Swapping Intra-photon entanglement to Inter-photon entanglement using linear optical devices
SSwapping Intra-photon entanglement to Inter-photon entanglement using linearoptical devices
Mohit Lal Bera ∗ , Abhishek Ghosh † , Asmita Kumari ‡ , and A. K. Pan § National Institute Technology Patna, Ashok Rajhpath, Patna, Bihar 800005, India
We propose a curious protocol for swapping the intra-photon entanglement between path and po-larization degrees of freedom of a single photon to inter-photon entanglement between two spatiallyseparated photons which have never interacted. This is accomplished by using an experimental setupconsisting of three suitable Mach-Zehnder interferometers along with number of beam splitters, po-larization rotators and detectors. Using the same setup, we have also demonstrated an interestingquantum state transfer protocol, symmetric between Alice and Bob. Importantly, the Bell-basisdiscrimination is not required in both the swapping and state transfer protocols. Our proposal canbe implemented using linear optical devices.
I. INTRODUCTION
Quantum physics emerges as a surprising yet natu-ral outgrowth of the revolutionary discoveries of physicsduring the first decade of twentieth century and has re-sulted in an extraordinary revision of our understandingof the microscopic world. Some quantum features canbe exploited for information processing tasks. In recentdecades a flurry of works have been performed, which in-cludes storage and distribution of information in betweennon interacting system (for reviews, see [1]). Quantumentanglement is a fundamental resource for performingmany information processing tasks including secret keydistribution [2] and dense coding [3]. In 1993, Bennettand colleagues [4] put forwarded a path breaking pro-tocol for transporting an unknown quantum state fromone location to a spatially separated one - a protocolnow widely known as quantum teleportation. A sharedentangled states between the two parties and a classi-cal communication channel are required to perform thequantum teleportation task. Right after this proposal,Bouwmeester et al . [5] and Boschi et al . [6] experimen-tally implemented the teleportation protocol using pho-tonic entangled state. Later, various other systems, suchas atoms [7–9], ions [10], electrons [11] and supercon-ducting circuits [12–14] have been used for experimen-tally demonstrating teleportation and interesting exten-sions were subsequently proposed, specially those regard-ing the teleportation of more than one qubit [15].By exploiting the notion of quantum teleportation afascinating consequence emerges known as entanglementswapping [16, 18]. In a swapping protocol, the entangle-ment can be generated between two photons which havenever interacted. If photon A entangled with photon Band C entangled with photon D, then the entanglementcan be created between A and D, although they never in- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] teracted in the past. However, the photons B and C needto be interacted with each other. The swapping of entan-glement has been extensively studied both theoretically[16, 17] and experimentally [18–21]. It is worthwhile tomention here that both the teleportation and entangle-ment swapping protocols require the Bell basis discrimi-nation which is practically a difficult task to achieve us-ing linear optical instruments. A number of experimenthave recently been conducted to perform the Bell basisanalysis using linear optical devices [22–28].The primary aim of the present paper is to demon-strate an interesting entanglement swapping protocol sothat the intra-photon entanglement between the two de-grees of freedoms of single photon is swapped to