Linear-Optical Hyperentanglement-Assisted Quantum Error-Correcting Code
LLinear-Optical Hyperentanglement-Assisted Quantum Error-Correcting Code
Mark M. Wilde
Center for Quantum Information Science and Technology, Department of Electrical Engineering,University of Southern California, Los Angeles, California 90089, USA
Dmitry B. Uskov
Department of Physics, Tulane University, New Orleans, Louisiana 70118, USA andHearne Institute for Theoretical Physics, Department of Physics and Astronomy,Louisiana State University, Baton Rouge, Louisiana 70803, USA (Dated: November 14, 2018; Received text; Revised text; Accepted text; Published text)We propose a linear-optical implementation of a hyperentanglement-assisted quantum error-correcting code. The code is hyperentanglement-assisted because the shared entanglement resourceis a photonic state hyperentangled in polarization and orbital angular momentum. It is possible toencode, decode, and diagnose channel errors using linear-optical techniques. The code corrects forpolarization “flip” errors and is thus suitable only for a proof-of-principle experiment. The encodingand decoding circuits use a Knill-Laflamme-Milburn-like scheme for transforming polarization andorbital angular momentum photonic qubits. A numerical optimization algorithm finds a unit-fidelityencoding circuit that requires only three ancilla modes and has success probability equal to 0.0097.
PACS numbers: 03.67.Hk, 03.67.Pp, 42.50.ExKeywords: entanglement-assisted quantum error correction, hyperentanglement, linear-optical gate optimiza-tion
I. INTRODUCTION
Quantum error correction plays an active role in thefuture realization of a quantum communication system[1, 2]. Several optical experiments have already imple-mented simple quantum error correction routines [3, 4, 5].The entanglement-assisted stabilizer formalism is a re-cent extension of the theory of quantum error correc-tion that incorporates entanglement shared between asender and receiver [6, 7]. A further extension of thistheory incorporates gauge qubits [8] and others give astructure appropriate for a stream of quantum informa-tion [9, 10, 11]. The likely candidate for implement-ing an entanglement-assisted code is photonics becausethe entanglement-assisted model is more appropriate forquantum communication rather than quantum comput-ing.In this article, we propose a linear-optical implementa-tion of a hyperentanglement-assisted quantum code. Ourcode is hyperentanglement-assisted because it exploits hy-perentanglement of two photons [12]. Two photons arehyperentangled if they have entanglement in multiple de-grees of freedom such as polarization and orbital angularmomentum (OAM) [13, 14]. The benefit of hyperentan-glement is that a linear-optical setup suffices to perform acomplete Bell-state analysis [15, 16, 17]. Our proposal forthe hyperentanglement-assisted code relies on the recentoptical realization [18] of the superdense coding proto-col [19] and the close connection between entanglement-assisted quantum error correction and superdense coding[7]. We also employ a recent numerical optimization al-gorithm [20] to find an encoding circuit and a decodingcircuit that has unit fidelity, success probability equal to0.0097, and requires only three ancilla modes. The cir- cuits act on both the polarization and OAM degrees offreedom of the photonic qubits.We structure this article as follows. The first sec-tion reviews hyperentangled states, the single-photonpolarization-OAM states, and mentions that it is possibleto distinguish the single-photon polarization-OAM stateswith linear optics. We then discuss the superdense codingprotocol for hyperentangled states and highlight the con-nection between superdense coding and entanglement-assisted quantum error correction. We give a brief de-scription of our code, its error analysis, and correctiveoperations. The final part of this article discusses thenumerical optimization technique for finding our code’sencoding circuit and decoding circuit.
