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Quantum key distribution rates from semidefinite programming

Computing the key rate in quantum key distribution (QKD) protocols is a long standing challenge. Analytical methods are limited to a handful of protocols with highly symmetric measurement bases. Numerical methods can handle arbitrary measurement bases, but either use the min-entropy, which gives a loose lower bound to the von Neumann entropy, or rely on cumbersome dedicated algorithms. Based on a recently discovered semidefinite programming (SDP) hierarchy converging to the conditional von Neumann entropy, used for computing the asymptotic key rates in the device independent case, we introduce an SDP hierarchy that converges to the asymptotic secret key rate in the case of characterised devices. The resulting algorithm is efficient, easy to implement and easy to use. We illustrate its performance by recovering known bounds on the key rate and extending high-dimensional QKD protocols to previously intractable cases. We also use it to reanalyse experimental data to demonstrate how higher key rates can be achieved when the full statistics are taken into account.

Quantum bit error rate timing jitter dependency on multi-mode fibers.

The full time-jitter response of a single-photon detector can make a significant contribution to the quantum bit error rate (QBER) of high repetition rate quantum key distribution (QKD) implementations. Although there have been studies into understanding the contribution for single-mode optical fiber coupled single-photon detectors, the contribution of larger, multimode core diameters to the QBER have not been explored in detail. With the growing importance of free-space QKD, which typically use multimode fibers to reduce coupling loss, it is vitally important to understand how the multimode fiber coupling will impact the total QBER. This work studies the impact of the time-jitter contribution to QBER when coupling a commercial off-the-shelf silicon single-photon avalanche diode with various multimode fibers while simulating operating at 1 GHz with empirical measurements taken at 1 MHz repetition rate. It was found that step-index multimode fibers can significantly increase the QBER, while graded-index fibers can provide an QBER contribution similar to a single-mode fiber. The results highlight that there is a significant benefit in using graded index multimode fibers for a free-space QKD receiver, particularly for high repetition rate applications.

Neural network-based prediction of the secret-key rate of quantum key distribution

Numerical methods are widely used to calculate the secure key rate of many quantum key distribution protocols in practice, but they consume many computing resources and are too time-consuming. In this work, we take the homodyne detection discrete-modulated continuous-variable quantum key distribution (CV-QKD) as an example, and construct a neural network that can quickly predict the secure key rate based on the experimental parameters and experimental results. Compared to traditional numerical methods, the speed of the neural network is improved by several orders of magnitude. Importantly, the predicted key rates are not only highly accurate but also highly likely to be secure. This allows the secure key rate of discrete-modulated CV-QKD to be extracted in real time on a low-power platform. Furthermore, our method is versatile and can be extended to quickly calculate the complex secure key rates of various other unstructured quantum key distribution protocols.

Quantum-key-expansion protocol based on number-state-entanglement-preserving tensor network with compression

Quantum-key-distribution (QKD) protocols allow exchanging a cryptographic key between two parties. With the development of the Internet, however, all forms of communications consume a huge number of cryptographic keys. Improving the secure key rate of QKD is becoming increasingly important. Quantum compression and key expansion are two ways to increase the secure key rate. In this paper, we propose a quantum-key-expansion (QKE) protocol. A series of singlet states are compressed by number-state-entanglement-preserving tensors and disentanglers to obtain a compressed two-body-entangled state. The cryptographic key is expanded by the number-state-entanglement-preserving tensor network. The advantages of the proposed QKE protocol over the classical-key-expansion algorithm in the key-expansion method, the key length, and the security are discussed in this study. Moreover, due to the structure of the number-state-preserving compression tensor network, our protocol can resist the intercept-resend attack, entanglement-and-measurement attack, and coherent attack.

Finite-key analysis of practical time-bin high-dimensional quantum key distribution with afterpulse effect

High-dimensional quantum resources provide the ability to encode several bits of information on a single photon, which can particularly increase the secret key rate rate of quantum key distribution (QKD) systems. Recently, a practical four-dimensional QKD scheme based on time-bin quantum photonic state, only with two single-photon avalanche detectors as measurement setup, has been proven to have a superior performance than the qubit-based one. In this paper, we extend the results to our proposed eight-dimensional scheme. Then, we consider two main practical factors to improve its secret key bound. Concretely, we take the afterpulse effect into account and apply a finite-key analysis with the intensity fluctuations. Our secret bounds give consideration to both the intensity fluctuations and the afterpulse effect for the high-dimensional QKD systems. Numerical simulations show the bound of eight-dimensional QKD scheme is more robust to the intensity fluctuations but more sensitive to the afterpulse effect than the four-dimensional one.

