Proposed optical realisation of a two photon, four-qubit entangled χ state
PProposed optical realization of a two photon,four-qubit entangled χ state Atirach Ritboon, Sarah Croke and Stephen M Barnett
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UKE-mail: [email protected]
Abstract.
The four-qubit states (cid:12)(cid:12) χ ij (cid:11) , exhibiting genuinely multi-partite entangle-ment have been shown to have many interesting properties and have been suggestedfor novel applications in quantum information processing. In this work we propose asimple quantum circuit and its corresponding optical embodiment with which to pre-pare photon pairs in the (cid:12)(cid:12) χ ij (cid:11) states. Our approach uses hyper-entangled photon pairs,produced by the type-I spontaneous parametric down-conversion (SPDC) process intwo contiguous nonlinear crystals, together with a set of simple linear-optical transfor-mations. Our photon pairs are maximally hyper-entangled in both their polarisationand orbital angular momentum (OAM). After one of these daughter photons passesthrough our optical setup, we obtain photon pairs in the hyper-entangled state (cid:12)(cid:12) χ (cid:11) ,and the (cid:12)(cid:12) χ ij (cid:11) states can be achieved by further simple transformations.PACS numbers: 42.50.Dv Keywords : entanglement, genuine entangled states, hyper-entangled states, OAM
Submitted to:
J. Opt. a r X i v : . [ qu a n t - ph ] M a r roposed optical realization of a two photon, four-qubit entangled χ state
1. Introduction
Quantum entanglement is known to be at the heart of quantum computation andquantum information, in which it is the fundamental resource of many informationprocessing tasks including quantum dense coding, quantum cryptography and quantumteleportation [1, 2, 3, 4, 5, 6, 7, 8]. It is also required for violation of the famous Bellinequality [9] and other demonstrations of quantum nonlocality [10]. While bipartiteentanglement is rather well understood [11, 12, 13], the properties and characteristicsof the various types of multipartite entanglement remain a topic of active research [14].In particular, there is much research to show that multipartite entanglement ishelpfully employed in several quantum communication protocols, including universalerror correction [15], quantum secret sharing [16], telecloning [17], and deterministicsecure quantum communication [18]. As one might expect, increasing the number ofentangled particles leads to stronger and more dramatic demonstrations of nonlocality,or we could say that entangling greater numbers of particles leads to a wider range ofnonclassical effects that can be observed [19, 20]. There are two well-known classes ofgenuinely tripartite entangled states: the Greenberger-Horne-Zeilinger (GHZ) state | GHZ (cid:105) = 1 √ | (cid:105) + | (cid:105) ) abc , (1)and the Werner (W) state | W (cid:105) = 1 √ | (cid:105) + | (cid:105) + | (cid:105) ) abc . (2)These are inequivalent under stochastic local operations and classical communication(SLOCC), which means they cannot be converted to each other under SLOCCoperations [21]. For this reason, each of these has distinct entanglement properties.With the GHZ state, the Bell-type inequality is maximally violated [22, 23], and it canbe employed for open-destination teleportation by using the protocol of A. Karlsson andM. Bourennane [24]. On the other hand, and in contrast to the GHZ state, losing oneof particles in the W state does not make its reduced state separable [25]. Each of thesemay be extended to more than three particles in a natural way: | GHZ (cid:105) N = 1 √ (cid:16) | (cid:105) ⊗ N + | (cid:105) ⊗ N (cid:17) , (3 a ) | W (cid:105) N = 1 √ N N (cid:88) i =1 | (cid:105) ⊗ ( i − | (cid:105) | (cid:105) ⊗ ( N − i ) . (3 b )In 2006 Yeo and Chua proposed a new type of four-qubit entangled state [26] (cid:12)(cid:12)(cid:12) χ (cid:69) = 1 √ (cid:12)(cid:12)(cid:12) ζ (cid:69) + (cid:12)(cid:12)(cid:12) ζ (cid:69) ) abcd , (4 a ) (cid:12)(cid:12)(cid:12) χ ij (cid:69) = σ i ⊗ σ j ⊗ I ⊗ I (cid:12)(cid:12)(cid:12) χ (cid:69) , (4 b )with (cid:12)(cid:12)(cid:12) ζ (cid:69) ≡
12 ( | (cid:105) − | (cid:105) − | (cid:105) + | (cid:105) ) , (4 c ) (cid:12)(cid:12)(cid:12) ζ (cid:69) ≡
