Patent Publication Number: US-2022216988-A1

Title: Quantum key distribution scheme using a thermal source

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT 
     This invention was made with government support under Contract No. DE-AC05-00OR22725 awarded by the U.S. Department of Energy. The government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     The present disclosure relates to generating shared random information, quantum communication, quantum cryptography, and quantum key distribution. The following publications by the inventors are each hereby incorporated by reference in their entirety:  Passive state preparation in the Gaussian - modulated coherent - states quantum key distribution , B. Qi, P. Evans and W. Grice, Physical Review A 97, 012317, published Jan. 16, 2018; and  Passive state preparation in continuous - variable quantum key distribution , B. Qi, P. Evans and W. Grice, in Conference on Lasers and Electro-Optics, OSA Technical Digest [online] (Optical Society of America, 2018), paper JTh2A.9. 
     BACKGROUND 
     Quantum key distribution (QKD) has drawn attention for its proven security against adversaries with unlimited computing power. QKD is a secure communication method that implements a cryptographic scheme (sometimes referred to as a protocol) involving quantum mechanics. In QKD, two remote legitimate clients (typically referred to as Alice and Bob or transmitter and receiver) can establish a secure key by transmitting quantum states through an insecure channel controlled by an adversary (typically referred to as Eve or eavesdropper). The security of the key is based on features of quantum physics, rather than assumptions regarding computationally difficult problems. QKD exploits quantum phenomena to enable communications that can only be intercepted by violating known laws of physics. Accordingly, any attacks by Eve will, with a high probability, disturb the transmitted quantum state, and thus can be detected. 
     Many practical QKD systems are based on a prepare-and-measure scheme, where Alice prepares quantum states and transmits them to Bob, who in turn performs measurements. The measurement results can act as a key or be used as inputs to a key generation algorithm that can enable secure communication between Alice and Bob. The quantum state preparation step is conventionally implemented in an active manner: Alice first generates truly random numbers using a quantum random number generator, which she uses to prepare a corresponding quantum state by performing modulations on the output of a single source, or switching among multiple sources. One well-known example is the decoy state BB84 QKD using phase randomized weak coherent 
     sources, where for each transmission, Alice randomly prepares one of the four BB84 states, randomly changes the average photon number (to generate either the signal state or one of the decoy states), and (in certain implementations) randomizes the global phase of the weak coherent state. As the transmission rate in QKD has been growing dramatically over the years, it is becoming more and more challenging to prepare quantum state precisely at the corresponding speed. 
     More recently, passive state preparation schemes have been proposed in QKD as an alternative approach. In this scheme, Alice uses intrinsic fluctuations of the source, or intentionally designs the source in a way such that certain parameters (for example, intensity) will present unpredictable fluctuations. Typically, two optical modes with correlated fluctuations are output from the source. By measuring one mode locally, Alice can determine the random noise carried by the other mode, which will be transmitted to Bob. This idea was initially proposed as a simple way to generate random intensity fluctuations in the decoy-state QKD schemes. Later on, it was also applied in preparing the four BB84 states approximately. So far, the passive state preparation scheme has only been developed in a discrete-variable (DV) QKD based on single photon detection. 
     Two families of QKD schemes are discrete-variable (DV) and continuous variable (CV). In essence, CV-QKD is “analog”, while discrete-variable (DV) QKD is “digital.” One CV-QKD scheme is the Gaussian-modulated coherent-states (GMCS) QKD scheme, which is illustrated in  FIG. 1  and has been demonstrated over practical distances. Known implementations of the GMCS QKD are based on an active state preparation scheme: for each transmission, Alice first generates a pair of Gaussian-distributed random numbers  104 , encodes them on a weak coherent state  102  using optical amplitude and phase modulators  106 , and then transmits the Gaussian-modulated weak coherent pulse to Bob over a quantum channel  108 . Since the modulation format is relatively complicated and the tolerable modulation error is small, high extinction ratio modulators with good stability are commonly required in the GMCS QKD scheme. 
     CV-QKD based on coherent detection is especially appealing for a number of applications because of its compatibility with standard telecom technologies. Nevertheless, high-speed Gaussian random number generators and high performance optical modulators are currently required to implement CV-QKD, impeding its practical application. 
     SUMMARY 
     A system and method are provided to yield a passive continuous variable quantum key distribution (CV-QKD) scheme using a thermal source. In one embodiment, a CV-QKD scheme is provided based on coherent detection using a broadband thermal source, such as a continuous wave or pulsed source. The CV-QKD scheme is passive because it operates without a random number generator and without an optical modulator, which simplifies the implementation of CV-QKD and makes it more practical relative to known active CV-QKD schemes and more practical relative to known passive DV-QKD schemes. 
     In one embodiment of the CV-QKD scheme with a passive state preparation scheme using a thermal source, the transmitter client splits the output of a thermal source into two spatial modes using a beam splitter. The transmitter client measures one mode locally using optical homodyne detectors, and transmits the other mode to the receiver client after applying optical attenuation. A secure key can be established based on the correlation between the transmitter client&#39;s measurement results and the receiver client&#39;s measurement results. Given the initial thermal state generated by the source being strong enough, this scheme can tolerate high detector noise at the transmitter side. Furthermore, the output of the source does not need to be single mode, since an optical homodyne detector can selectively measure a single mode determined by the local oscillator. 
     In summary, the present invention provides a simple passive state preparation scheme in CV-QKD, which can be implemented using off-the-shelf amplified spontaneous emission (ASE) sources. 