Investigation of Long-Term Stability of Single-Photon Quantum Key Distribution in a Polarization Coding Scheme

Experimental results demonstrating long-term stability of the operation of our atmospheric quantum cryptography setup using the BB84 protocol and polarization coding are presented. It was shown that the “sifted” quantum key distribution rate and the quantum bit error rate in the key remained constant for 1 hour and were equal to 10 kbit/s and 6.5 %, respectively, at a distance between the transmitter and the receiver equal to 20 cm. Theoretical dependences of the secret quantum key generation rate on a quantum channel transmission coefficient for single-photon detectors, which were used in this experiment, and for new detectors with a reduced level of dark pulses are given.

Effects of ocean turbulence on photon orbital angular momentum quantum communication

The effect of the turbulent motion of ocean on the quantum communication based on the orbital angular momentum in an underwater quantum channel is studied in this work. Based on the power spectrum model of ocean turbulence proposed by Elamassie, the quantitative relationships of different ocean turbulence parameters with the single photon detection probability of orbital angular momentum photons, the channel capacity, the key generation rate, the concurrence of two entangled photons are proposed. The maximum entanglement distance of the orbital angular momentum entangled photon-pairs in the ocean turbulence is further studied by the universal entanglement decay of the concurrence of entangled photon-pairs in the ocean turbulence. The results show that the detection probability of single photon, the channel capacity, the key generation rate, and the concurrence of entangled photon-pairs decrease with the increase of the dissipation rate of turbulent kinetic energy and the decrease of the rate of dissipation of mean-squared temperature. The influence of the temperature and salinity balance parameter of ocean turbulence on the performance of underwater quantum communication are significantly different under the condition of whether the stable stratification of seawater is assumed or not. In the ocean turbulent environment, the increasing of the initial orbital angular momentum quantum number of signal photons can improve the key generation rate of quantum key distribution and the resistance of entangled photons to entanglement decay.

Quantum Keyless Private Communication Versus Quantum Key Distribution for Space Links

We study information theoretical security for space links between a satellite and a ground-station. Quantum key distribution (QKD) is a well established method for information theoretical secure communication, giving the eavesdropper unlimited access to the channel and technological resources only limited by the laws of quantum physics. But QKD for space links is extremely challenging, the achieved key rates are extremely low, and day-time operating impossible. However, eavesdropping on a channel in free-space without being noticed seems complicated, given the constraints imposed by orbital mechanics. If we also exclude eavesdropper's presence in a given area around the emitter and receiver, we can guarantee that he has only access to a fraction of the optical signal. In this setting, quantum keyless private (direct) communication based on the wiretap channel model is a valid alternative to provide information theoretical security. Like for QKD, we assume the legitimate users to be limited by state-of-the-art technology, while the potential eavesdropper is only limited by physical laws: physical measurement (Helstrom detector) and quantum electrodynamics (Holevo bound). Nevertheless, we demonstrate information theoretical secure communication rates (positive keyless private capacity) over a classical-quantum wiretap channel using on-off-keying of coherent states. We present numerical results for a setting equivalent to the recent experiments with the Micius satellite and compare them to the fundamental limit for the secret key rate of QKD. We obtain much higher rates compared with QKD with exclusion area of less than 13 meters for Low Earth Orbit (LEO) satellites. Moreover, we show that the wiretap channel quantum keyless privacy is much less sensitive to noise and signal dynamics and daytime operation is possible.

Performance analysis of free-space quantum key distribution using multiple spatial modes.

In the diffraction-limited near-field propagation regime, free-space optical quantum key distribution (QKD) systems can employ multiple spatial modes to improve their key rate. This improvement can be effected by means of high-dimensional QKD or by spatial-mode multiplexing of independent QKD channels, with the latter, in general, offering higher key rates. Here, we theoretically analyze spatial-mode-multiplexed, decoy-state BB84 whose transmitter mode set is either a collection of phase-tilted, flat-top focused beams (FBs) or the Laguerre-Gaussian (LG) modes. Although for vacuum propagation the FBs suffer a QKD rate penalty relative to the LG modes, their potential ease of implementation make them an attractive alternative. Moreover, in the presence of turbulence, the FB modes may outperform the LG modes.

