Abstract:
The invention provides systems and methods enabling high fidelity quantum communication over long communication channels even in the presence of significant loss in the channels. The invention involves laser manipulation of quantum correlated atomic ensembles using linear optic components ( 110, 120 ), optical sources of low intensity pulses ( 10 ), interferers in the form of beam splitters ( 150 ), and single-photon detectors ( 180, 190 ) requiring only moderate efficiencies. The invention provides fault-tolerant entanglement generation and connection using a sequence of steps that each provide built-in entanglement purification and that are each resilient to realistic noise levels. The invention relies upon collective rather single particle excitations in atomic ensembles and results in communication efficiency scaling polynomially with the total length of the communication channel.

Description:
CLAIM OF PRIORITY 
       [0001]    This application is a divisional application of U.S. patent application Ser. No. 10/500,203, and claims the benefit of priority under 35 U.S.C. § 365 of International Patent Application Serial No. PCT/US02/15135, filed on May 20, 2002, designating the United States of America. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention relates to long-distance quantum communication systems and methods for entanglement-based quantum communication protocols, such as quantum teleportation and quantum cryptography. 
       BACKGROUND ART 
       [0003]    All current proposals for implementing quantum communication are based on photonic channels. The degree of entanglement generated between two distant quantum systems coupled by photonic channels decreases exponentially with the length of the connecting channel due to optical absorption and noise in the channel. This decrease represents a loss of “entanglement fidelity” in the communication system, which can be restored via “entanglement purification.” Entanglement purification (referred to as simply “purification” in the literature) means regaining a high degree of entanglement. See, e.g., Bennett, C. H. et al., “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722-725 (1991). The exponential decay of entanglement as a function of channel length requires an exponentially increasing number of partially entangled states to obtain one highly entangled state. 
         [0004]    The concept of quantum repeaters has been proposed to overcome the difficulty associated with the exponential decay of entanglement. See, e.g., Briegel, H.-J., Duer, W., Cirac, J. I. &amp; Zoller, P., “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932-5935 (1991). A quantum repeater using a cascaded entanglement purification protocol for a quantum communication system is proposed in the article by Knill, E., Laamme, R. &amp; Zurek, W. H., entitled “Resilient quantum computation,” Science 279, 342-345 (1998), and in the article by Preskill, J., entitled “Reliable quantum computers,” Proc. R. Soc. Lond. A 454, 385-410 (1998). The proposed quantum repeater includes many short segments, with the length of each segment being comparable to a channel attenuation length. The proposed quantum repeater is used by first generating entanglement and purifying the entanglement for each segment. The purified entanglement is then extended by connecting two adjacent segments through entanglement swapping, and so on. Each entanglement swapping, however, decreases the overall entanglement, which therefore requires a large number of iterations for sufficient purification. 
       SUMMARY OF THE INVENTION 
       [0005]    Implementing the long-distance quantum communication systems and methods of the present invention includes generating entanglement between distant quantum bits (qubits), storing them for sufficiently long time, and performing local collective operations on several of these qubits. Having quantum memory is essential for the system, since all purification protocols are probabilistic. When entanglement purification is performed for each segment of the channel, the quantum memory is used to keep the segment state if the purification succeeds, and to repeat the purification for only those segments where the previous attempt failed. Having quantum memory is important for the polynomial scaling of the communication efficiency because having no available memory means that the purifications for all the segments must succeed at the same time. Unfortunately, the probability of this happening decreases exponentially with the channel length. The requirement of quantum memory implies that local qubits are stored in atomic internal states instead of the photonic states, since it is difficult (to the point of impracticality) to store photons. If atoms are used as the local information carriers, it is difficult to implement quantum repeaters because of the technically challenging proposition of having to use strong coupling between atoms and photons with high-finesse cavities for atomic entanglement generation, purification, and swapping. 
         [0006]    An example embodiment of an apparatus of the present invention includes a light source, two cells each containing an atomic ensemble with a Lambda level configuration (examples are alkali atoms such as Sodium, Rubidium and Cesium), two filters, two channels, a beam splitter and two single-photon photodetectors. 
         [0007]    Physical systems that may be used for atomic ensembles of the invention include ensembles of atoms in similar potential energy traps, such as atoms in an ion trap, optical lattice position, or crystal lattice position, and which have collective excitations so that they may couple to photons, such as phonon excitations in solids. Preferably, these atomic systems have energy levels similar to the Lambda configuration such that their collective excitations can be coupled to flying qubits in the same matter as for alkali metal atoms. Additionally, ensembles of localized excitation states such as electron states in superlattice structures including quantum dot ensembles, and Bose-Einstein condensates, may be useful ensembles for implementing the invention, and are also included in the definition of “atomic ensemble” as the phrase is used herein. 
         [0008]    Another embodiment of the invention comprises a source, four atomic ensembles, two beam splitters, four single-photon detectors (SPDs), and two interferers. Additional elements comprise a synchronizer and single count photo detectors control system. Four atomic ensembles form two pairs of entangled atomic ensembles. Another method uses a preferred embodiment of the apparatus modified to be efficiently used in the quantum communication protocols, such as quantum cryptography, and for Bell inequality detection. 
         [0009]    Another embodiment of the invention comprises a source, six atomic ensembles, two beam splitters, and four single count photo detectors. Additional elements comprise a synchronizer and single count photo detectors control system. Six atomic ensembles form two pairs of entangled atomic ensembles and two pairs of atomic ensembles with collective atomic excitations. Another method uses a preferred embodiment of the apparatus modified for realization of the quantum teleportation of the atomic polarization state. 
         [0010]    The present invention relies upon collective rather than single particle excitations in atomic ensembles. It provides entanglement connection using simple linear optical operations, and it is inherently robust against the realistic imperfections. Here, “entanglement connection” means that two objects are connected through entangled systems. Consider the system of  FIG. 2  (described in more detail below) having two separate entangled bipartite systems. That is to say, the two pairs (L, I 1 ) and (R, I 2 ) where the subsystem L is entangled with subsystem I 1  and subsystem R is entangled with subsystem I 2 . There is a way to create an entangled state between the two subsystems L and R. Such a process is generally called “entanglement connection.” A simple example of entanglement connection is called “entanglement swapping.” Entanglement swapping is accomplished by performing a so-called Bell-measurement between the two other systems I 1  and I 2 . A successful Bell-measurement connects the two subsystems L and R into an entangled state. In linear optics, it is often the case that a Bell-measurement can be successfully performed only probabilistically. A method to implement such a probabilistic Bell-measurement with linear optics and single-photon detectors of finite efficiency is shown in  FIG. 2  and discussed below. 
