Entanglement and Nonlocality in a MicroMacroscopic system
Abstract
In recent years two fundamental aspects of quantum mechanics have attracted a great deal of interest, namely the investigation on the irreducible nonlocal properties of Nature implied by quantum entanglement and the physical realization of the ”Schrœdinger Cat”. The last concept, by applying the nonlocality property to a combination of a microscopic and of a macroscopic systems, enlightens the concept of the quantum state, the dynamics of large systems and ventures into the most intriguing philosophical problem, i.e. the emergence of quantum mechanics in the real life. Rather surprisingly these two aspects, which appeared in the same year 1935 by the separated efforts of Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) and of Erwin Schrœdinger Eins35 ; Schr35 appear not being generally appreciated for their profound interconnections that establish the basic foundations of modern science. Likely, this follows from the extreme difficulty of realizing a system which realizes simultaneously the following three conditions: (a) the quantum superposition of two multiparticle, mutually orthogonal states, call it a ”Macrosystem” (b) the quantum nonseparability of this superposition with a far apart singleparticle state, i.e. the ”Microsystem”, (c) the violation of the Bell’s inequality stating that by no means any hidden variable formulation can simulate the nonlocal correlations affecting the joint MicroMacro system Bell65 .
In the present work these crucial conditions are simultaneously
realized and experimentally tested. A Macro  state consisting of photons in a quantum
superposition and entangled with a far apart single  photon state
is generated. Then, the nonseparability of the overall
MicroMacro system is demonstrated and the corresponding Bell’s
inequalities are found to be violated. Precisely, an entangled
photon pair is created by a nonlinear optical process, then one
photon of the pair is injected into an optical parametric
amplifier (OPA) operating for any input polarization state, i.e. into a phasecovariant cloning machine. Such transformation
establishes a connection between the single photon and the multi
particle fields, as show in Figure 1. We then demonstrate the
nonseparability of the bipartite system by adopting a local
filtering technique within a positive operator valued measurement
(POVM) DeMa05b . The work shows that the amplification
process applied to a microscopic system is a natural approach to
enlighten the quantumtoclassical transition and to investigate
the persistence of quantum phenomena into the ”classical” domain
by measurement procedures applied to quantum systems of increasing
size Zure03 . Furthermore, since the generated MicroMacro
entangled state is directly accessible at the output of the
apparatus, the implementation of significant multi qubit logic
gates for quantum information technology can be achieved by this
method. At last, we demonstrate how our scheme may be upgraded
to an entangled Macro  Macro quantum system, by then establishing
a peculiar nonlocal correlation process between two spacelike
separated macroscopic quantum superpositions.
In recent years quantum entanglement has been demonstrated within a two photon system Kwia95 , within a single trappedion onephoton system Blin04 ; Volz06 , within a single photon and atomic ensemble Mats05 ; deRi06 , within atomic ensembles Juls01 ; Chou05 ; Mats06 ; Lan07 ; Moeh07 and superconducting qubits Berk03 . In addition, the observation of quantum features has been extended to ”cluster” entangled states involving four Pan01 , five Zhao05 and six particles Leib05 ; Lu07 and to a more complex architecture Hald07 .
The innovative character of the present work is enlightened by the diagrams reported in Figure 1. While, according to the 1935 proposal the nonlocal correlations were conceived to connect the dynamics of two ”microscopic” objects, i.e. two spins within the well known EPRBohm scheme here represented by diagram (a)Kwia95 , in the present work the entanglement is established between a ”microscopic” and a ”macroscopic”, i.e. multiparticle quantum object, via cloning amplification: diagram (b). The amplification is achieved by adopting a highgain nonlinear (NL) parametric amplifier acting on a singlephoton input carrier of quantum information, i.e., a qubit state: . This process, referred to as ”quantum injected optical parametric amplification” (QIOPA) DeMa98 ; DeMa05b turned out to be particularly fruitful in the recent past to gain insight into several little explored albeit fundamental, modern aspects of quantum information, as quantum cloning machines DeMa05b ; Pell03 ; DeMa05 , quantum UNOT gate DeMa02 , quantum nosignaling DeAn07 . Here, by exploiting the amplification process, we convert a single photon qubit into a Macroqubit involving a large number of photons. Let us venture in a more detailed account of our endeavor.
