Quantum communication with coherent states of light

Quantum communication with coherent states of light rsta.royalsocietypublishing.org

Imran Khan1,2 , Dominique Elser1,2 , Thomas Dirmeier1,2 , Christoph Marquardt1,2 and

Review Cite this article: Khan I, Elser D, Dirmeier T, Marquardt C, Leuchs G. 2017 Quantum communication with coherent states of light. Phil. Trans. R. Soc. A 375: 20160235. http://dx.doi.org/10.1098/rsta.2016.0235

Gerd Leuchs1,2 1 Max Planck Institute for the Science of Light, 91058 Erlangen,

Germany 2 Institute of Optics, Information and Photonics, Friedrich-Alexander-University Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany IK, 0000-0002-1376-2018

Accepted: 28 March 2017 One contribution of 7 to a discussion meeting issue ‘Quantum technology for the 21st century’. Subject Areas: quantum physics, optics Keywords: quantum communication, coherent states, continuous variables Author for correspondence: Christoph Marquardt e-mail: [email protected]

Quantum communication offers long-term security especially, but not only, relevant to government and industrial users. It is worth noting that, for the first time in the history of cryptographic encoding, we are currently in the situation that secure communication can be based on the fundamental laws of physics (information theoretical security) rather than on algorithmic security relying on the complexity of algorithms, which is periodically endangered as standard computer technology advances. On a fundamental level, the security of quantum key distribution (QKD) relies on the non-orthogonality of the quantum states used. So even coherent states are well suited for this task, the quantum states that largely describe the light generated by laser systems. Depending on whether one uses detectors resolving single or multiple photon states or detectors measuring the field quadratures, one speaks of, respectively, a discrete- or a continuous-variable description. Continuous-variable QKD with coherent states uses a technology that is very similar to the one employed in classical coherent communication systems, the backbone of today’s Internet connections. Here, we review recent developments in this field in two connected regimes: (i) improving QKD equipment by implementing front-end telecom devices and (ii) research into satellite QKD for bridging long distances by building upon existing optical satellite links. This article is part of the themed issue ‘Quantum technology for the 21st century’.

2017 The Author(s) Published by the Royal Society. All rights reserved.

1. Introduction

For pure quantum states such as single photon states or coherent states, the fundamental laws of physics guarantee secure QKD provided the correct quantum protocols are used. For optimum operation of a QKD protocol, one has to match the quantum state of light to the detector. Homodyne detection and coherent states typically match very well with continuous variables, whereas single photons and click detectors match with discrete quantum variables [4] (figure 1). It is sometimes more convenient to describe one particular detection scheme in one of the

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2. Coherent states and single photon states: perfect quantum states

rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 375: 20160235

One of today’s most widely used cryptographic protocols is the Advanced Encryption Standard (AES), which requires a shared and short secret key to be known by the communicating parties. This key is usually distributed using asymmetric cryptography such as the Rivest–Shamir– Adleman (RSA) or the Diffie–Hellman key exchange. The security of these protocols relies on the computational complexity of the mutual underlying mathematical problem. While this problem may be hard for a classical computer, it does not necessarily have to be hard for a quantum computer. In fact, quantum algorithms are already known to solve some of these problems efficiently. In general, one can say that security based on algorithmic complexity is always endangered by a correspondingly large increase in computational power even of a classical computer or by advances of classical algorithms. In the history of cryptography, the only information-theoretically secure way of encrypting information is by way of the one-time pad. As the key for this method has to be as long as its plain text, it has been challenging to provide a method to efficiently distribute keys between distant parties. Quantum key distribution (QKD) is a protocol that solves this problem by exploiting features of quantum mechanics to detect a potential eavesdropper during the key distribution stage. In comparison to conventional cryptographic protocols, the security proofs for QKD rely on a minimal set of assumptions and bound the security of the entire protocol in an information-theoretic manner on the basis of fundamental physical laws. While the security may be proved on paper, a technical realization of such a protocol requires careful implementation and characterization in order not to open up unforeseen loopholes. It is noteworthy to say that keys generated from QKD used in conjunction with a one-time pad provide the highest level of security. Depending on the security requirements, one can combine an AES-based protocol and initialize and frequently refresh the symmetric key with QKD. However, one should bear in mind that mathematically proved security is offered only by QKD in combination with Vernam’s one-time-pad protocol. Since the first invention of quantum cryptography, experiments have advanced from a proofof-principle level to readily available commercial products. Specifically, in optical fibre systems, continuous-variable quantum cryptography already reaches GHz speed and offers efficient integration with existing telecommunication techniques, especially in short links within a city or by connecting data centres [1–3]. Compact and efficient sending and receiving devices can be built using integrated optical components including quantum random number generators (QRNG). Alternatively, one may use free-space optical quantum communication (FSO-QC) to address challenges, such as ad hoc deployment in intra-city communication, air-to-ground or satellite-toground scenarios. FSO-QC is a reliable means to transmit classical and quantum information. In quantum communication, experimental effort has so far been devoted mostly to discrete variables such as the polarization state of single photons and using click detectors. We present experiments investigating free-space transmission of quantum continuous-variable states using homodyne measurement, rendering the quantum states immune to stray light and enabling daylight operation. As a next step, quantum communication with satellites offers a viable solution to bridge long distances. We will discuss our current joint project with Tesat-Spacecom/Airbus and the German Aerospace Center (DLR) in the development of QKD with coherent optical communication in satellite systems.

