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Issue #4/2024
D. N. Frolovtsev, A. V. Demin
Influence of Quantum State Fidelity of a Single Photon Source on the Bit Error Rate in Quantum Key Distribution
Influence of Quantum State Fidelity of a Single Photon Source on the Bit Error Rate in Quantum Key Distribution
DOI: 10.22184/1993-7296.FRos.2024.18.4.282.294
All-Russian Research Institute of Optical and Physical Measurements (VNIIOFI), Moscow, Russia
The paper analyzes the efficiency of a quantum state fidelity to assess the raw bit error value introduced by the polarization state of a single photon source. The analysis was performed for the quantum key distribution schemes BB84 and BBM92. It has been shown theoretically and experimentally that when the fidelity is decreased from 1 to 0 in the BB84 scheme, the raw bit error value linearly increases from 0 to 1, and in the BBM92 scheme – from 0 to 1/2. The experimental setups for determine the influence of fidelity on the bit error value are described in detail.
All-Russian Research Institute of Optical and Physical Measurements (VNIIOFI), Moscow, Russia
The paper analyzes the efficiency of a quantum state fidelity to assess the raw bit error value introduced by the polarization state of a single photon source. The analysis was performed for the quantum key distribution schemes BB84 and BBM92. It has been shown theoretically and experimentally that when the fidelity is decreased from 1 to 0 in the BB84 scheme, the raw bit error value linearly increases from 0 to 1, and in the BBM92 scheme – from 0 to 1/2. The experimental setups for determine the influence of fidelity on the bit error value are described in detail.
Теги: fidelity quantum key distribution quantum state metrology spontaneous parametric light scattering квантовое распределение ключа метрология квантовых состояний спонтанное параметрическое рассеяние света фиделити
Influence of Quantum State Fidelity of a Single Photon Source on the Bit Error Rate
in Quantum Key Distribution
D. N. Frolovtsev 1, A. V. Demin 2
Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia
All-Russian Research Institute of Optical and Physical Measurements (VNIIOFI), Moscow, Russia
The paper analyzes the efficiency of a quantum state fidelity to assess the raw bit error value introduced by the polarization state of a single photon source. The analysis was performed for the quantum key distribution schemes BB84 and BBM92. It has been shown theoretically and experimentally that when the fidelity is decreased from 1 to 0 in the BB84 scheme, the raw bit error value linearly increases from 0 to 1, and in the BBM92 scheme – from 0 to 1/2. The experimental setups for determine the influence of fidelity on the bit error value are described in detail.
Keywords: quantum state metrology, fidelity, quantum key distribution, spontaneous parametric light scattering
Article received: February 10, 2024
Article accepted: March 15, 2024
1. Introduction
The development of an optical platform for quantum computing [1, 2] and quantum technology [3–6] requires metrological support for its constituent elements: single photon sources [7], single photon detectors [8], interferometers [9], polarization plates, filters, etc. to assess the value of errors introduced by the system components. To ensure the specifications of the photon source tools in the quantum optical technologies [10], it is proposed to use the parameters g(2) [11] and Grangier’s α [12, 13] related to the statistics of source photons. This paper analyzes the source of errors in the case of quantum key distribution that is associated not with the photon statistics, but with the inaccurate рreparation of the quantum polarization state by the source and the “ideal” one.
In order to provide metrological support of quantum polarization states of the single photon sources in the photon pairs, it is proposed to use a quality measure for the source reproduction of the required quantum state, namely fidelity [14]. Fidelity is a proximity measure for two quantum states: the quantum state required in a practical problem and the state of photons generated by the source used in practice. Mathematically, the fidelity value F is determined by the following expression:
F = | ψt | ψr | 2,
where | ψt is the quantum photon state required from the source, | ψr is an actually generated quantum state. If the source generates photons in a mixed state ˆρ, then fidelity is determined as follows:
F = | ψt | ˆρ | ψr | 2.
The purpose of this paper is to show that the raw bit error rate introduced by the non-ideal quantum photon polarization state in the case of quantum key distribution can be estimated on the basis of a single parameter, namely fidelity.
We will briefly describe the application of fidelity in quantum communications to determine the bit error level in the quantum key distribution schemes. On a theoretical and experimental level, section 2 examines the use of fidelity in the BB84 scheme [15], and section 3 – in the BBM92 scheme [16]. Section 4 provides the summarization of research.