I Test of MicroMacro entanglement
An entangled pair of two photons in the singlet state =was produced through a Spontaneous Parametric DownConversion (SPDC) by the NL crystal 1 (C1) pumped by a pulsed UV pump beam: Fig.2. There and stands, respectively, for a single photon with horizontal and vertical polarization while the labels refer to particles associated respectively with the spatial modes and . Precisely, represent the two spacelike separated Hilbert spaces coupled by the entanglement. The photon belonging to , together with a strong ultraviolet (UV) pump laser beam, was fed into an optical parametric amplifier consisting of a NL crystal 2 (C2) pumped by the beam . More details on this setup and on its properties are given in the Appendix and in Naga07 . The crystal 2, cut for collinear operation, emitted over the two modes of linear polarization, respectively horizontal and vertical associated with . The interaction Hamiltonian of the parametric amplification acts on the single spatial mode where is the one photon creation operator associated with the polarization . The main feature of this Hamiltonian is its property of ”phasecovariance” for ”equatorial” qubits , i.e. representing equatorial states of polarization, , in a Poincaré sphere representation having and as the opposite ”poles” Naga07 . The equatorial qubits are expressed in terms of a single phase in the basis . Owing to the corresponding invariance under transformations, we can then rewrite: = where and . The generic polarization state of the injected photon state on mode evolves into the output state according to the OPA unitary transformation Naga07 . The overall output state amplified by the OPA apparatus is expressed, in any polarization equatorial basis , by the MicroMacro entangled state commonly referred to in the literature as the ”Schroedinger Cat State” Schl01
(1) 
where the mutually orthogonal multiparticle ”Macrostates” are:
(2)  
(3) 
with , , being the NL gain DeMa02 . There stands for a Fock state with photons with polarization and photons with over the mode . Most important, any injected singleparticle qubit is transformed by the information preserving QIOPA operation into a corresponding Macroqubit , i.e. a ”Schrœdinger Cat” like, macroscopic quantum superposition DeMa98 . The quantum states of Eq.(23) deserve some comments. The multiparticle states , are orthonormal and exhibit observables bearing macroscopically distinct average values. Precisely, for the polarization mode the average number of photons is for , and for . For the mode these values are interchanged among the two Macrostates. On the other hand, as shown by DeMa98 , by changing the representation basis from to , the same Macrostates, or are found to be quantum superpositions of two orthogonal states , which differ by a single quantum. This unexpected and quite peculiar combination, i.e. a large difference of a measured observable when the states are expressed in one basis and a small HilbertSchmidt distance of the same states when expressed in another basis turned out to be a useful and lucky property since it rendered the coherence patterns of our system very robust toward coupling with environment, e.g. losses. This was verified experimentally. The decoherence of our system was investigated experimentally and theoretically in the laboratory: cfr: DeMa05 ; Cami06 ; Naga07 . The above features are not present in atomic ensemble experiments, in which quantum phenomena usually involve microscopic fluctuations of a macroscopic system and the qubit states are encoded as collective spin excitations. ,
As shown in Figure 2, the single particle field on mode was analyzed in polarization through a BabinetSoleil phaseshifter (PS), i.e. a variable birefringent optical retarder, two waveplates and (ALICE box). The multiphoton QIOPA amplified field associated with the mode was sent, through a singlemode optical fiber (SM), to a measurement apparatus consisting of a set of waveplates and (BOB box). The output signals of the PM’s were analyzed by an ”orthogonality filter” (OF) that will be described shortly in this paper. , a (PBS) and two photomultipliers (PM) and polarizing beam splitter (PBS). It was finally detected by two singlephoton detectors
We now investigate the bipartite entanglement between the modes and . We define the spin Pauli operators for a single photon polarization state, where the label refer to the polarization bases: , , . Here are the right and left handed circular polarizations and . It is found where are the two orthogonal qubits corresponding to the basis, e.g., = , etc. By the QIOPA unitary process the singlephoton operators evolve into the ”Macrospin” operators: Since the operators are built from the unitary evolution of eigenstates of , they satisfy the same commutation rules of the single particle spin: where is the LeviCivita tensor density. The generic state is a Macroqubit in the Hilbert space spanned by , as said. To test whether the overall output state is entangled, one should measure the correlation between the single photon spin operator on the mode and the Macrospin operator on the mode . We then adopt the criteria for two qubit bipartite systems based on the spincorrelation. We define the ”visibility” a parameter which quantifies the correlation between the systems and Precisely where is the probability to detect the systems and in the states and , respectively. The value corresponds to perfect anticorrelation, while expresses the absence of any correlation. The following upper bound criterion for a separable state holds Eise04 :
(4) 
In order to measure the expectation value of a discrimination among the pair of states for the three different polarization bases is required. Consider the Macrostates , expressed by Equations 23, for and . In principle, a perfect discrimination can be achieved by identifying whether the number of photons over the mode with polarization is even or odd, i.e. by measuring an appropriate ”parity operator”. This requires the detection of the macroscopic field by a perfect photonnumber resolving detectors operating with an overall quantum efficiency 1, a device out of reach of the present technology.
