An improved statistical fluctuation analysis method for reference-frame-independent quantum key distribution protocol

Abstract

The application belongs to the fields of quantum communication and quantum information technology, and discloses an improved statistical fluctuation analysis method for a reference frame independent quantum key distribution protocol, which comprises the following steps: step 1, quantum state preparation; step 2, quantum state measurement; step 3, base vector comparison; step 4, parameter estimation; and step 5, code rate under failure probability after the McDiarmid inequality is adopted. The application can directly perform statistical fluctuation analysis on channel parameters and key rates, and can make the estimation of the channel parameters more compact and the final secure code rate higher by reducing the finite length effect of the measurement data without changing the reference frame independent quantum key distribution experimental setting and post-processing method.

Description

Technical Field

[0001] This invention belongs to the fields of quantum communication and quantum information technology, specifically relating to an improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols. Background Technology

[0002] In practical quantum key distribution (QKD) systems, both communicating parties need to calibrate their reference frames in real time. For example, in a satellite-to-ground polarization-coded QKD system, due to factors such as the Earth's rotation and the satellite's orbital motion, the satellite and ground receiving station need to continuously adjust the polarization angle to ensure the QKD functions correctly. Similarly, in phase coding, due to changes in ambient temperature and mechanical vibrations of the experimental platform, the difference in interferometer arm lengths between the transmitter Alice and receiver Bob will change, thus requiring them to actively compensate for the phase. The operation of Alice and Bob actively adjusting the polarization state and compensating for the phase is called reference frame calibration. Reference frame calibration consumes a significant amount of time, reducing the efficiency of the entire communication process and increasing the complexity and cost of the QKD system, potentially even posing security risks. In 2010, Laing A, Scarani V, Rarity JG, and others proposed the Reference-Frame-Independent Quantum Key Distribution (RFI-QKD) protocol in their paper "Reference-Frame-Independent Quantum Key Distribution," which effectively solves the problem of needing to calibrate the reference frame. The reference frame-independent quantum key distribution (RFI-QKD) protocol is immune to the effects of slow reference frame drift, effectively estimating the amount of information Eve steals, and ultimately generating a secure key.

[0003] However, in the reference frame-independent quantum key distribution (RFI-QKD) protocol, measurement data with three bases and different basis vector combinations are required. Therefore, it is severely affected by the finite length effect, which leads to non-tight channel parameter estimation and limits the final key rate and distance of the reference frame-independent quantum key distribution (RFI-QKD) system. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides an improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols. By using an improved McDiarmid inequality, statistical fluctuation analysis can be directly performed on channel parameters and the key rate R. This method, without altering the RFI-QKD experimental setup and post-processing methods, reduces the finite-length effect of measurement data, resulting in a more compact estimation of channel parameters and ultimately a higher secure key generation rate. Furthermore, this method can resist mass attacks and significantly improve the key rate while ensuring system security.

[0005] To achieve the above objectives, the present invention is implemented through the following technical solution:

[0006] This invention provides an improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols, comprising the following steps:

[0007] Step 1, Quantum State Preparation: Alice prepares quantum states with probability P. μ P ν 1-P μ -P ν Randomly modulated signal state, decoy state, and vacuum state pulses. For the signal state pulse, Alice uses conditional probability... Preparation of Z A X A and Y A Quantum states under the basis; for decoy state pulses, Alice respectively with probability and Preparation of Z A X A and Y A Quantum states under the basis.

[0008] Step 2, Quantum State Measurement: Bob uses probability and Randomly select Z B X B and Y B The base measures the received pulses and records the measurement results.

[0009] Step 3, Basis Alignment: Alice and Bob publish the basis vectors and intensity selection information through a certified classical channel, preserving the prepared measurement basis vector combination Z. A Z B X A X B X A Y B Y A X B and Y A Y B The data below is discarded, while data for other basis vector combinations are discarded. Alice and Bob randomly select bits from the sieved key to estimate the basis vector combination Z. A Z B Gain below And the total sub-bit error rate. Since the RFI-QKD protocol requires measurement data using three sets of basis vectors and different basis vector combinations, it is severely affected by the finite length effect. Therefore, the McDiarmid inequality is used for statistical fluctuation analysis.

[0010] Step 4, Parameter Estimation: The single-photon bit error rate and intermediate parameter C under each basis vector combination can be estimated. The amount of information stolen by Eve, I. E It is the combination of intermediate parameter C and basis vectors Z A Z B The upper bound of the single-photon bit error rate is a function of the base, and I is reduced using the McDiarmid inequality. E The finite-length effect makes channel parameter estimation more compact;

[0011] Step 5: Obtain the final failure probability as The formula for the code generation rate R is as follows.