theintra-photon entanglement between two spatially sepa-rated photons. Note that, the inter-photon entanglementis relatively more fragile than intra-photon one becausethe former is more prone to decoherence. In an interest-ing work [29], the swapping of this kind was proposed. Inthis work, we use a different and elegant setup than thatis used in [29] but similar to [30] to propose our entan-glement swapping protocol. The same setup can be usedto perform quantum state transfer which is technicallydifferent from the usual teleportation protocol. Both ofour swapping and state transfer protocols do not requireBell-basis discrimination. Although our protocol is quiteclose in terms of the spirit of the original swapping pro-tocol [16, 18], but instead of using four photons, we usetwo photons and the inter-photon entanglement betweenpath and polarization degrees of freedom of each of thephotons. A suitable experimental setup involving threeMach-Zehnder interferometers (MZIs) and a few otherlinear optical devices are used to accomplish this task.Curiously, the photons have never interacted with eachother during the whole process of swapping and statetransfer. However, the path degrees of freedom of an-other photon in one of the three MZIs plays a crucialrole.The paper is organized as follows. In Section II, wepropose an experimental setup of the entanglement swap-ping protocol by using simple linear optical devices whichallows to swap a path-polarization intra-photon entangle-ment of single photon onto the polarization-polarization a r X i v : . [ qu a n t - ph ] J un or path-path intra-photon entanglement between twospatially separated photons. We demonstrate the quan-tum state transfer protocol in Section III. We provide abrief summary of our results in Section IV. II. ENTANGLEMENT SWAPPING PROTOCOL
Our experimental setup consists of three suitable
M ZIs where
M Z and M Z belong to Alice and Bobrespectively, and the third interferometer M Z is sharedby both as shown in the Figure 1. Let us denote thephotons in M Z , M Z and M Z as ‘1’, ‘2’ and ‘3’ re-spectively. The entire setup consists of five
50 : 50 beamsplitters, five polarizing beam splitters, three polarizationrotators, eight detectors and two mirrors are denoted by BS i ( i = 1 , ... , P BS j ( j = 1 , ... , P R k ( k = 1 , , D l ( l = 1 , .. and M m ( m = 1 , respectively.This arrangement can be considered as a chainedHardy setup [31]. The well-known Hardy setup was orig-inally proposed for demonstrating the non-locality with-out inequalities. It uses two MZIs, one with electron andother with positron, coupled through a common beamsplitter. The positron and electron annihilate if they si-multaneously pass through that common beam splitter.This is crucial to produce the non-maximally entangledstate required for demonstrating Hardy non-locality. Oursetup (Figure 1) is a chained Hardy setups in the sensethat M Z and M Z share the BS , and M Z and M Z share the BS . If electrons pass through the M Z and M Z and positrons pass through M Z , then electronsand positrons annihilate at BS and BS . In our setup,we use photons for the implementation of our protocol inwhich an effect similar to annihilation at BS and BS is necessary for producing a suitable entangled state re-quired for our purpose. For the case of photons, sucheffect is obtained by using the bunching of indistinguish-able photons. This effect has been extensively discussedin the literature (see, for example,[32, 33]), and also in[34] verifying Hardy paradox experimentally.The task of our protocol is to generate a polarization-polarization or path-path entangled state between thephotons ‘1’ or ‘3’ entering M Z and M Z respectivelywhile ensuring that