12 ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) , (4 d ) roposed optical realization of a two photon, four-qubit entangled χ state σ j are the Pauli matrices and I = σ is the identity operator. These statescannot be transformed into the four-qubit forms of the GHZ and W states by SLOCCand so form a distinct class of entangled states. In fact the state | χ (cid:105) appeared alsoin an earlier study by Lee et al of entanglement teleportation [27], which showed thatthis state can be produced by a nonlocal transformation of the tensor product of EPRstates : | Φ + (cid:105) a (cid:48) b (cid:48) ⊗ | Φ + (cid:105) c (cid:48) d (cid:48) , where | Φ + (cid:105) = ( | (cid:105) + | (cid:105) ) / √ χ -type states forman orthonormal basis for the four-qubit space, they provide a new type of representationfor four-qubit systems [28].The novel properties of the χ states led, naturally, to interest in how they might beprepared efficiently. In 2009, Liu and Kuang proposed that these entangled states maybe generated in four atomic qubits by employing the interaction between light and fouratoms placed in four separate optical cavities [28]. In the same year, Wang and Zhangpublished a scheme with which to produce these states with a simple experimental setupemploying maximally and non-maximally polarization-entangled photons to encode thestate of the output photons [29].The aim of our work is to propose an alternative schemeto generate the states | χ ij (cid:105) by using two hyper-entangled photons, which are maximallyentangled in both their polarization and orbital angular momentum.The structure of this paper is as follows. Section 2 introduces a requiredtransformation that results in an entangled photon pair in the state | χ (cid:105) , along withits corresponding quantum circuit. In section 3 we explain how each optical elementaffects the composite state, polarization and orbital angular momentum, of a light beam.Finally, section 4 presents our proposed optical system for implementing the quantumcircuit and further transformations of the | χ (cid:105) to prepare any one of the states | χ ij (cid:105) .
2. Transformation of two entangled photons
In [30], photons that are hyper-entangled in both their polarization and orbital angularmomentum (OAM) are obtained by employing a 351 nm Argon ion laser with 120mW power pumping two connected β -barium borate (BBO) crystals whose optical axesperpendicularly aligned. Photon pairs generated from the first crystal are horizontallypolarized, while the second crystal produces the vertical polarization. As the non-linearcrystals are in close proximity, the spatial modes of the output photons originating ineach of the crystals are identical. Thus, the (unnormalized) state of the emitted photonpairs is ( | HH (cid:105) + | V V (cid:105) ) pAB ⊗ ( | RL (cid:105) + α | GG (cid:105) + | LR (cid:105) ) oAB (5) roposed optical realization of a two photon, four-qubit entangled χ state Figure 1.
The optical alignment to create hyper-entangled photon pairs by coherentsequential spontaneous parametric down-conversions (SPDC). where H and V denote horizontal and vertical polarizations, respectively, while R , L and G represent the modes with OAM +¯ h , − ¯ h and 0 respectively for each photon ‡ . Thesuperscripts p and o indicate the polarization and orbital angular momentum states,respectively, while the subscripts A and B indicate that it is the state of photon A or B respectively. The scalar quantity α is determined by mode-matching conditions. Thephoton pairs are also entangled in their emission times and frequencies, but in this workthis type of entanglement is important only in that it allows the use of arrival time atthe detectors to select uniquely photon pairs that are entangled. It is evident that wecan obtain the maximally hyper-entangled state if the | GG (cid:105) component is omitted. Thismay be achieved by spatial filtering to remove the beam centre or, more rigorously, byemploying a mode-sorter [33] to select only odd-valued OAM states. The selected stateof the emitted photon pairs then becomes (cid:12)(cid:12)(cid:12) Φ + (cid:69) pAB ⊗ (cid:12)(cid:12)(cid:12) Ψ + (cid:69) oAB = 12 ( | HH (cid:105) + | V V (cid:105) ) pAB ⊗ ( | RL (cid:105) + | LR (cid:105) ) oAB = 12 ( | (cid:105) + | (cid:105) ) pAB ⊗ ( | (cid:105) + | (cid:105) ) oAB (6)To obtain (6) we encode | H (cid:105) p ( | V (cid:105) p ) to be | (cid:105) p ( | (cid:105) p ) and | R (cid:105) o ( | L (cid:105) o ) to be | (cid:105) o ( | (cid:105) o ).We can finally rewrite the state in terms of the superposition of the product states ofthe photons A and B as | X (cid:105) AB = 12 ( | (cid:105) A | (cid:105) B + | (cid:105) A | (cid:105) B + | (cid:105) A | (cid:105) B + | (cid:105) A | (cid:105) B ) . (7)The first qubits of photons A and B , in this equation, now represent polarization statesof these photons while the second represent their OAM states, so that, for example, | (cid:105) A represents the composite state | H (cid:105) p | L (cid:105) o of the photon A . We note the symmetricproperty of the photon pair: swapping the states of photons A and B leaves the state ‡ The precise form of this state contains also states with higher orbital angular momentum, but therelative sizes of these contributions can be controlled[31, 32]. roposed optical realization of a two photon, four-qubit entangled χ state | p (cid:105) B H Z • ×| o (cid:105) B × Figure 2.