[0012] Furthermore, the McDiarmid inequality is used to reduce the finite-length effect in the code generation rate formula. When the failure probability is... The code generation rate is:

[0013]

[0014] Among them, I E It's the amount of information Eve stole. P1(μ) is the probability of a single photon with intensity μ. Since k = 3 (odd), we take k0 = 2. It is the variable j from k0 to k, but skips n. It is an intensity of μ n The gain. For all intensities of μ n Calculate the gain after averaging. It represents the number of bits used in Z-based preparation and measurement. Let χ be the probability of failure, and ε be 10. sec and ε cor are the security parameters for the privacy and correctness of the key ultimately shared between Alice and Bob, respectively. f is the error correction efficiency, and H(x) = -xlog2(x) - (1-x)log2(1-x) is the binary Shannon entropy function.

[0015] Furthermore, single-photon gain Statistical fluctuations caused by finite-length effects can be handled using Hoeffding's inequality for hypergeometric random variables. This is a special case of McDiarmid's inequality. Applying Hoeffding's inequality to hypergeometric random variables, we have: The probability of failure at this time does not exceed Here, χ is 10, and the function is defined. esssup is the essential maximum value, and essinf is the essential minimum value.

[0016] Furthermore, the McDiarmid inequality is used to reduce I. E The finite-length effect.

[0017]

[0018] in

[0019]

[0020]

[0021]

[0022] The single-photon bit error rate can be obtained under the following conditions:

[0023]

[0024]

[0025]

[0026] ξ A ξ B ∈{X,Y,Z}. and They are respectively ξ A ξ B Under basis vector combination, with intensity μ n Gain and average bit error rate at that time.

[0027] Furthermore, the central sequence This relates to the improved McDiarmid inequality, which is characterized as follows:

[0028] in: {w P(i)} is a decreasing sequence, allowing the set The k elements in the array are arranged in descending order as {w (1) ,w (2) ,……,w (k)}, n (i) Is with The number of corresponding Bell basis measurement events. And:

[0029]

[0030]

[0031]

[0032]

[0033] and They are respectively ξ A ξ B Conditional gain and bit error rate of a single photon under basis vector combination For ξ A ξ B Number of bits in a basis vector combination.

[0034] Furthermore, the parameters y, t, x, It is related to the single-photon count rate and bit error rate, and its characterization is as follows:

[0035]

[0036]

[0037]

[0038] From the above formula, we can obtain The minimum value.

[0039] The beneficial effects of this invention are as follows: The improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols described in this invention, using an improved version of the McDiarmid inequality, can directly perform statistical fluctuation analysis on channel parameters and key rate R. The RFI-QKD protocol requires measurement data using three bases and different basis vector combinations, thus being severely affected by the finite-length effect. This method, without changing the RFI-QKD experimental setup and post-processing methods, reduces the finite-length effect of the measurement data, resulting in a more compact estimation of channel parameters and ultimately a higher secure key generation rate. This method can resist mass attacks and significantly improves the key rate while ensuring system security. Furthermore, compared to other statistical fluctuation methods, the actual performance of quantum keys under this method is the best, exhibiting the best performance in terms of security assumptions, key generation rate, and transmission distance. Attached Figure Description

[0040] Figure 1 The solution of this invention is in the case of pulse number N=10 10 10 11 10 12 10 13 A comparison chart of fluctuations between time periods and those not recorded.

[0041] Figure 2 The solution of this invention is in the case of pulse number N=10 10 A comparison of key rates using the time-based statistical fluctuation method based on the Hoeffding inequality.

[0042] Figure 3 The solution of this invention is in the case of pulse number N=1011 A comparison of key rates using the time-based statistical fluctuation method based on the Hoeffding inequality.

[0043] Figure 4 The solution of this invention is in the case of pulse number N=10 12 A comparison of key rates using the time-based statistical fluctuation method based on the Hoeffding inequality.

[0044] Figure 5 This is a flowchart of the present invention. Detailed Implementation

[0045] To better illustrate the purpose, technical solution, and advantages of this invention, the following sections, in conjunction with specific embodiments and descriptions, will provide a more detailed explanation of the invention.

[0046] like Figure 5 As shown, this invention is an improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols, which includes the following steps:

[0047] Step 1, Quantum State Preparation: Alice prepares quantum states with probability P. μ P v 1-P μ -P ν Randomly modulated signal state, decoy state, and vacuum state pulses. For the signal state pulse, Alice uses conditional probability... Preparation of Z A X A and Y A Quantum states under the basis; for decoy state pulses, Alice respectively with probability and Preparation of Z A X A and Y A Quantum states under the basis.

