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5
i l St. Petersburg Polytechnic University Journal. Physics and Mathematics. 2022 Vol. 15, No. 3.2 Научно-технические ведомости СПбГПУ. Физико-математические науки. 15 (3.2) 2022
Conference materials UDC 04.056.55
DOI: https://doi.org/10.18721/JPM.153.212
QKD and phase modulator imperfections
Reutov А. A. 1H, Tayduganov A. S. 12 1 QRate, Skolkovo, Moscow, Russia; 2 NTI Center for Quantum Communications, National University of Science and Technology MISiS,
Moscow, Russia H aleksey.reutov@phystech.edu Abstract. Very often in practical schemes of quantum key distribution various realistic device imperfections are usually neglected. In this work we consider the imperfect phase-modulation encoding that might lead to a potential information leakage and study its effect on the secret key generation rate.
Keywords: quantum cryptography, QKD, state preparation
Citation: Reutov A. A., Tayduganov A. S., QKD and phase modulator imperfections, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. 15 (3.2) (2022) 65-68. DOI: https://doi.org/10.18721/JPM.153.212
This is an open access article under the CC BY-NC 4.0 license (https://creativecommons. org/licenses/by-nc/4.0/)
Материалы конференции УДК 04.056.55
DOI: https://doi.org/10.18721/JPM.153.212
КРК и неидеальности фазового модулятора
А. А. Реутов 1 е, А. С. Тайдуганов 12
1 QRate, Сколково, г. Москва, Россия; 2 НТИ Центр квантовых коммуникаций, Национальный университет науки и технологий МИСиС, г. Москва, Россия н aleksey.reutov@phystech.edu
Аннотация. Очень часто в практических схемах распределения квантового ключа обычно пренебрегают различными реальными несовершенствами устройств. В этой работе мы рассматриваем неидеальности фазово-модуляционного кодирования, которые могут привести к потенциальной утечке информации и изучаем их влияние на скорость генерации секретного ключа.
Ключевые слова: квантовая криптография, КРК, приготовление состояний
Ссылка при цитировании: Реутов А.А., Тайдуганов А.С. КРК и неидеальности фазового модулятора // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2022. Т. 15. № 3.2. С. 65-68. DOI: https://doi.org/10.18721/ JPM.153.212
Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https:// creativecommons.org/licenses/by-nc/4.0/)
Introduction
The quantum key distribution (QKD) is based on fundamental laws of quantum physics and theoretically provides security regardless eavesdropper's potential resources. However, the practical QKD implementations suffer from imperfections of realistic devices [1]. One of the sources of potential information leakage in practical BB84 [2] schemes is the imperfect state preparation by a phase modulator (PM) used for the qubit state encoding. This imperfection leads to an asymmetry between the bases, that can provide more efficient strategies for an eavesdropper [3]. In this work we consider the general quantum state description and introduce two possible phase modulation
© Reutov A. A., Tayduganov A. S., 2022. Published by Peter the Great St.Petersburg Polytechnic University.
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St. Petersburg Polytechnic University Journal. Physics and Mathematics. 2022 Vol. 15, No. 3.2
uncertainties. Using the density matrix formalism, we provide a metrics of distinguishability between the BB84 bases in our model and estimate its effect on the secret key rate.
Theory and Methods
In our setup the linearly polarized light is brought to the Alice's PM at an angle of 45° with respect to the crystal axes of the modulator with an error 50. In general, there is some phase difference 9 between the field amplitudes along the ordinary and extraordinary axes. By applying an electric voltage along one of the axes, an additional phase difference (with some uncertainty
) is created, which determines the basis and bit states.
basis
Without uncertainty 5%^ and error 80, the polarization state, prepared by Alice in X' and Y' bases, can be described on the Bloch sphere (Fig. 1) in the following form,
1°X (H+i t) •
S
|i1 (И+«* ""'It) • I«' ■>=-=■ (И+e"-'/2 It)),
(i)
V2
'(ф-п'/2
It).
where and ||) are the standard horizontal Fig. 1. Physical polarization states on the Bloch and vertical polarization states. The general sphere; experimental phase uncertainties are polarization state with PM imperfection schematically depicted as red and blue cones corrections is given by
f ) = cos
I ^ = cos
n 50 I. \ ---
v 4 2 11 '
n 50
■e'(ф+5фl'sin
К 0')
К')
= cos
= cos
л uo i| \ ---
v 4 2 } '
n 50 \ \
---+
4 2 Г '
e'(ф+П+5ф2' sin
(7-?) t.
n 50 11 x ---
v 4 2 11 '
e'(ф+п/2+5фз' sin
е'(ф-п/2+5ф4' sin
7-f) *»• t,
(2)
where 50 represents in Fig. 1 the deviation from the xy — plane on the Bloch sphere caused by e.g. the 45° fiber input error.
To describe the distinguishability between the states in two bases, the metric A called the imbalance of "quantum coin" [4] is used. It can be expressed in terms of fidelity F [5]:
A = -
1 -V F (pf', pY')
(3)
and fidelity F is given by
f (pf', py >f ъЩрf
(4)
© Реутов А. А., Тайдуганов А. С., 2022. Издатель: Санкт-Петербургский политехнический университет Петра Великого.