they never interact. Further, ourgoal is to transfer the polarization state | χ (cid:105) to Bob or | χ (cid:105) to Alice. Let three photons are allowed to inci-dent simultaneously on the beam splitters P BS , P BS and P BS are represented by the quantum states | ψ (cid:105) , | A (cid:105) and | φ (cid:105) respectively, so that, the initial state of thethree photons is | Ψ (cid:105) = | ψ (cid:105) ⊗ | A (cid:105) ⊗ | φ (cid:105) . We also as-sume that polarization states of the photons ‘1’, ‘2’ and‘3’ are | χ (cid:105) = a | H (cid:105) + b | V (cid:105) , | χ (cid:105) = √ ( | H (cid:105) + | V (cid:105) ) and | χ (cid:105) = c | H (cid:105) + d | V (cid:105) respectively, with | a | + | b | = | c | + | d | = 1 . However, | χ (cid:105) does not play any activerole in the present context. The total state of the pho-tons ‘1’, ‘2’ and ‘3’ entering the experimental setup isgiven by | Ψ (cid:105) = | ψ (cid:105) ⊗ | χ (cid:105) ⊗ | A (cid:105) ⊗ | χ (cid:105) ⊗ | φ (cid:105) ⊗ | χ (cid:105) andthe total state of the photons emerging from the P BS , P BS and P BS is given by | Ψ (cid:105) = ( a | ψ (cid:105)| H (cid:105) + ib | ψ (cid:105)| V (cid:105) ) ⊗ ( | A (cid:105)| H (cid:105) + i | A (cid:105)| V (cid:105) ) √ ⊗ ( c | φ (cid:105)| H (cid:105) + id | φ (cid:105)| V (cid:105) ) (1)Next, for understanding the operation M , BS , BS and M on photons let us rearrange Eq.( ) in the followingway | Ψ (cid:105) = 1 √ (cid:2) − b | ψ (cid:105)| V (cid:105)| A (cid:105)| V (cid:105) (cid:0) c | φ (cid:105)| H (cid:105) + id | φ (cid:105)| V (cid:105) (cid:1) (2a) + c (cid:0) a | ψ (cid:105)| H (cid:105) + ib | ψ (cid:105)| V (cid:105) (cid:1) | A (cid:105)| H (cid:105)| φ (cid:105)| H (cid:105) (2b) + id (cid:0) a | ψ (cid:105)| H (cid:105) + ib | ψ (cid:105)| V (cid:105) (cid:1) | A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) (2c) + ia | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105) (cid:0) c | φ (cid:105)| H (cid:105) + id | φ (cid:105)| V (cid:105) (cid:1)(cid:3) (2d) Figure 1: (color online) The setup for implementing theswapping of intra-photon path-polarization entanglement ofeach of the photons ‘1’ and ‘3’ to inter-photon polarization-polarization entanglement between ‘1’ and ‘3’ and for trans-ferring polarization state of photon ‘1’ to photon ‘3’. (Detailsare given in the text).
In Eq.( a ) two indistinguishable photons | ψ V (cid:105) and | A V (cid:105) from P BS and P BS respectively incident si-multaneously on BS (central beam splitter of M Z and M Z ), which results in bunching effect at BS like annihilation in the case of electron and positron, | ψ V (cid:105)| A V (cid:105) → i √ | ψ V (cid:105) + | A V (cid:105) . Similarly inEq.( b ) indistinguishable photons | A H (cid:105) and | φ H (cid:105) from P BS and P BS respectively simultaneouslybunches at BS (central beam splitter of M Z and M Z ), | A H (cid:105)| φ H (cid:105) → i √ | A H (cid:105) + | φ H (cid:105) . Thendue to bunching effect the terms | ψ V (cid:105)| A V (cid:105) and | A H (cid:105)| φ H (cid:105) are dropped and consequently Eq.( a ) andEq.( b ).Next, the term iad | ψ (cid:105)| H (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) in Eq.( c )got phase shift of − i due to three reflection at M , BS and M respectively. However, transmissionof | A (cid:105)| H (cid:105) at BS has been ignored, hence, theamplitude of iad | ψ (cid:105)| H (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) reduces withthe factor of / √ . On the other hand the term − bd | ψ (cid:105)| V (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) in Eq.