II. HYPERENTANGLED STATES
The standard hyperentangled state is a state of twophotons simultaneously entangled in polarization andOAM: 12 ( | HH (cid:105) + | V V (cid:105) ) ⊗ ( | (cid:9)(cid:8) (cid:105) + | (cid:8)(cid:9) (cid:105) ) . The symbols H and V represent horizontal and verticalpolarization respectively and the symbols (cid:9) and (cid:8) repre-sent paraxial Laguerre-Gauss spatial modes with + (cid:126) and − (cid:126) respective units of OAM [21]. Changing the polariza-tion degree of freedom of Alice’s photon in the above stateaccording to the four standard Pauli operators, whileleaving the OAM degree of freedom unchanged, gives theTypeset by REVTEX a r X i v : . [ qu a n t - ph ] F e b following four hyperentangled states: (cid:12)(cid:12) Φ ± (cid:11) ≡
12 ( | HH (cid:105) ± | V V (cid:105) ) ⊗ ( | (cid:9)(cid:8) (cid:105) + | (cid:8)(cid:9) (cid:105) ) , (cid:12)(cid:12) Ψ ± (cid:11) ≡
12 ( | HV (cid:105) ± | V H (cid:105) ) ⊗ ( | (cid:9)(cid:8) (cid:105) + | (cid:8)(cid:9) (cid:105) ) . (1)We can rewrite the above four states in terms of thesingle-photon polarization-OAM states φ ± , ψ ± [30]: (cid:12)(cid:12) Φ ± (cid:11) = 14 (cid:0) φ + A ⊗ ψ ± B + φ − A ⊗ ψ ∓ B + ψ + A ⊗ φ ± B + ψ − A ⊗ φ ∓ B (cid:1) , (cid:12)(cid:12) Ψ ± (cid:11) = 14 (cid:0) ± φ + A ⊗ φ ± B ∓ φ − A ⊗ φ ∓ B ± ψ + A ⊗ ψ ± B ∓ ψ − A ⊗ ψ ∓ B (cid:1) , where A and B label the first and second respective pho-tons and the single-photon polarization-OAM states φ ± and ψ ± are as follows: φ ± ≡ | H (cid:9) (cid:105) ± | V (cid:8) (cid:105)√ , ψ ± ≡ | H (cid:8) (cid:105) ± | V (cid:9) (cid:105)√ . The above “quad-rail” basis states | H (cid:9) (cid:105) , | H (cid:8) (cid:105) , | V (cid:9) (cid:105) , | V (cid:8) (cid:105) are four-mode single-photon states defined interms of the Fock basis: | H (cid:9) (cid:105) ≡ | (cid:105) H (cid:9) | (cid:105) H (cid:8) | (cid:105) V (cid:9) | (cid:105) V (cid:8) , | H (cid:8) (cid:105) ≡ | (cid:105) H (cid:9) | (cid:105) H (cid:8) | (cid:105) V (cid:9) | (cid:105) V (cid:8) , | V (cid:9) (cid:105) ≡ | (cid:105) H (cid:9) | (cid:105) H (cid:8) | (cid:105) V (cid:9) | (cid:105) V (cid:8) , | V (cid:8) (cid:105) ≡ | (cid:105) H (cid:9) | (cid:105) H (cid:8) | (cid:105) V (cid:9) | (cid:105) V (cid:8) . (2)Hyperentangled states are useful because a linear-optical analyzer distinguishes the single-photon polarization-OAM states φ ± , ψ ± and thus distinguishes the hyper-entangled states | Φ ± (cid:105) , | Ψ ± (cid:105) as well [15, 16, 17]. III. SUPERDENSE CODING ANDENTANGLEMENT-ASSISTED QUANTUMERROR CORRECTION
We briefly describe the superdense coding protocol forpolarization encoding and hyperentangled states [18]. Asender Alice and a receiver Bob share a hyperentangledstate | Φ + (cid:105) AB . Alice encodes one of four classical mes-sages (two classical bits) by applying one of four trans-formations to her half of | Φ + (cid:105) AB : (1) the identity, (2) | V (cid:105) → − | V (cid:105) , (3) | H (cid:105) ↔ | V (cid:105) , or (4) both | V (cid:105) → − | V (cid:105) and | H (cid:105) ↔ | V (cid:105) . Let Z denote the second operationand let X denote the third operation. The result is totransform the original state | Φ + (cid:105) AB to one of the fourstates | Φ ± (cid:105) AB , | Ψ ± (cid:105) AB . She then sends her half of theencoded | Φ + (cid:105) AB to Bob. Bob performs a single-photonpolarization-OAM state analysis in the basis φ ± , ψ ± oneach of the systems A and B to distinguish the messageAlice transmitted.In the above analysis, it is important to stress that thedense coding transformations affect only the polarization | ψ (cid:31) A NoiseEncoder Decoder Recovery R | Φ + (cid:31) A B FIG. 1: (Color online) The operation of thehyperentanglement-assisted quantum code. Red qubits(those labeled as A and A ) belong to the sender