[0048] Step 2, Quantum State Measurement: Bob uses probability and Randomly select Z B X B and Y B The received signal state pulses and decoy state pulses are measured, and the measurement results, i.e., the basis vector combination Z, are recorded. A Z B X A X B X A Y B Y A X B and Y A Y B The following data.

[0049] Step 3, Basis Alignment: After the signal transmission phase is complete, Alice and Bob publish the basis vectors and intensity selection information through a certified classical channel, retaining the prepared measurement basis vector combination Z. A Z B X A X B X A Y B Y A X B and Y A Y B The data from the sieved key is discarded, while data from other basis vector combinations are discarded. Alice and Bob randomly select bits from the sieved key to estimate the basis vector combination Z. A Z B X A X B X A Y B Y A X B and Y A Y B The theoretical calculation model for the gain and total sub-bit error rate under the given conditions is as follows:

[0050]

[0051]

[0052] Where P n Y represents the photon number distribution with average photon number λ∈{μ,ν,0}, where n∈{0,1,2,3...} represents the number of photons. n and e n Let ξ represent the count rate and bit error rate of the n-photon state, respectively. A ξ B This indicates that Alice chose ξ when preparing the quantum state. A ∈{X A Y A Z A Bob chose ξ as the basis when measuring the quantum state. B ∈{X B Y B Z B}base.

[0053] However, the number of pulses Alice transmits in a real system is finite. Therefore, in the practical security analysis of the RFI-QKD protocol, the impact of statistical fluctuations on parameter estimation needs to be considered. Therefore, this invention uses the McDiarmid inequality for statistical fluctuation analysis.

[0054] Single photon gain Statistical fluctuations caused by finite-length effects can be handled using Hoeffding's inequality for hypergeometric random variables, a special case of McDiarmid's inequality. Applying Hoeffding's inequality to hypergeometric random variables, we have: The failure probability at this point does not exceed ε / χ, where χ is 10. Define the function... esssup is the essential maximum value, and essinf is the essential minimum value. For all intensities of μ n Calculate the gain after averaging. This represents the number of bits used in the Z-based preparation and measurement. In this scheme, k = 3, μ1 = μ, μ2 = v, and μ3 = 0.

[0055] Step 4: Use single-photon counting rate Based on the aforementioned gain and qubit error rate, the basis vector combination ξ can be obtained. A ξ B Lower bound of single photon count rate and the upper bound of single-photon bit error rate

[0056]

[0057]

[0058]

[0059] in

[0060]

[0061]

[0062]

[0063] in, It is a central sequence, related to the improved McDiarmid inequality, and is expressed as:

[0064] in: {w P(i)} is a decreasing sequence, allowing the set The k elements in the array are arranged in descending order as {w (1) ,w (2) ,……,w (k)}, n (i) Is with The number of corresponding Bell basis measurement events. And:

[0065]

[0066]

[0067]

[0068]

[0069] in

[0070]

[0071]

[0072]

[0073] use as well as The intermediate parameter C can be obtained, and the calculation formula is as follows:

[0074]

[0075] The amount of information Eve stole: I E It is the combination of intermediate parameter C and basis vectors ξ A ξ B Upper bound of single-photon bit error rate The function can be represented as:

[0076]

[0077] in

[0078]

[0079]

[0080] H(x) = -x log2(x) - (1-x)log2(1-x) is the binary Shannon entropy function.

[0081] Step 5, Settings The final probability of failure is The following formula for code generation rate R:

[0082]

[0083] Appendix Figure 1 The solution of this invention is in the case of pulse number N=10 10 10 11 10 12 10 13 The comparison chart shows the fluctuations before and after the statistics. To simulate the actual situation, the simulation parameters are uniformly shown in Table 1.

[0084] Table 1 Simulation System Parameters

[0085]

[0086] in, This means Bob chose Z. B The probability of the basis, η Bob For detection efficiency, e d Where α is the background bit error rate, α is the fiber loss coefficient, f is the error correction probability, and Y0 is the dark count rate. Let e0 be the failure probability and e0 be the bit error rate of the vacuum pulse. This indicates that Alice chose Z under the condition of strength μ when preparing the quantum state. A The probability of the basis, P μ P represents the probability that Alice chooses the signal state. v Let β represent the probability that Alice chooses the decoy state, and β be the deflection angle.

[0087] from Figure 1 It can be seen that the fewer the number of pulses, the greater the impact of statistical fluctuations; conversely, the more pulses there are, the smaller the impact. When the number of pulses N is 10... 13 At this point, analysis using the improved McDiarmid inequality method reveals that the key rate is already very close to infinite. Statistical fluctuation analysis using the improved McDiarmid inequality method can yield higher key generation rates and longer transmission distances.