It is nevertheless possible to exploit, by a somewhat sophisticated electronic device dubbed ”Orthogonality Filter” (OF), the macroscopic difference existing between the functional characteristics of the probability distributions of the photon numbers associated with the quantum states . The measurement scheme works as follows: Figures 2 and 3. The multiphoton field is detected by two PM’s which provide the electronic signals corresponding to the field intensity on the mode associated with the components , respectively. By (OF) the difference signals are compared with a threshold . When the condition () is satisfied, the detection of the state () is inferred and a standard transistortransistorlogic (TTL) electronic squarepulse () is realized at one of the two output ports of (OF). The PM output signals are discarded for , i.e. in condition of low state discrimination. By increasing the value of the threshold an increasingly better discrimination is obtained together with a decrease of detection efficiency. This ”local distillation” procedure is conceptually justified by the following theorem: since entanglement cannot be created or enhanced by any ”local” manipulation of the quantum state, the nonseparability condition demonstrated for a ”distilled” quantum system, e.g., after application of the OFfiltering procedure, fully applies to the same system in absence of distillation Eise04 . This statement can be applied to the measurement of and for any pair of quantum states . This method is but an application of a Positive Operator Value Measurement procedure (POVM) Pere95 by which a large discrimination between the two states is attained at the cost of a reduced probability of a successful detection. A detailed description of the OF device and of its properties is found in the Appendix and in Ref.Naga07 .
The present experiment was carried out with a gain value leading to a number of output photons , after OF filtering. In this case the probability of photon transmission through the OF filter was: . A NL gain was also achieved with no substantial changes of the apparatus. Indeed, an unlimited number of photons could be generated in principle by the QIOPA technique, the only limitation being due to the fracture of the NL crystal 2 in the focal region of the laser pump. In order to verify the correlations existing between the single photon generated by the NL crystal 1 and the corresponding amplified Macrostate, we have recorded the coincidences between the single photon detector signal (or ) and the TTL signal (or ) both detected in the same basis : Figure 2. This measurement has been repeated by adopting the common basis .
Since the filtering technique can hardly be applied to the basis, because of the lack of a broader covariance of the amplifier, the small quantity could not be precisely measured. The phase between the components and on mode was determined by the BabinetSoleil variable phase shifter () Figure 4 shows the fringe patterns obtained by recording the rate of coincidences of the signals detected by the Alice’s and Bob’s measurement apparata, for different values of . These patterns were obtained by adopting the common analysis basis with a filtering probability , corresponding to a threshold about times higher than the average photomultiplier signals . In this case the average visibility has been found . A similar oscillation pattern has been obtained in the basis leading to: . Since always is , our experimental result = = implies the violation of the separability criteria of Equation (4) and then demonstrates the nonseparability of our MicroMacro system belonging to the spacelike separated Hilbert spaces and . By evaluating the experimental value of the ”concurrence” for our test, connected with the ”entanglement of formation”, it is obtained Benn96 ; Hill97 . This result again confirms the nonseparability of our bipartite system. Further details on the measurement can be found in the Appendix. A method similar to ours to test the nonseparability of a 2atom bipartite system was adopted recently by Moeh07 .
Ii Violation of the MicroMacro Bell’s inequalities
A further investigation on the persistence of quantum effects in large multiparticle systems has been carried out by performing a nonlocality test implying the violation of a Bell’s inequality Reid02 . To carry out such a test higher correlations among the fields realized on the and modes are required. Hence we adopted the QIOPA amplification of a 4photon entangled state. In this second experiment, the NL crystal 1 generated by SPDC two simultaneous entangled photon couples, i.e. a 4photon entangled state . The 4 photons were emitted over the two output modes () in the singlet state of two spin1 subsystems
(5) 
We note that the state keeps the same expression in any polarization basis owing to its rotational invariance. The 2 photons generated over the mode were injected into the collinear QIOPA amplifier: Figure 2. After the amplification process the overall output state can be expressed in any ”equatorial” polarization basis on the Poincaré sphere as
(6) 
where stands for the amplified field generated by injecting the 2photon state : .
Let us analyze the output field when the state was detected over the mode . This was done by the simultaneous measurement, by the singlephoton detectors and inserted on the two output modes of an ordinary beamsplitter (BS), of two photons emerging simultaneously from the same output port of a analyzer (PBS), inserted on the mode (see Figure 2inset). Each event of simultaneous detection by and was identified by the generation of a signal at the output port of an additional electronic coincidence device, not shown in Figure 2. In this condition the corresponding, correlated state was injected into the QIOPA amplifier on mode leading to the generation of . When detected in the polarization basis of the mode , the average photon number was found to depend on the phase as follows: with . Hence the output state analyzed in the polarization basis exhibits a fringe pattern of the field intensity depending on with a gaindependent visibility .When , the mode is injected by a two photon state, the output field is and . When the state is generated and a regime of spontaneous emission is established: . The two Macrostates were singled out, in our conditional experiment, by the simultaneous detection by the Alice’s Measurement apparatus of the correlated states . As said, the realization of the state was identified by the emission of the signal , having previously set the variable phaseshifter PS at the wanted phase value . The Macrostates exhibited observables macroscopically distinct, the difference being larger than the one observed in the case of the QIOPA amplification of a single photon state. Accordingly, the state distinguishability problem was found easier than in the single photon case because of a lower mutual overlap of the 2photon probability distributions. The number of photons emitted by QIOPA in this experiment, after OF filtering, was , with a corresponding transmission probability through the OF filter: .