2

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LO

signal

mirror

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Figure 1. Schematic set-up of balanced homodyne detection. A bright coherent beam, called local oscillator (LO), interferes with a signal beam at a beam splitter (BS). The difference current of the two detectors is the output signal of the homodyne detector. (Online version in colour.) theoretical descriptions rather than in the other one. However, both can in principle be used to characterize any type of quantum state. As a side remark, please note that a number of experiments with coherent states can be described classically without field quantization. But for any pure quantum state it is always possible to find experimental scenarios which can only be described with quantization. This also holds for coherent states which, for example, lead to a revival of Rabi oscillations in their interaction with an atom in the Jaynes–Cummings model. This effect can only be properly described with a quantized field [5,6]. Coming back to detection schemes, coherent states can be characterized with photon-numberresolving detectors. This is meaningful only if either the coherent states are very weak or the measurement time bins are very short, because the output of these detectors saturates above a certain photon number. In the theoretical description, one can use either discrete quantum variables by expansion in terms of Fock states or continuous quantum variables corresponding to the field quadratures (e.g. amplitude and phase quadrature). In this work, we focus on homodyne detection, where a bright local oscillator (LO) interferes with a signal at a beam splitter. Subtracting the photocurrents of two detectors after the beam splitter results in the output signal, which corresponds to the measurement of an electric field quadrature of the signal beam. Note that the measured quadrature depends on the relative phase between the signal and the LO field. This phase can be accurately controlled. As an example, if one uses homodyne detection on coherent states, the resulting measurement histogram will be of Gaussian shape. This is due to the Heisenberg uncertainty relation, governing the electric field quadratures of coherent states [7,8]. Other Gaussian states are, for example, squeezed and thermal states (for reviews, see [9,10]). Non-Gaussian states are either superpositions of coherent states, called cat states or Fock states. In the following, we compare the typical characteristics of homodyne and click detectors.

(a) Wavelength Using homodyne detection, interference with an LO effectively amplifies small signals with a sensitivity at the single photon level. Homodyning works whenever there is a coherent source of light at the desired wavelength. Thus, there is almost no limit in wavelength. In solid-state physics, using superconducting qubits one can even detect signals at the single microwave photon level using homodyne detection. Click detectors, in comparison, have a more restricted wavelength range. They work better the higher the optical frequency. In the higher wavelength regime (wavelength longer than 1.5 µm), there is a limit when the energy of a single photon becomes too small to trigger a detection event. An exception is transition-edge sensors, which are highly tunable in their wavelength range [11,12].

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(a) single photons – discrete variables

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Figure 2. Sequence of quantum states as seen by a click detector (a) and a homodyne detector (b), assuming losses of 75%. On average, the click detector randomly sees 25% of the quantum states in full and the rest not at all. The homodyne detector always measures an event, but each time with intensity reduced to 25%. (Online version in colour.)

