Entanglement conditions involving intensity correlations of optical fields: the case of multi-port interferometry
Junghee Ryu, Marcin Marciniak, Marcin Wieśniak, Dagomir Kaszlikowski, Marek Żukowski
aa r X i v : . [ qu a n t - ph ] M a r Entanglement conditions involving intensity correlations ofoptical fields: the case of multi-port interferometry
Junghee Ryu , , Marcin Marciniak , Marcin Wie´sniak , ,Dagomir Kaszlikowski , , and Marek ˙Zukowski , Centre for Quantum Technologies, National University of Singapore, 3 ScienceDrive 2, 117543 Singapore, Singapore Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics,Physics and Informatics, University of Gda´nsk, 80-308 Gda´nsk, Poland Institute of Informatics, Faculty of Mathematics, Physics and Informatics,University of Gda´nsk, 80-308 Gda´nsk, Poland Department of Physics, National University of Singapore, 2 Science Drive 3,117542 Singapore, Singapore Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna,AustriaNormalized quantum Stokes operators introduced in [Phys. Rev. A , 042113 (2017)] enable one to better observe non-classical correlationsof entangled states of optical fields with undefined photon numbers. Fora given run of an experiment the new quantum Stokes operators are de-fined by the differences of the measured intensities (or photon numbers)at the exits of a polarizer divided by their sum. It is this ratio that is tobe averaged, and not the numerator and the denominator separately, asit is in the conventional approach. The new approach allows to constructmore robust entanglement indicators against photon-loss noise, which candetect entangled optical states in situations in which witnesses using stan-dard Stokes operators fail. Here we show an extension of this approachbeyond phenomena linked with polarization. We discuss EPR-like exper-iments involving correlations produced by optical beams in a multi-modebright squeezed vacuum state. EPR-inspired entanglement conditions forall prime numbers of modes are presented. The conditions are much moreresistant to noise due to photon loss than similar ones which employ stan-dard Glauber-like intensity, correlations.(1) arxiv˙version printed on October 19, 2018 PACS numbers: 03.65.Ud, 42.50.-p, 03.67.Bg, 03.67.Mn
1. Introduction
A new approach to the analysis of polarization correlations was in-troduced in [1]. It involves a non-conventional definition of Stokes pa-rameters of quantum optical fields. The “textbook” quantum Stokes pa-rameters involve averaged differences of the measured intensities of lightexiting polarization analyzers, measured in three complementary arrange-ments (horizontal-vertical, diagonal-anti-diagonal and right-left-handed cir-cular polarizations), and the total average intensity. If one assumes that themeasured intensities are proportional to photon numbers, then the textbookStokes parameters read D ˆ Σ i E = Tr[( a † i a i − a † i ⊥ a i ⊥ ) ̺ ] , (1)where a i is the annihilation operator for photons of polarization i , a i ⊥ for theorthogonal polarization, and the index i denotes the three complementaryarrangements. One also has ˆ Σ = a † i a i + a † i ⊥ a i ⊥ = ˆ N a tot .However, this approach to Stokes parameters faces problems when theintensities of quantum optical fields are fluctuating. This is so for examplein the case for the optical fields generated multi-mode parametric down-conversion, especially for higher pump powers. This is due to the factthat the traditional Stokes parameters depend on intensity fluctuations.Experimental runs involving higher measured intensities contribute moreto their values. However, polarization of light is an intensity independentphenomenon.It turns out that one can revise the concept of quantum Stokes ob-servables, and remove this conceptual difficulty. The new approach allowsone to derive much more effective indicators of non-classicality for quantumoptical fields with undefined, fluctuating, intensities. To this end, Ref. [1]introduces normalized Stokes operators. They are based on the ratios of thenumbers of photons registered by one of the two