We address now the problem of the distinguishability between and , which are mutually orthogonal as shown by the quantum analysis reported in the Appendix. Figure 5 reports the 3dimensional representations of the two probability distributions of the number of photons which are associated with the two Macrostates These distributions are drawn as functions of the variables and which are proportional to the size of the electronic signals and generated by the PM’s . In analogy with the entanglement experiment accounted for in the previous section, a large discrimination between could be obtained by adopting the OFfilter. The output stations located in spaces and measured a dichotomic variable with eigenvalues . The correlation fringe patterns reported in Figure 6 were determined by simultaneous detection on the mode of the state for different values of the ”equatorial” phase and on the mode of the OFfiltered . Precisely, this was done by recording via the ”Coincidence Box” the rate of coincidences between the signal realized at Alice’s site and the TTL signal realized at one of the output ports of the OFfilter: Figure 2. The bestfit fringe pattern reported in Figure 6 was obtained by recording, for different values the coincidence rate between and the TTL signal realized at the output port of the OFfilter with a filtering probability The visibility of the fringe pattern was found , i.e. a large enough value that allows a MicroMacro non locality test for the entangled two spin1 system. An identical but complementary pattern, i.e. shifted by a phase was found by collecting the coincidences between and the TTL realized at the port of the OFfilter. This additional pattern is not shown in Fig. 6.
Let us briefly outline the inequality introduced by Clauser, Horne, Shimony, and Holt (CHSH) Bell65 . Each of two partners, (Alice) and (Bob) measures a dichotomic observable among two possible ones, i.e. Alice randomly measures either or while Bob measures or : Figure 7a. For any couple of measured observables , , we define the following correlation function
(7) 
where stands for the number of events in which the observables and have been found equal to the dichotomic outcomes and . Finally we define a parameter which takes into account the correlations for the different observables.
(8) 
Assuming a local realistic theory CHSH found that the relation holds.
To carry out a nonlocality test in the Micromacro” regime, we define the two sets of dichotomic observables for A and B which can be measured over different polarization basis sets. Both partners perform measurements in equatorial basis . A associates to the detection of the 2photons with polarization , i.e. of , the observable and to the detection of the 2photons with polarization , i.e. of , the observable The two possible output values correspond to and : Figure 7b
B associates to the detection of the the observable and to the detection of the the observable The two possible values correspond to and In order to achieve a high discrimination among the OFfilter is adopted.
Let us consider briefly the conceptual issues and the possible loopholes raised by the present nonlocality test. As shown in Figure 7, our system fully reproduces the standard BohmBell scheme for testing nonlocality for an entangled pair of spin1 particles Bouw02 . Two spacelike separated, uncorrelated ”black boxes” and receive the entangled photons from a common EPR source, i.e. involving the NL crystal 1 (C1). The box coincides with the ”ALICE Box” in Figure 2 while the box contains all measurement devices appearing in the”BOB Box” in Figure 2 and the QIOPA amplifier, i.e. the NL crystal 2 (C2). Assuming this interpretation, the following two conditions characterize our present nonlocality test: (1) No artificial selection, sampling or filtering whatsoever is made on the photon couples emitted by the EPR source, before the particles enter the boxes and . (2) Any external operation acting on the particles before measurement, e.g. amplification, loss due to reduced quantum efficiencies, OFfiltering etc., is a ”local operation” because it is exclusively attributable to the separated internal dynamics of the devices contained in the boxes and . On the basis of these premises any sampling or filtering made on the particles by our system must be defined a ”fair sampling” operation Bell65 . There ”fairness” is precisely implied by condition (1), i.e. stating that no externally biased perturbation is allowed to act on the only carriers of nonlocality connecting and : the EPR entangled particles. On these premises, any hidden variable analysis of the overall process will possibly identify new ”loopholes” in our test. Certainly the simple ”detection loophole” does not apply to our complex scheme Kwia94 . In this connection, we red that several nonlocality tests valid for postselected events have been conceived in the past. For instance, the ones adopting photonic GHZ states Bouw99 or 4 photon cluster states Kies05 , in which the tripartite entangled state is generated only when a detector fires on each output mode. We believe that our test is similar to the last condition, the only difference being due to the use made in either cases by the local postselection operation. While in the previous case it was instrumental to generate a quantum state, in our case it is used to properly improve a local measurement procedure, i.e. to make the quantum measurement sharper. A procedure somewhat similar to ours was adopted by Babichev et al. to carry out the nonlocality proof of single photon dualmode optical qubit Babi04 . We believe that a careful quantum analysis will be required to fully clarify the aspects of our POVM method in the context of any hidden variable theory Pear70 ; Huel95 .
Experimentally we obtained the following values by carrying out a measurement with a duration of hours and a statistics per setting equal to about 500 events:

which leads to
Hence a violation by more than standard deviation over the value is obtained. This experimental value is in agreement with an average experimental visibility of which should lead to
Iii Applications to Quantum Information. MacroMacro entanglement
We have experimentally demonstrated the quantum nonseparability of a
MicroMacrosystem. Furthermore, by increasing the size of the injected
”seed” state, namely by adopting at the outset a 4photon entangled state,
we have reported a violation of Bell inequalities. It is possible to
demonstrate that the methods adopted in present experiment can be
reformulated in the context of the ”continuous variable” approach (CV),
which is commonly used to analyze the quantum informational content of
”quadrature operators” for multiparticle systems Brau05 . Indeed
the intersection of the present QIOPA method and of the CV approach appears
to be an insightful and little explored field of research to which the
present work will contribute by eliciting an enlightening theoretical
endeavor.