     Before the embodiments of the invention are explained in detail, it is to be understood that the invention is not limited to the details of operation or to the details of construction and the arrangement of the components set forth in the following description or illustrated in the drawings. The invention may be implemented in various other embodiments and of being practiced or being carried out in alternative ways not expressly disclosed herein. Also, it is to be understood that the phraseology and terminology used herein are for the purpose of description and should not be regarded as limiting. The use of “including” and “comprising” and variations thereof is meant to encompass the items listed thereafter and equivalents thereof as well as additional items and equivalents thereof. Further, enumeration may be used in the description of various embodiments. Unless otherwise expressly stated, the use of enumeration should not be construed as limiting the invention to any specific order or number of components. Nor should the use of enumeration be construed as excluding from the scope of the invention any additional steps or components that might be combined with or into the enumerated steps or components. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a prior art block diagram of a Gaussian-modulated coherent-states quantum key distribution implementation. 
         FIG. 2  shows an embodiment of a passive state preparation of continuous variable quantum key distribution (CV-QKD) system using a thermal source. 
         FIG. 3  shows an embodiment of a passive state preparation of continuous variable quantum key distribution (CV-QKD) transmitter using a thermal source. 
         FIG. 4  shows an embodiment of a passive state preparation of continuous variable quantum key distribution (CV-QKD) system using a thermal source. 
         FIG. 5  shows a system for demonstrating the output of a source is a thermal state. 
         FIG. 6  shows a graph of simulation results of the secure key rate for three different average photon numbers of the thermal state. 
         FIG. 7  shows measurement results with a vacuum input and a thermal state input in phase space. 
         FIG. 8  shows a histogram of sample results with a thermal state input. 
         FIG. 9  shows one embodiment of a passive state preparation of continuous variable quantum key distribution (CV-QKD) method using a thermal source. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention is generally directed to a system and method of a passive state preparation scheme for quantum key distribution using a thermal source. In a Gaussian-modulated coherent-states (GMCS) QKD scheme, such as shown in  FIG. 1 , from Eve&#39;s and Bob&#39;s points of view, the quantum states sent by Alice are thermal. In the current embodiments, instead of preparing a thermal state from a coherent state by preforming Gaussian modulations, the transmitter client uses a thermal source. 
     A thermal source can include essentially any amplified spontaneous emission source, such as a Superluminescent Light Emitting Diode (SLED). The thermal source can be a single mode or multimode source and can be operated in a continuous wave or pulsed mode. One example of a thermal source is a fiber amplifier with a vacuum state input. One such fiber amplifier is the FA-30 fiber optic amplifier available from PRITEL, Inc. 
     As shown in  FIG. 2 , the transmitter client (i.e., Alice) splits the output of the thermal source  202  into two spatial modes using a beam splitter  204 . The transmitter side measures the X and/or P quadratures (i.e., quantum states) of one spatial mode using a detector system, such as an optical homodyne detector system  206 , in conjunction with a local oscillator  207  (for example a strong light pulse) and transmits the other mode to the transmitter client (i.e., Bob) over a quantum channel  214  after applying optical attenuation, for example with an attenuator  208  or an asymmetric beam splitter. To facilitate correlation with the receiver&#39;s quantum state measurements, a transmitter side controller  216  can estimate the quadrature values of the outgoing mode by scaling scale down its measurement results by the attenuation applied on the outgoing beam. 
     At the receiver client, similar measurements can be performed using a detector system, such as optical homodyne detector system  210 , in conjunction with a local oscillator  212  to determine the quadrature values (i.e. quantum states) of the received attenuated spatial mode. Under normal conditions, the transmitter client&#39;s measurement results can be correlated to the receiver client&#39;s, and a secure key can be established using conventional methods where a transmitter and receiver have shared random information. In this embodiment, the shared randomness originates from the intrinsic quadrature fluctuations of a thermal state. This is in contrast to previous CV-QKD schemes using a noisy coherent state or thermal state, where the super-Poissonian photon statistics of the source is regarded as excess noise in CV-QKD based on active state preparation. 
     In the current embodiment, the equivalent “preparation” noise in the transmitter can be suppressed by using a bright thermal source and then applying strong and “trusted” attenuation on the outgoing mode. Here, the term “trusted” means the attenuator is well calibrated and cannot be accessed by an eavesdropper. As shown in  FIG. 2 , noises presented in Alice&#39;s detector  206  increases the uncertainty of Alice&#39;s estimation on the quantum state of the outgoing mode, and appear as preparation noise. This preparation noise cannot be distinguished from the noises introduced by an eavesdropper and must be suppressed below certain threshold value to make any attack by an eavesdropper on the quantum channel detectable. The “trusted” attenuator  208  reduces the amplitude of the outgoing mode, thus reducing Alice&#39;s uncertainty on the mode after the attenuator. This effectively suppresses the equivalent preparation noise due to Alice&#39;s detector. In alternative embodiments, instead of a symmetrical beam splitter (50/50 splitting ratio) and a separate attenuator  308 , these components can be replaced with an asymmetrical beam splitter. For example, if an asymmetrical beam splitter has a 90/10 splitting ratio, when Alice measures X and P quantum states she can estimate the X and P measured by Bob by accounting for the splitting ratio in her estimate so that the shared key can be established based on the correlation between Alice&#39;s estimate and Bob&#39;s actual measurements of X and P. 
     The current embodiments of the CV-QKD scheme involve measuring a quantum state, such as an X and/or P quadrature of a spatial mode. X and P quadratures are a function of the amplitude, E, and phase, Phi, of classical electromagnetic waves. For example, the X and P qudratures can be represented by the following formulas: 
     
