Systems and methods for entanglement based CV-QKD with information reconciliation over entanglement assisted link

Abstract

An entanglement based continuous variable (CV) QKD scheme is proposed performing the information reconciliation over the entanglement assisted link. The same entanglement generation source can be used in both raw key transmission and information reconciliation. The entanglement generation source employs only the low-cost devices operated in the C-band. The proposed CV-QKD scheme with information reconciliation over the entanglement assisted link significantly outperforms corresponding CV-QKD scheme with information reconciliation over the authenticated public channel. It also outperforms the CV-QKD scheme in which classical free-space optical communication link is used to perform the information reconciliation.

Description

SYSTEMS AND METHODS FOR ENTANGLEMENT BASED CV-QKD WITH INFORMATION RECONCILIATION OVER ENTANGLEMENT ASSISTED LINKGOVERNMENT SUPPORT

[0001] This invention was made with government support under Grant No. 2244365 awarded by the National Science Foundation. The government has certain rights in the invention.CROSS REFERENCE TO RELATED APPLICATIONS

[0002] This is a PCT application that claims benefit to U.S. provisional application serial number 63 / 639,442 filed on April 26, 2024 which is incorporated by reference in its entirety.FIELD

[0003] The present disclosure generally relates to quantum technologies; and in particular to entanglement based continuous variable (CV) - quantum-key distribution (QKD) with information reconciliation over entanglement assisted link.BACKGROUND

[0004] With quantum communications technologies, a strong turbulence regime poses different technical challenges. This regime may occur in, e.g., terrestrial free-space optical links, in uplink to high-altitude platform systems (HAPS), uplink / downlink to / from satellites, near-Earth applications, underwater communications, to mention a few.

[0005] It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed.BRIEF DESCRIPTION OF THE DRAWINGS

[0006] FIG. 1 is an illustration of a proposed entanglement based CV-QKD scheme with information reconciliation over the entanglement assisted link.

[0007] FIG. 2 is an illustration of the reverse information reconciliation entanglement assisted testbed developed at the University of Arizona (campus).

[0008] FIG. 3 is an illustration of the entanglement based CV-QKD free- space optical testbed.

[0009] FIG. 4 is a graph of SKR vs. total channel attenuation experienced over the free-space optical (FSO) link for different information reconciliation (IR) schemes; the raw key signaling rate was 10Gb / s.

[0010] FIG. 5 is a histogram of received power during experiments.

[0011] FIG. 6 is an example process associated with the inventive concept decribed herein.

[0012] Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims.DETAILED DESCRIPTION

[0013] The present disclosure relates to systems and methods associated with an entanglement based continuous variable (CV) QKD scheme for performing the information reconciliation over the entanglement assisted link. The same entanglement generation source is used in both raw key transmission and information reconciliation. The entanglement generation source employs only the low-cost devices operated in the C-band. The proposed CV-QKD scheme with information reconciliation over the entanglement assisted link significantly outperforms corresponding CV-QKD scheme with information reconciliation over the authenticated public channel. It also outperforms the CV-QKD scheme in which classical free-space optical communication link is used to perform the information reconciliation. The experimental demonstration over the free- space optical testbed established at the University of Arizona campus indicates that the proposed CV-QKD can operate in strong turbulence regime. To improve the secret key rates performance further the adaptive optics is used.

[0014] 1. Introduction

[0015] With the help of entanglement one can: beat the classical channel capacities, achieve better than classical sensors sensitivity, and provide quantum mechanics based security. The security of quantum key distribution (QKD) is guaranteed by the quantum information processing theorems, such as no-cloning theorem and theorem on indistinguishability of arbitrary quantum states, rather than computational complexity. Different photon degrees of freedom can be utilized in QKD including polarization, time, frequency, phase, and orbital angular momentum. Among different QKD protocols, discrete variable (DV)-QKD and continuous variable (CV)-QKD are very popular ones. In DV-QKD schemes, a single photon detector (SPD) is applied, while in CV-QKD we rely on uncertainty principle. With CV-QKD we can achieve higher secret key rates (SKRs) compared to corresponding DV-QKD schemes thanks to its compatibility with the state-of-the-art telecom optical communications.