Aliceand blue qubits (the one labeled as B ) belong to thereceiver Bob (though all qubits belong to Bob after thenoisy channel). Alice and Bob share a hyperentangled state ˛˛ Φ + ¸ A B prior to quantum communication. Alice usesthe hyperentangled state to aid in encoding an informationphoton in the state | ψ (cid:105) A . Her encoding circuit consistsof one controlled-phase gate. She sends her photons overa noisy polarization-error quantum channel. Bob receivesthe photons, performs a decoding circuit, and performs twosingle-photon polarization-OAM analyses in the basis φ ± , ψ ± on the systems A and B to determine the error syndrome.Bob finally performs a recovery operation to obtain theinformation photon | ψ (cid:105) A that Alice first sent. degree of freedom of the hyperentangled state | Φ + (cid:105) AB .The classical information resides in a four-dimensionalsubspace of the 16-dimensional Hilbert space. The extradimensions help in single-photon polarization-OAM stateanalysis in order to distinguish the classical messages.Ref. [7] discusses the close relationship between su-perdense coding and entanglement-assisted quantum er-ror correction. In superdense coding, one exploits theclassical bits encoded in a Bell state so that Alice cantransmit classical information to Bob. In entanglement-assisted error correction, one exploits the encoded clas-sical bits for use as error syndromes. Another way ofthinking about this latter scenario is that Eve, the en-vironment, is superdense coding messages (errors) intothe Bell states. Bob can determine the errors that Eveintroduces by measuring in the Bell basis. IV. OPERATION OF THE CODE
The operation of our code begins with an initial, un-encoded state consisting of one information photon andone hyperentangled state: | ψ (cid:105) A (cid:12)(cid:12) Φ + (cid:11) A B . (3)The information photon is as follows: | ψ (cid:105) A ≡ α | H (cid:105) A + β | V (cid:105) A . The sender Alice possesses photons A and A and the receiver Bob possesses photon B . Theentanglement-assisted communication paradigm assumesthat Alice and Bob share the hyperentangled state priorto quantum communication. Figure 1 highlights the op-eration of our hyperentanglement-assisted quantum code.The sender Alice applies an encoding circuit consistingof one controlled-sign gate (we discuss this gate later inmore detail) so that the state shared between Alice andBob is the following unnormalized encoded state: | ψ (cid:105) A (cid:16)(cid:12)(cid:12) Φ − (cid:11) A B + (cid:12)(cid:12) Φ + (cid:11) A B (cid:17) + Z | ψ (cid:105) A (cid:16)(cid:12)(cid:12) Φ + (cid:11) A B − (cid:12)(cid:12) Φ − (cid:11) A B (cid:17) . (4)She sends her photons A and A over a noisypolarization-error channel (discussed below). Bob re-ceives the photons and we relabel them as B and B (cid:48) respectively. For now, suppose that the channel doesnot introduce an error. Bob finally applies the decodingcircuit (same as the encoding circuit) and the resultingdecoded state is as follows: | ψ (cid:105) B (cid:12)(cid:12) Φ + (cid:11) B (cid:48) B , (5)where the information photon appears in Bob’s system B . Bob performs two single-photon polarization-OAManalyses in the basis φ ± , ψ ± on the systems B (cid:48) and B to diagnose the channel error. The polarization-OAManalysis distinguishes the four states {| Φ ± (cid:105) , | Ψ ± (cid:105)} . Bobmeasures the result | Φ + (cid:105) when the channel does not in-troduce noise. The state | Φ + (cid:105) is a syndrome that deter-mines the channel error. Bob does not need to performa recovery operation in this case. V. ERROR ANALYSIS