       
         
           
             X 
             = 
             
               E 
               * 
               
                 Cos 
                 ⁡ 
                 
                   ( 
                   Phi 
                   ) 
                 
               
             
           
         
       
       
         
           
             P 
             = 
             
               E 
               * 
               
                 sin 
                 ⁡ 
                 
                   ( 
                   Phi 
                   ) 
                 
               
             
           
         
       
     
     Depending on the application, the system and method can include measuring a single quadrature value by using an optical homodyne detector system. Whether X or P is measured is dictated by the phase of the local oscillator (LO). A conjugate homodyne detector (C-HD), as shown in  FIGS. 3, 4, and 5 , can be used to measure X and P simultaneously. Inside an exemplary C-HD, there are two sets of optical homodyne detectors with a 90 degree phase shift between them. Perhaps this is best shown with reference to the C-HD  515  of  FIG. 5 . One of the sets of homodyne detectors measures X, and the other set of homodyne detectors measures P. Accordingly, the transmitter and receiver can be configured to each measure the same quantum state or to each measure multiple of the same quantum states. 
     Put another way, there are a variety of different possible measurement schemes, for example, a) Alice and Bob can both use C-HD to measure both X and P. In this case, they could generate a secure key from both X and P; b) One of the clients measures both X and P with a C-HD, while the other one uses a single homodyne detector to measure either X or P (that client can randomly choose for each measurement). After the quantum transmission stage, the client with the single homodyne detector unit can inform the other which quadrature has been measured, and they can generate a secure key from that quadrature; c) Both clients use a single homodyne detector unit to measure a randomly chosen quadrature. They can generate a secure key when they happen to measure the same quadrature. 
     In the protocol, a thermal state is employed as the source of randomness. If its quadrature value is measured with a homodyne detector repeatedly, the outputs are random numbers following Gaussian distribution. In order to generate shared randomness between two parties, a beam splitter is employed to split the thermal state into two output spatial modes. Each output mode is still a thermal state (with half of the power of the input state). The quadrature noises of the two output modes are highly correlated. In the protocol, Alice measures the quadrature values of one mode locally and transmit the other mode to Bob after introducing appropriate attenuation, who in turn measures the quadrature values of the receiving mode. After they have collected a large amount of data, the transmitter and receiver can establish an identical secure key, for example by conducting reconciliation and privacy amplification through a classical authentication channel. For example, as shown in  FIG. 2 , a transmitter side controller  216  (or other authentication equipment) and receiver side controller  218  (or other authentication equipment) can communicate over a classical authentication channel  220 . That is, authentication can be conducted over a standard communication channel between Alice and Bob using conventional techniques. Although  FIG. 2  shows a transmitter side controller and a receiver side controller, it should be understood that other devices commonly used in an optical communication system can implement functions such as synchronization, data acquisition, reconciliation, privacy amplification, or other algorithms involved in generating an identical secure key from shared random information between two or more clients. For example, by comparing a subset of data, Alice and Bob can quantify the correlation between their measurement results and estimate the properties of the quantum channel. If the observed correlation is above a certain threshold (or equivalently, the noise introduced by the quantum channel is below a certain threshold), they can further perform reconciliation/privacy amplification algorithm to generate the identical secure key. 
     One potential issue is that vacuum noise introduced by Alice&#39;s conjugate optical homodyne detection could ultimately prevent her from acquiring a precise estimation of the quadrature values of the outgoing mode. However, as discussed in more detail below, if the initial thermal state generated by the source is strong enough, the contribution of Alice&#39;s detector noise on the estimation error of the outgoing state can be reduced effectively by introducing high attenuation on the outgoing mode. 
     In practice, it may be difficult to prepare a single mode thermal state and match its spectral-temporal mode with that of the local oscillator (LO) used in homodyne detection. However, in the protocol of the current embodiment of the present invention, Alice&#39;s thermal source does not need to be single mode. The LO in homodyne detection can act as a mode “filter” and can selectively measure only one mode emitted by the source. The only requirement is that the LO at the transmitter&#39;s side should be in the same mode as the LO at the receiver&#39;s side. By using a multimode (broadband) source, it is also easy to align the central wavelength of the LO within the spectrum of the thermal source. 
     The passive state preparation using a thermal source of the present invention will now be discussed in more detail mainly with reference to  FIG. 3  and  FIG. 9 . While the protocol can be conveniently implemented with a multimode thermal source, for simplicity, in this section we assume the thermal source is single mode.  FIG. 9  illustrates a flowchart of one embodiment of a method of the present invention with the following steps: 
     (1) Splitting Thermal Source. For example, Alice can split the output of a thermal source  302  into two spatial modes (mod 1  and mod 2  in  FIG. 3 ) using a 50:50 beam splitter  304 . The average output photon number of the source is assumed to be n 0 . (Step  902 ). 
     (2) Attenuating First Spatial Mode. For example, Alice can attenuate the average photon number of mod 1  down to V A /2 by using an optical attenuator  306  and transmits it to Bob. Here V A &lt;n 0  is the desired modulation variance. (Step  904 ). 
     (3) Measuring Quantum State of Second Spatial Mode. For example, Alice can measure both X and P quadratures of mod 2  using a conjugate homodyne detection system  330 . As shown in  FIG. 5 , a local oscillator may work in conjunction with the conjugate homodyne detection system  515 ,  330 . The system  330  may include a beam splitter  308  for creating the 90 degree phase shift between the outputs to the two pairs of homodyne detectors  314 ,  316 . From her measurement results of {x 2 ,p 2 }, Alice can also estimate the quadrature values of the outgoing mode as 
     