The quantum circuit corresponding to the transformation of the state ofphoton B given in (8), where | p (cid:105) B and | o (cid:105) B are the polarization and OAM parts of thecomposite state of the photon. unchanged. To obtain | χ (cid:105) , the state of one of the photons should be transformed as | (cid:105) B → √ | (cid:105) − | (cid:105) ) B , | (cid:105) B → √ | (cid:105) − | (cid:105) ) B , | (cid:105) B → √ | (cid:105) + | (cid:105) ) B , | (cid:105) B → √ | (cid:105) + | (cid:105) ) B . (8)The above transformation is described by the quantum circuit shown in figure 2. Thefirst quantum gate in the circuit is the CNOT gate such that the target qubit will beflipped if the associated control qubit is | (cid:105) . If the swap gate is not included, we obtain (cid:12)(cid:12)(cid:12) χ (cid:69) = 1 √ (cid:12)(cid:12)(cid:12) λ (cid:69) + (cid:12)(cid:12)(cid:12) λ (cid:69) ) AB , (9 a )with (cid:12)(cid:12)(cid:12) λ (cid:69) ≡
12 ( | (cid:105) − | (cid:105) − | (cid:105) + | (cid:105) ) , (9 b ) (cid:12)(cid:12)(cid:12) λ (cid:69) ≡
12 ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) = (cid:12)(cid:12)(cid:12) ζ (cid:69) , (9 c )which is the state | χ (cid:105) proposed by Lee et al in 2002 [27]. Including the swap gate atthe end serves only to ensure that the OAM state of the photon B is realized to be thethird qubit while its polarization is the last. After the given transformation, the stateof the photon pairs turns out to be (cid:12)(cid:12)(cid:12) χ (cid:69) = 1 √ (cid:12)(cid:12)(cid:12) ζ (cid:69) + (cid:12)(cid:12)(cid:12) ζ (cid:69) ) AB , (10)as desired.
3. Optical realization
Before discussing the proposed optical system realizing the given transformation, wefirst explain how each of the optical elements we will employ act on the compositestate of the photons. Let us start with birefringent wave plates, the effect of whichon polarization of light is well known. Both half- and quarter- wave plates are waveretarders; their function is to delay the phase of the polarization component lying in roposed optical realization of a two photon, four-qubit entangled χ state π and π/ J q ( θ ) = (cid:32) cos θ − sin θ sin θ cos θ (cid:33) (cid:32) i (cid:33) (cid:32) cos θ sin θ − sin θ cos θ (cid:33) = (cid:32) cos θ + i sin θ (1 − i ) sin θ cos θ (1 − i ) sin θ cos θ sin θ + i cos θ (cid:33) , (11 a ) J h ( θ ) = (cid:32) cos θ − sin θ sin θ cos θ (cid:33) (cid:32) − (cid:33) (cid:32) cos θ sin θ − sin θ cos θ (cid:33) = (cid:32) cos 2 θ sin 2 θ sin 2 θ − cos 2 θ (cid:33) . (11 b )The angle θ gives the orientation, relative to the horizontal, of the fast axis [34, 35].As these birefringent wave plates affect only the polarization of photons while leavingthe OAM modes unaffected, the total effects of these wave plates on a composite statecorrespond to the transformation | p (cid:48) , o (cid:48) (cid:105) = J i ⊗ I | p, o (cid:105) with i = q , h , (12)where | p, o (cid:105) and | p (cid:48) , o (cid:48) (cid:105) are the composite states of a photon before and after passingthrough the optical elements.A Dove prism is an optical element frequently employed in optical orbital angularmomentum experiments, which acts to flip the sign of the orbital angular momentumof light. For example, in this work it converts the OAM mode from l = 1 to − J D = (cid:32) i (cid:33) . (13)Therefore, we can describe its effect on a composite state by | p (cid:48) , o (cid:48) (cid:105) = J D ⊗ σ x | p, o (cid:105) . (14)As a result, if one wishes only to invert the OAM state of an incident beam while leavingits polarization unaffected, one needs to add a quarter-wave plate before or after theDove prism to compensate its phase retardation effect.Another optical element that will be discussed is a polarizing beam splitter (PBS).Its function is rather obvious in its name as it splits an optical beam into two different roposed optical realization of a two photon, four-qubit entangled χ state Figure 3.