[0088] Figure 2 The solution of this invention is in the case of pulse number N=10 10 Comparison of key rates between time-based and Hoeffding's inequality statistical fluctuation method; Figure 3 The solution of this invention is in the case of pulse number N=10 11 Comparison of key rates between time-based and Hoeffding's inequality statistical fluctuation method; Figure 4 The solution of this invention is in the case of pulse number N=10 12 Comparison of key rates between time-based and Hoeffding's inequality statistical fluctuation method;

[0089] From the appendix Figures 2-4 It can be seen that when the number of pulses N = 10 10 10 11 10 12 At this time, the improved McDiarmid inequality method consistently achieves higher key rates and transmission distances than the Hoeffding inequality method. This is especially true when the pulse number is 10. 10 -10 12When using the improved McDiarmid inequality for statistical fluctuation analysis, the key rate is significantly improved. In contrast, the Hoeffding inequality shows a significant decrease in key rate when the number of transmit pulses is small, and is more sensitive to changes in the number of transmit pulses. Overall, using the McDiarmid inequality scheme for statistical fluctuation analysis in the three-strength decoy state RFI-QKD protocol yields the best actual performance.

[0090] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial results of the present invention. It should be understood that the current patent specification only introduces an improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols. For example, the method used in the specific embodiments of the present invention is also applicable to quantum key distribution systems based on other protocols and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols, characterized in that: The statistical fluctuation analysis method includes the following steps: Step 1, Quantum State Preparation: Alice prepared quantum states with probability. , , Randomly modulated signal state, decoy state, and vacuum state pulses. For the signal state pulse, Alice uses conditional probability... , , preparation , and Under the basis of quantum states, for decoy state pulses, Alice respectively with probability , and preparation , and Quantum states under the basis; Step 2, Quantum State Measurement: Bob uses probability , and Random selection , and The signal state pulses and decoy state pulses received by the basis measurement are measured, and the measurement results, i.e., basis vector combination, are recorded. , , , as well as The following data; Step 3, Basis Alignment: Alice and Bob publish the basis vectors and intensity selection information through a certified classical channel, preserving the basis vector combination for preparation. , , , as well as The data from the filtered key is discarded, and Alice and Bob randomly select bits from the filtered key to estimate the basis combinations. Gain below The total sub-bit error rate was analyzed using the McDiarmid inequality for statistical fluctuation analysis. Step 4, Parameter Estimation: Estimate the various basis vector combinations from Step 3. Single-photon bit error rate and intermediate parameters under the base The amount of information Eve stole It is an intermediate parameter Combination of base and vector The upper bound of the single-photon bit error rate is a function of the base, which is reduced using the McDiarmid inequality. The finite-length effect makes channel parameter estimation more compact; Step 5: The final failure probability after applying the McDiarmid inequality is... The code rate ;in, In step 3, the statistical fluctuation analysis is performed using the McDiarmid inequality, specifically as follows: Hoeffding's inequality for hypergeometric random variables is used to handle single-photon gain. Statistical fluctuations due to the finite-length effect: ， in, For all intensities Calculate the gain after averaging. yes The number of bits prepared and measured. It is the probability of failure. It is the strength of Gain, It's the amount of information Eve stole. = , It is the strength of The probability of a single photon; In step 4, the McDiarmid inequality is used to reduce... Finite-length effect: = ， in = ， = ， in, It is the binary Shannon entropy function.

2. The improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols according to claim 1, characterized in that: In step 4, intermediate parameters Represented as: C= + + + ， Using single-photon count rate The basis vector combination is obtained by combining the gain and qubit error rate estimated in step 3. single-photon bit error rate under basis vector combination : in, ， = in, , From arrive variables But skip , , , for Under the basis combination, the strength is Gain at time, for Under the basis combination, the strength is Bit error rate at time for Under the basis combination, the strength is Average bit error rate at time This is due to the deviation in the single-photon count rate caused by finite-length sampling. The central sequence.

3. The improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols according to claim 2, characterized in that: Central sequence Represented as: in, = , It is a decreasing sequence, making the set In The elements are arranged in descending order. , Is with The corresponding number of Bell basis measurement events, and: in, and They are respectively Conditional gain and bit error rate of a single photon under basis vector combination for Number of bits in a basis combination, parameters , , , The relationship between the single-photon count rate and the bit error rate is expressed as follows: Calculated from the above formula The minimum value.

4. The improved statistical fluctuation analysis method for reference frame-independent quantum key distribution protocols according to claim 3, characterized in that: In step 5, the McDiarmid inequality is used to reduce the finite-length effect in the code generation rate formula, with a failure probability of... The code rate for: in, , and These are security parameters for the privacy and correctness of the key ultimately shared between Alice and Bob. To improve error correction efficiency.

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