detectors, to the number ofphotons counted by both detectors for a given run (not averages). In newformalism the redefined, normalized Stokes parameters read D ˆ S i E = Tr " Π a † i a i − a † i ⊥ a i ⊥ ˆ N a tot Π ̺ , (2)where Π denotes a projection written ˆ I − | Ω ih Ω | with the identity operatorˆ I and the vacuum state | Ω i for modes i and i ⊥ , which simply removes thevacuum terms in the state ̺ , and makes the operator in the numerator well rxiv˙version printed on October 19, 2018 defined. Note that in case of vacuum events, that is, measurements showing N tot = 0, we have no contribution to the redefined Stokes parameters (andneither to the standard ones). The ˆ Σ parameter is replaced by ˆ S = Π,and simply gives the probability of a non-vacuum event.The redefined Stokes operators allow a derivation of new quantum op-tical Bell inequalities [2], which are an improvement of standard ones in-troduced for general optical intensities in [3]. The new Bell inequalities arebased only on the standard Bell assumptions of realism, locality and freewill, and nothing more. They do not require any specific additional “rea-sonable” assumptions on the form of the hidden variable theories, which arenecessary to derive the standard ones of Ref. [3]. Most importantly the new“theoretical-loophole” free Bell inequalities constructed for the redefinedStokes operators can be violated by classes of entangled optical fields forwhich the standard ones are not violated. E.g., this is the case for four-mode(bright) squeezed vacuum (BSV), generated in type-II parametric down con-version for stronger pump powers. Note that this is a typical example ofa laboratory situation, where often parametric down conversion is used toget entanglement effects. For stronger laser pumping the squeezed vacuumhas many contributing Fock components with different photon numbers -we have exactly the aforementioned situation of undefined photon numbers.The new Stokes operators also allow to re-formulate any known en-tanglement conditions [1]. In case of the four-mode, BSV states, Ref. [1]reformulates an entanglement indicator of Ref. [4], which reads X i (cid:28)(cid:16) Σ ai + Σ bi (cid:17) (cid:29) sep ≥ D ˆ N a tot + ˆ N b tot E sep , (3)where h· · · i sep denotes an average over a separable state. Here, a and b denote two beams (defined by propagation directions). The lower bound isproportional to the averaged total photon numbers, i.e., the sum of aver-aged total intensities of beams a and b . The idea of the indicator is basedon the Einstein-Podolsky-Rosen (EPR)-like condition which is satisfied bythe rotationally invariant bright squeezed vacuum: for this state one has azero value for the left hand side expression in (3). Such a state is essen-tially a super-singlet with undefined photon numbers, which for identicalpolarization settings at two separated observation stations, always givesanti-correlated results.In Ref. [1], the photon number operators, like e.g., a † i a i , were in theleft hand side in (3) replaced by the rates, respectively Π a † i a i ˆ N a tot Π. For therotationally invariant squeezed vacuum, we still have the EPR condition. arxiv˙version printed on October 19, 2018
However one can show that for a separable state one has X i (cid:28)(cid:16) ˆ S ai + ˆ S bi (cid:17) (cid:29) sep ≥ * Π a N a tot Π a + sep + * Π b N b tot Π b + sep . (4)The new condition turns out to be more robust with respect to photonlosses, modeled in [1] by inefficient detectors.Generally, re-formulating the separability conditions with new Stokes op-erators enable a much better entanglement detection in case of BSV states.New condition is more robust against photon losses (or non-perfect detec-tion efficiency). The previous condition (3) fails to detect the entanglementof the BSV state for the detection efficiency lower than 1 / d -mode beams.For technical reasons, which will be discussed below, we assume that d isa prime. After finishing this manuscript we have found a condition whichworks for all d , which are powers of a prime number. This will be reportedelsewhere.