2. Quantum key transfer
using the BB84 scheme
The concept of quantum cryptography was first proposed by C. Bennett and G. Brassard in 1984 [15]. The aim of the scheme is to generate an identical random bit sequence between two placeholders (called Alice and Bob). The communication channel security is determined by the fact that when listening to the quantum channel, the placeholders can reliably register the sequence compromise by a third participant, usually called Eve, based on the bit error rate [17]. Let us recall that the scheme can be divided into three stages.
At the first stage, Alice prepares the single photons. To do this, she randomly selects a coding basis: a laboratory or diagonal one. Then she selects a random number (“0” or “1”) and prepares a single photon with polarization, a random number. In the laboratory basis, “0” and “1” are encoded by the horizontal | H and vertical | V photon polarization, and in the diagonal basis – by the states of photon linear polarization ±45° to the horizontal: “0” corresponds to the polarization state | D = | H + | V —2 , and “1” corresponds to the polarization state | A = | H − | V —2 . The random selection procedure for a polarization basis and an information bit is repeated for each photon prepared by Alice.
At the second stage, Alice sends the prepared photons through a quantum channel to Bob, who measures their polarization in a randomly selected basis. Then the subscribers use an open channel to inform each other in which basis (laboratory or diagonal) they have performed the measurements, and keep only the results in the matching basis. In this way, a raw key is generated for the subscribers. If the devices used by Alice and Bob are ideal and there is no eavesdropping, the subscribers’ raw keys are identical. As follows from the no-cloning theorem [18] of a quantum state, listening to the quantum channel leads to the available errors in the raw key. For example, if some curious person (called Eve) is constantly listening while taking measurements of the photon polarization state in the quantum channel in a randomly selected basis and transmitting the measurement results to Bob instead of the original photon, then Alice and Bob should register an increase in the error up to 25%. If Eve measures not every photon, but only a part of the flow sent by Alice to Bob, then the introduced error value is decreased, although Eve learns less information about the key. The permissible amount of information that is fundamentally accessible to the attackers, as well as the methods applied to enhance the key secrecy [9, 19], determine the maximum permissible error level.
Any errors in the key generation process occur not only due to the eavesdropping, but also due to the inevitable equipment imperfection. For example, if Alice uses a source that produces photons with an error, then Bob will receive erroneous bits in the raw key as a result of measurements.
Let Alice’s photon source produce the single photons in a polarization state | ψ = α | H − β | V instead of | H . When registering the photon polarization in a laboratory basis with the probability of, where F is fidelity, Bob will obtain the correct measurement resultL the photon has horizontal polarization. If the probability is | V | ψ | 2 = | β | 2 = 1 − F, then Bob will register the vertical photon polarization, and an error will appear in the key. Similar relations between fidelity and the bit error value are developed when Alice prepares the remaining states | V , | D and | A. Thus, as fidelity decreases, the probability of error is increased.
The experimental setup layout to determine the influence of the quantum state preparation accuracy by the source on the bit error value is shown in Fig. 1. The setup simulates the operation of a quantum key distribution system using the single photons according to the BB84 scheme from Alice to Bob. The single photons are prepared using the spontaneous parametric light scattering (SPLS) [20] in a ppKTP crystal. A laser with a wavelength of 405 nm is used for pumping. As a result of SPLS, a pair of photons with a wavelength of 810 nm is generated, one of which is reflected by a polarizing beam splitter and recorded by a single photon detector. This event notifies that the Alice’s source has prepared a single photon. Using a half-wave plate, Alice sets the polarization direction of a single photon (Table 1). To simulate the non-ideal preparation of the polarization state by the single photon source, the Alice’s half-wave plate is additionally rotated by a certain angle θ, and the polarization plane is rotated by an angle 2θ. As a result, the fidelity of the prepared polarization state takes on the value F = cos2 2θ.
Bob detects the single photons in the laboratory and diagonal basis while selecting the measurement basis using his half-wave plate. For each basis, the measurements are performed at two positions of the wave plate, in which the detectors “swap their places” that makes it possible to consider their various quantum efficiencies in the calculations. The relevant positions of the wave plate are given in Table 2.