However, in spite of these appealing
perspectives, we believe that the QIOPA approach is more directly
applicable to the field of Quantum Information and Computation in virtue of
the intrinsic informationpreserving property of the QIOPA dynamics. Indeed, this property implies the direct realization of the
quantum map connecting
any singleparticle qubit to a corresponding Macroqubit, by then allowing
the direct extension to the multi particle regime of most binary logic
methods and algorithms. A simple example may be offered by the
implementation of several universal 2qubit logic gates such as the CNOT or
the phasegate. Consider a 2qubit phase gate in which the controltarget
interaction is provided by a Kerrtype optical nonlinearity. It is well
known that the strength of this nonlinearity is far too small to provide a
sizable interaction between the”control” and the ”target”
singleparticle qubits, even in the special case of the atomic quasi
resonant electromagnetic induced transparency configuration (EIT) Hau99 ; Flei05 . However, by replacing these qubits by the corresponding
Macroqubits associated with photons, the NL interaction strength can be
enhanced by a large factor since the 3dorder NL polarization
scales as . By assuming the values of the NL gain realized
experimentally in the present work, the value of this factor can be as large
as for and for . Such large enhancement of the NL interaction strength may
represent the key solution towards the technical implementation of
the most critical optical components of a quantum computer.
Of course, all these applications are made possible by an
important property of the QIOPA scheme: there the multiparticle
entangled Macrostates are directly accessible at the
output of the QIOPA apparatus. In other words, in our systems the
many photons involved in the quantum superpositions are not sealed
or trapped in hardly accessible electromagnetic cavities, nor
suffer from decoherence processes due to handling, storing or
measurement procedures.
At last, the MicroMacro experimental method demonstrated in this work can be upgraded in order to achieve an entangled MacroMacro system showing again marked nonlocality features. Such scheme could exploit an ”entanglement swapping” protocol Pan98 as shown in Fig.8(a). There the final entangled state is achieved through a standard intermediate Bell measurement carried out on the Microstates. A similar process has been suggested in different contexts, for instance to entangle micromechanical oscillators Pira06 . As an alternative approach, the single photon states on mode and could be amplified by two independent QIOPA’s :Fig.8(b). Another appealing perspective is the lightmatter entanglement, consisting of the coupling, either linear or nonlinear, of the multiphoton generated QIOPA fields with the mechanical motion of atomic systems. As an example, the coupling of a Bose Einstein Condensed (BEC) assembly of Rb atoms with the entangled multiparticle field generated by our apparatus is presently being investigated in our Laboratories Cata07 . All this can open interesting scenarios in modern science and technology. On a very fundamental side, the observation of quantum phenomena with an increasing number of particles can shed light on the elusive boundary between the ”classical” and the ”quantum” worlds, and provide new paths to investigate the notion of quantum wavefunction and its intriguing ”collapse” Guer07 .
Acknowledgements.
We acknowledge technical support from Sandro Giacomini, and Giorgio Milani and experimental collaboration with Eleonora Nagali,Tiziano De Angelis and Nicolo’ Spagnolo. This work was supported by the PRIN 2005 of MIUR and project INNESCO 2006 of CNISM.Appendix A Experimental setup
We provide here more structural details on the apparatus shown in Figure 2 of the main text (MT) . The excitation source was a Ti:Sa Coherent MIRA modelocked laser amplified by a Ti:Sa regenerative REGA device operating with pulse duration , repetition rate and average output power: . The pumping laser of the amplifier was a laser Coherent Verdi operating at a continuouswave power level . The output beam, frequencydoubled by secondharmonic generation, provided the OPA excitation field beam at the UV wavelength (wl) with power: . The UV beam, splitted in two beams through a waveplate (wp) and a polarizing beam splitter (PBS), excited two BBO (barium borate) NL crystals cut for type II phasematching. Crystal 1, (C1) excited by the beam , was the SPDC source of entangled photon couples with wl , emitted over the two output modes () in the singlet state =. The power of beam was set at a low enough level to generate with negligible probability more than two simultaneous pairs of photons. The photon associated with mode , hereafter referred to as trigger mode, was coupled into a single mode fiber and excited through a PBS one of the single photon counting modules (SPCM) , while the single photon state generated over the mode was injected, together with a strong UV pump beam (mode ), into the NL crystal 2 (C2) and stimulated the simultaneous emission of large number of photon pairs. The measurement was carried out on the mode by adopting a set of two waveplates, + , a SoleilBabinet variable phaseshifter () and a polarizing beamsplitter . By a delay () the time superposition in the OPA of the excitation UV pulse and of the injection photon wavepacket was provided. The injected single photon and the UV pump beam were superimposed by means of a dichroic mirror () with high reflectivity at and high transmittivity at . The output state of the crystal with wl was spatially separated by the fundamental UV beam again by a dichroic mirror () and spectrally filtered by an interference filter () with bandwidth , transmittivity () and coupled to a single mode fiber (), polarization analyzed and then detected by 2 equal photomultipliers (PM) and . The SPCM detectors were singlephoton modules Perkin Elmer type AQR14FC. The PM’s were Burle A02 with a GaAs photocathode having a detector quantum efficiency .
