2018 Munich Workshop on Information Theory of Optical Fiber
Machine Learning in Heterodyne Quantum Receivers
Christian G. Schaeffer, Max Rückmann, Sebastian Kleis, Darko Zibar cgs@hsu‐hh.de
FKZ: 16KIS0490
x Motivation: Why Physical Layer Security?
Public key method Quantum key distribution (QKD)
► Logical layer ► Physical layer
Simple to implement Unconditional security
Computational secure Attacker has to break the system when it is used Vulnerable to quantum computers Complex and costly
Threat of State of the art key rate "store now, break later" 10 bit/symbol @ 90 km1
1D. Huang et al., Nature Scientific Reports, 2016, doi:10.1038/srep19201 23.11.2018 Christian G. Schaeffer 2 Outline
1. The QKD principle 2. Promises and challenges of coherent detection for QKD 3. Coherent quantum PSK
► Mutual information
► Key rate optimization
► Excess noise
► Experimental setup 4. DSP design for coherent quantum communications 5. Bayesian Inference & laser phase noise 6. Conclusion & Outlook
23.11.2018 Christian G. Schaeffer 3 The QKD Principle
Quantum state transmission
Reconciliation (classical) Error correction, privacy amplification
Secret key
Encrypt public channel
► Information advantage based on quantum properties ● Non‐orthogonality of coherent states (Heisenberg uncertainty) (CV) ● Single photon or entanglement (DV)
► A key is not transmitted but generated after the quantum state transmission by interactive reconciliation via the classical channel
23.11.2018 Christian G. Schaeffer 4 Secret Key Rate
► Key rate equals information advantage
∙ , Reconciliation efficiency • 0 1 Eve‘s maximum Mutual information of information Alice and Bob • Depends on signal • Depends on signal power, channel power and receiver attenuation and excess • 0 log noise ′ [bit/symbol] • ′ 0 [shot noise units]
► : Unexplained noise power in the received signal ● Assumed to be introduced by Eve
► For maximum key rate, the optimum signal power should be found
► Usually ≪ photon per symbol
23.11.2018 Christian G. Schaeffer 5 The Coherent Quantum Channel I Heterodyne Detection
exp 2 /
: channel transmittance
: Sent photons/symbol
► Attenuation increases Heisenberg uncertainty
► Here: coherent ‐PSK
| |
► After quantum state transmission: Estimation of necessary
23.11.2018 Christian G. Schaeffer 6 The Optical Coherent Quantum Channel II Heterodyne Detection
Promises Challenges
► High quantum efficiency ► Local oscillator required
► Spectral efficiency ► Phase noise compensation
► Standard telecom ► Frequency estimation components ► Synchronization ► Great selectivity, WDM tolerance due to LO ► Complex reconciliation procedure
► Challenges not solved yet
► To date, only prototype systems for coherent QKD do exist
23.11.2018 Christian G. Schaeffer 7 Typical Experimental Results on Mutual Information (Back to Back)
1 photon/symbol 2 measured ► Lowest power dependent on pilot ideal 4-PSK signal power ratio (18 dB) 1 0 -15 -10 -5 0 5 10 15 ► Experimental evaluation of , 3 ● Penalty: ~2 dB 2 1 ● 1 dB due to Rx quantum efficiency [bit/symbol] 8-PSK 0 ● 0,5 dB due to electronic noise -15 -10 -5 0 5 10 15 4 , ► Experimental raw key rate 2 ● Optimize Alice's power level 16-PSK 0 considering receiver characteristics -15 -10 -5 0 5 10 15 photons ● Found MI penalty serves as worst 10 log 0.2 symbol case estimate for key rate penalty 0.1 0 -14 -12 -10
23.11.2018 Christian G. Schaeffer 8 Properties of Quantum PSK
Signal power influence Optimum signal power
► Optimization of optical power ● Beam splitter attack Very weak signal ● Hard decision at long distances ● Ideal reconciliation
► SNR
23.11.2018 Christian G. Schaeffer 9 Excess Noise Estimation
► Key rate: ∙ , ′ Alice's symbols ► Excess noise determines Eve's max. Information Bob's noisy symbols detector quantum efficiency ► Alice reveals part of her symbols channel transmittance
► Power components of the received signal
total power estimation signal power estimation shot noise, electronic noise Excess noise ∗ Cov , calibrated before transmission Residual power
8‐PSK, 0.95
Key rate and achievable distance very sensitive to ′!