         [0011]    Quantum communication protocols of the present invention include quantum teleportation, cryptography, and Bell inequality detection. “Perfect entanglement” means that interacting physical systems are fully correlated. Increasing the communication length in embodiments of the present invention only increases overhead in the communication time polynomially. 
         [0012]    An example embodiment of the present invention is implemented with only linear optical components such as beamsplitters, and imperfect single-photon detectors. Yet, it allows for intrinsic fault-tolerance. The invention is probabilistic. Many of the sources of errors do not affect the resulting state, even though they affect the probability of success. Some of the errors only affect the coefficients of the successful state. Such modified coefficients can be easily compensated for or taken into account by system design without affecting polynomial scaling. Thus, one advantage of the present invention is that it allows for of long-distance quantum communication or entanglement with resources that only scale polynomially with the communication distance. 
         [0013]    The present invention also provides stable local information carriers, i.e., stable quantum memory, due to the stability of collective excitations, resulting in relatively long decoherence times. In addition, relatively large cross sections exist between certain photons and certain collective excitations that provide a substantial probability of collective excitation generation per incident photon. These probabilities can be as great as approaching 1, preferably at least 0.5, and less preferably 0.25 or 0.1. 
         [0014]    The inventors recognized that the light source may excite a number of other optical modes besides the desirable collective mode discussed below. The effects of the other optical modes can be computed by standard perturbation theory in quantum mechanics. Therefore, it is simplest to understand the present invention by focusing on the desirable collective mode only. For this reason, a one-dimensional light propagation model and only the desirable collective mode are discussed herein below, and all other optical modes are ignored. However, the invention applies to more general configurations and is operable in the presence of other optical modes. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]    Embodiments of the invention are described below in conjunction with the following Figures, in which like or corresponding elements are given the same reference number. 
           [0016]      FIG. 1  is a schematic diagram of an example embodiment of an apparatus according to the present invention that is used to create entanglement between two atomic ensembles; 
           [0017]      FIG. 2  is a schematic diagram of an example embodiment of an apparatus according to the present invention that employs two pairs of atomic ensembles for performing entanglement swapping; 
           [0018]      FIG. 3  is a schematic diagram of another example embodiment of an apparatus according to the present invention that employs two pairs of atomic ensembles; and 
           [0019]      FIG. 4  is a schematic diagram of an apparatus according to the present invention that is adapted for performing probabilistic quantum teleportation of an atomic polarized state. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0020]    In some formulas below, the “proportional to” sign           is replaced by a box symbol, such as “quadrature,” which also means “proportional to.” 
         [0021]      FIG. 1  is a schematic diagram of an example embodiment of an apparatus  1  for performing quantum entanglement according to the present invention. Apparatus  1  includes a light source  10 . Light source  10  may include a pulser or amplitude modulator to convert continuous light to pulsed light. Source  10  may also include electronics for synchronizing laser pulses. Light source  10  emits either continuous light or pulses of light. In an example embodiment, light source  10  is laser. 
         [0022]    Apparatus  1  also includes a light pulse synchronizer  20  optically coupled to light source  10  and from which emanates light beams (pulses) P 1  and P 2  along optical paths  30  and  40 . Synchronizer  20  is adapted to receive light P 0  from light source  10  and generate synchronized output beams (e.g., synchronized output pulses P 1  and P 2 ) along optical paths  30  and  40 . Apparatus  1  also includes two atomic ensemble cells  50  and  60  containing atomic ensembles L, R, respectively. Cells  50  and  60  are arranged along optical paths  30  and  40 , respectively, so as to receive synchronized light pulses P 1  and P 2  from the synchronizer. In an example embodiment, atomic ensembles L and R emit Stokes light pulses SL and SR in response to being irradiated by the synchronized laser pulses. In an example embodiment, each cell contains an alkali gas sample containing about 10 12  atoms of an alkali element. The cells may be, for example, glass cells with an interior antireflection coating. To facilitate enhanced coupling to optical pulses P 1  and P 2 , the atomic ensembles are preferably optically thick along the direction of light propagation. This is achieved, for example, by providing a pencil-shaped cell. As mentioned above, the term “atomic ensemble” is not limited to atoms per se, and the phrase as used herein is to be construed broadly to include other forms of matter capable of supporting a collective excitation and emitting Stokes light pulses, such as molecules, ensembles of localized excitation states such as electron states in superlattice structures including quantum dot ensembles, Bose-Einstein condensates, and the like. 
         [0023]    Apparatus  1  also includes filters  70  and  80  arranged downstream of and optically coupled to cells  50  and  60 , respectively. Filters  70  and  80  are structured to select (i.e., transmit) a specified polarization and frequency corresponding to Stokes light pulses SL and SR emanating from cells  50  and  60 , respectively, and to reject (i.e., reflect) all other light along optical paths  90  and  100 , respectively. 
         [0024]    Further included in apparatus  1  are transmission channels  110  and  120  arranged downstream of and optically coupled to filters  70  and  80  via optical paths  130  and  140 , respectively. In an example embodiment, transmission channels  110  and  120  are structures, such as optical fibers, through which propagate Stokes pulses SL and SR transmitted by filters  70  and  80 . 
         [0025]    Apparatus  1  further includes a beamsplitter  150  arranged at the intersection of transmission channels  110  and  120 . Beam splitter  150  is preferably a mirror that provides 50% incident intensity reflection and transmission. In alternative embodiments, beam splitter  150  is or includes a 50:50 fiber coupler. Light exiting beamsplitter  150  can travel along two optical paths  160  and  170 . Generally, beam splitter  150  is adapted to interfere the Stokes light pulses SL and SR from channels  110  and  120  with a 50:50 ratio. That is, the beam splitter serves as an “interferer” because it provides for interference of the two Stokes light pulses. 
         [0026]    Apparatus  1  also includes single-photon detectors (SPDs)  180  and  190 , which in an example embodiment are conventional SPDs capable of receiving and detecting light propagated along respective optical paths  160  and  170  from beam splitter  150 . Additional elements of apparatus  1  include a photodetector control system  200  operably coupled to SPDs  180  and  190 . Photodetector control system  200  includes, for example, electronics for controlling the operation of the SPDs. Apparatus  1  also includes electronics (not shown), such as a controller, for controlling the operation of light source  10  and/or synchronizer  20  for synchronizing laser pulses P 1  and P 2  delivered to cells  50  and  60 . 
         [0027]    In operation, light source  10  generates synchronized laser pulses P 0  and synchronizer  20  creates therefrom synchronized light pulses P 1  and P 2  that propagate along optical paths  30  and  40  and into atomic ensembles L and R contained in cells  50  and  60 . Stokes light pulses SL an SR are emitted from the atomic ensembles L and R in cells  50  and  60  in response to being irradiated by synchronized laser pulses P 1  and P 2 . For the generation of entanglement using Stokes light pulses generated in the cells, light source  10  must operate at the wavelength that matches the transition frequency of the atomic ensembles in each cell. Light sources  10  capable of matching the transition frequency of the atomic ensembles contained in cells  50  and  60  include certain coherent and partially coherent sources, such as certain lasers, including solid state, semiconductor, dye, free electron, and gas lasers. 