2
4
Simulation of physical processes
where the density matrices pX' and pr are given by pbasis can be written (with help (2)) as:
: (| vtT) |+1 vb™) (■vbrs I) /
2 and
p '2
PY =
{ 1 + sin 50 1 e'"9 (- e^92) cos 50^
1 e1'9 (e1'59 - e1'59) cos 50
1 + sin 50
— e19 (e'593 - e'5pp4) cos 50
2
1 — sin 50
( e-'593 — e-5p4) cos 50 1 — sin 50
(5)
It's leads us to the following result for fidelity:
„/ X' Y'\ 1 + sin250 cos2 50 5p12 5p34 cos250
F(pX ,pY ) =-+-cos ——12cos ——34 +-
V / 0 O O 0 s
y.'=1,2 (—1) sin5p.., (6)
j=3,4
where designation 89 . = 5(p - 89. is used. The minimum of (6) is achieved at 50 = 0 and
5^1(3) = - 892(4) = ± Wi ax, .
2(4)
F = 1 + cos2 5Pmax
(7)
Results and Discussion
Using explicit state parametrization (2), we provide the analytical computation of fidelity (6) and estimate the single-photon phase error rate correction according to [4]:
E
phase = Eb— + 4A ' (1 — A') (1 — 2Eb— ) + 4 (1 — 2A')^A '(1 — A') E\u (1 — E\u ),
where the single-photon yield Y1 and bit error rate E11 are determined via decoy-state technique taking into account the finite-key-size effects [6]. Here we consider the worst case, when only a part of the pulses reached to Bob due to losses, but all the different (in the sense of a quantum coin) and useful for eavesdropper pulses were not lost during transmission over the channel. That's way we use A' instead of A:
a' = A ,
Y1 '
E,phase is used for the secret key rate calculation [7],
r =
sec
Q
a
1 — h2 (Efase)] — fech2 (Ej,
(8)
where Q1 is single-photon gain, Q is signal gain and E is measured quantum bit error. Then we study the effects of 80 and 59^" uncertainties on rs'ec.
Our main result is presented in Fig. 2. One can see that for the 100 km—long optical line the critical (i.e. when rs'ec < 0 no secret key can be distributed) value of a phase disturbance is about 1.5°. For the length of 50 km our result is more promising — the key rate vanishes only for the phase error is about 5°.
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^St. Petersburg Polytechnic University Journal. Physics and Mathematics. 2022 Vol. 15, No. 3.2
— 50 km — 100 km H = 0.3 ui = ttf 1 1*1 = (//100 15 = 20% № = 10"" Papt - 1%
0 2 3 4 5
<Vmax [deg]
Fig. 2. Minimized secret key rate (7) (per bit) lower bound as function of maximum phase modulation error 59max. The parameters Vj, v2, n, Pdc, Popt are signal, week decoy and vacuum decoy intensities, quantum efficiency, dark count rate probability and probability of the optical error respectively. The simulation is run for the 50 km (orange line) and 100 km (blue line) optical fiber distances.
Conclusion
From our research we conclude that rather precise (~ 1° for short distances and ~ 0.1° for large distances) fine tuning of PM is required in order to have a reasonable secret key generation rate.
Acknowledgments
The authors would like to acknowledge Arina Gavrilovich for thoughtful and productive discussions.
REFERENCES
1. Bennett, C. H., Brassard G., Quantum Cryptography: Public-Key Distribution and Coin Tossing, In: Proc. Of the IEEE Int. Conf. On Comp. Sys. And Sign. Process. (IEEE, 1984), 175-79.
2. Gottesman D., Lo H.-K., Luttkenhaus N., Preskill J., Security of Quantum Key Distribution with Imperfect Devices, Quant. Inf. Comput. 4 (2004) 325-60.
3. Jozsa R., Fidelity for Mixed Quantum States, Journal of Modern Optics 41 (12) (1994) 2315-23.
4. Lo H.-K., Ma X., Chen K., Decoy State Quantum Key Distribution, Phys. Rev. Lett. 94 (2005) 230504.
5. Lo H.-K., Preskill, J., Security of Quantum Key Distribution Using Weak Coherent States with Nonrandom Phases, Quantum Info. Comput., 7 (5) (2007) 431-58.
6. Trushechkin A. S., Kiktenko E. O., Fedorov A. K., Practical Issues in Decoy-State Quantum Key Distribution Based on the Central Limit Theorem, Phys. Rev. A 96 (2017) 022316.
7. Xu F., Ma X., Zhang Q., Lo H.-K., Pan J.-W., Secure Quantum Key Distribution with Realistic Devices, Rev. Mod. Phys. 92 (May) (2020) 025002.
THE AUTHORS
REUTOV Aleksei A. TAYDUGANOV Andrey S.
aleksey.reutov@phystech.edu a.tayduganov@goqrate.com
ORCID: 0000-0001-5838-4770
Received 15.08.2022. Approved after reviewing 16.08.2022. Accepted 08.09.2022.
© Peter the Great St. Petersburg Polytechnic University, 2022