For the measurements in a laboratory basis, the half-wave plate is installed with a rotation angle 0° or 45°. In the first case, the photon passed through the Bob’s polarization beam splitter corresponds to the “0” bit, and the photon reflected by the beam splitter corresponds to the “1” bit. When the plate is positioned at 45° the bit values are opposite: the transmitted photon corresponds to “1”, and the reflected photon corresponds to “0”.
Bob makes measurements in the diagonal basis when the half-wave plate is position at an angle of 22.5° or (22,5° + 45°). When the plate is positioned at the photon passed through the Bob’s beam splitter corresponds to the “1” bit, and the reflected photon corresponds to the “0” bit. When the plate is positioned at (22.5° + 45°) the photon passed through the beam splitter corresponds to the “0” bit, and the reflected photon corresponds to “1”.
The measurement of the bit error rate depending on the fidelity value was performed as follows. In accordance with the selected fidelity value, the angle value θ was determined. Then Alice prepared the polarization state of single photons by installing her half-wave plate in accordance with Table 1. For each polarization state prepared by Alice, Bob made a bit error measurement. To do this, Bob installed his half-wave plate to measure polarization in a basis that coincided with the Alice’s basis, and measured the error probability (receipt of a bit value opposite to that set by Alice) for 20 seconds. The counting rate for the single photons correlated with the trigger Alice’s photocount by the Bob’s detectors is ~500 photocounts per second. The measurement results for four bit error values (for the laboratory and diagonal basis, the measurements in each basis were performed for two positions of the Bob’s half-wave plate) were averaged, and the bit error probability was calculated perr:
NL1err + NL2err + ND1err + ND2err
perr –––,
NL1norm + NL2norm + ND1norm + ND2norm
where NerrL1, 2/D1, 2 is the number of bits erroneously received by Bob, and NnormL1, 2/D1, 2 is the total number of bits received by Bob (including erroneous and correct bits). The indices L1, L2, D1, D2 correspond to the Bob’s measurement in the laboratory (L) and diagonal (D) basis, the numbers 1 and 2 correspond to various plate positions according to Table 2.
The measured value of the bit error depending on the fidelity is given in Fig. 2. The experimental data obtained confirm that fidelity makes it possible to unambiguously determine the bit error value introduced by the source in the BB84 scheme.
3. Quantum key distribution
using the BBM92 scheme
We will consider the influence of fidelity value on the bit error value during the quantum key distribution using the BBM92 scheme [16]. The scheme uses the pairs of quantum entangled particles. For the sake of argument, we will assume that the pairs of particles in the Bell polarization state are used | Ф(+) = | HH + | VV —2 . Alice and Bob each receive one of the entangled photons. The source of entangled photons can be external, or located at one of the subscribers. After the subscribers have each received one photon from the entangled pair, Alice and Bob measure the photon polarization. To do this, each subscriber has a meter similar to the meter used by Bob in the BB84 scheme discussed above. With the probability of 1 / 2, Alice measures the photon polarization in the laboratory basis, and with the probability of 1 / 2 – in the diagonal basis. Bob performs the same activities independently. Then Alice and Bob tell each other the basis in which the measurements have been made, but do not provide the specific measurement results.
Any errors in the developed key occur due to the eavesdropping, as well as in the case of using a low-quality source of entangled photons that generates the photon pairs in a state different from | Ф(+) determined by the density matrix ˆρ. Let us analyze the errors occurred due to the quantum state deviation of photon pairs from | Ф(+) .
To do this, we represent the density matrix of the source polarization state as follows:
ˆρ = F | Ф(+) Ф(+) | + ( 1 − F ) ˆρ⊥, (4)
where F = Ф(+) | ˆρ | Ф(+) is the fidelity value, and ˆρ⊥ meets the conditions of Ф(+) | ˆρ⊥ | Ф(+) = 0, Tr ˆρ⊥ = 1 and quasi-positive determinacy.
To obtain the error probability, we express the quantum states relevant to the error event(| HV , | VH , | AD and | DA ) in the basis of Bell states:
| HV = | ψ(+) + | ψ(−) .
| VH = | ψ(+) − | ψ(−) .
| AD = | Ф(−) + | ψ(−) .
| DA = | Ф(−) − | ψ(−) . (5)
Having determined the probabilities of state measurements in (5) using the density matrix (4), we obtain the raw bit error value:
per = pHV + pVH + pAD + pDA =
= 1 – F 1 + ψ(−) | ˆρ⊥ | ψ(−) 2. (6)
Due to the fact that Tr ˆρ⊥ = 1 and the element values on the matrix diagonal ˆρ⊥ lie in the range from 0 to 1, then 0 ≤ ψ(−) | ˆρ⊥ | ψ(−) ≤ 1. Thus,
≤ per ≤ 1 – F, (7)
that is fidelity determines the range of possible bit error values in the BBM92 scheme.