[0016] In the present disclosure, one inventive concept focuses with the entanglement based free-space optical (FSO) continuous variable (CV) QKD scheme in which the reverse information reconciliation is performed over the entanglement assisted communication link. The entanglement assisted communication is based onprinciples described in the following references, incorporated by reference in their entireties: Djordjevic, I. B., On entanglement assisted classical optical communication with transmitter side optical phase-conjugation. IEEE Access 2021 , 9, 168930 - 168936; Nafria, V., Djordjevic, I. B. Entanglement Assisted Communication over the Free-Space Optical Link with Azimuthal Phase Correction for Atmospheric Turbulence by Adaptive Optics. Optics Express 2023, 31 (24), 39906-39916; and Djordjevic, I. B., Nafria, V. Entanglement Based Detection, Networking, Sensing, and Radars. IEEE Photonics Journal 2024, 16 (1 ), 7300610.

[0017] To reduce the system complexity and cost the same entanglement generation source is used for both raw key transmission and information reconciliation. Instead of using high-cost 780 nm pump laser to implement the entangled source, an entanglement generation source was developed using only low-cost telecom devices operated in C-band. To experimentally evaluate the proposed CV-QKD system, a free- space optical testbed at the University of Arizona campus was developed with propagation path length of 1 .5 km. It was experimentally demonstrated that in strong turbulence regime the proposed CV-QKD scheme with information reconciliation over entanglement assisted link significantly outperforms the corresponding conventional scheme performing the information reconciliation over the authenticated classical channel. To improve the secret key rates performance the adaptive optics is used.

[0018] The invention is organized as follows. In Section 2 the proposed entanglement CV-QKD scheme with information reconciliation over the entanglement assisted link is described. In Sec. 3 the terrestrial FSO CV-QKD testbed is described that was developed at the University of Arizona campus. Experimental results are provided in Sec. 3. Some important concluding remarks are given in Sec. 4.

[0019] 2. Proposed Entanglement based CV-QKD with InformationReconciliation over the Entanglement Assisted Links

[0020] The proposed entanglement based CV-QKD scheme with information reconciliation over the entanglement assisted link is provided in FIG. 1. The entanglement generation source, based on the parametric down conversion (PDC), is placed on Alice side. The PDC entangled source generates the two-mode squeezedvacuum (TMSV) states, which can be represented in the number (Fock) states basis as follows:

[0021] the mean photon number corresponding to either Alice (A) or Bob (B) qubits. The Alice and Bob photon creation (annihilation) operators are described by aA(aA~) andThe phase-sensitive crosscorrelation (PSCC) coefficientis related to the Alice-Bob photon pair entanglement.

[0022] The Wigner covariance matrix of the pure maximally entangled zero-mean Gaussian TMSV state is given by:

[0023] where 1 denotes the identity matrix and Z is the Pauli Z-matrix. Evidently, in regime with Ns« 1, the phase-sensitive cross-correlation (PSCC) coefficient is (dAaB) « N and in comparison with the classical limit Ns, and we have that ■ NS' » Ns.

[0024] By using the electro-optical l / Q modulator, Alice randomly selects the point in the signal-space (phase space). With optical switch in position 1 , Alice performs the homodyne balanced detection on her qubit photons by mixing them with the local oscillator (LO) photons using directional coupler based balanced detector. By setting the phase-shift after the LO oscillator to either 0 or n / 2, Alice selects to measure either in-phase or quadrature component. The Bob’s qubit photons at the output of the entangled source are transmitted over the quantum channel. With optical switch in position 1 , Bob’s randomly measures either in-phase on quadrature component with the help of his homodyne balanced detector. By selecting the instances when Alice and Bobmeasured the same component, after corresponding analog-to-digital converters (ADCs), Bob and Alice obtain the raw keys x and y respectively.