In general, the channel introduces errors on the pho-tons that Alice transmits. We assume in this arti-cle that the channel is a noisy polarization-error chan-nel (analogous to the classical bit-flip channel.) A noisypolarization-error channel independently applies a polar-ization error X that flips the horizontal and vertical po-larization bases. We assume that this channel affectsthe polarization degree of freedom only and does not af-fect the OAM degree of freedom. Although this channelmay not be entirely realistic, it provides a platform fora proof-of-principle demonstration of the operation of anentanglement-assisted quantum code [31].The code protects against a single polarization erroron either of the two photons A or A that Alice sends.It also protects against a double polarization error onboth photons. Suppose that a polarization error occurson photon A . After Bob applies the decoding circuit,the state becomes X | ψ (cid:105) B | Φ − (cid:105) B (cid:48) B . Bob measures thephotons B (cid:48) and B , determines they are in the state | Φ − (cid:105) , and flips the polarization of photon B to recoverthe initial information photon | ψ (cid:105) . Table I summarizesthe other cases. Error Recovery Syndrome
I I Φ + X A X Φ − X A Z Ψ + X A X A ZX Ψ − TABLE I: The table details the results of Bob’s Bell stateanalysis. The states in the third column are syndromes thatdetermine the channel error (“Error”) and the recovery op-eration (“Recovery”) that Bob should perform to recover theinitial information photon.
VI. OPTICAL ENCODING AND DECODINGCIRCUIT
The seminal paper of Knill, Laflamme, and Milburn(KLM) showed how to perform two-qubit interactionswith linear-optical devices [22, 23]. Their method isa measurement-assisted scheme: it first mixes a set of“computational” modes and ancilla modes in a linear-optical device and then counts the photons in the an-cilla modes. The optical transformation acts on thecomputational modes. The ancilla modes help performthis transformation and we measure them at the end ofthe measurement-assisted scheme. The gates exploit theHong-Ou-Mandel quantum interference of indistinguish-able photons [24]. These measurement-assisted trans-formations are heralded, non-deterministic, and non-destructive. A destructive gate involves a measurementon the computational modes—the informational statecollapses to one of the states in the measurement ba-sis even when the gate succeeds. A non-destructive gaterequires a measurement only on the ancilla modes—theresult is that all the information encoded in the stateremains intact when the gate succeeds [23, 25].The measurement-assisted scheme is useful for dual-rail encoded qubits and even polarization-encoded qubits[26], but until now, no one has considered its appli-cation to “quad-rail” encoded quantum information inpolarization-OAM states.The implementation of a polarization-OAMmeasurement-assisted scheme requires unconven-tional linear-optical elements analogous to beamsplittersand other tools of linear optics acting on OAM states ofphotons. Holographic elements suffice for this purposebecause they act on OAM components [18]—similarlyto the action of polarization beamsplitters on photonpolarization. The extension of the measurement-assistedscheme to OAM states is a generalization of the idea inRef. [26]. There, the authors extended the measurement-assisted scheme to polarization-encoded qubits. Here,we use a similar idea to extend the measurement-assistedscheme to OAM states.The encoding transformation corresponding to ourcode generates the encoded state in (4). It is a controlled-sign gate that acts on the four-dimensional Hilbert space H A ⊗ H A of the information photon A and the polar-ization subspace of Alice’s part A of the hyperentangledstate in (3). The gate acts on the polarization degrees offreedom and leaves the OAM degrees of freedom invari-ant. It is a linear-optical transformation on a set of sixmodes—two modes for the A system and four modes forthe A system. The gate leaves the following