Let the BBM92 scheme use the so-called double-crystal source of photon pairs [21], generating the photon pairs in a polarization state
| ψ = cos θ0 | HH + eiϕ sin θ0 | VV , (8)
where θ0 and ϕ are the parameters specified by the polarization ellipse direction and the pumping polarization ellipticity. Fidelity for a given quantum state is equal to the following:
F = 1 + cos 2θ0 cos ϕ , (9)
and the optimal values are θ0 = π4 and ϕ = 0, at which F = 1. The error probability of, as can be verified by direct substitution, is equal to per = 1 − F2.
Due to the availability of decoherence mechanisms [22], a double-crystal source can generate the photon pairs in a mixed state. Such a source is described by the density matrix [23] as follows:
ˆρ = F | Ф(+) Ф(+) | + ( 1 − F )| Ф(−) Ф(−) |, (10)
and the bit error value is again equal to per = 1 − F2. Thus, a double-crystal circuit with the errors in its settings introduces the minimum possible bit error value allowed by the expression (6).
The setup layout to study the fidelity influence of the source quantum state on the bit error value in the case of quantum key distribution using the BBM92 scheme is shown in Fig. 3. The double-crystal circuit is used as a source of polarization-entangled photon pairs. A continuous wave laser emitting at a wavelength of 405 nm passes sequentially through a pump beam shape compensator, half-wave and quartz plates, and passes through an interference filter. The resulting radiation is incident on a double crystal. The pairs of polarization-entangled photons are propagated at an angle 3° of relative to the pump.
One of the received photons is sent to the Alice’s setup, and the other one is sent to the Bob’s setup. The subscribers have the wave plates and a polarization beam splitter, allowing them to select a basis and make the polarization measurements. The single photons are recorded by the single photon detectors.
The fidelity value was experimentally adjusted by tilting the plate P, changing the phase in a state (8). The value was θ0 = π4. At each established position of the plate P, a quantum tomography procedure of the polarization state of photon pairs was performed [24]. The duration of one tomographic measurement was 60 seconds, the total counting rate of correlated photons (without any polarization filtering) was ≈300 photocounts per second. Based on the density matrix ˆρ reconstructed by the likelihood function method, the formula (2) was used to calculate the fidelity value of the source state ˆρ and state | Ф(+) . Then the bit error value was measured using a procedure similar to BB84, with additional averaging based on the Alice’s measurements.
To measure the bit error probability, the Alice’s and Bob’s waveplates were set up to measure the polarization state of single photons in the matching bases (laboratory or diagonal). For four possible combinations of plate positions given in Table 2, the probability of receiving mismatched bits was measured (within 10 seconds each). The resulting error probability measurements were then averaged across the plate combinations and across two matching bases to obtain the BBM92 bit error probability.
Figure 4 shows the experimentally measured dependence of the bit error on fidelity. The figure shows that the fidelity value can be used as the basis for possible prediction of contribution to the bit error made by the quantum light source.
4. Conclusion
On the theoretical and experimental level, the paper shows that the fidelity measurement makes it possible to determine the value of one of the key parameters of a quantum key distribution system, namely the raw bit error rate introduced by the source.
For the BB84 scheme, it has been theoretically and experimentally shown that as the fidelity is decreased from 1 to 0, the raw bit error rate increases linearly from 0 to 1.
For the BBM92 scheme, it has been theoretically shown that the bit error value is within the range from 1 − F2 to 1 − F. It has been experimentally demonstrated that when using a double-crystal source of polarization-entangled photon pairs, the raw bit error value takes on the minimum value allowed by the fidelity, and when the fidelity decreases from 1 to 0, the raw bit error is linearly increased from 0 to 1 / 2.
Thus, application of the fidelity criterion as a standard for the quantum state generated by a source of entangled photons is appropriate.