In a first experiment with no quantum injection, we measured the gain value of the optical parametric process and the overall detection efficiency of the detection apparatus. The average signal amplitude of was measured for different values of the UV power. The gain value of the process was obtained by fitting the experimental data Eise04 ; Cami06 with an exponential function, leading to , corresponding to an overall mean photon number per mode . A gain , was also realized with no substantial changes in the apparatus. The exponential growth demonstrates the multiple generation of photon pairs by a selfstimulation process within the NL crystal. The overall efficiency on the mode, was determined by fiber coupling (), transmittivity and detector .
a.1 Orthogonality Filter
Here we give more details of the ”Orthogonality filter” (OF), a device adopted to discriminate among the quantum states . This operation is realized by exploiting the different functional characteristics existing between the corresponding photon number distribution patterns and corresponding to the Macro states expressed respectively by Equations(23) of MT for and : and . Precisely, since the singlephoton resolving detection is beyond the reach of present technology and , we may consider that the two Fock components of the Macro states, and belonging respectively to the expressions of and , generate the same signals at the output of any couple of PM detectors: . However a discrimination may still be carried out efficiently between the above orthogonal Macro states by exploiting the different ”shape” of the probability distributions and for large values of . For this purpose, we introduce an appropriate threshold for signal discrimination. By analyzing the two probability distributions we infer that when the signals can be attributed to the state with a fidelity that increases with the value of . In this case the measured eigenvalue of is found = +1. Likewise, when the signals is inferred to correspond to the state with eigenvalue = 1. The events which satisfy the inequality are discarded since there the two distributions approximately overlap impairing a reliable state discrimination. This technique can be adopted even with a low value of detection efficiency () since it is found that the functional characteristics of the distributions which are pertaining to the discrimination of the states are preserved under the signal propagation over a lossy channel. This important point has been carefully investigated theoretically and experimentally in our laboratory Naga07 .
The measurement scheme just described has been physically implemented by the OF shown in Figure 9, an electronic device by which the pulse heights of the couple of input signals provided by two PM’s are summed with opposite signs by a balanced linear amplifier LA) with ”gain” (chip National LM733). Each of the two signals realized at the two symmetric outputs of (LA) feeds an independent electronic discriminator (AD9696) set at a common threshold level . Owing to previous considerations the two discriators never fire simultaneously and each of them provides, when activated, a standard transistortransistorlogic (TTL) square signal at its output port. As said, when the condition , i.e. is satisfied, a TTL signal is generated and then the realization of the state is inferred. Likewise, when a TTL signal is generated and the realization of the state is inferred. The events that are discarded for: correspond to the ”inconclusive” outcomes of any POVM Pere95 .
The OF device has been tested and characterized in condition of spontaneous emission, i.e., in absence of any quantum injection into C2. In this condition the output TTL signals () were measured by sending only the signal () as input and by varying the threshold . In this regime the number of photons generated per mode should exhibit a thermal probability distribution: with average photon number per mode. Hence the probability to detect a signal above the threshold is: We have experimentally checked the dependence on the threshold of the number of counts, which is expressed by , being the repetition rate of the source. The experimental data shown by Figure 10 represent a fair support of the expected exponential behavior.
a.2 Entanglement Test
To carry out the measurement of the degree of the entanglement ALICE and BOB adopt a common polarization basis . In the basis the measurements were carried out by setting the and waveplates, shown in the ALICE Box in Figure 2, with the optical axes making angles respect to the vertical direction equal to and , respectively. An identical setting was adopted for the and wp’s in the BOB Box in Figure 2. Likewise, in the basis the measurements were carried out by setting the two sets of
The experimental ”Visibility” were measured by recording the coincidence patterns recorded for both spaces and in the basis Eise04 . According to the standard definition: where are respectively the maximum and the minimum values of the detected signals corresponding to the value of phase . The visibility values reported in the Main Text were recorded by averaging over a time hours the results of events for each experimental datum shown in Figure 4, for each basis.