[0025] In conventional scheme Alice and Bob will further perform the information reconciliation over the public channels to which Eve has access to. In our proposed solution, we perform the reverse information reconciliation over the entanglement assisted system as shown in FIG. 1 by employing the same entangled source used for raw key transmission. With both optical switches in position 2, Bob performs LDPC encoding to get the parity bits s. With the help of phase modulator, the parity bits are imposed on Bob’s qubit photons, and transmitted over the same quantum channel in the opposite direction. Alice then performs the homodyne balanced detection on received photons from Bob by using her qubit’s photons as the reference photons. Given that Bob’s and Alice’s photons are entangled, the information reconciliation is performed over the entanglement assisted system. Following the balanced homodyne detection, Alice performs the LDPC decoding to get the correct key identical to Bob one. Finally, Alice and Bob perform the privacy amplification to remove any correlation with Eve and thus get the secure key.

[0026] We can calculate the normalized secret key rate (SKR) as follows: r = pi(A; B) -X(B,- E), (3)

[0027] where is the reconciliation efficiency, / (A; B) is the mutual information between Alice and Bob, and / (B;E) is the Holevo trans-information between Bob and Eve, denoted as / (B; E). The mutual information between Alice and Bob is identical for both individual and collective attacks and is given by:

[0028] where v is the variance of the source, while the variance of the total noise, denoted as totai, is obtained by the summing up the variance of the channelnoise xUneand the homodyne detection noise Xhomodyne , which can be expressed by referring to the channel input by:

[0029] with T being the transmittance of the channel, vetis photodiodes’ electrical noise, J? is the detector efficiency, and £ is the excess noise that accounts for the modulation imperfections, the phase noise, the relative intensity noise (RIN) of the LO reference signal, etc. The Holevo trans-information between Bob and Eve is determined by:

[0030] are the symplectic eigenvalues of corresponding covariance matrices determined by:

[0031] with corresponding parameters A, B, C, and D being defined by:

[0032] The reconciliation efficiency in the proposed CV-QKD scheme is determined by:

[0033] where R is the code rate of the LDPC code used in the reverse information reconciliation (RIR), while / (B; ) is the mutual information between Bob and Alice channel used in the RIR, which is different from one used in Eq. (4), as discrete modulation is used in the RIR, while Gaussian modulation in raw key transmission.

[0034] 3. Description of Terrestrial FSO Testbed to Study the ProposedEntanglement based CV-QKD Scheme

[0035] To evaluate the SKR performance of the proposed entanglement based CV-QKD scheme, we have developed the free-space optical (FSO) testbed at the University of Arizona campus. In FIG. 2, the reverse reconciliation entanglement assisted (EA) communication testbed is illustrated and described that is composed of the following stages: 1 ) entanglement generation source, 2) the WDM demultiplexerbased stage to separate the signal and idler photons, 3) modulation stage, 4) transmission stage, 5) beam collection and compression stage, 6) stage to delay the idler photons, 7) homodyne balanced detection stage, and 8) BER computing stage. The experimental setup is located in ECE Room 549 of the ECE building at University of Arizona, where Quantum Communication (QuCom) Lab is located. The entanglement generation source employs only low-cost telecom devices operated in the C-band. The tunable laser set to 1545.9 nm is used as the pump laser, whose output is amplified by the high-power EDFA and split into two parts using a 50:50 beam splitter. The top beam splitter output is used as the input to a type-0 periodically poled lithium niobate (PPLN) waveguide. In this PPLN waveguide the entangled photon pairs are generated by processes of secondary harmonic generation (SHG), followed by the spontaneous parametric down-conversion (SPDC). The entanglement pair we selected are 1550 nm as signal photons and 1541 .8 nm as idler photons. The signal photons are modulated using a phase modulator, which is modulated by the RF signal from an arbitrary waveform generator (AWG), set to 10 Gb / s, in which the LDPC encoded BPSK information sequence is recorded. We then transmit the phase modulated signalphotons out of the Lab towards a retro reflector placed around 750 meters away on the roof top of Optical Sciences Meinel building. By using the mirror in the middle of the link we can establish connection when there is no line-of-sight between Alice and Bob. The reflected beam at ECE 549 lab window, after a round trip of ~1 .5 km, is collected by the periscope and compressing telescope. On the other hand, we perform the optical phase-conjugation on idler photons by mixing them with 1545.9 nm amplified pump signal and passing them through the bottom PPLN waveguide, thus performing the difference frequency generation. The output of the bottom (phase-conjugation) PPLN waveguide is passed through WDM demultiplexer, we select 1550nm output, and the phase-conjugated photons are propagated over 1 km of SMF, which serves as the optical delay line (ODL). After collecting the signal photons by a compressing telescope, we pass them over the optical bench with an adaptive optics setup and finally couple it in an optical fiber. The signal photons and phase-conjugated idler photons are fed to a balanced homodyne detector and the RF output of the balanced detector is recorded by a real-time oscilloscope running at sampling rate of 100 GSa / s. The recorded waveforms are then processed by the PC to perform uncoded BER calculation, LDPC decoding, post-FEC BER calculation, and reverse information reconciliation efficiency calculation. Further, to provide the improvements in BER the adaptive optics (AO) setup is used where the beam after the compressing telescope goes through AO setup, which is composed of a deformable mirror (DM) and a wavefront sensor (WFS) operated in a servo loop.