It should be noted that the results obtained are applicable for the BB84 and BBM92 schemes, based not only on the application of the polarization freedom degree of the photon, but also when using another degree of freedom with two discrete basis states, for example, in the case of phase encoding [25] or encoding with two projection values of the orbital light moment [26].
Acknowledgments
The authors express their gratitude to A. S. Chirkin for discussion of this paper and valuable comments.
SUPPORT
The paper has been prepared with the financial support in the form of a grant from the Russian Science Foundation, unique RSF project number No. 21-12-00155.
AUTHORS
Frolovtsev D. N., Researcher, Department of Physics, Lomonosov Moscow State University, Moscow, Russia
Demin A. V., Cand. of Technical Sciences, Researcher, FGBI “VNIIOFI”, Moscow, Russia.
Personal contribution
of the authors
D. N. Frolovtsev – 70% (theoretical calculations, conducting the experiment, processing the results of the experiment, analyzing the results obtained, writing the text of the article); A. V. Demin – 30% (analyzing the results obtained, writing the text of the article).
CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.
in Quantum Key Distribution
D. N. Frolovtsev 1, A. V. Demin 2
Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia
All-Russian Research Institute of Optical and Physical Measurements (VNIIOFI), Moscow, Russia
The paper analyzes the efficiency of a quantum state fidelity to assess the raw bit error value introduced by the polarization state of a single photon source. The analysis was performed for the quantum key distribution schemes BB84 and BBM92. It has been shown theoretically and experimentally that when the fidelity is decreased from 1 to 0 in the BB84 scheme, the raw bit error value linearly increases from 0 to 1, and in the BBM92 scheme – from 0 to 1/2. The experimental setups for determine the influence of fidelity on the bit error value are described in detail.
Keywords: quantum state metrology, fidelity, quantum key distribution, spontaneous parametric light scattering
Article received: February 10, 2024
Article accepted: March 15, 2024
1. Introduction
The development of an optical platform for quantum computing [1, 2] and quantum technology [3–6] requires metrological support for its constituent elements: single photon sources [7], single photon detectors [8], interferometers [9], polarization plates, filters, etc. to assess the value of errors introduced by the system components. To ensure the specifications of the photon source tools in the quantum optical technologies [10], it is proposed to use the parameters g(2) [11] and Grangier’s α [12, 13] related to the statistics of source photons. This paper analyzes the source of errors in the case of quantum key distribution that is associated not with the photon statistics, but with the inaccurate рreparation of the quantum polarization state by the source and the “ideal” one.
In order to provide metrological support of quantum polarization states of the single photon sources in the photon pairs, it is proposed to use a quality measure for the source reproduction of the required quantum state, namely fidelity [14]. Fidelity is a proximity measure for two quantum states: the quantum state required in a practical problem and the state of photons generated by the source used in practice. Mathematically, the fidelity value F is determined by the following expression:
F = | ψt | ψr | 2,
where | ψt is the quantum photon state required from the source, | ψr is an actually generated quantum state. If the source generates photons in a mixed state ˆρ, then fidelity is determined as follows:
F = | ψt | ˆρ | ψr | 2.
The purpose of this paper is to show that the raw bit error rate introduced by the non-ideal quantum photon polarization state in the case of quantum key distribution can be estimated on the basis of a single parameter, namely fidelity.
We will briefly describe the application of fidelity in quantum communications to determine the bit error level in the quantum key distribution schemes. On a theoretical and experimental level, section 2 examines the use of fidelity in the BB84 scheme [15], and section 3 – in the BBM92 scheme [16]. Section 4 provides the summarization of research.
2. Quantum key transfer
using the BB84 scheme
The concept of quantum cryptography was first proposed by C. Bennett and G. Brassard in 1984 [15]. The aim of the scheme is to generate an identical random bit sequence between two placeholders (called Alice and Bob). The communication channel security is determined by the fact that when listening to the quantum channel, the placeholders can reliably register the sequence compromise by a third participant, usually called Eve, based on the bit error rate [17]. Let us recall that the scheme can be divided into three stages.
At the first stage, Alice prepares the single photons. To do this, she randomly selects a coding basis: a laboratory or diagonal one. Then she selects a random number (“0” or “1”) and prepares a single photon with polarization, a random number. In the laboratory basis, “0” and “1” are encoded by the horizontal | H and vertical | V photon polarization, and in the diagonal basis – by the states of photon linear polarization ±45° to the horizontal: “0” corresponds to the polarization state | D = | H + | V —2 , and “1” corresponds to the polarization state | A = | H − | V —2 . The random selection procedure for a polarization basis and an information bit is repeated for each photon prepared by Alice.