2. EPR-like optical experiments involving pairs of multi-portbeam splitters
We shall consider here a class of entanglement experiments analyzed inRef. [8], allowing for EPR-like correlations [9], which involve measurementbasis transformations defined by two local multi-port interferometers (beam-splitter). Such interferometers were first studied in [10], however not in thecontext of entanglement. rxiv˙version printed on October 19, 2018 It is known that d -input-port- d -output-port interferometers can performany finite dimensional unitary d × d transformations of single particle statesdescribed by a d dimensional Hilbert space, see Ref. [11]. Such interferom-eters consist of interconnected beam splitters and phase shifters. The sameblueprint for multi-port devices allows for optical modes coupling resultingin a unitary relation way between the input and output modes. For in-stance, consider a single photon prepared in input mode i = 0 , , ..., d − | φ in i i . We assume throughout that the input and output modesare fully distinguishable. The multi-port action can be described by theunitary transformation which gives as output states | φ out k i = P i U ki | φ in i i ,where U is unitary matrix. The equivalent mode transformation for the sec-ond quantized description can be written down in terms of photon creationoperators (related with the ‘in’ and ‘out’ modes, or beams, i = 0 , , ..., d − a † out k = X i U ki a † in i . (5)Beacause of the assumed distinguishability among the ‘in’ modes and amongthe ‘out’ modes, we assume that [ a in i , a † in j ] = δ ij , and [ a in i , a in j ] = 0, andsimilar relations for the ‘out’ modes.With the advent of integrated optics, which allows for stable compli-cated interferometers, two multi-port experiments, such as the ones sug-gested in [8], are becoming feasible. Recently, the work [12] tested exactlysuch configurations. The schemes discussed here involve parametric down-conversion for higher pump powers, in the case of which we have superposi-tions of multi-pair emissions. Thus new phenomena can be expected, whichcan be both decremental or beneficial for possible quantum communicationexperiments. At least one should check to what extent the features of two-photon correlations related with entanglement and the EPR paradox, arestill present in the case of stronger fields.We consider quantum optical states produced by multi-mode emissionsin the parametric down-conversion process [8, 13]. Due to the phase match-ing conditions, the emissions from a parametric down-conversion source aredirectionally correlated. For example, type-I parametric down-conversionprocess generates the pairs of photons of the same frequency with emissiondirections which form a cone. One can register coincidences into pairs of“conjugated” directions along the cone which lay in the same plane as thepump field, for details see [13]. The directions of such pairs satisfy the phasematching condition. We can select an arbitrary number of such pairs of thedirections, and collect their radiation. In such a case the description of thecrystal-field interaction leading to the process can be given by an interaction arxiv˙version printed on October 19, 2018 Hamiltonian of the form: H = i γ d − X i =0 a † i b † i + h.c., (6)where a † i and b † i are the creation operators of i -th conjugate signal-idler modepair, and γ is a coupling constant proportional to the pumping power. Themode operators a i and b i refer to two conjugate directions.Notice that the Hamiltonian in (6) can be transformed into the followingform: H = i γ d − X k =0 a † out k b † out k + h.c., (7)where a † out k = P j U kj a † j , and b † out k = P i U ∗ ki b † i with a d × d unitary matrix U . Due to this symmetry of the Hamiltonian H , one has the perfect EPRcorrelations for the emitted photons. EPR correlations must occur for atlease two complementary operational situations, thus to see that the sym-metry implies, it is enough to consider a transformation U which leads toa complementary mode basis. This is when the observation bases at lo-cation a and b are allowing to measure respectively a † i a i and b † j b j , where i, j = 0 , , . . . , d − d pairs of Schmidt modes, the emitted photon pairs are pre-pared in the following entangled state: | BSV i = 1cosh d Γ ∞ X n =0 s ( n + d − n !( d − n Γ | ψ n i , (8)where | ψ n i = s n !( d − n + d − X p + ··· + p d = n | p i a · · · | p d i a d | p i b · · · | p d i b d . (9)The sum is taken over all combinations of nonnegative integers p i . Theparameter Γ describes the gain and is dependent on coupling constant γ and the interaction time (which can be put as equal to the length of thenon-linear crystal, along the propagation direction of the laser field, dividedby the speed of light).The local measurement devices which we consider consist of an unbi-ased or “symmetric” [8], multi-port beam-splitter and d detectors in itsoutput ports, which are by assumption capable to resolve photon numbers. rxiv˙version printed on October 19, 2018 M P B S M P B S d − d d − d S Source
Fig. 1. Schematic diagram of the experiment. The local measurement stationsconsist of a d -input- d -output multi-port beam-splitter (denoted by MPBS) anddetectors. The interaction in the source S , given by the Hamiltonian (6), generatesa d -mode bright squeezed vacuum state (8). This entangled state leads to perfectEPR correlations between the local conjugate modes. An unbiased multi-port beam-splitter is defined as a d -input and d -outputinterferometric device which realizes a mode transformation defined by aunitary matrix U which links two unbiased orthonormal bases in the d di-mensional Hilbert space. In the case of such transformations a single photonentering through a single port can be detected at any of the output portswith the same probability of 1 /d . A simple analytic formula holds for thevalues of matrix elements for such unitary transformations only for d whichis prime. This is the reason why we concentrate here on this case. It isknown that d + 1 mutually unbiased bases exist for all Hilbert spaces of di-mensions which are powers of primes [14, 15]. In other cases, including thesimplest one d = 6, the number of possible unbiased bases is still an openquestion. Finally, note that if two bases are unbiased this means that theydescribe two perfectly complementary operational measurement situations.In [7], the separability conditions for the three-output case are studied.We here shall derive the separability conditions for arbitrary d which isprime. We shall also use a different approach.
