigenvalue of ρ T B = ( | Φ + (cid:105)(cid:104) Φ + | ) T B to be | η (cid:105) = 1 √ (cid:16) | j, k (cid:105) − | k, j (cid:105) (cid:17) , (13)which produces a linear witness W Φ + L = 12 −
10 1 0 00 0 1 0 − . (14) In the case of two qubits, the nonlinear improvement onthe linear witness is [32] w Φ + = Tr( ρW Φ + L ) − | Tr( ρ ( ρ η U ) T B ) | . (15)We choose the operator U to be − σ z ⊗ σ z , which is equalto U = − − , (16)and ρ η = | η (cid:105)(cid:104) η | is equal to ρ η = 12 − − . (17)This nonlinear witness can be iterated to further improvethe strength of the witness [33]. When the number of it-erations goes to infinity, the following witness is obtained[32]: w Φ + ∞ ( ρ ) = Tr( ρW Φ + L ) − | Tr( ρ ( ρ η U ) T B ) | − | Tr( ρρ T B η ) − Tr( ρ ( ρ η U ) T B )Tr( ρU T B ) | − | Tr( ρU T B ) | . (18)To calculate w Φ + ∞ ( ρ ), we require knowledge of W Φ + L , U and ρ η . However, the choice of the form of the unitaryoperator U results in properties that minimise the num-ber of required measurements for the nonlinear witness.Since U = − σ z ⊗ σ z , ρ η does not change when multipliedby U ; it follows that( ρ η U ) T B = ρ T B η = W Φ + L = 12 −
10 1 0 00 0 1 0 − . (19)This means that equation (18) can be simplified to w ∞ ( ρ ) = Tr( ρW Φ + L ) − | Tr( ρW Φ + L ) | (20) − | Tr( ρW Φ + L ) − Tr( ρW Φ + L )Tr( ρU ) | − | Tr( ρU ) | . We now see that the expectation values Tr( ρW Φ + L ) andTr( ρU ) are the only measurements that are required forthe nonlinear witness. The operator W L , which is thestandard linear witness, can be decomposed into six lo-cal measurements, and the operator U requires four localmeasurements. Two of the four measurements ( | j, k (cid:105)(cid:104) j, k | and | k, j (cid:105)(cid:104) k, j | ) are in both W Φ + L and U ; therefore, thenonlinear witness requires a total of eight measurements,which is approximately half the number required for to-mography [16].We note that for our choice of U , we obtainTr( ρU T B ) = − ε = 1, resulting in zero in the denominator. However,in reality, the state we detect is not pure; that is, thephotons are in a state described by ρ ψ = | ψ (cid:105)(cid:104) ψ | p + (1 − p ) / , (21) where p indicates the purity of the state and repre-sents the four-dimensional identity matrix. In this case,Tr( ρ ψ U T B ) (cid:54) = − p (cid:54)(cid:54)