At the second stage, Alice sends the prepared photons through a quantum channel to Bob, who measures their polarization in a randomly selected basis. Then the subscribers use an open channel to inform each other in which basis (laboratory or diagonal) they have performed the measurements, and keep only the results in the matching basis. In this way, a raw key is generated for the subscribers. If the devices used by Alice and Bob are ideal and there is no eavesdropping, the subscribers’ raw keys are identical. As follows from the no-cloning theorem [18] of a quantum state, listening to the quantum channel leads to the available errors in the raw key. For example, if some curious person (called Eve) is constantly listening while taking measurements of the photon polarization state in the quantum channel in a randomly selected basis and transmitting the measurement results to Bob instead of the original photon, then Alice and Bob should register an increase in the error up to 25%. If Eve measures not every photon, but only a part of the flow sent by Alice to Bob, then the introduced error value is decreased, although Eve learns less information about the key. The permissible amount of information that is fundamentally accessible to the attackers, as well as the methods applied to enhance the key secrecy [9, 19], determine the maximum permissible error level.
Any errors in the key generation process occur not only due to the eavesdropping, but also due to the inevitable equipment imperfection. For example, if Alice uses a source that produces photons with an error, then Bob will receive erroneous bits in the raw key as a result of measurements.
Let Alice’s photon source produce the single photons in a polarization state | ψ = α | H − β | V instead of | H . When registering the photon polarization in a laboratory basis with the probability of, where F is fidelity, Bob will obtain the correct measurement resultL the photon has horizontal polarization. If the probability is | V | ψ | 2 = | β | 2 = 1 − F, then Bob will register the vertical photon polarization, and an error will appear in the key. Similar relations between fidelity and the bit error value are developed when Alice prepares the remaining states | V , | D and | A. Thus, as fidelity decreases, the probability of error is increased.
The experimental setup layout to determine the influence of the quantum state preparation accuracy by the source on the bit error value is shown in Fig. 1. The setup simulates the operation of a quantum key distribution system using the single photons according to the BB84 scheme from Alice to Bob. The single photons are prepared using the spontaneous parametric light scattering (SPLS) [20] in a ppKTP crystal. A laser with a wavelength of 405 nm is used for pumping. As a result of SPLS, a pair of photons with a wavelength of 810 nm is generated, one of which is reflected by a polarizing beam splitter and recorded by a single photon detector. This event notifies that the Alice’s source has prepared a single photon. Using a half-wave plate, Alice sets the polarization direction of a single photon (Table 1). To simulate the non-ideal preparation of the polarization state by the single photon source, the Alice’s half-wave plate is additionally rotated by a certain angle θ, and the polarization plane is rotated by an angle 2θ. As a result, the fidelity of the prepared polarization state takes on the value F = cos2 2θ.
Bob detects the single photons in the laboratory and diagonal basis while selecting the measurement basis using his half-wave plate. For each basis, the measurements are performed at two positions of the wave plate, in which the detectors “swap their places” that makes it possible to consider their various quantum efficiencies in the calculations. The relevant positions of the wave plate are given in Table 2.
For the measurements in a laboratory basis, the half-wave plate is installed with a rotation angle 0° or 45°. In the first case, the photon passed through the Bob’s polarization beam splitter corresponds to the “0” bit, and the photon reflected by the beam splitter corresponds to the “1” bit. When the plate is positioned at 45° the bit values are opposite: the transmitted photon corresponds to “1”, and the reflected photon corresponds to “0”.
Bob makes measurements in the diagonal basis when the half-wave plate is position at an angle of 22.5° or (22,5° + 45°). When the plate is positioned at the photon passed through the Bob’s beam splitter corresponds to the “1” bit, and the reflected photon corresponds to the “0” bit. When the plate is positioned at (22.5° + 45°) the photon passed through the beam splitter corresponds to the “0” bit, and the reflected photon corresponds to “1”.