(b) Detection bandwidth Both in discrete as well as in continuous variables, bandwidths in the GHz regime can be reached [13]. The highest bandwidths of shot-noise-limited continuous-variable detection are in the range of 10 GHz [14].

(c) Spectral filtering In homodyne detection, when beating the signal with an LO, only that part which matches the potentially extremely narrow spectrum of the LO will be amplified. Therefore, filtering is extremely narrowband. Even the worst case of a quantum detector looking directly into the Sun is possible (unless overall saturation of the detector occurs). The reason is that in bright sunlight, the number of photons per detection mode is much less than one [15]. By contrast, click detectors have to be used in conjunction with additional spectral filters to achieve this feature.

(d) Losses Click detectors have the advantage of a built-in perfect post-selection. This means that, for each signal pulse, the detector either sees the signal in full or it does not see it at all. A homodyne detector gives a measurement result for every signal pulse but with reduced quality (figure 2).

(e) Technology Homodyne detection is used in standard optical telecommunication technology, which can be a bonus when working with existing satellites. Thus, this technique is readily available also for quantum communication. Click detectors and photon-number-resolving detectors on the other hand are a rather new technology best suited when Fock state encoding is used.

3. Post-selection for continuous variables We have seen in figure 2 that losses lead to a reduction of signal-to-noise ratio in homodyne detection. At first glance, this seems to be a showstopper for QKD where typically all losses are attributed to eavesdroppers. As soon as losses exceed transmission, an eavesdropper would have an information advantage. A simplified beam splitter scenario is depicted in figure 3 where one of the two detectors belongs to the legitimate receiver Bob and the other detector belongs to the eavesdropper Eve. The part of the signal that is not measured by Bob is available to Eve. An interesting and nice property of coherent states is that, after splitting a coherent state into two parts, both outputs are again coherent states and are therefore pure quantum states. This is equivalent to stating that there is no correlation between the two output states. This property allows Bob to gain an information advantage by post-selecting on measurement events where uncertainty has randomly produced a favourable outcome [16]. Measurement events with too

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(b) coherent states – continuous variables

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Figure 3. A beam splitter divides a coherent state into two parts. Each of the parts, which are again coherent states, is detected by one of the detectors. As coherent states are pure quantum states, the measurement outcomes in the two detectors are uncorrelated. In QKD, this feature can be harnessed in order to increase the information of Bob above Eve’s information. (Online version in colour.) Xmax prepared by Alice

P (x)

Eve Bob

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Figure 4. Principle of continuous variable post-selection [16]: Bob builds up a double Gaussian histogram by measuring ensembles of the field quadratures of two coherent states, prepared by Alice. By keeping measurement events on the outer edges of the double Gaussian distribution, Bob increases his information advantage over Eve, at the cost of discarding all other measurement events. This is due to Eve’s measurement results being uncorrelated to Bob’s, making it impossible for her to guess which distribution it belongs to. (Online version in colour.) high ambiguity are discarded. Effectively, this leads to an information advantage for Bob, at the cost of key rate as usual in QKD. The B92 QKD protocol [17] as an example is based on two non-orthogonal states such as two coherent states, which is closely related to binary phase shift keying (BPSK), a technique used in classical communication. To be useful for QKD, the overlap of the two coherent states has to be large such that they are nearly indistinguishable per se (figure 4). By repeated measurements on the two signal states, histograms corresponding to the measurement probability distribution can be built up. A single measurement, however, leads to a random result within a Gaussian-shaped uncertainty distribution. Bob can choose to keep only large events and to discard any other events. When there is no correlation between Eve’s and Bob’s measurement results, Eve has a low chance to obtain a significant measurement value conditioned on Bob having a significant measurement value. Coherent states are thus perfect quantum states and therefore qualify for quantum protocols such as QKD in optical fibre and free-space quantum repeaters and random number generation.