       
         
           
             
               
                 x 
                 A 
               
               = 
               
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           η 
                           A 
                         
                       
                       
                         η 
                         D 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     x 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     p 
                     A 
                   
                 
                 = 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           η 
                           A 
                         
                       
                       
                         η 
                         D 
                       
                     
                   
                   ⁢ 
                   
                     p 
                     2 
                   
                 
               
             
             , 
           
         
       
     
     where η A  is the transmittance of the optical attenuator  406  and η D  is the efficiency of Alice&#39;s detectors  314 ,  316 . The transmittance and efficiency values are represented by η  310 ,  312  in  FIG. 3 . (Step  910 ). 
     (4) Measuring Quantum State of First Spatial Mode. For example, Bob can measure both the X and P quadratures of the received attenuated spatial mode of the thermal source by performing conjugate homodyne detection. The measurement results can be referenced as {x B , p B }. (Step  908 ). 
     (5) Repeat 1-4. Alice and Bob can optionally repeat the above process many times. 
     (6) Establishing Shared Key. Alice and Bob can establish a shared key based on the measured quantum states. For example, they can perform reconciliation and privacy amplification on the raw data {x A , p A } and {x B , p B }. Given the observed noise is below a certain threshold, they can establish a secure key using conventional methods where a transmitter and receiver have shared random information. For example, devices commonly used in an optical communication system can implement functions such as synchronization and data acquisition and classical authentication schemes can be applied, perhaps as best shown and described in connection with  FIG. 2 . (Step  912 ). 
     From Eve&#39;s point of view, the quantum state sent by Alice in this passive state preparation scheme is the same as the one in a conventional active state preparation scheme. So, the well-established security proofs of the GMCS QKD can be applied directly to this scheme. The GMCS QKD is described in more detail in the following publication, which is incorporated by reference in its entirety, JOUGUET, P. et al., “Experimental demonstration of long-distance continuous-variable quantum key distribution”, available at https://arxiv.org/pdf/1210.6216, published Oct. 23 2012. 
     The secure key rate can be determined accounting for how much additional noise is introduced by the passive state preparation scheme. As discussed below, given the thermal state generated by the source is bright enough, the scheme can tolerate high noise and low efficiency of Alice&#39;s detector. In alternative embodiments, the combination of BS 1    304  and Att.  306  in  FIG. 3  could be replaced by an asymmetric beam splitter. 
     An exemplary determination of the X quadrature or X quantum state is provided below. The P quadrature or P quantum state can be determined in a similar way. The X quadrature of the outgoing state to Bob is given by 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       1 
                     