basis states | H (cid:105) A | H (cid:9) (cid:105) A , | H (cid:105) A | H (cid:8) (cid:105) A , | H (cid:105) A | V (cid:9) (cid:105) A , | H (cid:105) A | V (cid:8) (cid:105) A , | V (cid:105) A | H (cid:9) (cid:105) A , | V (cid:105) A | H (cid:8) (cid:105) A , (6)invariant and adds a phase to the remaining basis states: | V (cid:105) A | V (cid:9) (cid:105) A → − | V (cid:105) A | V (cid:9) (cid:105) A , | V (cid:105) A | V (cid:8) (cid:105) A → − | V (cid:105) A | V (cid:8) (cid:105) A . (7)We make a statement about the mathematical struc-ture of the Hilbert space of polarization-OAM states. Itis possible to decompose any Hilbert space with a tensorproduct structure. E.g.,span {| H (cid:9) (cid:105) , | V (cid:9) (cid:105) , | H (cid:8) (cid:105) , | V (cid:8) (cid:105)} = span {| H (cid:105) , | V (cid:105)} ⊗ span {| (cid:9) (cid:105) , | (cid:8) (cid:105)} . While in the paraxial approximation, local opticaltransformations on the subspaces span {| H (cid:105) , | V (cid:105)} andspan {| (cid:9) (cid:105) , | (cid:8) (cid:105)} respect the tensor-product decompositionof the full four-dimensional space. However, the tensor-product notation in (1) from Ref. [18] may be some-what misleading in the context of a measurement-assistedtransformation (which the authors of Ref. [18] do notdiscuss). A qubit-coupling measurement-assisted trans-formation, based on the mixing of creation operators inseparate modes, does not respect such a tensor-productdecomposition in general. Instead such an operation actsnaturally on a space constructed as a direct sum, e.g.,span {| H (cid:9) (cid:105) , | V (cid:9) (cid:105)} ⊕ span {| H (cid:8) (cid:105) , | V (cid:8) (cid:105)} . The aboverestriction places a constraint on the form of the linear-optical encoding circuit and decoding circuit.Knill devised an optimal solution for the controlled-sign gate [27]. We could use Knill’s two dual-rail qubitgate for an implementation of the transformation in (7).It requires the combination of two separate transforms:the first acts on the (cid:9) OAM states and the secondacts on the (cid:8)
OAM states. Each transformation re-quires two ancilla modes and has a success probabilityof 2 /
27 [27]. This “Knill combination” scheme thus re-quires four ancilla modes with a success probability of(2 / ≈ . ordered success probability o p t i m i z a t i o n n u m b e r M a x i m a l s u c c e s s p r o b a b i l i t y f o r a c o m b i n a t i o n o f t w o C S g a t e s
FIG. 2: The figure displays the results of the numericalgate optimization algorithm for our case of six computationalmodes and three ancilla modes. The algorithm starts an op-timization at a randomly selected point in the space of 6 × > − − . The algorithm finds an encoding cir-cuit with a success probability almost twice that of the Knillcombination scheme. N × N matrix, where N is the number of optical modes,completely characterizes an optical transformation. Thenumerical implementation of the optimization algorithmperforms a gradient search in the space of these matrices.The algorithm first finds a matrix that guarantees a unittransformation fidelity and then performs a constrainedoptimization of success probability in the unit fidelitysubspace. Each optimization cycle ends at some localmaximum of success probabilitiy. The resulting Kraustransformation, or contraction map, acting on the com-putational modes matches the desired target transforma-tion in the case of a successful measurement outcome onthe ancilla modes.We now describe the results of the procedure for ourcase where the encoding circuit acts on the modes for Al-ice’s systems A and A and three ancilla modes. We per-formed the optimization in the 81-dimensional complexspace of a GL(9) matrix (six modes for Alice’s systems A and A and three ancilla modes). Such a matrix