         [0028]    As mentioned above, in an example embodiment, light source  10  is laser. More particularly, in an example embodiment, light source  10  is a diode laser emitting pulses P 0  at a wavelength of 852 nm with a pulse duration of 0.1 microsecond and a repetition rate of at least 1 MHz. This wavelength matches the absorption energy of Cesium atom ensembles, at approximately room temperature, of 852 nm. “Approximately room temperature” in this context means temperatures between about 20 and about 30 Celsius. Preferably, each cell  50  and  60  contains approximately N identical alkali atoms. 
         [0029]    Preferably, the atomic ensembles Land R in cells  50  and  60 , respectively, comprise atoms having a Lambda-level structure. For instance, an example atomic ensemble includes Rubidium atoms in vapor form at around 70 to 90 degree Celsius, with atomic densities of about 10 11  to 10 12  per cubic centimeter, used in combination with a light source  10  in the form of a diode laser with a frequency close to the atomic level transition frequency for Rubidium. In one example embodiment, the alkali atom ensembles are at low temperature. For instance, in one embodiment, approximately 11 million sodium atoms are cooled to about 0.9 micro-Kelvin, just above the critical temperature for Bose-Einstein condensation. The cell defines the atomic ensemble to have a length of about 350 microns in the longitudinal direction as traveled by the light and about 55 microns in the transverse directions, and a peak atomic density of about 11 atoms per cubic micron. Generally, atomic densities in the range of 1 to 100 atoms per cubic centimeter can be used. 
         [0030]    A desirable longitudinal dimension for cells  50  and  60  on the order of 3 cm. However, if a larger dimension is employed, the overall decoupling strength between the source signal and the atom ensemble is increased. Depending on the identity and density of the medium making up the atomic ensembles, in some applications, a larger cell dimension may be desirable. 
         [0031]    The relevant level structure of the atomic ensembles contains |g&gt; the ground state, |s&gt;, the metastable state for storing a qubit, and |e&gt;, the excited state. The transition |g&gt;→|e&gt; is coupled by the classical laser with the Rabi frequency, and the forward scattering Stokes light comes from the transition |e&gt;→|s&gt;. An off-resonant coupling with a large detuning Δ is assumed. The two atomic ensembles L and R are pencil shaped and illuminated by the synchronized classical laser pulses P 1  and P 2 . The forward-scattering Stokes pulses  130  and  140  are transmitted through polarization and frequency selective filters  70  and  80 , propagate through channels  110  and  120  and interfere at 50:50 beam splitter  150 . Finally, single photons propagating through beam splitter  150  are detected in SPDs  180  and  190 . If there is a click in either SPD  180  or  190 , the process is finished and the entanglement is generated between the atomic ensembles L and R. Otherwise, a repumping pulse is first applied to the transition 
         [0000]    |s&gt;→|e&gt; on the ensembles L and R to set the state of the ensembles back to the ground state 
         [0000]      |0           a   L           |0           a   R , 
         [0000]    then the same classical laser pulses as in the first round are applied to the transition |g&gt;→|e&gt; and the forward-scattering Stokes pulses are generated. This process is repeated until finally there is a click in either SPD  180  or  190 . 
         [0032]    A pair of metastable lower states |g&gt; and |s&gt; corresponds e.g., to hyperfine or Zeeman sublevels of electronic ground state of alkali atoms. All the atoms are initially prepared in the ground state |g&gt;. A sample is illuminated by a short, off-resonant laser pulse that induces Raman transitions into the states |s&gt;. The forward-scattered Stokes light that is co-propagating with the laser is uniquely correlated with the excitation of the symmetric collective atomic mode S given by 
         [0000]        S= (1/√{square root over ( N   a )})Σ i   |g                       s|,    
         [0000]    where the summation is taken over all the atoms (or molecules, etc.) in the atomic ensemble. In particular, an emission of the single Stokes photon in a forward direction results in the state of atomic ensemble given by 
         [0000]      S         |0 a             
         [0000]    where the ensemble ground state is given by 
         [0000]      |0           a ≡           i   |g             i . 
         [0033]    The light-atom interaction time t Δ  is short enough so that the mean photon number in the forward scattered Stokes pulse is much smaller than 1. An effective single-mode bosonic operator a can be defined for this Stokes pulse with the corresponding vacuum state denoted by |0&gt; p . The whole state of the atomic collective mode and the forward-scattering Stokes mode is written in the following form: 
         [0000]      |φ         =|0           a |0           p +√{square root over ( p   c )}S         a          |0           a |0           p   +o ( p   c )  (1) 
         [0000]    where p c  is the small excitation probability and o(p c ) represents the terms with more excitations whose probabilities are equal or smaller than p 2   c . A fraction of light is emitted in other directions due to the spontaneous emissions. However, whenever N a  is large, the contribution to the population in the symmetric collective mode from the spontaneous emissions is small. As a result there is a large signal-to-noise ratio for the processes involving the collective mode, which greatly enhances the efficiency of the present methods. 
         [0034]    The methods of the present invention generate entanglement between two distant atomic ensembles L and R using apparatus  1  of  FIG. 1 . Two laser pulses excite both atomic ensembles L and R simultaneously and the whole system is described by the state 
         [0000]      |φ&gt; L           φ           R , where |φ           L  and |φ           R    
         [0000]    are given by Equation (1) with all the operators and states distinguished by the subscript L or R. The forward scattered Stokes light (i.e., Stokes pulses  130  and  140 ) from both ensembles is combined at beam splitter  150  and a click in either SPD  180  or  190  measures the combined radiation from two samples, 
         [0000]      a +           a + or a −           a −  with 
         [0000]        a   ± =( a   L   ±e             a   R )/√{square root over (2)} 
         [0035]    Here, φ denotes an unknown difference of the phase shifts in the two-side channels  110  and  120 . The variable φ has an imaginary part to account for the possible asymmetry of the setup, which is corrected automatically. The asymmetry can be easily made very small, and φ is assumed real in the following. Conditional on the detector click, a + or a − are applied to the whole state 
         [0000]      |φ           L           |φ           R . 
         [0036]    This provides the projected state of the atomic ensembles L and R being nearly maximally entangled with the form (neglecting the high-order terms o(p c ): 
         [0000]      |Ψ φ             LR   ± =( S   L             ±e   iφ   S   R           )/√{square root over (2)}| 0   a             L |0 a             R .  (2) 
         [0037]    The probability for getting a click is given by p c  for each round, so the process must be repeated about 1/p c  times for a successful entanglement preparation, and the average preparation time is given by 
         [0000]      T 0 ˜t Δ /p c . 
         [0000]    The states 
         [0000]      |Ψ φ             LR   +  and 
         [0000]      |Ψ φ             LR   −   
         [0000]    are easily transformed to each other by a local phase shift. Without loss of generality, it is assumed in the following that the entangled state 
         [0000]      |Ψ φ             LR   +   
         [0000]    is generated. 
         [0038]    The presence of the noise modifies the projected state of the ensembles to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ρ 
                       LR 
                     