                     = 
                     
                       
                         
                           
                             
                               η 
                               A 
                             
                             2 
                           
                         
                         ⁢ 
                         
                           x 
                           in 
                         
                       
                       + 
                       
                         
                           
                             1 
                             - 
                             
                               
                                 η 
                                 A 
                               
                               2 
                             
                           
                         
                         ⁢ 
                         
                           x 
                           
                             υ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where x in  stands for the X quadrature of the output of the source, η A  is the transmittance of the optical attenuator, and x v1  represents vacuum noise introduced by beam splitter BS 1    304  and the attenuator  306 . Similarly, Alice&#39;s measurement result of the X quadrature is given by 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       2 
                     
                     = 
                     
                       
                         
                           
                             
                               η 
                               D 
                             
                             4 
                           
                         
                         ⁢ 
                         
                           x 
                           in 
                         
                       
                       + 
                       
                         
                           
                             1 
                             - 
                             
                               
                                 η 
                                 D 
                               
                               4 
                             
                           
                         
                         ⁢ 
                         
                           x 
                           
                             υ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                       + 
                       
                         N 
                         el 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where η D  and N el  are the efficiency and noise of Alice&#39;s detector; x v2  represents vacuum noise due to the two 50:50 beam splitters BS 1    304  and BS 2    308  and the loss of the x detector  310 . N el  is assumed to be Gaussian noise with zero mean and a variance of υ el . All of the noise variances are defined in the shot-noise unit. Alice can estimate x 1  from her measurement result x 2  using 
     
       
         
           
             
               
                 
                   
                     x 
                     A 
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           
                             η 
                             A 
                           
                         
                         
                           η 
                           D 
                         
                       
                     
                     ⁢ 
                     
                       
                         x 
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Using (1)-(3), Alice&#39;s uncertainty on x 1  is given by 
     
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     
                       〈 
                       
                         
                           ( 
                           
                             
                               x 
                               A 
                             
                             - 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         2 
                       
                       〉 
                     
                     = 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               η 
                               A 
                             
                           
                           
                             η 
                             D 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               υ 
                               el 
                             
                             - 
                             
                               
                                 η 
                                 D 
                               
                               2 
                             
                           
                           ) 
                         
                       
                       + 
                       1. 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     From (4), the excess noise (the noise above vacuum noise) due to the passive state preparation scheme is given by 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     A 
                   
                   = 
                   
                     
                       Δ 
                       - 
                       1 
                     
                     = 
                     
                       
                         
                           2 
                           ⁢ 
                           η 
                         
                         
                           η 
                           D 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               υ 
                               el 
                             
                             - 
                             
                               
                                 η 
                                 D 
                               
                               2 
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     From (5), by increasing the attenuation on the outgoing mode (decreasing η A ), the excess noise ε A  can be effectively reduced. The maximum attenuation Alice can apply is constrained by the average photon number n 0  of the thermal state produced by the source and the desired modulation variance V A . Using the relation V A =η A n 0 , (5) can be revised as 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     A 
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           V 
                           A 
                         
                       
                       
                         
                           η 
                           0 
                         
                         ⁢ 
                         
                           η 
                           D 
                         
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             υ 
                             el 
                           
                           - 
                           
                             
                               η 
                               D 
                             
                             2 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     From (6), given a desired modulation variance V A , the brighter the source, the smaller the excess noise introduced by Alice. A typical homodyne detector in the GMCS QKD can achieve η D =0.5 and υ el =0.1. For a typical value of V A =1, to reduce the excess noise E A  below 0.01, the required average photon number of the source is about 340 (per spatial-temporal mode), which can be satisfied by a practical ASE source. 
       FIG. 4  illustrates an embodiment of the passive CV-QKD scheme of the present invention. The transmitter includes a thermal source  402 , a beam splitter  404 , an attenuator  406 , a beam combiner  412 , a conjugate homodyne detector  430 , a local oscillator  407 , a beam splitter  408 , a delay unit  410 , and a mirror  414 . The receiver includes a conjugate homodyne detector  450 , a local oscillator  462 , a beam splitter  460 , mirrors  456 ,  458 , a delay unit  454 , and a beam splitter  452 . Essentially any phase recovery scheme can be utilized to establish a phase relation between the two local oscillators  407 ,  462 . One possible phase recovery scheme, which is illustrated in  FIG. 4 , that can be utilized to establish a phase relation between the two local oscillators is discussed in U.S. Pat. No. 9,768,885 to Qi, entitled Pilot-aided feedforward data recovery in optical coherent communications, filed on Sep. 10, 2015, which is hereby incorporated by reference in its entirety. Another scheme to establish a phase reference between transmitter and receiver homodyne detection systems is where Alice generates two strong LOs from the same laser, uses one of them in her local measurement, and sends the second one to Bob to be used as a LO in his measurement. 
     A simulation of t secure key rates of the passive state preparation scheme can be conducted. The asymptotic secure key rate of the GMCS QKD, in the case of reverse reconciliation, is given by: 
     