admits adecomposition into a sequence of operations where eachoperation corresponds to a standard optical element [28].We were able to simplify the transformation even moreby using a technique from Ref. [27]. The optimal trans-formation actually acts on three modes only: the verticalpolarization of Alice’s photon A , the vertical polarizationand (cid:8) OAM of Alice’s photon A , and the vertical polar-ization and (cid:9) OAM of Alice’s photon A . The reductionof the optimal solution to a three-mode operation shouldmake it easier to implement experimentally. The follow-ing equations illustrate the reduced transformation: | n (cid:105) AV | n (cid:105) A V (cid:8) → ( − f ( n ,n ) | n (cid:105) AV | n (cid:105) A V (cid:8) , | n (cid:105) AV | n (cid:105) A V (cid:9) → ( − f ( n ,n ) | n (cid:105) AV | n (cid:105) A V (cid:9) , where n and n are photon numbers equal to either zeroor one and the function f is equal to one if n = n = 1and is equal to zero otherwise. One can verify that theimplementation of the above reduced transformation isequivalent to the full transformation in (7). Figure 2illustrates a sample distribution of success probabilitiesin increasing order.The best found solution provides a success probabil-ity of 0.00974276 at a fidelity of 1 − (cid:0) × − (cid:1) . Thissolution, most likely, is the optimal global solution, orat least close to the global optimum. One cannot verifythe global character of a solution by numerical tools only.However, we have found that further optimization withthe current scheme does not improve the result. Thesuccess probability shows a slight increase for non-unitfidelity: at the level of 0.99 fidelity, we have found a so-lution with success probability of 0.011. Further relaxingthe unit fidelity requirement does not lead to any signif-icant increase of the success probability.We now address the issue of implementing ourhyperentanglement-assisted quantum code. As describedabove, our encoding circuit solution corresponds to someoptical transformation matrix. The first question arisingin connection with the possible experimental implemen-tation of such an optical transformation is whether theobtained matrix is unitary. Currently, we use an opti-mization algorithm that searches for an arbitrary ma-trix, one not necessarily restricted to the subspace ofunitary matrices. Therefore, one may need to apply aunitary dilation procedure [27, 29] to embed the matrixinto a larger unitary matrix. However, a solution cor-responding to a maximal success probability is a matrixrequiring minimal dilation, due to the singular behaviorof the gradient of the success probability function on themanifold of unitary matrices. Indeed the two best solu-tions (see insert in Figure 2) correspond to matrices withthe following singular values { ., ., ., ., ., . } . Oneneeds to introduce only one additional vacuum mode toembed the matrix into an SU(7) matrix in order to im-plement the transformation in the form of beamsplittersand phase shifters. Ref. [28] suggests the scheme of such adecomposition. Mathematically, it corresponds to a fac-torization of an arbitrary SU( N ) matrix into a product ofSU(2) matrices. The implementation in Ref. [28] requires N ( N − / VII. CONCLUSION
We have presented an optical implementation of anentanglement-assisted quantum code that should be re-alizable with current technology. The code encodes oneinformation photon with the help of a hyperentangledstate. To our knowledge, this proposal is the first sugges-tion for an implementation of an entanglement-assistedquantum code.
VIII. ACKNOWLEDGEMENTS
The authors thank Todd A. Brun for useful discussions.M.M.W. acknowledges support from NSF Grant 0545845and D.B.U. acknowledges support from the Army Re-search Office and the Intelligence Advanced ResearchProjects Activity.