Upper Bounds on Device-Independent Quantum Key Distribution

Quantum key distribution (QKD) is a method that distributes a secret key to a sender and a receiver by the transmission of quantum particles (e.g., photons). Device-independent quantum key distribution (DIQKD) is a version of QKD with a stronger notion of security, in that the sender and receiver base their protocol only on the statistics of input and outputs of their devices as inspired by Bell's theorem. We study the rate at which DIQKD can be carried out for a given bipartite quantum state distributed between the sender and receiver or a quantum channel connecting them. We provide upper bounds on the achievable rate going beyond upper bounds possible for QKD. In particular, we construct states and channels where the QKD rate is significant while the DIQKD rate is negligible. This gap is illustrated for a practical case arising when using standard postprocessing techniques for entangled two-qubit states.

Quantum key distribution over free space optic (FSO) channel using higherorder Gaussian beam spatial modes

Quantum key distribution(QKD) has emerged as a secure solution of secret key distribution utilizing the well established theories of modern physics. Since its introduction in 1984, many interesting and innovative ideas have been proposed for QKD in order to improve the security and efficiency of the scheme keeping in view of its applications and practicalimplementation. High error rate QKD scheme for long distance communication-theso-called KMB09 protocol is one such scheme which was designed to achieve longer communication distance in QKD, without compromising its security,by allowing the utilisation of higher dimensional photon states which is not possible with standard BB84 scheme. However the practical implementation of KMB09 protocol has not been presented yet because of its unique design. In this paper, we propose a framework for the practical implementation of QKD system that runs KMB09 protocol in two or more dimensions of photon states. We present the KMB09 based QKD system design and its simulation for practical implementation based on the encoding of secret bits in higher order Gaussian beam spatial modes. The proposed framework is specifically evaluated in terms of efficiency or success rate with two and four dimensions of photon states. We find that the simulation results of the proposed framework are inline with the numerical and analytical results of the same QKD model presented earlier.

Emulation of up-conversion based quantum key distribution scheme using active pump-controlled basis selection and adaptive thresholding

We emulate a method for quantum key distribution (QKD) that employs up-conversion detection. The scheme uses differential quadrature phase shift modulation and actively chooses the measurement basis by applying a dynamic phase-shift on the pump of the up-conversion interaction. The QKD rate of the quantum-hacking resistant system over an emulated 7.5 km marine-like turbulent channel is evaluated including the impact of thresholding the received data so as to ignore signals that are below a tolerable signal-to-noise ratio. The emulated system is well suited to operate over marine free space optical links coupled to single mode fiber.

A Reconciliation Strategy for Real-Time Satellite-Based QKD

In practical Quantum Key Distribution (QKD) deployments we would like to design QKD solutions that provide for a target QKD key rate, in bits/pulse, at a specified upper-limit on the failure probability. However, in the finite-signalling setting, in which all real-world QKD systems exist, the common practice of achieving such a solution fails to deliver the maximum throughput rate of the classical decoder. This in turn means that the possibility that classical reconciliation becomes the bottleneck of the entire QKD protocol is not minimised. A design strategy that minimises this latter possibility, whilst achieving a target QKD rate with a target ceiling on the failure probability has not been developed – a situation we remedy here. Although our new design strategy detailed here is for LDPC codes and applied to two specific QKD protocols, the same strategy is generally applicable to all classical decoders and all QKD protocols. It is also deployable even in circumstances where the quantum bit error is variable, such as in satellite QKD systems.

Backflashes from fast-gated avalanche photodiodes in quantum key distribution

InGaAs single-photon avalanche photodiodes (APDs) are key enablers for high-bit rate quantum key distribution. However, the deviation of such detectors from ideal models can open side-channels for an eavesdropper, Eve, to exploit. The phenomenon of backflashes, whereby APDs reemit photons after detecting a photon, gives Eve the opportunity to passively learn the information carried by the detected photon without the need to actively interact with the legitimate receiver, Bob. While this has been observed in slow-gated detectors, it has not been investigated in fast-gated APDs where it has been posited that this effect would be lessened. Here, we perform the first experiment to characterize the security threat that backflashes provide in a GHz-gated self-differencing APD using the metric of information leakage. We find that, indeed, the information leakage is lower than that reported for slower-gated detectors, and we show that its effect on the secure key rate is negligible. We also relate the rate of backflash events to the APD dark current, thereby suggesting that their origin is the InP multiplication region in the APD.