       
         
           
             
               
                 
                   
                     R 
                     = 
                     
                       
                         fI 
                         AB 
                       
                       - 
                       
                         χ 
                         BE 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     where I AB  is the Shannon mutual information between Alice and Bob; f is the efficiency of the reconciliation algorithm; χ BE  is the Holevo bound between Eve and Bob. I AB  and χ BE  can be determined from the channel loss, observed noises, and other QKD system parameters. 
     In an exemplary embodiment where the quantum channel between Alice and Bob is telecom fiber with an attenuation coefficient of γ, the channel transmittance is given by 
     
       
         
           
             
               
                 
                   
                     T 
                     = 
                     
                       10 
                       
                         
                           - 
                           yL 
                         
                         10 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where L is the fiber length in kilometers. In the case of conjugate homodyne detection, the noise added by Bob&#39;s detector (referred to Bob&#39;s input) is given by 
     
       
         
           
             
               
                 
                   
                     
                       χ 
                       het 
                     
                     = 
                     
                       
                         [ 
                         
                           1 
                           + 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 η 
                                 D 
                               
                             
                             ) 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               υ 
                               el 
                             
                           
                         
                         ] 
                       
                       / 
                       
                         η 
                         D 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     where we have assumed that Bob&#39;s detector has the same performance as Alice&#39;s. The channel-added noise referred to the channel input is given by 
     
       
         
           
             
               
                 
                   
                     
                       χ 
                       line 
                     
                     = 
                     
                       
                         1 
                         T 
                       
                       - 
                       1 
                       + 
                       
                         ɛ 
                         E 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where ε E  is the excess noise due to Eve&#39;s attack. In practice any untrusted noise from the QKD system can be included into ε E . ε E  can be separated into two terms, 
     
       
         
           
             
               
                 
                   
                     
                       ɛ 
                       E 
                     
                     = 
                     
                       
                         ɛ 
                         A 
                       
                       + 
                       
                         ɛ 
                         θ 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     where ε A  is the excess noise due to the passive state preparation scheme as given in (6). ε 0  represents other sources of untrusted noise. The overall noise referred to the channel input is given by 
     
       
         
           
             
               
                 
                   
                     χ 
                     tot 
                   
                   = 
                   
                     
                       χ 
                       line 
                     
                     + 
                     
                       
                         
                           χ 
                           het 
                         
                         T 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Since both quadratures can be used to generate the secure key, the mutual information between Alice and Bob can be determined by 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       AB 
                     
                     = 
                     
                       
                         log 
                         2 
                       
                       ⁢ 
                       
                         
                           V 
                           + 
                           
                             χ 
                             tot 
                           
                         
                         
                           1 
                           + 
                           
                             χ 
                             tot 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     where V=V A +1. 
     To estimate χ BE , a realistic noise model where Eve cannot control the loss inside Bob&#39;s system can be adopted, and the detector noise from Bob can be assumed to be trusted. Under this model, the Holevo bound of the information between Eve and Bob is given by: 
     
       
         
           
             
               
                 
                   
                     
                       𝒳 
                       BE 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           2 
                         
                         ⁢ 
                         
                           G 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   λ 
                                   i 
                                 
                                 - 
                                 1 
                               
                               2 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             3 
                           
                           5 
                         
                         ⁢ 
                         
                           G 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   λ 
                                   i 
                                 
                                 - 
                                 1 
                               
                               2 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       λ 
                       
                         1 
                         , 
                         2 
                       
                       2 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           A 
                           ± 
                           
                             
                               
                                 A 
                                 2 
                               
                               - 
                               
                                 4 
                                 ⁢ 
                                 B 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     = 
                     
                       
                         
                           V 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               2 
                               ⁢ 
                               T 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         T 
                       
                       + 
                       
                         
                           
                             T 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               V 
                               + 
                               
                                 𝒳 
                                 line 
                               
                             
                             ) 
                           
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     = 
                     
                       
                         
                           T 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               V 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 𝒳 
                                 line 
                               
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                       2 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       λ 
                       
                         3 
                         , 
                         4 
                       
                       2 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           C 
                           ± 
                           
                             
                               
                                 C 
                                 2 
                               
                               - 
                               
                                 4 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     = 
                     
                       
                         1 
                         
                           
                             ( 
                             
                               T 
                               ⁡ 
                               
                                 ( 
                                 
                                   V 
                                   + 
                                   
                                     𝒳 
                                     tot 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               A 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 𝒳 
                                 het 
                                 2 
                               
                             
                             + 
                             B 
                             + 
                             1 
                             + 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 𝒳 
                                 het 
                               