( c ) got phase shiftof − i due to three reflections at BS , BS and M butthe amplitude is reduced by the factor of 1/2 due to igno-rance of transmissions of | ψ (cid:105)| V (cid:105) and | A (cid:105)| H (cid:105) at BS , BS respectively. Hence the terms in Eq.( c ) evolves to id (cid:0) a | ψ (cid:105)| H (cid:105) + ib | ψ (cid:105)| V (cid:105) (cid:1) | A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) (3) → (cid:2) ad √ | ψ (cid:105)| H (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) + ibd | ψ (cid:105)| V (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) (cid:3) Similarly the term iac | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105)| φ (cid:105)| H (cid:105) inEq.(2d) got phase shift of − i after three reflections at M , BS and BS . However, due to ignorance of trans-missions of | A (cid:105)| V (cid:105) and | φ (cid:105)| H (cid:105) at BS and BS re-spectively overall amplitude is reduced by factor / . Onthe other hand the term − ad | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105)| φ (cid:105)| V (cid:105) inEq.(2c) shifted by the phase of − i due to three reflec-tion at M , BS and M , however the amplitude of thisterm is reduced by / √ due to ignorance of transmissionof | A (cid:105)| V (cid:105) at BS . The terms of Eq.(2d) after passingthrough M , BS , BS and M evolves to ia | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105) (cid:0) c | φ (cid:105)| H (cid:105) + id | φ (cid:105)| V (cid:105) (cid:1) (4) → (cid:2)(cid:0) ac | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105)| φ (cid:105)| H (cid:105) + iad √ | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105)| φ (cid:105)| V (cid:105) (cid:1)(cid:3) Now, Eq.(2a-2d) after passing through M , BS , BS and M is given by | Ψ (cid:105) = N (cid:2) ad √ | ψ (cid:105)| H (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) + ibd | ψ (cid:105)| V (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) + ac | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105)| φ (cid:105)| H (cid:105) + iad √ | ψ (cid:105)| H (cid:105)| A (cid:105)| V (cid:105)| φ (cid:105)| V (cid:105) (cid:3) where N = ( a c + 4 a d + b d ) − / is normalized con-stant. Using the polarization rotator P R before BS we flip the vertical polarization | V (cid:105) to | H (cid:105) , so that finalstate is given by | Ψ (cid:105) = N (cid:2) ad √ | ψ (cid:105)| H (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) (5) + ibd | ψ (cid:105)| V (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) + ac | ψ (cid:105)| H (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| H (cid:105) + iad √ | ψ (cid:105)| H (cid:105)| A (cid:105)| H (cid:105)| φ (cid:105)| V (cid:105) (cid:3) Let us now consider two cases:(i) When the state of the photon in the interferometer
M Z before BS is | A (cid:105)| H (cid:105) = ( i | A (cid:105) + | A (cid:105) ) | H (cid:105) / √ which results in a detection in D .(ii) When the state of the photon in M Z before BS is | A (cid:105)| H (cid:105) = ( i | A (cid:105) − | A (cid:105) ) | H (cid:105) / √ which results in adifferent detector at D .Ideally, a PBS can be used in place of BS . But, thepolarization | χ (cid:105) has no role in the protocol, so a normalbeam splitter can serve our purpose. In case (i), we endup with a four-qubit GHZ type entangled state of pathand polarization degrees of freedom of the photons ‘1’and ‘3’. The reduced state of the photons ‘1’ and ‘3’ canthen be written as, | Ψ (cid:105) = N ( ac | ψ (cid:105)| H (cid:105)| φ (cid:105)| H (cid:105) + bd | ψ (cid:105)| V (cid:105)| φ (cid:105)| V (cid:105) ) (6)where, N = ( a c + b d ) − / .We thus prepared an entangled state between the fourdegrees of freedoms of two photons by introducing con-strains in photons path and using a suitable projectivemeasurement on the photon ‘2’ in M Z . It is to be notedthat during the whole process, the photons ‘1’ and ‘3’ in M Z and M Z respectively have never interacted witheach other.Similarly, for the case(ii), the resulting reduced stateof the photons ‘1’ and ‘3’ can be written as | Ψ (cid:48) (cid:105) = N (cid:48) ( ac | ψ H (cid:105)| φ H (cid:105) + bd | ψ V (cid:105)| φ V (cid:105) (7) − i √ ad | ψ H (cid:105)| φ V (cid:105) ) where N (cid:48) = ( a c + b d + 8 a d ) − / . We do notfurther use the state in Eq.