[0036] The corresponding entanglement based FSO testbed for raw key transmission study is provided in FIG. 3.

[0037] Similarly to FIG. 2, the same entangled source is used for raw key transmission. Bob’s qubit photons at 1550 nm are transmitted over the 1.5 km FSO link. Alice’s 1541.8 nm qubit photons are modulated by the l / Q modulator. The ODL matches the FSO propagation time of Bob’s photos. Alice mixes her qubit photons on optical hybrid before balanced detector (not shown) with LO photons and measures randomly either in-phase or quadrature component by the homodyne balanced detector. Bob mixes his photons received by the compressing telescope on optical hybrid and selects to measure either the in-phase or quadrature component of the received signal. Bobannounces his selections of the measured component but keeps the results of the measurement in private. Bob and Alice keep the instances when they measured the same component as the raw keys x and y, respectively.

[0038] Bob’s encodes his raw key by the LDPC code, selected based on FSO channel conditions, imposes such encoded BPSK sequence by phase modulator and transmits it over entanglement assisted scheme shown in FIG. 2. The rest of the protocol is the same as described in previous section. Corresponding experimental results are provided in incoming Section.

[0039] 4. Experimental Results

[0040] The secret-key rate results are summarized in FIG. 4 for raw key rate of 10 Gb / s. The calculations are based on Eqs. (3)-(9). The histogram of the received power, provided in FIG. 5, has the Rayleigh distribution which indicates that the experimental demonstrations have been conducted in strong turbulence regime [13,14], The system parameters have been found to be= 0.8, Vei= 0.0321, and E = 0.011. We compare the SKRs for different reverse information reconciliation (IR) schemes: the proposed EA-based IR scheme with and without adaptive optics, the IR based on classical communication scheme operated over the FSO link, and the IR over the authenticated public channel. Clearly, the proposed EA-based IR scheme outperforms all other schemes for the range of total FSO channel attenuation found for the duration of the experiments. The achievable reconciliation efficiencies are provided in the Figure. For the total channel loss of 10 dB, the SKR of the proposed scheme is 0.92 Gb / s, while the SKR of IR over the classical FSO link is 0.39 Gb / s. The SKR of corresponding scheme with IR over the authenticated public channel is only 0.21 Gb / s. Therefore, the CV-QKD with the IR over the EA link significantly outperforms the corresponding conventional scheme with IR over the authenticated channel. In strong turbulence regime, the adaptive optics does not provide significant improvement in the SKR.