APPENDIX A
Below we list the matrix that implements the trans-formation for the encoding circuit. We do not provideit as one large matrix because the individual entries aretoo large, but instead provide it a few columns at a time.The first three columns are as follows: − . i . − . i . . i . − . i . . − i . − . i . − . − i . . i . . i . . i . . − i . − . i . . The second three columns are as follows: − . − i . . i . α − . i . α − . i . . − i . . i . , where α = − . − i . . i . . − i . − . − i . . − i . − . − i . . i . − . − i . − . − i . . − i . − . i . . i . − . i . . The last column is as follows: − . i . − . i . . − i . − . − i . − . − i . − . i . . [1] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A.Sloane, IEEE Trans. Inf. Theory , 1369 (1998).[2] D. Gottesman, Ph.D. thesis, California Institute of Tech-nology (arXiv:quant-ph/9705052) (1997).[3] J. L. O’Brien, G. J. Pryde, A. G. White, and T. C. Ralph,Phys. Rev. A , 060303(R) (2005).[4] T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys.Rev. A , 052332 (2005).[5] Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J.Briegel, and J.-W. Pan, Nature , 54 (2004).[6] T. A. Brun, I. Devetak, and M.-H. Hsieh, Science ,436 (2006).[7] T. A. Brun, I. Devetak, and M.-H. Hsieh, arXiv:quant-ph/0608027v2 (2006).[8] M.-H. Hsieh, I. Devetak, and T. Brun, Phys. Rev. A ,062313 (2007).[9] M. M. Wilde, H. Krovi, and T. A. Brun, arXiv:0708.3699(2007).[10] M. M. Wilde and T. A. Brun, arXiv:0712.2223 (2007).[11] M. M. Wilde and T. A. Brun, arXiv:0807.3803 (2008).[12] P. G. Kwiat, Journal Mod. Opt. , 2173 (1997).[13] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G.Kwiat, Phys. Rev. Lett. , 260501 (2005).[14] G. Molina-Terriza, J. P. Torres, and L. Torner, NaturePhysics , 305 (2007).[15] P. G. Kwiat and H. Weinfurter, Phys. Rev. A , R2623(1998).[16] C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter,Phys. Rev. Lett. , 190501 (2006).[17] M. Barbieri, G. Vallone, P. Mataloni, and F. DeMartini,Phys. Rev. A , 042317 (2007). [18] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, NaturePhysics , 282 (2008).[19] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. ,2881 (1992).[20] D. B. Uskov, L. Kaplan, A. M. Smith, S. D. Huver, andJ. P. Dowling, arXiv:0808.1926 (2008).[21] L. Allen, S. M. Barnett, and M. J. Padgett, eds., Opti-cal Angular Momentum (Institute of Physics Publishing,Bristol, 2003).[22] E. Knill, R. Laflamme, and G. J. Milburn, Nature ,46 (2001).[23] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. Dowl-ing, and G. Milburn, Rev. Mod. Phys. , 135 (2007).[24] C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. , 2044 (1987).[25] X.-H. Bao, T.-Y. Chen, Q. Zhang, J. Yang, H. Zhang,T. Yang, and J.-W. Pan, Physical Review Letters ,170502 (pages 4) (2007).[26] F. M. Spedalieri, H. Lee, and J. P. Dowling, Phys. Rev.A , 012334 (2006).[27] E. Knill, Phys. Rev. A , 052306 (2002).[28] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani,Phys. Rev. Lett. , 58 (1994).[29] N. M. VanMeter, P. Lougovski, D. B. Uskov, K. Kiel-ing, J. Eisert, and J. P. Dowling, Physical Review A ,063808 (2007).[30] Ref. [18] refers to these states as spin-orbit Bell states,but this name is inappropriate given that the states areencoded in polarization and orbital angular momentumand it is not clear whether these states could violate aBell’s inequality.,063808 (2007).[30] Ref. [18] refers to these states as spin-orbit Bell states,but this name is inappropriate given that the states areencoded in polarization and orbital angular momentumand it is not clear whether these states could violate aBell’s inequality.