                      
                     
                       ( 
                       
                         
                           c 
                           0 
                         
                         , 
                         ϕ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           c 
                           0 
                         
                         + 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             c 
                             0 
                           
                            
                           
                             
                                
                               
                                 
                                   0 
                                   a 
                                 
                                  
                                 
                                   0 
                                   a 
                                 
                               
                               〉 
                             
                             LR 
                           
                            
                           
                             〈 
                             
                               
                                 0 
                                 a 
                               
                                
                               
                                 0 
                                 a 
                               
                             
                              
                           
                         
                         + 
                         
                           
                             
                                
                               
                                 Ψ 
                                 ϕ 
                               
                               〉 
                             
                             LR 
                             * 
                           
                            
                           
                             〈 
                             
                               Ψ 
                               ϕ 
                             
                              
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the “vacuum” coefficient c 0  is determined by the dark count rates of the photon detectors. It will be seen below that any state in the form of Equation (3) will be purified automatically to a maximally entangled state in the entanglement-based communication schemes. Therefore, this state is called an “effective maximally entangled” (EME) state with the vacuum coefficient c 0  determining the purification efficiency. 
         [0039]    Each step of the process of entanglement generation and corresponding applications contains built-in entanglement purification, which makes the method resilient to realistic noise and imperfections. In entanglement generation, the dominant noise is photon loss noise, which includes the contributions from the channel attenuation, the spontaneous emissions in the atomic ensembles (which results in the population of the collective atomic mode with the accompanying photon going to other directions), the coupling inefficiency of the Stokes light into and out of the channels, and the inefficiency of the SPDs. The loss probability is denoted by 1−η p  with the overall efficiency 
         [0000]      η p =η′ p e −L     0     L     att   , 
         [0000]    where the channel attenuation 
         [0000]      e −L     0     /L     att      
         [0000]    is separated from other noise contributions η p ′ with η p ′ independent of the communication distance L 0  (L att  is the channel attenuation length). The photon loss decreases the success probability for getting a detector click from p c  to η p p c , but it has no influence on the resulting EME state. In this sense, the entanglement purification process is built-in and the process is intrinsically fault-tolerant. Due to this noise, the entanglement preparation time is replaced by 
         [0000]      T 0 ˜t Δ /(η p p c ). 
         [0040]    In non-technical terms, many types of errors are in some sense, “self-correcting” in the process. Note that the method of the invention is probabilistic in that it depends on the occurrence of a pulse being detected, where each outcome of an event is known to be either a “success” or a “failure,” e.g., by the existence or non-existence of a pulse detection. Errors that are self-correcting affect the probability of success and the coefficients of the resulting states, but they do not, a priori, lead to a corrupted state, when the protocol indicates a “success” in the process. The change in coefficients in the process due to errors can be either pre-computed or observed in a real experiment. Therefore, those effects can be compensated or accounted for without affecting the main spirit of the protocol, e.g., the ability to communicate. The built-in nature of entanglement purification and, thus, the intrinsic fault-tolerant nature of the entanglement purification process is discussed in more detail below. In summary, the present invention has built-in entanglement purification and is intrinsically fault-tolerant. 
         [0041]    The second source of noise comes from the dark counts of SPDs  180  and  190 . The dark count gives a detector click, but without population of the collective atomic mode, so it contributes to the vacuum coefficient in the EME state. If the dark count comes up with a probability p c  for the time interval t Δ , the vacuum coefficient is given by 
         [0000]        c   0   =p   dc /η p   p   c , 
         [0000]    which is typically much smaller than 1 because the Raman transition rate is much larger than the dark count rate. 
         [0042]    The final source of noise, which influences the fidelity to get the EME state, is caused by an event in which more than one atom is excited to the collective mode S whereas there is only one click in either SPD  180  or  190 . The conditional probability for that event is given by p c , so we can estimate the fidelity imperfection ΔF 0 ≡1−F 0  for the entanglement generation by 
         [0000]      ΔF 0 ˜p c   (4) 
         [0043]    By decreasing the excitation probability p c , it is possible to make the fidelity imperfection closer and closer to zero with the price of a longer entanglement preparation time T 0 . This is the basic idea of the entanglement purification. In this scheme, the confirmation of the click from the single-photon detector generates and purifies entanglement at the same time. 
         [0044]    In entanglement swapping, the dominant noise is loss, such as associated with SPD inefficiency, the inefficiency of the excitation transfer from the collective atomic mode to the optical mode, and the small decay of the atomic excitation during the storage. By introducing the detector inefficiency, the imperfection is automatically taken into account by the fact that SPDs  180  and  190  cannot distinguish between a single photon and two photons. The overall efficiency in the entanglement swapping is denoted by η s . The loss in the entanglement swapping gives contributions to the vacuum coefficient in the connected EME state, since in the presence of loss a single detector click might result from two collective excitations in the ensembles I 1  and I 2 , and in this case, the collective modes in the atomic ensembles L and R have to be in a vacuum state. After taking into account the realistic noise, it is possible to specify the success probability and the new vacuum coefficient for the i th  entanglement connection by the recursion relations 
         [0000]    
       
         
           
             
               
                 p 
                 i 
               
               ≡ 
               
                 
                   f 
                   1 
                 
                  
                 
                   ( 
                   
                     c 
                     
                       i 
                       - 
                       1 
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   η 
                   s 
                 
                  
                 
                   ( 
                   
                     1 
                     - 
                     
                       
                         η 
                         s 
                       
                       
                         2 
                          
                         
                           ( 
                           
                             
                               c 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
               / 
               
                 ( 
                 
                   
                     c 
                     
                       i 
                       - 
                       1 
                     
                   
                   + 
                   1 
                 
                 ) 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 c 
                 i 
               
               ≡ 
               
                 
                   f 
                   2 
                 
                  
                 
                   ( 
                   
                     c 
                     
                       i 
                       - 
                       1 
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 2 
                  
                 
                   c 
                   
                     i 
                     - 
                     1 
                   
                 
               
               + 
               1 
               - 
               
                 
                   η 
                   s 
                 
                 . 
               
             
           
         
       
     
         [0000]    The coefficient c 0  for the entanglement preparation is typically much smaller than 1−η s , c i ≈(2′−1)(1−η s )=(L i /L 0 −1)(1−η s ),
 
where L i  denotes the communication distance after i times entanglement connection. With the expression for the c i , the probability p i  and the communication time T n  , are evaluated for establishing an EME state over the distance L n =2 n L 0 . After the entanglement connection, the fidelity of the EME state also decreases, and after n times connection, the overall fidelity imperfection is given by:
 