[0041] 5. Concluding Remarks

[0042] The entanglement based CV-QKD scheme has been proposed performing the information reconciliation over the entanglement assisted communication link. In the proposed scheme the same entanglement generation source, developed byemploying the spontaneous parametric down conversion, has been used for dual purpose: raw key transmission and information reconciliation. The developed entanglement generation source employs the low-cost telecom devices operated in the C-band. To evaluate the proposed CV-QKD scheme, the free-space optical testbed has been developed at the University of Arizona campus with propagation path of length of 1 .5 km. Experimental verification has shown that in the strong turbulence regime the proposed CV-QKD scheme with information reconciliation over the entanglement assisted link has been capable of significantly outperforming the corresponding CV- QKD scheme performing the information reconciliation over the authenticated public channel. The proposed scheme has also been shown to outperform the CV-QKD scheme in which classical FSO communication link has been used to perform the information reconciliation. It has been found that in the strong turbulence regime the adaptive optics does not provide significant improvement in the secret key rates.

[0043] It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto.

Claims

CLAIMSWhat is claimed is:

1. A method, comprising: transmitting a first set of qubit photons from an entangled source over a quantum channel from a first apparatus to a second apparatus; accessing, at the second apparatus, a second set of qubit photons associated with the first set of qubit photons; and applying information reconciliation between the first apparatus and the second apparatus over an entanglement-assisted link by: performing, at the second apparatus, Low-Density Parity Check (LDPC) encoding resulting in a set of parity bits; imposing, at the second apparatus, the set of parity bits on the second set of qubit photons resulting in a set of parity-imposed qubit photons; transmitting, at the second apparatus, the set of parity-imposed qubit photons to the first apparatus over the quantum channel; applying, at the first apparatus, a homodyne balanced detection operation on the set of parity-imposed qubit photons using the first set of qubit photons as reference photons; and performing, at the first apparatus, Low-Density Parity Check (LDPC) decoding resulting in a corrected key that matches a second raw key associated with the second apparatus.

2. The method of claim 1 , further comprising: applying, at the second apparatus and at the first apparatus, a privacy amplification operation to remove correlation with a public channel to get a secure key.

3. The method of claim 1 , wherein the step of applying information reconciliation between the second apparatus and the first apparatus over the entanglement- assisted link includes bypassing an information reconciliation mechanism associated with a public channel.

4. The method of claim 1 , further comprising: generating, at the first apparatus, the first set of qubit photons at a parametric down conversion source; randomly selecting, at the first apparatus, a point in signal phase space for the first set of qubit photons; and applying, at the first apparatus, a homodyne detection operation on the first set of qubit photons by mixing with local oscillator photons using a directional coupler-based balanced detector.

5. The method of claim 1 , further comprising: selecting, at the second apparatus, a phase shift associated with the first set of qubit photons to measure an in-phase component or a quadrature component; and randomly measuring, at the second apparatus, an in-phase component or a quadrature component associated with the second set of qubit photons using a homodyne balanced detector.

6. The method of claim 5, further comprising: obtaining, by the first apparatus, the first raw key associated with measurement of the in-phase component or the quadrature component; and obtaining, by the second apparatus, a second raw key associated with measurement of the in-phase component or the quadrature component; where the first raw key and the second raw key correlate with instances where the first apparatus and the second apparatus both measure the in-phase component or where the first apparatus and the second apparatus both measure the quadrature component.

7. The method of claim 4, generating the entangled photons by: employing a pump laser operating in S- / C- / L-band and periodically polling a lithium niobate (PPLN) waveguide of sufficient length to perform cascaded second harmonic generation and difference frequency generation, and separating the entangled photons with the help of a WDM demultiplexer.

8. The method of claim 7, further comprising performing the phase-conjugation on idler photons with the help of a second PPLN waveguide by: mixing the half of the pump signal and idler photons before applying to the PPLN waveguide input to generate the phase-conjugated photons having the same wavelength as the signal photons so that the homodyne detector can be used as an entanglement assisted detector.

9. The method of claim 3, wherein the second apparatus modulates the signal photons by the parity-check bits and sends them over turbulent free-space optical channel, and the first apparatus demodulates the received signal photons by performing the homodyne detection in the balanced detector with the phase- conjugated photons serving the role of the LO laser signal.

10. The method of claim 9, wherein the first apparatus is performing LDPC decoding by using the demodulated samples of received parity-check bits with information bits generated by the first apparatus and obtaining the corrected key after the LDPC decoding, and performing further privacy amplification on the corrected key to get the secure key.

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