4. Quantum random number generation from measuring the quantum mechanical vacuum state Random numbers are required for classical cryptography to generate passwords, certificates and private/public key pairs. Also some simulations, such as the Monte Carlo algorithm, require

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Bob

(a) Terrestrial free-space quantum communication Using a variation of continuous-variable protocols, we can encode quantum states in two orthogonal polarization modes [24] and analyse them by Stokes-type measurements [25], named after Sir George Gabriel Stokes. This can be depicted in the form of a Poincaré sphere (figure 5). This type of encoding allows one to combine LO and signal in one spatial mode that is sent from Alice to Bob. Therefore, atmospheric beam and wavefront distortions are the same for signal and LO, leading to a nearly perfect spatial overlap and a high homodyne detection efficiency at the receiver. This protocol has been verified for the first time in an urban point-to-point link across Erlangen (figure 6). Using binary and quadrature phase shift encoding, it was demonstrated that the quantum properties of coherent states survive propagation through a real-world atmospheric channel [26]. Moreover, polarization encoding protects sensitive squeezed states from atmospheric disturbance [27]. Squeezed states have the potential to lead to a performance increase with respect to QKD [28].

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5. Free-space quantum communication using continuous variables

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random numbers as an initial value (also called seed value). Finally, QKD depends on random numbers and their quality directly influences the secret key rate. So-called pseudo-random numbers attempt to fill this need by generating random numbers using sophisticated algorithms. As algorithms are intrinsically deterministic, these random numbers produce a sequence that looks random, but is actually reproducible once the seed is known. While pseudo-random numbers may be sufficient for a certain class of tasks, they still require some amount of initial randomness to generate the seed. A remedy can be found in physical random number generators, where random numbers are generated by measuring a well-characterized physical process. An implementation can be as simple as measuring the thermal noise of a resistor. These numbers may seem random, but the actual physics to describe the thermal noise in the resistor is complex. Thus, a powerful adversary might be able to predict the measurement outcome without being noticed. True randomness can only be found in the quantum mechanical measurement process which is the basis for QRNG. Here, we focus on a QRNG based on measuring the quantum mechanical vacuum state. One particular coherent state is the one with amplitude zero, the quantum mechanical vacuum state. To measure this state, one can block the signal input of a homodyne detector. The vacuum state is a pure quantum state and thus cannot have any correlation with the outside world [18]. As the Gaussian distribution results from the quantum mechanical uncertainty, any measurement outcome is truly random and unique. A simple method to extract random numbers from the Gaussian probability distribution is to divide it into two parts and assign a bit value to each part. Higher extraction rates can be achieved by increasing the number of bins and, therefore, the number of bits per measurement. Previous experimental realizations have been demonstrated in principle [19] and with high speed in the GHz regime [20,21]. With current developments in integrated photonics, it is only natural to take this type of QRNG to the next level by integrating it onto a photonic chip. Currently, research is being carried out on two technological platforms: silicon (Si) and indium phosphide (InP) [22]. While it is clear that Si-based photonic chips have the advantage of being compatible with existing Si-based semiconductor technology, InP offers a broader variety and better quality of optical structures [23]. In particular, the toolbox of InP-based photonic chips ranges from lasers and modulators up to switches, optical amplifiers and detectors, including all components required for a QRNG. In an ongoing collaboration with the Austrian Institute of Technology and the company Roithner Lasertechnik, both in Vienna, we are building and characterizing such an InP photonically integrated QRNG.

S1

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Figure 5. Poincaré sphere for the representation of quantum polarization states [24] in terms of Stokes variables S1 , S2 and S3 . For circular polarization, the Stokes vector points to the pole. Taking a bird’s eye view on the pole of the sphere, the large excitation in S3 is not visible, and locally S1 and S2 span a two-dimensional phase space distribution at the centre resembling the vacuum state. Small deviation in S1 and S2 can be seen as a quadrature phase modulation that can be characterized by interference with the bright S3 component acting in a way as an LO, very closely related to homodyning. (Online version in colour.)

Figure 6. Free-space continuous-variable quantum link between the old building of the Max Planck Institute for the Science of Light (Alice) and the computer science building of the Friedrich-Alexander University Erlangen (Bob) (picture of Erlangen: Google). (Online version in colour.)