         [0000]      ΔF n ˜2 n ΔF 0 ˜(L n /L 0 )ΔF 0 . 
         [0000]    ΔF n  must be fixed to be small by decreasing the excitation probability p c  in Equation (4). 
         [0045]    The entanglement connection scheme also has built-in entanglement purification function. This can be understood as follows: Each time entanglement is connected, the imperfections of the setup decrease the entanglement fraction 1/(c i +1) in the EME state. However, the entanglement fraction decays only linearly with the distance (the number of segments), which is in contrast to the exponential decay of the entanglement for the connection schemes without entanglement purification. The reason for the slow decay is that in each time of the entanglement connection, the protocol must be repeated until there is a detector click, and the confirmation of a click removes part of the added vacuum noise since a larger vacuum component in the EME state results in more repetitions. The built-in entanglement purification in the connection scheme is essential for the polynomial scaling law of the communication efficiency. 
         [0046]    Entanglement-based applications of the invention, such as long-distance quantum communication systems, also have built-in entanglement purification which makes them resilient to realistic sources of noise. Firstly, the vacuum components in the EME states are removed from the confirmation of the detector clicks and thus have no influence on the fidelity of all the application schemes. Secondly, if the SPDs and the atom-to-light excitation transitions in the application schemes are imperfect with the overall efficiency, then these imperfections only influence the efficiency to get the detector clicks with the success probability replaced by 
         [0000]        p   a =η a /[2( c   n +1) 2 ], 
         [0000]    and have no effect on the communication fidelity. Finally, the phase shifts in the stationary channels and the small asymmetry of the stationary setup are removed automatically when the EME state is projected onto the PME state, and thus have no influence on the communication fidelity. The noise not correctable by this method includes the detector dark count in the entanglement connection and the non-stationary channel noise and set asymmetries. The resulting fidelity imperfection from the dark count increases linearly with the number of segments L n /L 0 , and form the non-stationary channel noise and set asymmetries increases by the random-walk law (L n /L 0 ) 1/2 . For each time of entanglement connection, the dark count probability is about 10 −5  if a typical choice is made that the collective emission rate is about 10 MHz and the dark count rate is 10 2  Hz. This noise is negligible even if for communications over a long distance. The non-stationary channel noise and setup asymmetries can also be safely neglected for such a distance. For instance, it is relatively easy to control the non-stationary asymmetries in local laser operations to values below 10 −4  with the use of accurate polarization techniques for Zeeman sublevels. 
         [0047]    Each entanglement generation and connection method of the present invention has built-in entanglement purification. As a result of this property, the communication fidelity can be fixed to be nearly perfect, and the time for preparing the system for communication increases only polynomially with the operating distance over which communication takes place. To communicate over a distance L=L n =2 n L 0 , the overall fidelity imperfection must be fixed to be a desired small value ΔF n , in which case the entanglement preparation time is 
         [0000]      T 0 ˜t Δ /(η p ΔF 0 )˜(L n /L 0 )t Δ /(η p ΔF n ). 
         [0048]    For an effective generation of the PME state, the total communication time 
         [0000]    
       
         
           
             
               T 
               tot 
             
             ∼ 
             
               
                 T 
                 n 
               
               / 
               
                 p 
                 a 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               T 
               n 
             
             ∼ 
             
               
                 T 
                 0 
               
                
               
                 
                   ∏ 
                   
                     i 
                     = 
                     1 
                   
                   n 
                 
                  
                 
                   
                     ( 
                     
                       1 
                       / 
                       
                         p 
                         i 
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
         [0049]    The total communication time scales with the distance by the law 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     tot 
                   
                   ∼ 
                   
                     2 
                      
                     
                       
                         
                           ( 
                           
                             L 
                             / 
                             
                               L 
                               0 
                             
                           
                           ) 
                         
                         2 
                       
                       / 
                       
                         ( 
                         
                           
                             η 
                             p 
                           
                            
                           
                             η 
                             a 
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           
                             F 
                             T 
                           
                            
                           
                             
                               ∏ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               p 
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the success probabilities p i , p a  for the i th  entanglement connection and for the entanglement application have been specified before. The expression (5) confirms that the communication time T tot  increases with the distance L only polynomially. In the first limiting case, the inefficiency 1−η s  or the entanglement swapping is assumed to be negligibly small. It is deduced from Equation (5) that in this case the communication time 
         [0000]      T tot ˜T con (L/L 0 ) 2 e L     0     /L     att,      
         [0000]    with the constant 
         [0000]        T   con =2t Δ /(η′ p η a   ΔF   T ) 
         [0000]    being independent of the segment and the total distances L 0  and L. The communication time T tot  increases with L quadratically. In the second case, it is assumed that the inefficiency 1−η s  is considerably large. The communication time in this case is approximated by 
         [0000]      T tot ˜T con (L/L 0 ) [log     2     (L/L     0     )+1]/2+log     2     (1/η     s     −1)+2 e L     0     /L     att,      
         [0000]    which increases with L still polynomially (or sub-exponentially in a more accurate language, but this makes no difference in practice since the factor log 2 (L/L 0 ) is well bounded from above for any reasonably long distance). If T tot  increases with L/L 0  by the m th  power law (L/L 0 ) m , there is an optimal choice of the segment length to be L 0 =mL att  to minimize the time T tot . As an estimation of the improvement in the communication efficiency, it is assumed that the total distance L is about 100L att , for a choice of the parameter η s ≈⅔, the communication time T tot /T con ˜10 6  with the optimal segment length L 0 ˜5.7L att . This result is a dramatic improvement compared with the direct communication case, where the communication time T tot  for getting a PME state increases with the distance L by the exponential law 
         [0000]      T tot ˜T con e L/L     att.      
         [0050]    For the same distance L 0 ˜100L att , one needs T tot ˜10 43  for direct communication, which means that for this example the present scheme is 10 37  times more efficient.  FIG. 2  is a schematic diagram of a quantum repeater apparatus  201  similar to that of  FIG. 1 , but that includes two pairs of atomic ensembles. Apparatus  201  includes source  10  and synchronizer  20 . Apparatus  201  also includes a first pair of optically coupled cells  230  and  241  that contain respective atomic ensembles I 1  and L, and a second pair of optically coupled cells  231  and  242  that contain respective atomic ensembles I 2  and R. Cells  230  and  231  are arranged downstream of and are optically coupled to synchronizer  20  via respective optical paths  250  and  251 . Filters  260  and  261  are arranged downstream of respective cells  230  and  231  and serve the same purpose as filters  70  and  80  in apparatus  1  of  FIG. 1 . Likewise, a beam splitter  270  is arranged at the intersection of optical paths  254  and  255  that are followed by Stokes pulses SI 2  and SI 1  that emanate from cells  231  and  230 . Optical paths  254  and  255  optically couple filters  261  and  260  to respective SPDs  280  and  281 . A photodetector control system  290  is coupled to SPDs  280  and  281  controls the operation of the SPDs. 
         [0051]    In the operation of apparatus  201 , light pulses P 0  emitted by source  10  pass through synchronizer  20 , which creates synchronized light pulses P 1  and P 2  that traverse paths  250  and  251  to atomic ensembles I 1  and I 2  contained in respective cells  230  and  231 . In atomic ensembles I 1  and I 2 , the photons in the light pulse incident on the atomic ensembles either undergo a collective Stokes scattering from the atoms of the ensemble, or they do not. If they do not, they are reflected by filters  260  and  261  along optical paths  252  and  253 . On the other hand, if the photons are Stokes scattered, they are converted to a different frequency (and/or k vector) and are transmitted by filters  260  and  261  along optical paths  254  and  255  as Stokes pulses SI 1  and SI 2 . The Stokes pulses (photons) then interfere at beam splitter  270 . SPDs  280  and  281  receive photons propagating along respective optical paths  254  and  255 , and each SPD generates a detection signal (“click”) when it detects a photon. Photodetector control system  290  receives detection signals S 280  and S 281  from SPDs  280  and  281  and processes these signals. 
         [0052]    Each of the atomic ensemble-pairs, i.e., pair L and I 1  and pair I 2  and R, is prepared in an EME state in the form of Equation (3). The excitations in the collective modes of atomic ensembles I 1  and I 2  are transferred simultaneously to the optical excitations by the repumping pulses applied to the atomic transition |s&gt;→|e&gt;. The stimulated optical excitations, after a 50%-50% beam splitter, are detected by the SPDs  180  and  190 . If either SPD  180  or  190  generates a click, the protocol is successful and an EME state is established between atomic ensembles L and R, thereby doubling the effective communication distance. If no SPD generates a click, the previous entanglement generation and swapping process is repeated until a click is generated by SPD  180  or  190 , that is, until the protocol succeeds. 
         [0053]    In an example embodiment of apparatus  201 , the two intermediated atomic ensembles I 1  and I 2  are replaced by one ensemble having two metastable states to store two different collective modes. The beam splitter  270  operation is replaced by its functional equivalent, a so-called π/2 pulse. In the language of quantum field theory, let the collective modes be described by the two collective mode annihilation operators, a 1  and a 2 . Such a π/2 pulse maps a 1 , and a 2  into their linear combinations in the following manner: 
         [0000]      a 1  into (a 1 +ia 2 )/√2 and a 2  into (a 1 −ia 2 )/√2  Eq. (A). 
         [0054]    Physically, such a π/2 pulse may be implemented by an external laser (not shown in  FIG. 2 ) that emits a pulse onto the ensemble containing the two said metastable states. The ensemble absorbs the π/2 laser pulse and changes its own state according to the transformation of the two said collective mode annihilation operators described by Eq. (A). 
         [0055]    The embodiment in  FIG. 2  provides two pairs of the entangled ensembles described by the state 
         [0000]    ρ LI     1             ρ I     2     R , where 
         [0000]      ρ LI     1    and ρ I     2     R    
         [0000]    are given by Equation (3). In the ideal case, the system shown in  FIG. 2  measures the quantities corresponding to operators 
         [0000]        S   ±             S   ± with    S   ± =( S   I     1     ±S   I     2   )/√{square root over (2)}. 
         [0056]    If the measurement is successful (i.e., one of the detectors registers one photon), the ensembles L and R are prepared into another EME state. 
         [0057]    The new φ-parameter is given by φ 1 +φ 2 , where φ 1  and φ 2  denote the old φ-parameters for the two segment EME states. In the presence of the realistic noise and imperfections, an EME state is still created after a detector click. Noise only influences the success probability to get a click and the new vacuum coefficient in the EME state. In general the success probability p 1  is expressed and the new vacuum coefficient c 1 , and 
         [0000]        p   1 =ƒ 1 ( c   0 ) and 
         [0000]        c   1 =ƒ 2 ( c   0 ), 
         [0000]    where the functions ƒ 1  and ƒ 2  depend on the particular noise properties. 
         [0058]    The above method for connecting entanglement between atomic ensembles can be cascaded to arbitrarily extend the effective communication distance. For the 
         [0000]      i th (i=1, 2, . . . , n) 
         [0000]    entanglement connection, two pairs of atomic ensembles are first prepared in parallel in the EME states with the same vacuum coefficient c 1  and the same communication length L i−1 . Entanglement swapping is then performed as described above in connection with  FIG. 2 . This entanglement swapping attempt succeeds with a probability 
         [0000]        p   i =ƒ 1 ( c   i−1 ). 
         [0059]    After a successful detector click, the communication length is extended to L i =2L i−1 , and the vacuum coefficient in the connected EME state becomes 
         [0000]        c   i =ƒ 2 ( c   i−1 ). 
         [0060]    Since the i th  entanglement connection needs to be repeated in average 1/p i  times, the total time needed to establish an EME state over the distance L n =2 n L 0  is given by 
         [0000]    
       