                             
                           
                           , 
                           
                             
 
                           
                           ⁢ 
                           
                             
                               × 
                               
                                 ( 
                                 
                                   
                                     V 
                                     ⁢ 
                                     
                                       B 
                                     
                                   
                                   + 
                                   
                                     T 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         V 
                                         + 
                                         
                                           𝒳 
                                           line 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 T 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       2 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     D 
                     = 
                     
                       
                         ( 
                         
                           
                             V 
                             + 
                             
                               
                                 B 
                               
                               ⁢ 
                               
                                 𝒳 
                                 bet 
                               
                             
                           
                           
                             T 
                             ⁡ 
                             
                               ( 
                               
                                 V 
                                 + 
                                 
                                   𝒳 
                                   tot 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     λ 
                     5 
                   
                   = 
                   1. 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Simulation parameters can be summarized as follows: γ=0.2 dB/km, ε 0 =0.01, υ el =0.1, η D =0.5, and f=0.95. The modulation variance V A  can be numerically optimized at different fiber lengths. 
       FIG. 6  shows exemplary relations of the secure key rate and the fiber length for three different average photon numbers η 0 . As shown in  FIG. 6 , a thermal source with an average output photon number above 100 can be employed to implement the passive CV QKD scheme efficiently. 
     Previous studies have shown that the ASE noise generated by a fiber amplifier is thermal. Conjugate homodyne detection can be conducted to verify the photon statistics of a single-mode component (selected by the LO) of an ASE source follows a Bose-Einstein distribution, as expected from a single-mode thermal state. Nevertheless, the average photon number of the thermal state is relatively low (about 15). The output of a commercial ASE source operated at higher output power can also be characterized in this way. 
       FIG. 5  shows an exemplary setup that demonstrates that the output of a practical source is a thermal state. Accordingly, this demonstration setup does not illustrate a complete quantum key distribution system, but rather a portion of one sufficient to demonstrate the thermal state. The system includes a fiber amplifier (PriTel) with vacuum state input as a broadband thermal source  502 . A 0.8-nm optical bandpass filter  504  centered at 1542 nm placed after the amplified spontaneous emission source to reduce the power of unused light. To select out a single polarization mode, a fiber pigtailed polarizer  506  can be employed. A continuous-wave (CW) laser source with a central wavelength of 1542 nm, such as the Clarity-NLL-1542-HP from Wavelength Reference, can be employed as the local oscillator (LO)  508  in coherent detection. It is not necessary to stabilize the laser wavelength, which can never drift out the above 0.8-nm range under normal operation. A variable optical attenuator  510  can be used to adjust the LO power, and a fiber polarization controller  512  can be used to match the polarization of the LO with that of the thermal source  502 . Conjugate optical homodyne detection can be performed with a commercial 90° optical hybrid  514 , available from Optoplex, and two balanced amplified photodetectors  516 , available from Thorlabs. The outputs of the two balanced photodetectors can be sampled by a real time oscilloscope  520 , or other measurement device. 
     The noise of the two balanced detectors  516  of the depicted embodiment are 0.37 and 0.35 in the shot-noise unit. The overall detection efficiency (taking into account the loss of the 90 optical hybrid  614  and the quantum efficiency of the photodiodes  516 ) is about 0.5.  FIG. 7  shows the distributions of the measurement results with either a vacuum input or a thermal input. By normalizing the quadrature variances of the thermal state to the vacuum noise, the average photon number (per mode) of the thermal state can be determined to be about 800. Such a thermal source is bright enough to implement the passive CV-QKD scheme, discussed above.  FIG. 8  shows a two-dimensional histogram of the measured data when the input is a thermal state. The small deviation from a perfect two-dimensional Gaussian distribution can be due to the non-uniform bin size of the 8-bit analog-to-digital converter of the oscilloscope. 
     The single-time second-order correlation function g (2) (0) is a parameter to characterize a photon source. g (2) (0) can be conveniently calculated from the statistics of the conjugate homodyne measurement using 
     
       
         
           
             
               
                 
                   
                     
                       
                         g 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             Z 
                             2 
                           
                           ) 
                         
                         - 
                         
                           4 
                           ⁢ 
                           
                             ( 
                             Z 
                             ) 
                           
                         
                         + 
                         2 
                       
                       
                         
                           ( 
                           
                             
                               ( 
                               Z 
                               ) 
                             