This figure shows the interferometer that realizes the CNOT gate for thecomposite state by adding an M-shaped Dove prism and a quarter-wave plate with itsfast axis at angle π/ distinct beams whose travelling paths depend on the optical polarization of the beam.In this work all PBSs are considered to transmit horizontal and reflect vertical polarizedbeams. Therefore, we can treat each type of optical beams separately, and it allows usto realize control gates for composite states such that the polarization and OAM statesare the control and the target qubits respectively.As an illustration of the operation of these devices, let us consider the interferometergiven in figure 3. The first PBS separates the incident beam into two different paths.The horizontally and vertically polarized beams travel along the internal paths 1 and2 of the interferometer respectively, and they are combined again at the second PBS.Therefore, if we introduce a Dove prism together with a quarter-wave plate into one ofthese paths, the overall effect will be to flip the OAM state of the composite systemfor just one component of polarisation - that corresponding to the path in which theseoptical elements are put. For example, in figure 3, we put a Dove prism and a quarter-wave plate into the path 2, so that if the incident beam is horizontally polarized, itwill be forced to travel along the path 1 and not encounter any elements affecting itscomposite state, and its composite state is untouched. On the other hand, if the beamis vertically polarized, it will be only permitted to go along the internal path 2 andencounter both the M-shaped Dove prism and the quarter-wave plate. The OAM partof the composite state is flipped in this case. This means that for a composite state ofpolarization and OAM the interferometer given in this figure acts as a CNOT gate suchthat if the polarization qubit, the control qubit in this case, is | (cid:105) p the OAM qubit willbe inverted, while it is left unchanged if the control qubit is | (cid:105) p . Devices built on thisprinciple have been used, successfully, to measure both the OAM and the spin for lightat the single photon level [33, 37, 38].At this point, one can notice that, according to (11 b ), single-qubit gates, such as roposed optical realization of a two photon, four-qubit entangled χ state π/ σ z σ x = i σ y which means we have to use two half-wave plates with differentorientation to realize the Y-gate.The complex amplitude of a Laguerre-Gaussian (LG) beam has the phase dependentterm, exp(i lφ ) where l is the orbital angular momentum quantum number of the lightbeam [39]. Thus when one rotates the beam by an angle α this term will becomeexp(i l ( φ + α )), or in other words rotation of an LG beam contributes a phase shift of∆ ψ = lα [40, 33]. An optical beam can be rotated by suitably oriented Dove prisms. Anon-rotated Dove prism gives a non-rotated, reflected image, and when the Dove prismis rotated by an angle β , the reflected image is rotated through 2 β . Non-rotated Doveprisms act as an X-gate for OAM qubits as they change the OAM state of an opticalbeam to be the opposite state, from | l (cid:105) to |− l (cid:105) . The single-qubit Y- and Z-gates forOAM can be implemented as follows: a Dove prism rotated by π/ | (cid:105) o → e − i π/ | (cid:105) o , | (cid:105) o → e i π/ | (cid:105) o , (15)when l = ±
1. This is exactly the transformation corresponding to acting with a Y-gateon a qubit. As we know that σ x σ y = i σ z , one can implement the Z-gate of OAM qubitsby using two Dove prisms: non-rotated and rotated by π/ θ on a composite qubit can thus be written as | p (cid:48) , o (cid:48) (cid:105) = J q ( θ ) ⊗ (cid:32) θ e − θ (cid:33) | p, o (cid:105) . (16)To compensate for this a quarter-wave plate should be added before or after each Doveprism. The physical realizations of the Pauli gates are depicted in figure 4.All effects of optical elements we discussed above on composite qubits aresummarized in table 1. Table 1.
Summary of the effects of optical elements on composite qubits
Optical Elements Effects on composite qubitsQuarter-wave plate fast axis at angle θ U q ( θ ) = J q ( θ ) ⊗ I Half-wave plate with fast axis at angle θ U h ( θ ) = J h ( θ ) ⊗ I Rotated M-shaped Dove prism at angle θ U D ( θ ) = J q ( θ ) ⊗ (cid:32) θ e − θ (cid:33) roposed optical realization of a two photon, four-qubit entangled χ state Figure 4.