         
           
             
               
                 T 
                 n 
               
               = 
               
                 
                   T 
                   0 
                 
                  
                 
                   
                     ∏ 
                     
                       i 
                       = 
                       1 
                     
                     n 
                   
                    
                   
                     ( 
                     
                       1 
                       / 
                       
                         p 
                         i 
                       
                     
                     ) 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where L 0  denotes the distance of each segment in the entanglement generation. 
         [0061]    Entanglement can be attempted at a rate up to the pulse rate of light source  10 . Typically available light sources operate up to the megahertz frequencies. In principle, pulse rates in the gigahertz frequencies are attainable, and would result in a sub-nanosecond time T 0 . Therefore, the average total time needed to establish entanglement in a system of a relatively large number of atomic ensembles using the systems and methods of the present invention can be shorter than a microsecond. 
         [0062]    In an example embodiment of the invention, the EME is used in the communication protocols, such as quantum teleportation, cryptography, and Bell inequality detection.  FIG. 3  is a schematic diagram of another example embodiment of an entanglement-based apparatus  301  that employs two pairs of atomic ensembles L 1  and R 1  and L 2  and R 2 . Apparatus  301  can be used, for example, in a QKD system or for testing Bell inequalities. Apparatus  301  includes source  10  and synchronizer  20 , as described above, with cells  330  and  331  arranged downstream of and optically coupled to synchronizer  20 . Cells  330  and  331  contain atomic ensembles L 1  and R 1 , respectively. Apparatus  301  further includes cells  340  and  341 , which respectively contain atomic ensembles L 2  and R 2 , optically coupled to respective cells  300  and  331 . Further, cell  330  and  340  are optically coupled via optical path  342 , while cells  331  and  341  are optically coupled via optical path  344 . A first beam splitter  350  is arranged at the intersection of optical paths  353  and  354  leading from cells  330  and  340  to SPDs  370  and  371 , respectively. A second beam splitter  351  is arranged at the intersection of optical paths  357  and  358  leading from cells  331  and  341  to SPDs  380  and  381 , respectively. A first phase shifter  360  is arranged in optical path  353  between cell  340  and first beam splitter  350 . A second phase shifter  361  is arranged in optical path  357  between cell  341  and second beam splitter  361 . Photo detector control system  290  is operably coupled to SPDs  370 ,  371 ,  380  and  381 . 
         [0063]    In apparatus  301 , each pair of atomic ensembles—for example, pair (L 1 , R 1 )—can be the result of a concatenated application of apparatus  201  of  FIG. 2  employing entanglement swapping and entanglement purification, as discussed above. Additional laser sources  10  and synchronization systems  20  may be needed to set up the initial state for atomic ensemble pairs (L 1 , R 1 ) and also (L 2 , R 2 ). In apparatus  301  of  FIG. 3 , it is assumed that the preparation of the initial states of atomic ensemble pairs (L 1 , R 1 ) and (L 2 , R 2 ) has already been performed. Formally, the state of the atomic ensemble pair, say (L 1 , R 1 ), is described as an EME state defined earlier. The arrow  385  indicates that the atomic ensemble pair (L 1 , R 1 ) has been prepared in an EME state and is quantum mechanically correlated with entanglement. Similarly, the arrow  386  indicates that the atomic ensemble pair (L 2 , R 2 ) has been prepared an EME state and is quantum mechanically correlated with entanglement. 
         [0064]    In  FIG. 3 , each atomic ensemble pair (L 1 , R 1 ) may be separated spatially by a very long distance, say 100, 1000, or even 10,000 km. However, atomic ensembles L 1  and L 2  are supposed to be spatially located much closer than the distance between atomic ensembles L 1  and R 1 . A typical distance between atomic ensembles L 1  and L 2  may be 10 m or less, although a much longer distance of say one km may also be acceptable. Similarly, atomic ensemble R 1  and R 2  are supposed to be spatially located much closer than the distance between atomic ensembles L 1  and R 1 . Two pairs of atomic ensembles are needed because a pair of photons is needed for an experiment in quantum key distribution, whereas a pair of atomic ensemble typically generates only a single photon. The beam splitters  350  and  351  allow for entanglement to occur between the two pairs of atomic ensembles, thus generating a maximally entangled state between them. 
         [0065]    In an example embodiment, the photodetector control system  290  includes two separate detector/control subsystems (not shown) located near atomic ensemble pairs (L 1 , L 2 ) and (R 1 , R 2 ), respectively. The detector/control subsystem located near to atomic ensemble pair (L 1 , L 2 ) controls the setting of phase shifter  360  and, thus decides the type of measurement to be performed on atomic ensembles (L 1 , L 2 ). Similarly, the detector/control subsystem located near (R 1 , R 2 ) controls the setting of phase shifter  361  and decides on the type of measurement to be performed on the atomic ensembles (R 1 , R 2 ). The output of apparatus  301  is governed by how many and which of the SPDs  370 ,  371 ,  380  and  381  have detected a photon. 
         [0066]    As shown in  FIG. 3 , one optical path  342  traverses in sequence atomic ensembles L 1  and L 2 , while another optical path  344  traverses atomic ensembles R 1  and R 2 . Photons generated by the collective excitations in either atomic ensemble L 1  or L 2  propagate to beam splitter (interferer)  350 . The relative angles of optical paths  342  and  344  and the relative angles of optical paths  353 ,  354  and  357  and  358  are drawn for convenience of illustration. For example, the photons generated by collective excitations in an atomic ensemble may propagate in the same direction as incident photons, and subsequently be diverted by a wavelength-dependent reflector disposed on a far side of the ensemble. Detection at SPD  370  or  371  indicates a collective excitation in atomic ensemble pair (L 1 , L 2 ), which indicates that atomic ensembles L 2  and R 2  are entangled with atomic ensembles L 1  and R 1 . 
         [0067]    A more detailed and more mathematical description of  FIG. 3  is as follows. The pairs of atomic ensembles (L 1 , R 1 ) and (L 2 , R 2 ) (or alternatively two pairs of metastable states) are each prepared in EME states. The collective atomic excitations on each side are transferred to the optical excitations, which, respectively after a relative phase shift φ L  or φ R , and a 50%-50% beam splitter, are detected by the single-photon detectors 
       D 1   L ,D 2   L  and 
     D 1   R ,D 2   R . 
       [0068]    There are four possible coincidences of 
         [0000]    D 1   L ,D 2   L  with D 1   R ,D 2   R ,
 