The measurement of the bit error rate depending on the fidelity value was performed as follows. In accordance with the selected fidelity value, the angle value θ was determined. Then Alice prepared the polarization state of single photons by installing her half-wave plate in accordance with Table 1. For each polarization state prepared by Alice, Bob made a bit error measurement. To do this, Bob installed his half-wave plate to measure polarization in a basis that coincided with the Alice’s basis, and measured the error probability (receipt of a bit value opposite to that set by Alice) for 20 seconds. The counting rate for the single photons correlated with the trigger Alice’s photocount by the Bob’s detectors is ~500 photocounts per second. The measurement results for four bit error values (for the laboratory and diagonal basis, the measurements in each basis were performed for two positions of the Bob’s half-wave plate) were averaged, and the bit error probability was calculated perr:
NL1err + NL2err + ND1err + ND2err
perr –––,
NL1norm + NL2norm + ND1norm + ND2norm
where NerrL1, 2/D1, 2 is the number of bits erroneously received by Bob, and NnormL1, 2/D1, 2 is the total number of bits received by Bob (including erroneous and correct bits). The indices L1, L2, D1, D2 correspond to the Bob’s measurement in the laboratory (L) and diagonal (D) basis, the numbers 1 and 2 correspond to various plate positions according to Table 2.
The measured value of the bit error depending on the fidelity is given in Fig. 2. The experimental data obtained confirm that fidelity makes it possible to unambiguously determine the bit error value introduced by the source in the BB84 scheme.
3. Quantum key distribution
using the BBM92 scheme
We will consider the influence of fidelity value on the bit error value during the quantum key distribution using the BBM92 scheme [16]. The scheme uses the pairs of quantum entangled particles. For the sake of argument, we will assume that the pairs of particles in the Bell polarization state are used | Ф(+) = | HH + | VV —2 . Alice and Bob each receive one of the entangled photons. The source of entangled photons can be external, or located at one of the subscribers. After the subscribers have each received one photon from the entangled pair, Alice and Bob measure the photon polarization. To do this, each subscriber has a meter similar to the meter used by Bob in the BB84 scheme discussed above. With the probability of 1 / 2, Alice measures the photon polarization in the laboratory basis, and with the probability of 1 / 2 – in the diagonal basis. Bob performs the same activities independently. Then Alice and Bob tell each other the basis in which the measurements have been made, but do not provide the specific measurement results.
Any errors in the developed key occur due to the eavesdropping, as well as in the case of using a low-quality source of entangled photons that generates the photon pairs in a state different from | Ф(+) determined by the density matrix ˆρ. Let us analyze the errors occurred due to the quantum state deviation of photon pairs from | Ф(+) .
To do this, we represent the density matrix of the source polarization state as follows:
ˆρ = F | Ф(+) Ф(+) | + ( 1 − F ) ˆρ⊥, (4)
where F = Ф(+) | ˆρ | Ф(+) is the fidelity value, and ˆρ⊥ meets the conditions of Ф(+) | ˆρ⊥ | Ф(+) = 0, Tr ˆρ⊥ = 1 and quasi-positive determinacy.
To obtain the error probability, we express the quantum states relevant to the error event(| HV , | VH , | AD and | DA ) in the basis of Bell states:
| HV = | ψ(+) + | ψ(−) .
| VH = | ψ(+) − | ψ(−) .
| AD = | Ф(−) + | ψ(−) .
| DA = | Ф(−) − | ψ(−) . (5)
Having determined the probabilities of state measurements in (5) using the density matrix (4), we obtain the raw bit error value:
per = pHV + pVH + pAD + pDA =
= 1 – F 1 + ψ(−) | ˆρ⊥ | ψ(−) 2. (6)
Due to the fact that Tr ˆρ⊥ = 1 and the element values on the matrix diagonal ˆρ⊥ lie in the range from 0 to 1, then 0 ≤ ψ(−) | ˆρ⊥ | ψ(−) ≤ 1. Thus,
≤ per ≤ 1 – F, (7)
that is fidelity determines the range of possible bit error values in the BBM92 scheme.
Let the BBM92 scheme use the so-called double-crystal source of photon pairs [21], generating the photon pairs in a polarization state
| ψ = cos θ0 | HH + eiϕ sin θ0 | VV , (8)
where θ0 and ϕ are the parameters specified by the polarization ellipse direction and the pumping polarization ellipticity. Fidelity for a given quantum state is equal to the following:
F = 1 + cos 2θ0 cos ϕ , (9)
and the optimal values are θ0 = π4 and ϕ = 0, at which F = 1. The error probability of, as can be verified by direct substitution, is equal to per = 1 − F2.