Its worth noting that the above procedure is ”phase covariant”, i.e. the overall quantum efficiency of the electronic filter OF is independent of the phase . In other words,and most import in the present context, every state produced by QIOPA, has the same overall probability to be filtered by the electronic filter and to produce a TTL signal at one output port, either or . This feature was experimentally verified by measuring the OFfiltering probability for different phase values of the injected qubit. In particular, assumes the same value when the measurement is applied to the state and states.
a.3 CHSH tests
Let us now account for the nonlocality test by considering the measurement carried out within the two Boxes A and B shown in Figure 7. (A) Measurement at the A Box. The angles respect to the vertical made by the optical axes of the waveplates and , respectively. The phase shifter (PS) was set in two different positions to obtain and . (B) Measurement at the B Box. The basis corresponding to was obtained by setting the angles respect to the vertical made by the optical axes of the wp’s and , respectively. The basis corresponding to was obtained by setting the angles respect to the vertical made by the optical axes of the same wp’s and , respectively. As shown in Figure 2, in the B Box the above phase changes were made by acting on the Macrostates, i.e., on the output multiparticle beam emerging from the QIOPA. at the values shown in the BOB Box in Figure 2, at the values shown in the ALICE Box in Figure 2 were set at
Appendix B 2photon wavefunction
The Macrostate generated by QIOPA on mode when a 2photon state is injected, is expressed by
(9) 
While, the following Macrostate is generated if a state is injected:
(10) 
The photon number probability distribution for the Macrostate is reported in Figure 11.
References
 (1) Einstein, A., Podolsky,B., & Rosen, N., Can QuantumMechanical Description of Physical Reality Be Considered Complete? , Phys. Rev. 47, 777780 (1935).
 (2) Schroedinger, E., Naturwissenschaften 23, 807 812, 823 828, 844 849 (1935). English translation 1983 in Quantum Theory and Measurement, edited by J. A. Wheeler and W. H. Zurek (Princeton University, Princeton, NJ), p. 152.
 (3) Bell, J.S., Physics (Long Island, N.Y.) 1, 195 (1965); Clauser, J.F., Horne M.A., Shimony, A., & Holt, R.A., Proposed Experiment to Test Local HiddenVariable Theories, Phys. Rev. Lett. 23, 880 (1969); Aspect, A., Grangier, P., and Roger, G., Experimental Tests of Realistic Local Theories via Bell’s Theorem, Phys. Rev. Lett. 47, 460 (1981).
 (4) De Martini, F., et al., Nonlinear Parametric Processes in Quantum Information, Prog. in Quantum Electronics 29, 165256 (2005).
 (5) Zurek, W.H., Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715775 (2003); Schlosshauer, M., Decoherence, the measurement problem, and interpretations of quantum mechanics, Rev. Mod. Phys. 76, 12671305 (2005).
 (6) Kwiat, P.G., et al., New highintensity source of polarizationentangled photon pairs, Phys. Rev. Lett. 75, 4337 4341 (1995).
 (7) Blinov,B.B., Moehring, D.L., Duan, L.M. , & Monroe, C., Observation of entanglement between a single trapped atom and a single photon, Nature (London) 428, 153157 (2004).
 (8) Volz, J., et al., Observation of Entanglement of a Single Photon with a Trapped Atom, Phys. Rev. Lett. 96, 030404 (2006).
 (9) Matsukevich, D. N. , et al., Entanglement of a Photon and a Collective Atomic Excitation, Phys. Rev. Lett. 95, 040405 (2005).
 (10) de Riedmatten, H., et al., Direct Measurement of Decoherence for Entanglement between a Photon and Stored Atomic Excitation, Phys. Rev. Lett. 97, 113603 (2006)
 (11) Matsukevich, D.N., et al., Entanglement of Remote Atomic Qubits , Phys. Rev. Lett. 96, 030405 (2006).
 (12) Lan, S.Y. et al., DualSpecies Matter Qubit Entangled with Light, Phys. Rev. Lett. 98, 123602 (2007)
 (13) Julsgaard, B., Kozhekin, A., & Polzik, E.S., Experimental longlived entanglement of two macroscopic objects, Nature 413 400403 (2001).
 (14) Moehring, et al., Entanglement of singleatom quantum bits at a distance, Nature(London)449, 6871 (2007).
 (15) Chou, C. W., et al., Measurementinduced entanglement for excitation stored in remote atomic ensembles, Nature 438, 828832 (2005).
 (16) Berkley, A.J., et al., Entangled macroscopic quantum states in two superconducting qubits (phase qubit), www.sciencexpress.org/15 May 2003/Page 1/10.1126/science.1084528.
 (17) Pan, J.W., et al. Experimental demonstration of fourphoton entanglement and highfidelity teleportation, Phys. Rev. Lett. 86, 4435 4438 (2001).
 (18) Zhao, Z., et al., Experimental demonstration of fivephoton entanglement and opendestination teleportation, Nature (London) 430, 54 58 (2004).
 (19) Leibfried, D., et al., Creation of a sixatom ’Schr dinger cat’ state, Nature (London) 438, 639 642 (2005).
 (20) Lu, C.Y., et al., Experimental entanglement of six photons in graph states, Nature Physics 3, 9195 (2007).
 (21) Halder, M., et al., Entangling independent photons by time measurement, Nature Physics 3, 692695 (2007).
 (22) De Martini, F., Amplification of Quantum Entanglement, Phys. Rev. Lett. 81, 28422845 (1998); De Martini, F., Quantum Superposition of Parametrically Amplified Multiphoton Pure States, Phys. Lett. A 250,1519 (1998).