which are functions of the phase difference φ L −φ R . Depending on the choice φ L  and φ R , this setup can realize both the quantum cryptography and the Bell inequality detection.
 
         [0069]    It is not obvious that the EME state (3), which is entangled in the Fock basis, is useful for these tasks since in the Fock basis it is experimentally hard to perform certain single-bit operations. EME states can be used to realize all these protocols with simple experimental configurations. 
         [0070]    The state of the two pairs of ensembles is expressed as 
         [0000]      ρ L     1     R     1             ρ L     2     R     2   , where 
         [0000]      ρ L     i     R     i   ( i =1,2) 
         [0000]    denote the same EME state with the vacuum coefficient c n  if entanglement connection has been done n times. The φ-parameters in 
         [0000]      ρ L     i     R     i   ( i =1,2) 
         [0000]    are the same provided that the two states are established over the same stationary channels. Only the coincidences of the two-side detectors are registered, so the protocol is successful only if there is a click on each side. Under this condition, the vacuum components in the EME states, together with the state components 
         [0000]      S L     1             S L     2             |vac &gt; and S R     1             S R     2             |vac         , where 
         [0000]      |vac          denotes the ensemble state 
         [0000]      |0 a 0 a 0 a 0 a             L     1     R     1     L     2     R     2,      
         [0000]    have no contributions to the experimental results. So, for the measurement apparatus of  FIG. 3 , the ensemble state 
         [0000]      ρ L     1     R     1             ρ L     2     R     2      
         [0000]    is effectively equivalent to the following “polarization” maximally entangled (PME) state 
         [0000]      |ψ         PME=( S   L     1               S   R     2               =S   L     2               S   R     1             )/√{square root over (2)}|vac           
         [0071]    The success probability for the projection from 
         [0000]      ρ L     1     R     1             ρ L     2     R     2    to  |ψ               PME      
         [0000]    (i.e., the probability to get a click on each side) is given by 
         [0000]        p   a =1/[2( c   n +1) 2 ]. 
         [0072]    In  FIG. 3 , the phase shift 
         [0000]      ψ Λ (Λ=L or R) 
         [0000]    together with the corresponding beam splitter operation are equivalent to a single-bit rotation in the basis 
         [0000]      {|0           Λ   ≡S   Λ     1             |0 a 0 a             Λ     1     Λ     2   ,|1           Λ   ≡S   Λ     2             |0 a 0 a             Λ     1     Λ     2}     
         [0000]    with the rotation angle θ=ψ Λ /2. Now, there is the PME state and it is possible to perform the desired single-bit rotations in the corresponding basis, and thus to implement quantum cryptography and Bell inequality detection. For instance, to distribute a quantum key between the two remote sides, we simply choose randomly from the set {0, π/2} with an equal probability, and keep the measurement results (to be 0 if D 1   L , clicks, and 1 if D 1   R  clicks) on both sides as the shared secret key if the two sides become aware that they have chosen the same phase shift after the public declare. This is exactly the Ekert scheme and its absolute security is already proven. For the Bell inequality detection, we infer the correlations 
         [0000]      E(ψ L ,ψ R )≡P D     1       L     D     1       R+P     D     2       L     D     2       R−P     D     1       L     D     2       R−P     D     2       L     D     1       R=cos(ψ     L −ψ R ) from the 
         [0000]    measurement of the coincidences 
         [0000]      P D     1       L     D     1       R   , etc. 
         [0073]    For the apparatus of  FIG. 3 , the output is 
         [0000]      | E (0,ρ/4)+ E (ρ/2,ρ/4)+ E (ρ/2,3ρ/4)− E (0,3ρ/4)|=2√{square root over (2)} 
         [0000]    whereas for any local hidden variable theories this value should be below 2. 
         [0074]    A third method of using the preferred embodiment realizes the setup for probabilistic quantum teleportation of the atomic “polarization” state. Similarly to the second method, two pairs of ensembles L 1 , R 1  and L 2 , R 2  are prepared in the EME states. An atomic “polarization” state 
         [0000]      (d 0 S I     1             +d 1 S I     2             )|0 a 0 a             I     2     I     2.      
         [0000]    can be teleported with unknown coefficients d 0 , d 1  from the left to the right side, where S I     1             ,S I     2             
 
denote the collective atomic operators for the two ensembles I 1  and I 2  (or two metastable states in the same ensemble). The collective atomic excitations in the ensembles I 1 , L 1  and I 2 , L 2  are transferred to the optical excitations, which, after a 50%-50% beam splitter, are detected by the single-photon detectors
 