ECOC 20117: HSU P2.SC6.26 Influence of the SNR of Pilot Tones on the Carrier Phase Estimation in Coherent Quantum Receivers, Sebastian Kleis; AITR P2.SC6.10 High-Rate Continuous-Variables Quantum Key Distribution with Piloted- Disciplined Local Oscillator,Bernhard Schrenk 23.11.2018 Christian G. Schaeffer 10 General Coherent Quantum System
► Major challenges: Laser phase noise and clock synchronization
,
Rec. Symbols
,
► Remote LO is a common approach but problematic ● Eve has access to the LO ● Limited reach due to attenuated LO
► Our approach: Heterodyne with real LO ● The DSP has to compensate laser freqency noise and perform clock recovery! 23.11.2018 Christian G. Schaeffer 11 Experimental Heterodyne Quantum PSK System2
expj expj
80 MHz 40 MHz 40 MBd
► Bob's LO and ADC are free running
► 2 pilot signals multiplexed in frequency domain ● Differential frequency provides clock information
► Power ratio between pilots and signal limited by dynamic range of the components (DAC, modulator, balanced Rx, ADC) ● Pilots exhibit low SNR, too
2S. Kleis and C. G. Schaeffer, Optics Letters, 2017, doi:10.1364/OL.42.001588 23.11.2018 Christian G. Schaeffer 12 Details of The Received Signal
► Received optical signal before balanced detection
► Pilots are equal in power
► The pilot to signal power ratio (PSPR) is the power ratio between one pilot and the quantum signal
► Pilot 2 provides clock information
23.11.2018 Christian G. Schaeffer 13 Design of DSP for Ultra Low SNR
► No known algorithms can deal with such low SNR → pilot signals necessary!
► Frequency estimation ● Classical system: Coarse estimation only, residual offset is corrected by carrier phase estimation ● Quantum system: Critical problem, residual offsets directly translate into phase errors
► Carrier phase estimation ● Classical system: Based on modulated signal, e. g. "Viterbi & Viterbi" ● Quantum system: Based on pilot signals, accuracy very important for the key rate
► Clock/timing recovery ● Classical system: Based on modulated signal, e. g. "filter and square" ● Quantum system: Pilot signals must contain clock information, precision critical for the key rate
23.11.2018 Christian G. Schaeffer 14 Experimental Results at Different Fiber Lengths
16‐PSK – 2 symbols – pilot to signal power ratio: 30 dB
► Here, no influence of fiber length → CD compensa on not necessary
► Penalty of < 2 dB (thermal noise, quantum efficiency)
► Less than 10 photons per symbol detectable!
► Setup shows great stability, repeatability of results
23.11.2018 Christian G. Schaeffer 15 Impact of a Phase Error in the Received Symbols
► With phase/frequency distortion: Estimated quantum ► Underestimation of ⇒ Overestimation of ′ Signal power Δ Signal power underestimation factor 2 1 Alice's power in photons/symbol 2 Δ Phase error 2 1 cos Excess noise
Resulting when is Gaussian distributed
20 km
70 km 140 km
23.11.2018 Christian G. Schaeffer 16 Experimental heterodyne quantum communication system[1]
Experimental parameters: Format 8‐PSK ► Bob's LO and ADC are free running Symbol rate 40 MBd
40 MHz ► Two pilots provide frequency, phase and clock information 120 MHz ● Fixed pilot to signal power ratio (PSPR) ● Limited by linearity and dynamic range of components ● The pilots should be weak, too!