3. Entanglement indicators for prime d Consider d + 1 unitary transformations which lead to the unbiased (com-plementary) bases in a d -dimensional Hilbert space. It is known that whenthe dimension d of a Hilbert space is an integer power of a prime number,the number of mutually unbiased bases is given by d + 1 [14, 15]. We put U ( d ) = ˆ I , while the others, indexed with m = 0 , , . . . , d −
1, have matrix arxiv˙version printed on October 19, 2018 elements which lead to the following transformations of the bases [14, 15]: U ( m ) js = 1 √ d ω js + ms , (10)where ω = exp(2 π i /d ). With such transformations one can relate a multi-port beam-splitter which couples the creation operators of input beams, a † s ,with the output ones, a † j ( m ) in the following way: a † j ( m ) = 1 √ d d − X s =0 ω js + ms a † s , (11)and we define a † j ( m = d ) = a † j . The photon number operator of j th exitmode of an U ( m ) multi-port reads ˆ n j ( m ) = a † j ( m ) a j ( m ). Note that forbeams b we have a conjugate unitary relation, compare relation (7) and itsdiscussion.For the (bright) squeezed vacuum, | BSV i , for d pairs of modes ( d is aprime), its perfect correlations give us the following relation: * d X m =0 d X j =1 (cid:2) ˆ n Aj ( m ) − ˆ n Bj ( m ) (cid:3) + BSV = 0 , (12)where indices A and B denote operators for Alice and Bob, respectively.The aim now is to find the lower bound of such an expression for anyseparable state. As separable states are convex combinations of productones, it is enough to find the minimum of the expression (12) for a productstate ̺ A ⊗ ̺ B instead of BSV, where both ̺ A and ̺ B are pure states. Wehave * d X m =0 d X j =1 (cid:2) ˆ n Aj ( m ) − ˆ n Bj ( m ) (cid:3) + ̺ A ⊗ ̺ B = X m,j (cid:10) ˆ n Aj ( m ) (cid:11) ̺ A + X m,j (cid:10) ˆ n Bj ( m ) (cid:11) ̺ B − X m,j (cid:10) ˆ n Aj ( m ) (cid:11) ̺ A (cid:10) ˆ n Bj ( m ) (cid:11) ̺ B ≥ X m,j (cid:10) ˆ n Aj ( m ) (cid:11) ̺ A + X m,j (cid:10) ˆ n Bj ( m ) (cid:11) ̺ B − X m,j (cid:10) ˆ n Aj ( m ) (cid:11) ̺ A / X m,j (cid:10) ˆ n Bj ( m ) (cid:11) ̺ B / . (13) rxiv˙version printed on October 19, 2018 Let us first consider one of the factors of the term in the last line of theinequality (13) (we drop below the index numbering the measuring stations
A, B ): d X m =0 d X j =1 h ˆ n j ( m ) i = d X j =1 h ˆ n j i + d − X m =0 d X j =1 h ˆ n j ( m ) i . (14)Let us now consider the second term in the above equation, involving sum-mation over m only up to d −
1. Using (11), we get the following: d − X m =0 d X j =1 h ˆ n j ( m ) i = 1 d d − X m =0 d X j =1 *X s,t ω j ( s − t )+ m ( s − t ) a † s a t + = 1 d X m,j X s ,t X s ,t ω j ( s − t + s − t )+ m ( s − t + s − t ) D a † s a t E D a † s a t E = 1 d X s ,t X s ,t D a † s a t E D a † s a t E X j ω j ( s − t + s − t ) X m ω m ( s − t + s − t ) = 1 d X s = t X s = t + X s = t X s = t + X s = t X s = t + X s = t X s = t × D a † s a t E D a † s a t E X j ω j ( s − t + s − t ) X m ω m ( s − t + s − t ) = 1 d d X s ,s h ˆ n s i h ˆ n s i + X s = t X s = t D a † s a t E D a † s a t E × X j ω j ( s − t + s − t ) X m ω m ( s − t + s − t ) . (15)The last equality follows from the fact that s − t + s − t = 0 when s = t , s = t or s = t , s = t .Now, we will show that for the case s = t , s = t , the expression P j ω j ( s − t + s − t ) P m ω m ( s − t + s − t ) is nonzero if and only if s = t and s = t . To this end, observe