

Feature Articles: Toward Quantum Technology Innovation Vol. 21, No. 6, pp. 48–52, June 2023. https://doi.org/10.53829/ntr202306fa6 Improving the Performance of Quantum Key DistributionAbstractEncryption is an essential technology for secure communications. Quantum key distribution (QKD) can enable ultimately secure cryptographic communications by using quantum mechanics. Toward secure networks using QKD, NTT has been conducting various studies from theoretical security analysis to experimental control of optical quantum states. In this article, we introduce our recent activities on QKD using multivalued information (highdimensional QKD) and a scalable measurement device for improving its error robustness. Keywords: quantum key distribution, high dimension, error robustness 1. Basics of quantum key distribution and research in NTTIt is necessary to encrypt information for secure communications via the Internet. For example, malicious eavesdropper can read information about a credit card if we send it without encryption. Publickey cryptography, such as Rivest–Shamir–Adleman (RSA), is used for this purpose. Its security is based on a problem that is difficult to solve using a modern digital computer, so called computational security. However, a quantum computer having error correction capability can break RSA efficiently. There is another encryption called onetime pad, which is a commonkey cryptography. This cryptography uses the fact that a third party does not have enough information to break the encryption. Such an informationtheoretic security cannot be broken by any type of computer. Despite its strong encryption, it has a serious disadvantage in that it is difficult to create a situation in which the third party does not have enough information. We use a bit represented by 0 and 1 in a communication system. To make onetime pad ultimately secure, only the legitimate users for the communications need to share random numbers^{*1} (secret key), the length of which is equal to the bit length of the information they really want to send. If we can send the secret key securely, onetime pad looks useless because, in a rough consideration, we can directly send the information using such a secure method used for sharing the secret key instead of onetime pad. Quantum key distribution (QKD) is a technique that solves this problem. QKD uses the fact that an attempt to clone a quantum state changes the original state (Fig. 1). By monitoring such a change in the state, we can estimate how much information was leaked during communications. We cannot directly send a massage we want to encrypt because we cannot prevent the message from being leaked. However, if we send random numbers having no meaningful information, we can generate a secret key by erasing the leaked information from the shared random numbers after the communication. By using this secret key for onetime pad, we can enable unbreakable secret communications.
While we often hear about QKD on the news, it has been studied since its invention in 1984 [1]. NTT has continued its longterm research on QKD [2], such as security analysis, experimental demonstration, and the proposal of our unique QKD protocol called differential phase shift QKD. Let us introduce our recent activities on highdimensional QKD and a technology for improving its error robustness.
2. Highdimensional QKD and scalable measurement deviceHighdimensional QKD uses a highdimensional quantum state to enhance the secretkey rate, which corresponds to the communication speed of QKD. A conventional QKD system uses a quantum bit (qubit) representing 0 or 1. In our research group, we use a timebin qubit, 0 or 1 of which are represented by two temporal positions of a photon^{*2} (Fig. 2 left). If we can encode multiple values, such as 0, 1, 2, …, instead of a bit value, we can increase the amount of information per photon (Fig. 2 right). A highdimensional state is such a state representing multiple values and its concept is similar to pulse amplitude modulation (PAM)^{*3} and quadrature amplitude modulation (QAM)^{*4} used in modern optical communications. A very highspeed secretkey generation of 26.2 Mbit/s has been reported by using fourdimensional timebin states represented by four temporal positions [3].
As we explained in the previous section, it is important to estimate how much information of random numbers was leaked for generating a secure secret key. For this purpose, QKD uses superposed states. A superposed state is a state in which we cannot essentially determine 0 or 1 for a qubit and 0, 1, 2, … for a highdimensional state (Fig. 3). We can estimate the amount of leaked information by using superposed states that satisfy a special relation called mutually unbiased. In a ddimensional system, we can use at most (d + 1) measurements, which satisfy such a special relation. An experiment using two types of measurements to estimate the amount of leaked information achieved 26.2Mbit/s secret key generation [3]. By using (d + 1) measurements, we can more precisely evaluate the change in the quantum state during communications. Therefore, we can more precisely estimate the amount of leaked information, which results in an improvement in the secretkey rate. In other words, multiple measurements make a highdimensional QKD system more robust against errors.
NTT developed and demonstrated an implementation of a scalable measurement device of (d + 1) measurements for highdimensional timebin states [4]. A measurement for timebin states uses delay MachZehnder interferometers (MZIs)^{*5} and singlephoton detectors. In an implementation of a previous measurement device, (d − 1) MZIs and (d + 1) singlephoton detectors were required even for two measurements [3]. By using the above scalable measurement device, all the (d + 1) measurements for d = 2^{N} can be implemented using N MZIs and three singlephoton detectors independent of d (Fig. 4). We conducted experiments involving five measurements for fourdimensional timebin states. The experimental results indicated error rates lower than the threshold required to generate a secret key (Fig. 5). Hence, we expect that this device can be used for a highdimensional QKD system that is more robust against errors.
3. Extension of the security proofIn the previous section, we explained that (d + 1) measurements can be used for a robust highdimensional QKD system. However, a rigorous security of this QKD protocol was proven only if d is a prime number (2, 3, 5, …). Therefore, we cannot directly apply that security proof for a fourdimensional system with the above scalable measurement device. To circumvent this problem, we also extended the previous security proof [5] for prime power dimensions (e.g., d = 2, 4, 8, and 3, 9, 27). In this security proof, we use operators that describe operations on a quantum state. Corresponding to the operators used in the previous security proof, there is a generalized operator using the Galois field^{*6}, which is used in, e.g., coding theory [6]. Because we can use the Galois field as long as d is a prime power, we could extend the previous security proof by using these generalized operators for primepower dimensions. Therefore, we can use the scalable measurement device for a fourdimensional QKD system to ensure rigorous security.
4. Toward a practical QKD systemWe introduced highdimensional QKD and a scalable measurement device to enhance the error robustness as recent activities in NTT to improve the performance of QKD. We conducted a proofofprinciple experiment on generations and measurements of highdimensional quantum states for such a QKD system. To implement a practical QKD system, we also need to consider other problems, for example, a finite key analysis which is a detailed treatment of statistical errors due to a finite number of measurement results. It is also important to explore other applications of the scalable measurement device because the mutually unbiased relation can be found in quantum communications and information processing other than QKD. Although we introduced only the approach of using a highdimensional state in this article, NTT will continue to conduct research on other approaches toward practical quantum information technologies. References