[1] S. Kleis, C. G. Schaeffer, "Continuous variable quantum key distribution with a real local oscillator using simultaneous pilot signals", Optics Letters 42(8), 2017. 23.11.2018 Christian G. Schaeffer 17 Digital Signal Processing Routine
► Block wise procedure for signals of arbitrary length
Estimated frequencies
Δ Estimated initial pilot phase ► Pilot SNR improvement by coherent superposition Zero roll‐off Nyquist filter Combined pilot ► Approach for phase estimation EKF/ Extended Kalman Filter/ ● Previously: Nyquist filter with optimized bandwidth EKS/ Extended Kalman Smoother/ PS Particle Smoother ● Novel: Bayesian inference Resampling factor ● Methods can be switched for comparison
23.11.2018 Christian G. Schaeffer 18 Bayesian Inference Methods [2]
► Purpose is to compute the state of a phenomenon when only the measurements are observed ‐th estimate ‐th measurement ► Recursive methods: ,
► The information of a new measurement is used to update the old information
Prior knowledge of state Physical model Measurement yn
predict update ′ Output estimate of state predict update ′
time step [2] S. Särkkä, Bayesian filtering and smoothing. Cambridge University Press, 2013, vol. 3. 23.11.2018 Christian G. Schaeffer 19 Bayesian Inference Methods [2]
► Recursive methods: , ‐th estimate ► Provide optimal estimates if the state space model is exact ‐th measurement
► Filters: To estimate , measurements : are taken into account
► Smoothers: For , all measurements : are taken into account 0 n N ► Extended Kalman filter/smoother Prediction ● Analytical method Filtering ● Extension of Kalman filter to non‐linear models Smoothing ● Approximation by linearization estimate ► Particle filter/smoother ● Statistical method (Particles are randomly chosen) ● No approximation involved ● Converges to optimum for large number of particles (computationally heavy) [2] S. Särkkä, Bayesian filtering and smoothing. Cambridge University Press, 2013, vol. 3. 23.11.2018 Christian G. Schaeffer 20 Measured Laser Phase Noise Characteristics
Frequency noise PSD (FNPSD) Power spectrum (self‐heterodyne)
► White frequency noise ⟺ Lorentzian line, not here!
► Strong 1/ frequency noise component
► Three parameters describe the frequency noise: , ,
► These parameters should be included in the state space model for Bayesian inference
23.11.2018 Christian G. Schaeffer 21 State space model including noise
► Process Ω : Frequency to include 1/ ‐ noise Ω 10Ω , : Signal phase 11 , , : WGN processes ► Exact model for ‐ noise ( 2) ► ‐noise with 2not feasible due to an infinite number of poles [3] ● Results in an infinite number of state space variables ● Measured: 1.23 ● To reduce the impact of that mismatch, ≔ ,
► Measurement of the pilot quadratures with a coherent receiver
y , cos , , sin , [3] N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/ power law noise generation,” Proceedings of the IEEE, vol. 83, no. 5, pp. 802–827, 1995. 23.11.2018 Christian G. Schaeffer 22 Simulation Model for
► Block diagram Nyquist‐shaped 8‐PSK signal
Power estimator for quantum signal
Δ Signal power underestimation factor
White Gaussian noise PNR Pilot to noise ratio (SNR of )
► Only additive Gaussian noise and phase noise included
► Signal constellations after phase correction
23.11.2018 Christian G. Schaeffer 23 Experimental and Simulation Results
Simulations Ref. method Bayes methods
f is omitted, set manually
~20%
► It is beneficial to include in the model ● Better performance / no manual adjustment necessary
► The EKS method outperforms the reference by 20% S. Kleis, C. G. Schaeffer, "Improving the Secret ► PS and EKS show same performance Key Rate of Coherent Quantum Key Distribution with Bayesian Inference", → Indicates op mum for given state space model JLT October 2018. 23.11.2018 Christian G. Schaeffer 24 Impact on the Secret Key Rate
► The realized PSPR has a very strong influence
► The EKS method improves the system ● Significantly higher key rate ● Extended reach ● More efficient tuning (no optimization required)
A. Becir et al., “Continuous‐Variable Quantum Key Distribution Protocols With Eight‐State Discrete Modulation,” International Journal of Quantum Information, vol. 10, no. 1, p. 1250004, 2012. 23.11.2018 Christian G. Schaeffer 25 Combining the Quantum Channel with Infinera‘s Commercial WDM System
► L‐Band configuration Blue Red WDM WDM
► S‐Band configuration Blue Red WDM WDM
► Prior to experiments, noise contributions are measured with an OSA to predict the performance
► Quantum signal is Gaussian modulated!