that the nonzero value occurs only if s − t + s − t = 0 , (16) s − t + s − t = 0 . (17) arxiv˙version printed on October 19, 2018 It follows from (16) that s − t = − ( s − t ) = 0. Thus, the second equationimplies s + t − s − t = 0 . (18)Equations (16) and (18) lead to the conditions s = t and s = t . Finally,we observe that if these conditions are satisfied, then X j ω j ( s − t + s − t ) X m ω m ( s − t + s − t ) = d . Thus, the formula (14) is reduced to X j h ˆ n j i + X s ,s h ˆ n s i h ˆ n s i + 1 d X s = t d D a † s a t E D a † t a s E = X j h ˆ n j i + D ˆ N E + X s = t h a s ψ | a t ψ i h a t ψ | a s ψ i≤ X j h ˆ n j i + D ˆ N E + X s = t || a s ψ || || a t ψ || = X j h ˆ n j i + D ˆ N E + X s = t h ˆ n s i h ˆ n t i = 2 D ˆ N E , (19)where | ψ i is a normalized vector which determines the pure state ̺ X ( X = A, B ), | a t ψ i = a t | ψ i , and || a t ψ || is its norm. Finally, we have d X m =0 d X j =1 h ˆ n j ( m ) i ≤ D ˆ N E . (20) rxiv˙version printed on October 19, 2018 In case of the first two terms in (13), we have X j ˆ n j + 1 d d − X m =0 d X j =1 X s ,t ,s ,t ω j ( s − t + s − t )+ m ( s − t + s − t ) a † s a t a † s a t = X j ˆ n j + 1 d X s ,t ,s ,t X j ω j ( ··· ) X m ω m ( ··· ) ! a † s a t a † s a t = X j ˆ n j + X s ,s ˆ n s ˆ n s + X s = t a † s a t a † t a s = X j ˆ n j + X s ,s ˆ n s ˆ n s + X s = t a † s a s ( a † t a t + 1)= X j ˆ n j + X s ,s ˆ n s ˆ n s + X s = t ˆ n s ˆ n t + ( d − X s ˆ n s = ˆ N + ( d −
1) ˆ N + ˆ N = 2 ˆ N + ( d −
1) ˆ N (21)To obtain this, we used again the observation from the paragraph following(15). With the help of the results of (19) and (21), finally one derives theseparability condition (13) in the form of * d X m =0 d X j =1 (cid:2) ˆ n Aj ( m ) − ˆ n Bj ( m ) (cid:3) + ̺ A ⊗ ̺ B = 2 D ˆ N A E + ( d − D ˆ N A E + 2 D ˆ N B E + ( d − D ˆ N B E − D ˆ N A E D ˆ N B E ≥ ( d − (cid:16)D ˆ N A E + D ˆ N B E(cid:17) . (22)To get an analog conditions for the rates, one has to retrace the above deriva-tion, replacing the number operators by the rates ˆ r j ( m ) ≡ Π ˆ n j ( m ) N Π.The condition for the rates reads: * d X m =0 d X j =1 [ˆ r Aj ( m ) − ˆ r Bj ( m )] + sep ≥ ( d − (cid:28) Π A N A Π A (cid:29) sep + (cid:28) Π B N B Π B (cid:29) sep ! . (23)For d = 3, this condition is equivalent to the one derived in [7], and thusas shown there it is more robust with respect to noise related with photonlosses. Our numerical studies show that the condition (23) outperforms (22)also for higher d ’s. arxiv˙version printed on October 19, 2018
4. Complementarity relations
As a by-product, a kind of complementarity relations for arbitrary prime d follow from the relations (19), which read P m P j h ˆ n j ( m ) i ≤ D ˆ N E , andtheir analog for the rates ˆ r j ( m ) reads d X m =0 d X j =1 h ˆ r j ( m ) i ≤ . (24)Thus, if e.g., h ˆ r ( m ) i = 1, then P m P j =1 h ˆ r j ( m ) i ≤ . As a matter offact, h ˆ r ( m ) i = 1 implies that the state in question describes all photonsexiting via beam 1 (or exit 1) of the multi-port beam-splitter related withthe complementary situation m , and also no vacuum component in the state.This implies that in such a case for all the other complementary situations m ′ = m , and each j th exit, one has h ˆ r j ( m ′ ) i = 1 /d . Thus the relation (24) isa form of the usual property of mutually unbiased bases, for complementaryinterferometers and arbitrary optical fields.