                             - 
                             1 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     where Z=X 2 +P 2 . 
     The g (2) (0) of one embodiment of an ASE source is 2.012, which is reasonably close to the theoretical value of 2 for a perfect thermal source. 
     In the GMCS QKD system, the quantum state preparation step is implemented using a random number generator, a weak coherent source, and high performance optical modulators. Quantum random number generation schemes are discussed in U.S. patent application Ser. No. 16/057,345 to Qi, entitled Quantum Random Number Generator, filed Aug. 7, 2018; and QI, B., “True randomness from an incoherent source”, Review of Scientific Instruments 88, 113101, published Nov. 1, 2017, both of which are incorporated by reference in their entirety. In the present invention, a passive state preparation scheme is described where Alice and Bob generate shared randomness by measuring correlated thermal states split from a common thermal source. This scheme significantly simplifies the implementation of CV-QKD and makes it more practical. 
     There are two potential types of imperfections on the transmitter side. One is associated with the thermal source itself, and the other is associated with transmitter side detector. 
     To deal with the imperfection of the thermal source itself, precise quantum state tomography can be performed to quantify the deviation of the output of the source from a perfect thermal state. There should be a lack of correlation between quadrature values of different modes. Otherwise, Eve may gain information without introducing noise by measuring modes not detected by Alice and Bob. Once the imperfection of the source has been quantified, it should be taken into account in the secure proof and key rate calculation. The output of a practical light source may drift with time, and the state tomography process may be repeated. In the conventional active state preparation scheme, both the laser source and the modulators need to be calibrated over time for similar reasons. A thermal source operated in continuous wave mode may show better stability than optical modulators operated at high speed. 
     Alice&#39;s detector noise can be untrusted and thus the corresponding excess noise can be attributed to Eve&#39;s attack. In practice it is unlikely that Eve can access Alice&#39;s system, so Alice&#39;s detector noise could be trusted, just like the noise from Bob&#39;s detector. Under reverse reconciliation, the trusted noise from Alice&#39;s detector will reduce the mutual information I AB  but will not change Eve&#39;s information χ BE . This trusted noise model can tolerate higher detector noise and work with a lower photon number of the source. To justify the trusted noise model in practice, a specially designed monitoring system may be utilized to prevent Eve from manipulating the detector performance. 
     The randomness is generated from a thermal source. While quantum randomness is ultimately connected to quantum superposition states, in the trusted device scenario, the state received by the detector does not need to be a pure state. For example, trusted randomness can be generated by measuring only one photon from an entangled photon pair, given Eve cannot access the other photon. In the current embodiments, photons from the thermal source are generated through spontaneous emission processes and are entangled with the atoms inside the source. Given the source itself is protected from Eve, true randomness can be generated. 
     In practice, embodiments can be implemented using a broadband source operated in continuous wave or pulsed mode (a multimode source), thanks to the mode-selective feature of coherent detection. Since the randomness carried by different modes of the source are independent of each other, to generate shared randomness Alice&#39;s and Bob&#39;s detectors can measure the same mode of the source. Furthermore, Alice and Bob have a scheme to establish a phase reference between their homodyne detection. Two schemes have been developed in CV QKD based on active state preparation, and either of them can be adopted in the passive state preparation protocol. In the first scheme, Alice generates two strong LOs from the same laser, uses one of them in her local measurement, and sends the second one to Bob to be used as a LO in his measurement. As discussed in U.S. Pat. No. 9,768,885, which was previously incorporated by reference, in a second scheme, both Alice and Bob generate LO signals from their own local lasers. The phase relation between their measurement bases can be determined by sending a relatively weak phase reference pulse between Alice and Bob. 
     Directional terms, such as “vertical,” “horizontal,” “top,” “bottom,” “upper,” “lower,” “inner,” “inwardly,” “outer” and “outwardly,” are used to assist in describing the invention based on the orientation of the embodiments shown in the illustrations. The use of directional terms should not be interpreted to limit the invention to any specific orientation(s). 
     The above description is that of current embodiments of the invention. Various alterations and changes can be made without departing from the spirit and broader aspects of the invention as defined in the appended claims, which are to be interpreted in accordance with the principles of patent law including the doctrine of equivalents. This disclosure is presented for illustrative purposes and should not be interpreted as an exhaustive description of all embodiments of the invention or to limit the scope of the claims to the specific elements illustrated or described in connection with these embodiments. For example, and without limitation, any individual element(s) of the described invention may be replaced by alternative elements that provide substantially similar functionality or otherwise provide adequate operation. This includes, for example, presently known alternative elements, such as those that might be currently known to one skilled in the art, and alternative elements that may be developed in the future, such as those that one skilled in the art might, upon development, recognize as an alternative. Further, the disclosed embodiments include a plurality of features that are described in concert and that might cooperatively provide a collection of benefits. The present invention is not limited to only those embodiments that include all of these features or that provide all of the stated benefits, except to the extent otherwise expressly set forth in the issued claims. Any reference to claim elements in the singular, for example, using the articles “a,” “an,” “the” or “said,” is not to be construed as limiting the element to the singular. Any reference to claim elements as “at least one of X, Y and Z” is meant to include any one of X, Y or Z individually, and any combination of X, Y and Z, for example, X, Y, Z; X, Y; X, Z; and Y, Z.