(b) Satellite quantum communication For quantum communication over very long distances, existing telecom fibre infrastructure is not suitable because the losses in optical fibres become too high. Other than in classical telecommunication, intermediate signal amplification is not suitable for quantum communication because this destroys the quantum properties. Quantum repeaters that would preserve quantum properties are currently under development and still at the stage of fundamental research. In outer space, on the other hand, quantum states can propagate without distortion, only affected by noise-free diffraction, for which the signal scales as the inverse squared distance, in contrast with absorption and scattering losses in a fibre, for which the signal decreases exponentially with the distance. Therefore, a satellite-based relay could provide the currently missing links for global quantum communication [29]. Furthermore, transmitting quantum states over large distances and through Earth’s gravitational potential and atmosphere is of great interest in fundamental research.

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1000 500 0

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Figure 8. Typical homodyne measurement histograms of a BPSK signal together with a homodyne measurement of the quantum vacuum state for comparison. (Online version in colour.)

In a collaboration with the company Tesat-Spacecom in Backnang, Germany, and the German Aerospace Center (DLR) in Bonn, Germany, we are performing quantum-limited measurements of signals from the Alphasat in geostationary Earth orbit (GEO), 36 000 km above ground (figure 7). Alphasat is equipped with a Laser Communication Terminal (LCT), developed by Tesat-Spacecom and DLR. The LCT is designed for classical BPSK modulation of coherent states and homodyne detection. Therefore, the LCT technology uses the same technology that we used in our terrestrial link as mentioned before. The main difference is the state overlap, which needs to be sufficiently small for data communication to distinguish between the states with low error. For quantum communication, on the other hand, the state overlap should be large in order to be able to harness quantum uncertainty for achieving security. On the long path from GEO to Earth, only the comparatively thin atmospheric layer could potentially disturb the quantum states, especially when using phase encoding. However, it turns out that such disturbances are bounded to a regime where quantum communication is feasible [30]. This verification was performed at the Transportable Adaptive Optical Ground Station of Tesat, tested on Tenerife, using quantum-noise-limited homodyne detection. Typical homodyne measurement histograms of a BPSK signal are shown in figure 8, together with a homodyne measurement of the quantum vacuum state for comparison. In accordance with theory, all histograms are of Gaussian shape and the variance of the signal states equals the variance of the

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Figure 7. A Laser Communication Terminal (LCT) on the geostationary spacecraft Alphasat links to the Transportable Adaptive Optical Ground Station (TAOGS), currently located in the Teide Observatory on Tenerife, Spain (picture of Alphasat: ESA; picture of the TAOGS: Tesat-Spacecom, Synopta, DLR). (Online version in colour.)

Continuous-variable quantum technology has been shown to be a viable alternative to discretevariable approaches in certain scenarios. Terrestrial as well as satellite free-space quantum communication can be efficient and economic, building upon coherent optical communication technology, which is well developed and frequently used. However, there is some room for improvement regarding the security proofs for continuous-variable QKD protocols, which need to be adapted to more practical modulation formats (discrete and differential phase shift keying). It would likewise be beneficial to study the combination of the aforementioned technologies in a real quantum communication network environment in order to ensure compatibility with existing infrastructure and to define the necessary interfaces between classical cryptography systems and novel quantum devices. Data accessibility. This article has no supporting data. Competing interests. We declare we have no competing interests. Funding. The Laser Communication Terminal (LCT) and the Transportable Adaptive Optical Ground Station (TAOGS) are supported by the German Aerospace Center (DLR), Space Administration, with funds from the Federal Ministry for Economic Affairs and Energy according to a decision of the German Federal Parliament.

Acknowledgements. We thank Kevin Günthner for preparation of the figures and our colleagues at the FAU computer science building for their kind support and for hosting our receiver set-up. We also thank Norbert Lütkenhaus for valuable discussions.

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6. Conclusion

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quantum vacuum state. This confirms that the detected signals are very close to perfect quantum coherent states. Reducing the intensity at the sender would allow one to increase the overlap into a regime suitable for quantum communication. We note that losses per se are nearly negligible on the path from outer space to Earth under good weather conditions. However, even though the LCT beam is very close to being diffractionlimited, there is significant beam expansion on the long propagation path and, close to Earth, beam spreading from refraction by atmospheric density fluctuations. Therefore, only a small portion of the beam can be captured by the aperture of the ground station. Still, a performance and cost analysis shows that QKD is economically feasible for approximately $1 per key of length 256 bit [31,32].

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