(7) in this paper.In order to achieve the path-path or polarization-polarization entanglement between the photons ‘1’ and‘3’, we need to invoke a suitable disentangling processwhich again requires no direct interaction between thephotons in M Z and M Z . For this, we consider therecombination of | ψ (cid:105) and | ψ (cid:105) by the beam splitter BS , so that | ψ (cid:105) = ( | ψ (cid:105) + i | ψ (cid:105) ) / √ and | ψ (cid:105) =( i | ψ (cid:105) + | ψ (cid:105) ) / √ . The state after BS can then be writ-ten as, | Ψ (cid:105) = N √ | ψ (cid:105) [ ac | H (cid:105)| φ (cid:105)| H (cid:105) + ibd | V (cid:105)| φ (cid:105)| V (cid:105) ]+ | ψ (cid:105) [ iac | H (cid:105)| φ (cid:105)| H (cid:105) + bd | V (cid:105)| φ (cid:105)| V (cid:105) ]) (8)Similarly, the beam splitter BS recombines the twopaths | φ (cid:105) = ( | φ (cid:105) + i | φ (cid:105) ) / √ and | φ (cid:105) = ( i | φ (cid:105) + | φ (cid:105) ) / √ . Then, the joint state of the photons ‘1’and‘3’ after the BS becomes, | Ψ (cid:105) = N [ | ψ (cid:105)| φ (cid:105) ( ac | H (cid:105)| H (cid:105) − bd | V (cid:105)| V (cid:105) ) (9) + i | ψ (cid:105)| φ (cid:105) ( ac | H (cid:105)| H (cid:105) + bd | V (cid:105)| V (cid:105) )+ i | ψ (cid:105)| φ (cid:105) ( ac | H (cid:105)| H (cid:105) + bd | V (cid:105)| V (cid:105) ) − | ψ (cid:105)| φ (cid:105) ( ac | H (cid:105)| H (cid:105) − bd | V (cid:105)| V (cid:105) )] Depending on a suitable joint path measurement chosenby Alice and Bob the following polarization-polarizationintra-photon entangled state | Ψ (cid:105) = ac | H (cid:105)| H (cid:105) − bd | V (cid:105)| V (cid:105) can be generated. When Alice and Bob chose | ψ (cid:105) and | φ (cid:105) , then an additional gate operation ˆ σ z isrequired for the path | ψ (cid:105) or | φ (cid:105) in order to obtain thestate | Ψ (cid:105) .Hence, using our setup we have generated apolarization-polarization entanglement between the pho-tons ‘1’ and ‘3’ even when they have never interacted witheach other. It is important to note that, both the pho-tons contain an intra-photon path-polarization entangle-ment that is swapped to the inter-photon entanglementbetween them. Thus, the protocol differs from the usualswapping protocols in the literature and also from [29].The same setup can also be used to create path-path andpath-polarization hybrid entanglement between the twophotons. For this, a few small changes need to be ade- quately incorporated in the setup. The same argumentof swapping can be drawn by using the state given byEq.( ). However, we have not explicitly shown it here. III. QUANTUM STATE TRANSFER
As mentioned before, our setup can also be used fordemonstrating the teleportation of an unknown quantumstate. One may say that it is an obvious fact once wehave generated the entangled state | Ψ (cid:105) , the teleporta-tion is one more step. For this, one more qubit needs tobe brought either by Alice or Bob followed by a relevantBell-basis measurement. However, it seems interesting ifthe polarization state | χ (cid:105) belongs to Alice or | χ (cid:105) be-longs to Bob can be teleportated without introducinganother qubit state and Bell-basis analysis. We exactlyprovide such a scheme of state transfer.In order to demonstrate such a state transfer pro-tocol, let us invoke two polarization rotators P R and P R along the path | φ (cid:105) and | φ (cid:105) respectively. So thatthe states | H (cid:105) and | V (cid:105) can be transformed as, | H (cid:105) = ( | H (cid:105) + | V (cid:105) ) / √ and | V (cid:105) = ( | H (cid:105) − | V (cid:105) ) / √ . After thetwo rotations, the state given by Eq.