 (23) Pelliccia, D., Schettini, V., Sciarrino, F., Sias, C. & De Martini, F., Contextual realization of the universal quantum cloning machine and of the universalnot gate by quantuminjected optical parametric amplification, Phys. Rev. A 68, 042306 (2003); De Martini, F., Pelliccia, D., & Sciarrino, F., Contextual, Optimal, and Universal Realization of the Quantum Cloning Machine and of the NOT Gate. Phys. Rev. Lett. 92, 067901 (2004).
 (24) De Martini, D., Sciarrino, F., & Secondi, V., Realization of an Optimally Distinguishable Multiphoton Quantum Superposition, Phys. Rev. Lett. 95, 240401 (2005).
 (25) De Martini, F., Buzek, V., Sciarrino, F., & Sias, C., Nature (London) 419, 815 (2002).
 (26) De Angelis, T., Nagali, E., Sciarrino F., & De Martini, F., Experimental test of the nosignaling theorem, Phys. Rev. Lett. 99, 193601 (2007).
 (27) Nagali, E., De Angelis, T., Sciarrino, F., & De Martini, F., Experimental realization of macroscopic coherence by phasecovariant cloning of a single photon, Phys. Rev. A 76, 042126 (2007).
 (28) Schleich, W.P., Quantum Optics in Phase Space (Wiley, New York, 2001), Chaps. 11 and 16.
 (29) Caminati, M., De Martini, F., Perris, R., Sciarrino, F., & Secondi, V., Entanglement, EPR correlations and mesoscopic quantum superposition by the highgain quantum injected parametric amplification, Phys. Rev. A 74, 062304 (2006).
 (30) Eisenberg, H.S.,Khoury, G., Durkin, G.A., Simon, C., & Bouwmeester, D., Quantum Entanglement of a Large Number of Photons, Phys. Rev. Lett. 93, 0193901 (2004); Durkin, G.A., Ph.D. Thesis, Light & Spin Entanglement (2004).
 (31) A. Peres, Quantum Theory: Methods and Concepts (Kluwer Academic Publishers, Dordrecht,1995).
 (32) Bennett, C.H., Di Vincenzo, D.P:, Smolin, J.A., & Wootters, W.K:, Mixedstate entanglement and quantum error correction, Phys. Rev. A 54, 38243851 (1996).
 (33) Hill, S. & Wootters, W.K., Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78, 50225025 (1997).
 (34) Reid, M.D., Munro, W.J., & De Martini, F., Violation of multiparticle Bell inequalities for low and highflux parametric amplification using both vacuum and entangled input states, Phys. Rev. A 66, 033801 (2002).
 (35) Howell, J.C., LamasLinares, A., & Bouwmeester, D., Experimental Violation of a Spin1 Bell Inequality Using Maximally Entangled FourPhoton States, Phys. Rev. Lett. 88, 030401 (2002)
 (36) Kwiat, P. et al., Proposal for a loopholefree Bell inequality experiment, Phys. Rev. A 49, 3209 (1994)
 (37) Bouwmeester,D., et al., Observation of ThreePhoton GreenbergerHorneZeilinger Entanglement, Phys. Rev. Lett. 82, 1345 (1999)
 (38) Kiesel, CN., et al., Experimental Analysis of a FourQubit Photon Cluster State, Phys. Rev. Lett. 95, 210502 (2005)
 (39) Babichev, S.A., Appel, J., & Lvovsky, A.I., Homodyne Tomography and Characterization and Nonlocality of a DualMode Optical Qubit, Phys. Rev. Lett. 92, 0193601 (2004).
 (40) Pearle, P., HiddenVariable Example Based upon Data Rejection, Phys. Rev. D 2, 1418  1425 (1970)
 (41) Huelga, S.F. , Ferrero, M. & Santos, E., Loopholefree test of the Bell inequality, Phys. Rev. A 51, 5008 (1995)
 (42) Braunstein, S.L., & and Loock, P.v., Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005)
 (43) Hau, L.V., Harris, S E., Dutton, Z, & Behroozi, C.H., Light speed reduction to 17 metres per second in an ultracold atomic gas, Nature (London) 397, 594 (1999).
 (44) Fleischhauer, M., Imamoglu, A., &, Marangos J.P., Electromagnetically induced transparency: Optics in coherent media, Rev. Mod. Phys. 77, 633673 (2005).
 (45) Pan, J., et al., Experimental Entanglement Swapping: Entangling Photons That Never Interacted, Phys. Rev. Lett. 80, 38913894 (1998).
 (46) Pirandola, S., et al., Macroscopic Entanglement by Entanglement Swapping, Phys. Rev. Lett. 97, 150403 (2006).
 (47) Cataliotti, F., & De Martini, F., Macroscopic quantum entanglement in light reflection from BoseEinstein Condensates, (in preparation).
 (48) Guerlin, C., et al., Progressive fieldstate collapse and quantum nondemolition photon counting, Nature (London) 448, 889893 (2007).