Security proof for quantum key recycling with noise

Quantum Key Recycling aims to re-use the keys employed in quantum encryption and quantum authentication schemes. QKR protocols can achieve better round complexity than Quantum Key Distribution. We consider a QKR protocol that works with qubits, as opposed to high-dimensional qudits. A security proof was given by Fehr and Salvail in the case where there is practically no noise. A high-rate scheme for the noisy case was proposed by \v{S}kori\'{c} and de Vries, based on eight-state encoding. However, a security proof was not given. In this paper we introduce a protocol modification and provide a security proof. The modified protocol has high rate not only for 8-state encoding, but also 6-state and BB84 encoding. Our proof is based on a bound on the trace distance between the real quantum state of the system and a state in which the keys are completely secure. It turns out that the rate is higher than suggested by previous results. Asymptotically the rate equals the rate of Quantum Key Distribution with one-way postprocessing.

Attitude measurement and simulation of two-dimensional position sensitive detector

Aiming at the problems of high error rate of quantum key distribution of physical star and inaccurate location information of motion platform in space, using partial differential equations and finite element tool values to analyze the potential field distribution of PSD. Combined with MATLAB software theory derivation and model simulation for the application of the lateral photoelectric effect of two-dimensional photoelectric position sensor in three-dimensional attitude measurement, especially deepening the theoretical and simulation aspects of lateral photoelectric effect. The experimental data analysis shows that the 2D-PSD detector can realize the displacement resolution precision of XY axis 2um and the polarization angle measurement accuracy within 0.5°of the working interval, which proves the feasibility of 2D attitude measurement by 2D-PSD.

Fundamental limit and trade-off between security and secure key generation rate in quantum key distribution

Fundamental limit and trade-off between security and secure key generation rate in quantum key distribution

Repeaters for continuous-variable quantum communication

Optical telecommunication is at the heart of today's internet and is currently enabled by the transmission of intense optical signals between remote locations. As we look to the future of telecommunication, quantum mechanics promise new ways to be able to transmit and process that information. Demonstrations of quantum key distribution and quantum teleportation using multi-photon states have been performed, but only over ranges limited to one hundred kilometers. To go beyond this, we need repeaters that are compatible with these quantum multi-photon continuous variables pulses. Here we present a design for a continuous variable quantum repeaters that can distribute entangled and pure two-mode squeezed states over arbitrarily long distances with a success probability that scales only polynomially with distance. The proposed quantum repeater is composed from several basic known building blocks such as non-Gaussian operations for entanglement distillation and an iterative Gaussification protocol (for retaining the Gaussian character of the final state), but complemented with a heralded non-Gaussian entanglement swapping protocol, which allows us to avoid extensive iterations of quantum Gaussification. We characterize the performance of this scheme in terms of key rates for quantum key distribution and show a secure key can be generated over thousands of kilometers.

Round-robin-differential-phase-shift quantum key distribution based on wavelength division multiplexing

Most quantum communications are resilient against eavesdropping attacks. But most protocols fail to address the problems of the poor rates of data transmission and key generation with the constraints of transmission distance and misattribution errors. To address these issues, we propose a new round-robin-differential-phase-shift (RRDPS) quantum key distribution (QKD) protocol to improve the transmission and key generation rates, secure transmission distance and misattribution errors. The new protocol named WDM-OAM-MDI-RRDPS QKD protocol is based on a combination of the wavelength division multiplexing (WDM) technique, Fibonacci-valued orbital angular momentum (OAM) theory, decoy state and measurement-device independent (MDI) method. It utilizes a series of coherent weak sources simultaneously and high-capacity Fibonacci-valued encoding technology. In our protocol, WDM technology can greatly increase the amount of information transmitted in optical fiber communication, and improve the transmission efficiency, the total secure bit rate and the key generation rate of QKD. The OAM theory can achieve basis independence and increase the information capacity per photon from one to five. Fibonacci-valued OAM states can be used to reduce misattribution errors. The decoy state and MDI method can be used to increase the transmission distance resilient to the Photon-number-splitting and detector blinding attacks.

Reliable numerical key rates for quantum key distribution

In this work, we present a reliable, efficient, and tight numerical method for calculating key rates for finite-dimensional quantum key distribution (QKD) protocols. We illustrate our approach by finding higher key rates than those previously reported in the literature for several interesting scenarios (e.g., the Trojan-horse attack and the phase-coherent BB84 protocol). Our method will ultimately improve our ability to automate key rate calculations and, hence, to develop a user-friendly software package that could be used widely by QKD researchers.