(9) can be writtenas | Ψ (cid:105) = N √ | ψ (cid:105) [( ac | H (cid:105) − bd | V (cid:105) ) | φ (cid:105)| H (cid:105) + ( ac | H (cid:105) + bd | V (cid:105) ) | φ (cid:105)| V (cid:105) ] (10) + i | ψ (cid:105) [( ac | H (cid:105) + bd | V (cid:105) ) | φ (cid:105)| H (cid:105) + ( ac | H (cid:105) − bd | V (cid:105) ) | φ (cid:105)| V (cid:105) ]+ i | ψ (cid:105) [( ac | H (cid:105) + bd | V (cid:105) ) | φ (cid:105)| H (cid:105) + ( ac | H (cid:105) − bd | V (cid:105) ) | φ (cid:105)| V (cid:105) ] − | ψ (cid:105) [( ac | H (cid:105) − bd | V (cid:105) ) | φ (cid:105)| H (cid:105) + ( ac | H (cid:105) + bd | V (cid:105) ) | φ (cid:105)| V (cid:105) ]) Bob now measures on his photon ‘3’ by using
P BS and P BS and detects the photon in four detectors D , D , D and D . For four outcomes of Bob yield eight differentpossibilities at Alice’s end. The states of the Bob’s pho-ton corresponding to the detectors D , D , D and D are | φ (cid:105)| H (cid:105) , | φ (cid:105)| V (cid:105) , | φ (cid:105)| H (cid:105) and | φ (cid:105)| V (cid:105) respectively.The measurements at Bob’s end produce the followingstates at Alice’s end are given by | Ψ D (cid:105) = | ψ (cid:105) ( ac | H (cid:105) − bd | V (cid:105) ) (11a) + | ψ (cid:105) ( ac | H (cid:105) + bd | V (cid:105) ) | Ψ D (cid:105) = | ψ (cid:105) ( ac | H (cid:105) + bd | V (cid:105) ) (11b) + | ψ (cid:105) ( ac | H (cid:105) − bd | V (cid:105) ) | Ψ D (cid:105) = | ψ (cid:105) ( ac | H (cid:105) + bd | V (cid:105) ) (11c) + | ψ (cid:105) ( ac | H (cid:105) − bd | V (cid:105) ) | Ψ D (cid:105) = | ψ (cid:105) ( ac | H (cid:105) − bd | V (cid:105) ) (11d) + | ψ (cid:105) ( ac | H (cid:105) + bd | V (cid:105) ) Note here that | Ψ D (cid:105) = | Ψ D (cid:105) and | Ψ D (cid:105) = | Ψ D (cid:105) . Letus now assume that a = b = 1 / √ . Then after the detec-tion of photon ‘3’ in four different detectors ( D , D , D and D ), Bob needs to send the information througha classical communication channel. Following Bob’s in-struction, Alice performs suitable gate operations to ob-tain the desired state | χ (cid:48) (cid:105) = c | H (cid:105) + d | V (cid:105) as given inthe Table-1. Then, whenever Bob gets photon ‘3’ in D or in D , he asks Alice to use a Pauli gate ˆ σ z in thechannel | ψ (cid:105) . If he gets the photon in D or in D , Al-ice has to use the ˆ σ z in the channel | ψ (cid:105) . Hence, wedemonstrated a state transfer protocol from Bob to Al-ice without any direct interaction between photons ‘1’and ‘3’ in two interferometers M Z and M Z . Note thatthe success probability of teleportation in this case is / ,i.e., the cost of the state transfer is larger than the orig-inal teleportation protocol. Importantly, no Bell-basismeasurement is required in the whole process. Bob’s detection Alice’s operationon | ψ (cid:105) on | ψ (cid:105) D ˆ σ Z ˆ I D ˆ I ˆ σ Z D ˆ I ˆ σ Z D ˆ σ Z ˆ I Table I: Alice’s unitary rotation on the path | ψ (cid:105) and | ψ (cid:105) upon receiving instructions from Bob. IV. DISCUSSION
We have demonstrated an interesting swapping proto-col using simple linear optical devices where the intra-photon entanglement between path and polarization de-grees of freedom of a single photon is swapped to polarization-polarization entanglement of two spatiallyseparated photons. Note that, those photons have neverinteracted during the whole process. We have furthershown how the same setup can be used for the purposeof a curious quantum state transfer. Both the protocolsavoid Bell basis discrimination which is taken care by ex-ploiting the actions of the path degrees of freedoms in