05.12.2018 Sebastian Kleis 26 Experimental Results for S‐Band –„RedWDM“ – ‐3dBm/ch 50 km 25 km
x2.33
► High tolerance also at large number of channels! Parameters for 25 km (50 km) 2.42 ph/sym ● More than 56 of the 96 C‐Band channels at 25km! Launch power (1.99) ● High flexibility for the C‐Band DWDM system Reconciliation eff. 0.95 ● Advantage compared to [Eriksson2018], where a El. Noise power 0.18 SNU narrow local minimum is used Key rate w/o 5.3x10‐2 bit/sym excess noise (1.4x10‐2) No. evaluated qu. [Eriksson2018]: 18ch, 13.7dBm, 10km symbols per data 1.5 x 108 This work: 56ch, 14.5dBm, 25km point
05.12.2018 Sebastian Kleis 27 Conclusion
► Quantum communications can be realized very similar to classical systems
► The DSP must perform the same tasks but under significantly different conditions
► Shown: Feasibility for signal powers lower than 10 photons per symbol with standard components only
► Laser phase noise is a limiting factor for the achievable reach in CV‐QKD systems
► Bayesian smoothers can improve the system ● Better performance in terms of key rate and reach ● No try and error optimization required ● EKS and PS show best performance ● EKS is preferred due to lower computational complexity
► Coexistence of QKD and WDM investigated
23.11.2018 Christian G. Schaeffer 28 Thanks for your attention.
Some references: Zibar, Darko ; Piels, Molly ; Jones, Rasmus Thomas ; Schaeffer, C. G.: „ Machine Learning Techniques in Optical Communication“, 41st European Conference on Optical Communication (ECOC), paper Th.2.6.1
S. Kleis, C. G. Schaeffer, "Improving the Secret Key Rate of Coherent Quantum Key Distribution with Bayesian Inference", IEEE JLT October 2018.
S. Kleis, C. G. Schaeffer, "Continuous variable quantum key distribution with a real local oscillator using simultaneous pilot signals", Optics Letters 42(8), 2017.
23.11.2018 Christian G. Schaeffer 29 Quantum Computers
► They are coming: Google unveiled Bristlecone, The next phase of quantum with 72 quantum qubits Computing: Rigetti 128 https://www.rigetti.com/
23.11.2018 Christian G. Schaeffer 30 Key Rate Optimization Details
► Key rates are calculated according to 8‐PSK security proof [3]
● Optimization of required Alice’s photons/symbol ● Key rate also depends on: Δ , , , , Signal power Δ underestimation factor ● Δ depends on and the PNR → Polynomial fits are included in the optimization Receiver efficiency Reconciliation efficiency Fiber length
Electronic noise power PNR Pilot to noise ratio
Scenario parameters fiber loss 0.2 dB/km 0.95
0.2 0.51
[3] A. Becir et al., “Continuous‐Variable Quantum Key Distribution Protocols With Eight‐State Discrete Modulation,” International Journal of Quantum Information, vol. 10, no. 1, p. 1250004, 2012. 23.11.2018 Christian G. Schaeffer 31 The Extended Kalman Filter algorithm
► Prediction : Measured phase : Filtered phase , : Covariance matrices of, , : Jacobian matrix of with respect to around ► Update
► Currently in progress: Integration into existing DSP procedure
23.11.2018 Christian G. Schaeffer 32 Strongest Passive Attack5
► Beam splitter attack: Eve replaces channel by loss‐less beam splitter
| | 1 |
: transmittane; : complex amplitude | |
► Direct reconciliation:
, 1 , | | → no key for 0.5
► Reverse reconciliation:
, 1 , | | → key for any
5G. van Assche, “Quantum Cryptography and Secret Key Distillation“, Cambridge University Press, 2006 23.11.2018 Christian G. Schaeffer 33 Eve‘s Information
► Scenario: ● Beam splitter attack ● Eve uses a perfect coherent receiver (with measurement outcome ) ● Bob applies hard decision
, log log |
, , d
23.11.2018 Christian G. Schaeffer 34 General Active Attack Scenario (with Excess Noise)
► Entanglement cloner attack2
► Generalization of the beam splitter attack
| |
| 1 |
► Eve increases by introducing one half of an entangled pair | , generated by her.
► This additional correlation between Eve and Bob induces excess noise in Bob‘s received signal
2G. van Assche, “Quantum Cryptography and Secret Key Distillation“, Cambridge University Press, 2006 23.11.2018 Christian G. Schaeffer 35