The particular alignments of M-shaped Dove prisms together with quarter-wave plates, to compensate their polarization effect, which provide physical realizationsof the Pauli gates for OAM qubits when l = ±
1. The dashed arrows represent the fastaxes of the quarter-wave plates.
4. Optical system
In this section, we seek an optical system that transforms the composite state ofthe photon B so as to prepare photons A and B in any one of the entangled states | χ ij (cid:105) . We start with the optical transformation corresponding to the quantum circuitin figure 2. This quantum circuit includes two different CNOT gates, two single-qubitgates, Hadamard and Y-gates, and the swap gate at the end. We use optical elementscorresponding to each of these.The principle of the first CNOT gate in the circuit is that if the control qubit,the polarization qubit, is in the state | (cid:105) p , then the target qubit, the OAM qubit,will be flipped, while it is left unchanged if the control qubit is | (cid:105) p . As mentionedin the previous section, the CNOT gates in the quantum circuit can be realized byinterferometers with a M-shaped Dove prism in one of their internal paths. Recall thatthe logical qubits, | (cid:105) p and | (cid:105) p , are encoded as horizontal and vertical polarizationstates, | H (cid:105) and | V (cid:105) , respectively. This means we can realize the first CNOT gate by thesame interferometer as given in figure 3, but the Dove prism must be in the internal path1, the path associated with the horizontally polarized component of the input beam,rather than path 2. As the Dove prism leaves horizontal polarization unaffected, we donot need to add a quarter-wave plate in this case. The second CNOT gate is slightlydifferent from the first one as it will flip the target qubit if the control qubit is in thestate | (cid:105) p , and do nothing if the control qubit is | (cid:105) p . This means the second CNOT roposed optical realization of a two photon, four-qubit entangled χ state Figure 5.
This figure shows our proposed optical system corresponding to thequantum circuit given in figure 2. The lengths of the pieces of glass in thetwo interferometers are selected so as to compensate for the delay associated withpropagation through the Dove prisms. The quarter wave plate in the secondinterferometer can be removed if the interferometer path length is suitably adjusted. gate can be exactly realized as the interferometer in figure 3.According to (11 b ), we can explicitly see that the Jones matrices of half- waveplates with optical axes in the horizontal plane and at the angle π/ | χ (cid:105) . The swap gate of the quantum circuit can be implemented easily byrelabelling the composite state of photon B as mentioned earlier. With this swap gate,Yeo’s version of the | χ (cid:105) state is finally obtained. The optical system corresponding tothe quantum circuit given in figure 2 is illustrated in figure 5.With (4 b ), once we obtain | χ (cid:105) , any of the 15 other χ -type four-party entangledstates can be generated using only local Pauli operations: (cid:12)(cid:12)(cid:12) χ ij (cid:69) = σ i ⊗ σ j ⊗ I ⊗ I (cid:12)(cid:12)(cid:12) χ (cid:69) AB . At this point our task is to implement Pauli gates for both polarization and OAMstates. As mentioned in section 3, the Pauli gates of polarization and OAM qubits canbe implemented by half-wave plates and Dove prisms (together with quarter-wave platesto compensate the polarization effect of these Dove prisms) respectively. For example,the tensor product of the Pauli X and Y operators for polarization and OAM qubits, roposed optical realization of a two photon, four-qubit entangled χ state σ xp ⊗ σ yo , can be realized by a half-wave plate with fast axis at angle π/ π/ π/ | χ ij (cid:105) in the set of χ -type states can be realized by applying birefringent wave platesand M-shaped Dove prisms with specified orientations after the optical system shownin figure 5.
5. Conclusion
We have presented the operation required to transform the maximally hyper-entangledstate of a photon pair, obtained from SPDC process, into | χ (cid:105) , a state with genuinefour-party entanglement. We have shown the effect of each optical element we useon the composite state, and an optical system suitable for preparing | χ (cid:105) has beenproposed. To obtain any other of the χ -type states | χ ij (cid:105) , further simple transformationsare required and these may be realized using birefringent wave plates and M-shapedDove prisms.As the proposed optical system requires only readily available linear opticalcomponents, preparation of the desired states should be possible using currenttechnology. Unlike previous work the proposed scheme does not require any postselection, so the efficiency of successful transformation does not depend, intrinsically,on the efficiency of photon detectors. We hope that our scheme for the production ofelements of this class of multipartite entangled states may be realized experimentallyand that doing so will give us better insight of multipartite entanglement, and enablethe demonstration of novel quantum information protocols. Acknowledgments
A Ritboon and S M Barnett acknowledge support from the Development and Promotionof Science and Technology Talents Project (DPST), Thailand, and the Royal Society(RP150122) respectively.
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