M Z and M Z . We believe that the proposed setup canbe experimentally implemented with the existing tech-nology that uses linear optical devices. Acknowledgments
AG and MLB acknowledge the local hospitality of NITPatna during their visits. AKP acknowledges the supportfrom Ramanujan Fellowship research grant (SB/S2/RJN-083/2014). [1] M. Nielsen and I. Chuang,
Quantum Computation andQuantum Information (Cambridge University Press,Cambridge, 2000); D. Bouwmeester, A. Ekert and A.Zeilinger (Editors),
The Physics of Quantum Informa-tion: Quantum Cryptography Quantum TeleportationQuantum Computation (Springer-Verlag, Berlin, 2001);R. Horodecki et al ., Rev. Mod. Phys. , 865 (2009); O.Gühne and G. Toth, Phys. Rep. , 1 (2009); N. Brun-ner et al ., Rev. Mod. Phys. , 419 (2014).[2] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[3] C. H. Bennett and S. Wiesner, Phys. Rev. Lett. ,2881 (1992); D. Home, Conceptual Foundations of Quan-tum Physics: An overview from Modern Perspectives (Plenum, New York, 1997); H. K. Lo, S. Popescu and T.Spiller (Editors),
Introduction to Quantum Computationand Information (World Scientific, Singapore, 1998).[4] C. H. Bennett and G. Brassard,
Proceedings of IEEE In-ternational Conference on Computers, Systems and Sig-nal Processing , (Bangalore, India, 1984), p. 175.[5] D. Bouwmeester et al ., Nature , 575 (1997).[6] D. Boschi et al ., Phys. Rev. Lett. , 1121 (1998).[7] J. I. Cirac and A. S. Parkins, Phys. Rev. A , 50 (1994).[8] M. Riebe et al ., N. J. Phys. , 211 (2007).[9] S. Q. Tang, C. J. Shan and X. X. Zhang, Int. J. Theor.Phys. , 1899 (2010).[10] S. Olmschenk et al ., Science , 486 (2009).[11] De R. L. Visser and M. Blaauboer, Phys. Rev. Lett. ,246801 (2006).[12] M. Baur et al ., Phys. Rev. Lett. , 040502 (2012).[13] J. Joo and E. Ginossar, Scientific Rep. , 26338 (2016).[14] T. Li and Z. Q. Yin, Sci. Bull. , 163 (2016).[15] D. Bouwmeester, J.-W. Pan, H. Weinfurter, and A.Zeilinger, J. Mod. Opt. , 279 (2000).[16] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert , Phys. Rev. Lett. , 4287 (1993).[17] S. Bose, V. Vedral and P. L. Knight, Phys. Rev. A ,882 (1998); Phys. Rev. A , 1 (1999).[18] J. W. Pan, D. Bouwmeester, H. Weinfurter and A.Zeilinger, Phys. Rev. Lett. , 3891 (1998).[19] T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger,Phys. Rev. Lett. , 017903 (2001).[20] C. Schmid et al ., New J. Phys. , 033008 (2009).[21] X. S. Ma et al ., Nature Phys. , 479 (2012).[22] J. Calsamiglia and N. Lütkenhaus, Appl. Phys. B , 67(2001).[23] C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter,Phys. Rev. Lett. , 190501 (2006).[24] T. Jennewein, R. Ursin, M. Aspelmeyer and A. Zeilinger,arXiv:quant-ph/0409008v1.[25] W. P. Grice, Phys. Rev. A , 042331 (2011).[26] S. W. Lee and H. Jeong, Proceedings of the First In-ternational Conference on Entangled Coherent State andIts Application to Quantum Information Science , (Tam-agawa University, Tokyo, (2012), p. 46.[27] L. Zhou and Y. B. Sheng, Scientific Rep. , 20901 (2016).[28] F. Wang et al ., Optica , 1462 (2017).[29] S. Adhikari, A. S. Majumdar, D. Home and A. K. Pan,EPL, , 10005 (2010).[30] B. de Lima Bernardo, Ann. Phys. , 80 (2016).[31] L. Hardy, Phys. Rev. Lett, 68, 2981 (1992); 71, 1665(1993).[32] C. K. Hong, Z. Y. Ou and L. Mandel, Phys. Rev. Lett. , 2044 (1987).[33] J. W. Pan et al ., Rev. Mod. Phys. , 777 (2012).[34] W. T. M. Irvine, J. F. Hodelin, C. Simon, and D.Bouwmeester, Phys. Rev. Lett.95