5. Summary and closing remarks
For 2 × d -mode (where d is a prime) quantum optical fields of undefinedintensities, we formulate series of separability criteria based on observed in-tensities (22), and observed rates (23). As an example, we consider a d -modebright squeezed vacuum state. Such optical states have a EPR-like correla-tions of numbers of photons registered in conjugated modes and thereforethey violate the conditions (22) and (23). With the help of multi-portintegrated optics beam-splitter techniques, observation of such correlationbecomes feasible. As the critical efficiencies for our entanglement conditionsare quite moderate, for | BSV i as our numerical studies show they are below1 / ( d + 1), the conditions can find application in analysis of experiments.The condition (23) is capable to detect entanglement, in situations inwhich the one based on intensities, given by (22), fails. All this concurswith our conjecture that the correlation functions involving rates ratherthan intensities can become a useful tool in quantum optics. We expectthat one can find benefits by using the rates in various cases, e.g., quantumsteering, and etc., see our forthcoming manuscripts.The results can be generalized to all d for which d + 1 mutually unbiasedbases are known to exits. See a forthcoming paper. Acknowledgments.
The works was initiated as part of the BRISQ2 FP7-ICT EU grant no. 308803. MZ acknowledges COPERNICUS DFG/FNPaward-grant (2014-2018). The team of authors was additionally supported rxiv˙version printed on October 19, 2018 by TEAM project of FNP. JR acknowledges the National Research Foun-dation, Prime Ministers Office, Singapore and the Ministry of Education,Singapore under the Research Centres of Excellence programme, and DZacknowledges National Research Foundation and Ministry of Education inSingapore. MW acknowledges UMO-2015/19/B/ST-2/01999.REFERENCES [1] M. ˙Zukowski, W. Laskowski, and M. Wie´sniak, Phys. Rev. A , 042113(2017).[2] M. ˙Zukowski, M. Wie´sniak, and W. Laskowski, Phys. Rev. A , 020102(R)(2016).[3] M. D. Reid and D. F. Walls, Phys. Rev. A , 1260 (1986); D. F. Walls andG. J. Milburn, Quantum Optics (Springer, Berlin, 1994).[4] C. Simon and D. Bouwmeester, Phys. Rev. Lett. , 053601 (2003).[5] K. Rosolek, K. Kostrzewa, A. Dutta, W. Laskowski, M. Wie´sniak, and M.˙Zukowski, Phys. Rev. A , 042119 (2017).[6] C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. , 2044 (1987).[7] J. Ryu, M. Marciniak, M. Wie´sniak, and M ˙Zukowski, J. Opt. , 044002(2018).[8] M. ˙Zukowski, A. Zeilinger, and M. A. Horne, Phys. Rev. A , 2564 (1997).[9] A. Zeilinger, H. J. Bernstein, D. M. Greenberger, H. A. Horne, M. ˙Zukowski,Controlling Entanglement in Quantum Optics, in Quantum Control and Mea-surement, H. Ezawa, Y. Murayama (Editors), (Elsevier Sci. Publ. B. V. , 1993);A. Zeilinger, M. ˙Zukowski, M. A. Horne, H. J. Bernstein, D. M. Greenberger,Einstein-Podolsky-Rosen correlations in higher dimensions, in FundamentalAspects of Quantum Theory, Eds. F. DeMartini, G. Denardo, A. Zeilinger,(Singapore, World Scientific, 1993).[10] N. G. Walker and J. E. Carroll, Opt. Quantum Electron. , 355 (1986); N.G. Walker, J. Mod. Opt. , 15 (1987).[11] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Phys. Rev. Lett. ,58 (1994).[12] C. Schaeff, R. Polster, M. Huber, S. Ramelow, A. Zeilinger, Optica , 523(2015), see also arXiv:1502.06504; for early multiport experiments see K. Mat-tle, M. Michler, H. Weinfurter, A. Zeilinger, and M. ˙Zukowski, Applied PhysicsB-Lasers and Optics , S111-S117 (1995).[13] J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. ˙Zukowski,Rev. Mod. Phys. , 777 (2012).[14] W. K. Wootters and B. D. Fields, Ann. Phys. , 363 (1989)[15] I. D. Ivanovic, J. Phys. A14