Due to the availability of decoherence mechanisms [22], a double-crystal source can generate the photon pairs in a mixed state. Such a source is described by the density matrix [23] as follows:
ˆρ = F | Ф(+) Ф(+) | + ( 1 − F )| Ф(−) Ф(−) |, (10)
and the bit error value is again equal to per = 1 − F2. Thus, a double-crystal circuit with the errors in its settings introduces the minimum possible bit error value allowed by the expression (6).
The setup layout to study the fidelity influence of the source quantum state on the bit error value in the case of quantum key distribution using the BBM92 scheme is shown in Fig. 3. The double-crystal circuit is used as a source of polarization-entangled photon pairs. A continuous wave laser emitting at a wavelength of 405 nm passes sequentially through a pump beam shape compensator, half-wave and quartz plates, and passes through an interference filter. The resulting radiation is incident on a double crystal. The pairs of polarization-entangled photons are propagated at an angle 3° of relative to the pump.
One of the received photons is sent to the Alice’s setup, and the other one is sent to the Bob’s setup. The subscribers have the wave plates and a polarization beam splitter, allowing them to select a basis and make the polarization measurements. The single photons are recorded by the single photon detectors.
The fidelity value was experimentally adjusted by tilting the plate P, changing the phase in a state (8). The value was θ0 = π4. At each established position of the plate P, a quantum tomography procedure of the polarization state of photon pairs was performed [24]. The duration of one tomographic measurement was 60 seconds, the total counting rate of correlated photons (without any polarization filtering) was ≈300 photocounts per second. Based on the density matrix ˆρ reconstructed by the likelihood function method, the formula (2) was used to calculate the fidelity value of the source state ˆρ and state | Ф(+) . Then the bit error value was measured using a procedure similar to BB84, with additional averaging based on the Alice’s measurements.
To measure the bit error probability, the Alice’s and Bob’s waveplates were set up to measure the polarization state of single photons in the matching bases (laboratory or diagonal). For four possible combinations of plate positions given in Table 2, the probability of receiving mismatched bits was measured (within 10 seconds each). The resulting error probability measurements were then averaged across the plate combinations and across two matching bases to obtain the BBM92 bit error probability.
Figure 4 shows the experimentally measured dependence of the bit error on fidelity. The figure shows that the fidelity value can be used as the basis for possible prediction of contribution to the bit error made by the quantum light source.
4. Conclusion
On the theoretical and experimental level, the paper shows that the fidelity measurement makes it possible to determine the value of one of the key parameters of a quantum key distribution system, namely the raw bit error rate introduced by the source.
For the BB84 scheme, it has been theoretically and experimentally shown that as the fidelity is decreased from 1 to 0, the raw bit error rate increases linearly from 0 to 1.
For the BBM92 scheme, it has been theoretically shown that the bit error value is within the range from 1 − F2 to 1 − F. It has been experimentally demonstrated that when using a double-crystal source of polarization-entangled photon pairs, the raw bit error value takes on the minimum value allowed by the fidelity, and when the fidelity decreases from 1 to 0, the raw bit error is linearly increased from 0 to 1 / 2.
Thus, application of the fidelity criterion as a standard for the quantum state generated by a source of entangled photons is appropriate.
It should be noted that the results obtained are applicable for the BB84 and BBM92 schemes, based not only on the application of the polarization freedom degree of the photon, but also when using another degree of freedom with two discrete basis states, for example, in the case of phase encoding [25] or encoding with two projection values of the orbital light moment [26].
Acknowledgments
The authors express their gratitude to A. S. Chirkin for discussion of this paper and valuable comments.
SUPPORT
The paper has been prepared with the financial support in the form of a grant from the Russian Science Foundation, unique RSF project number No. 21-12-00155.
AUTHORS
Frolovtsev D. N., Researcher, Department of Physics, Lomonosov Moscow State University, Moscow, Russia
Demin A. V., Cand. of Technical Sciences, Researcher, FGBI “VNIIOFI”, Moscow, Russia.
Personal contribution
of the authors
D. N. Frolovtsev – 70% (theoretical calculations, conducting the experiment, processing the results of the experiment, analyzing the results obtained, writing the text of the article); A. V. Demin – 30% (analyzing the results obtained, writing the text of the article).
CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.
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