       D 1   I , D 1   L  and D 2   I , D 2   L . 
       [0075]    If there are a click in D 1   I  or D 1   L  and a click in D 2   I  or D 2   L , the protocol is successful. A φ-phase rotation is then performed on the collective mode of the ensemble R 2  conditional on that the two clicks appear in the detectors D 1   L , D 2   L  or D 1   R , D 2   R . The collective excitation in the ensembles R 1  and R 2 , if appearing, would be found in the same “polarization” state 
         [0000]      d 0 S R     1             +d 1 S R     2             )|0 a 0 a             R     1     R     2      
         [0076]    The established long-distance EME states can be also used for faithful transfer of unknown quantum states through quantum teleportation, with the apparatus of  FIG. 4 . 
         [0077]      FIG. 4  is a schematic diagram of another example embodiment of an entanglement-based apparatus  401  according to the present invention that includes two pairs of atomic ensembles (L 1 , R 1 ) and (L 2 , R 2 ). Apparatus  401  includes a secondary control system  495  that includes light source  10  and synchronizer  20 . Secondary control system  495  is operably coupled to photodetector control system  490  via a communication link  496 . Secondary control system  495  is also operably coupled to cells  430  and  431  that contain atomic ensembles R 1  and R 2 , respectively, via respective optical channels  497  and  498 . Apparatus  401  also includes cells  440  and  441  that respectively contain atomic ensembles L 1  and L 2  and that are optically coupled to cells  430  and  431 , respectively, as indicated by arrows  485  and  486 . 
         [0078]    Apparatus  401  further includes cells  442  and  443  that contain respective atomic ensembles I 1  and I 2 . Optical paths  550  and  552  optically couple cells  440  and  442  to a first beam splitter (interferer)  450  and then to SPDs  470  and  471 . Likewise optical paths  551  and  553  optically couple cells  441  and  443  to a second beam splitter (interferer)  451  and then to SPDs  480  and  481   
         [0079]    In operation of apparatus  401  of  FIG. 4 , photodetector control system  490  receives detection signals (“clicks”) from SPDs  470  and  471 , and  480  and  481 . Based upon the received detection signals, photodetector control system  490  sends optical pulse generation control data in the form of signal S 490  via communication link  496  to secondary control system  495 . Based upon the optical pulse generation control data, secondary control system  495  controls the generation of optical pulses P 1  and P 2  for transmission to atomic ensemble R 1  or atomic ensemble R 2 . The two-headed arrow denoting optical paths  485  and  486 , respectively, indicates that atomic ensembles R 1  and L 1  are entangled, and that atomic R 2  and L 2  are entangled. 
         [0080]    Apparatus  401  of  FIG. 4  can be used for probabilistic quantum teleportation. Consider the atomic ensembles (L 1 , R 1 ) and (L 2 , R 2 ). Each atomic ensemble pair—for example, (L 1 , R 1 )—can be the result of a concatenated application of apparatus  201  of  FIG. 2 , which employs entanglement swapping and entanglement purification, as discussed above. Additional laser sources  10  and any synchronization systems  20  may be needed to set up the initial state of atomic ensemble pairs (L 1 , R 1 ) and also (L 2 , R 2 ). However, it is assumed that the preparation of the initial states of atomic ensemble pairs (L 1 , R 1 ) and (L 2 , R 2 ) has already been performed. Formally, the state of a atomic ensemble pair, say (L 1 , R 1 ), is described as an EME state defined earlier. The double arrow for optical path  485  indicates that the atomic ensemble pair (L 1 , R 1 ) has been prepared in an EME state and is, thus, quantum mechanically correlated with entanglement. Similarly, the double arrow for optical path  486  indicates that the atomic ensemble pair (L 2 , R 2 ) has been prepared an EME state and is thus, quantum mechanically correlated with entanglement. 
         [0081]    In apparatus  401 , each atomic ensemble pair (L 1 , R 1 ) may be separated spatially by a very long distance, say 100, 1000, or even 10,000 km. However, atomic ensembles L 1  and L 2  are supposed to be spatially located much closer than the distance between atomic ensembles L 1  and R 1 . A typical distance between atomic ensembles L 1  and L 2  may be 10 meters or less, although a much longer distance of say one km may also be acceptable. Similarly, atomic ensembles R 1  and R 2  are supposed to be spatially located much closer than the distance between atomic ensembles L 1  and R 1 . Note that two pairs of atomic ensembles are needed because the experiment concerns a state of a pair of polarization entangled photons, whereas a pair of atomic ensembles typically gives out only a single photon. 
         [0082]    In apparatus  401 , the pair of atomic ensembles (I 1 , I 2 ) is in some unknown quantum state. The goal of apparatus  401  is to teleport this unknown state from atomic ensemble pair (I 1 , I 2 ) to the remote atomic ensemble pair (R 1 , R 2 ). Here, atomic ensemble I 1  is located spatially close to atomic ensemble L 1 , and atomic ensemble I 2  is spatially close to L 2 . A typical distance between atomic ensembles I 1  and L 1  may be 10 meters or less, although a much longer distance of say one km may also be acceptable. The same is true for atomic ensembles I 2  and L 2 . Here, a photodetector control system  490  can be used for controlling SPDs  470 ,  471 ,  480  and  481 . Effectively, atomic ensemble pair (L 1 , L 2 ) constitutes a logical qubit (call it qubit A) and atomic ensemble pair (I 1 , I 2 ) constitutes an unknown input qubit (call it qubit E). In apparatus  401 , a probabilistic Bell-measurement is performed on the joint system of qubits (A,E). The atomic ensemble pair (R 1 , R 2 ) represents a third qubit, qubit B. Photodetector control system  490  obtains the outcome of the probabilistic Bell-measurement and sends it via the communication link  496  via signal S 490  to the secondary control system  495 . Conditional on the information in the signal it has received, secondary control system  495  may apply one of a number of possible operations on the atomic ensemble R 1  and R 2  via optical paths  497  and  498  to regenerate the state of teleported qubit (qubit E). In other words, after the re-generation, the state of the qubit B, which is stored in the physical system of the atomic ensemble pair (R 1 , R 2 ), now becomes the state of the original qubit E. Therefore, the quantum signal has been sent over long distances of 100, 1000 or even 10000 km. 
         [0083]    A mathematical and more detailed description of apparatus  401  of  FIG. 4  is as follows: In apparatus  401 , if two SPDs click on the left side (i.e., SPDs  471  and  480 ), there is a significant probability that there is no collective excitation on the right side of the apparatus, since the product of the EME states 
         [0000]      ρ L     1     R     1             ρ L     2     R     2      
         [0000]    contains vacuum components. However, if there is a collective excitation appearing from the right side, its “polarization” state would be exactly the same as the one input from the left. The teleportation here is probabilistic and needs posterior confirmation; but if it succeeds, the teleportation fidelity would be nearly perfect since in this case the entanglement is equivalently described by the PME state (6). The success probability for the teleportation is also given by 
         [0000]        p   a =1/[2( c   n +1) 2 ], 
         [0000]    which determines the average number of repetitions for a successful teleportation. 
         [0084]    The foregoing describes the theory of the invention and practical applications based upon collective excitation of atomic ensembles, and provides some specific examples of long-distance signal communication systems. However, collective excitations in many types of correlated systems exist, and many of those systems may be found suitable for use as a correlated system of the invention. Thus, while only certain features of the present invention have been outlined and described herein, many modifications and variations will be apparent to those skilled in the art. Therefore, the claims appended hereto are intended to cover all such modifications and equivalents that fall within the broad scope of the invention.