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arXiv:1507.03740 · Physical Review A

Quantum key distribution using qudits that each encode one bit of raw key

H. F. Chau

Published Dec 11, 2015• 32 citations
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Abstract

All known qudit-based prepare-and-measure quantum key distribution (PMQKD) schemes are more error resilient than their qubit-based counterparts. Their high error resiliency comes partly from the careful encoding of multiple bits of signals used to generate the raw key in each transmitted qudit so that the same eavesdropping attempt causes a higher bit error rate (BER) in the raw key. Here I show that highly-error-tolerant PMQKD schemes can be constructed simply by encoding one bit of classical information in each transmitted qudit in the form $(|i\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|j\ensuremath{\rangle})/\sqrt{2}$, where $|i\ensuremath{\rangle}$'s form an orthonormal basis of the ${2}^{n}$-dimensional Hilbert space. Moreover, I prove that these schemes can tolerate up to the theoretical maximum of a 50% BER for $n\ensuremath{\ge}2$ provided the raw key is generated under a certain technical condition, making them extremely-error-tolerant PMQKD schemes involving the transmission of unentangled finite-dimensional qudits. This shows the potential of processing quantum information using lower-dimensional quantum signals encoded in a higher-dimensional quantum state.

DOI
10.1103/physreva.92.062324

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All known qudit-based prepare-and-measure quantum key distribution (PMQKD) schemes are more error resilient than their qubit-based counterparts. Their high error resiliency comes partly from the careful encoding of multiple bits of signals used to generate the raw key in each transmitted qudit so that the same eavesdropping attempt causes a higher bit error rate (BER) in the raw key. Here I show that highly-error-tolerant PMQKD schemes can be constructed simply by encoding one bit of classical information in each transmitted qudit in the form (∣i\ensuremath⟩\ifmmode±\else\textpm\fi∣j\ensuremath⟩)/2(|i\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|j\ensuremath{\rangle})/\sqrt{2}(∣i\ensuremath⟩\ifmmode±\else\textpm\fi∣j\ensuremath⟩)/2​, where ∣i\ensuremath⟩|i\ensuremath{\rangle}∣i\ensuremath⟩'s form an orthonormal basis of the 2n{2}^{n}2n-dimensional Hilbert space. Moreover, I prove that these schemes can tolerate up to the theoretical maximum of a 50% BER for n\ensuremath≥2n\ensuremath{\ge}2n\ensuremath≥2 provided the raw key is generated under a certain technical condition, making them extremely-error-tolerant PMQKD schemes involving the transmission of unentangled finite-dimensional qudits. This shows the potential of processing quantum information using lower-dimensional quantum signals encoded in a higher-dimensional quantum state.
Recently, Chau [Phys. Rev. A 92, 062324 (2015)] introduced an experimentally feasible qudit-based quantum-key-distribution (QKD) scheme. In that scheme, one bit of information is phase encoded in the prepared state in a 2n{2}^{n}2n-dimensional Hilbert space in the form $(|i\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|j\ensuremath{\rangle})/\sqrt{2}$ with $n\ensuremath{\ge}2$. For each qudit prepared and measured in the same two-dimensional Hilbert subspace, one bit of raw secret key is obtained in the absence of transmission error. Here we show that by modifying the basis announcement procedure, the same experimental setup can generate nnn bits of raw key for each qudit prepared and measured in the same basis in the noiseless situation. The reason is that in addition to the phase information, each qudit also carries information on the Hilbert subspace used. The additional $(n\ensuremath{-}1)$ bits of raw key comes from a clever utilization of this extra piece of information. We prove the unconditional security of this modified protocol and compare its performance with other existing provably secure qubit- and qudit-based protocols on market in the one-way classical communication setting. Interestingly, we find that for the case of n=2n=2n=2, the secret key rate of this modified protocol using nondegenerate random quantum code to perform one-way entanglement distillation is equal to that of the six-state scheme.
Abstract High-dimensional quantum key distribution (QKD) based on entangled qudits has emerged as a promising route for advancing the performance and practicality of quantum communication. By extending beyond the qubit framework, qudit entanglement enables the encoding of multiple bits per particle and offers greater robustness against noise, loss, and eavesdropping. In this review, we present qudit-based entanglement QKD protocols and their experimental demonstrations. We first introduce the basic concepts of qudits and provide an overview of foundational qubit-based entanglement QKD schemes, such as Ekert '91 (E91) and Bennett–Brassard–Mermin '92 (BBM92), which establish the core principles of entanglement-based secure key generation. Subsequently, we examine the generalization of these protocols to entangled qudits, highlighting how higher dimensionality enhances secure key rates and noise tolerance. Finally, we survey experimental demonstrations of qudit-based entanglement QKD across photonic platforms and outline prospects, showing that high-dimensional entanglement can be reliably generated, distributed, and measured for QKD with high stability and scalability. Remarkably, these achievements not only verify the feasibility of practical high-dimensional QKD, but also demonstrate clear advantages over qubit-based systems. Overall, theoretical frameworks and experimental realizations reviewed here position photonic qudit-based entanglement QKD as a promising resource for scalable, high-speed, and secure quantum communication networks, unveiling its potential as a central component in the future of quantum information technologies.
Employing the fundamental laws of quantum physics, Quantum Key Distribution (QKD) promises the unconditionally secure distribution of cryptographic keys. However, in practical realisations, a QKD protocol is only secure, when the quantum bit error rate introduced by an eavesdropper unavoidably exceeds the system error rate. This condition guarantees that an eavesdropper cannot disguise his presence by simply replacing the original transmission line with a less faulty one. Unfortunately, this condition also limits the possible distance between the communicating parties, Alice and Bob, to a few hundred kilometers. To overcome this problem, we design a QKD protocol which allows Alice and Bob to distinguish system errors from eavesdropping errors. If they are able to identify the origin of their errors, they can detect eavesdropping even when the system error rate exceeds the eavesdropping error rate. To achieve this, the proposed protocol employs an alternative encoding of information in two-dimensional photon states. Errors manifest themselves as quantum bit and as index transmission errors with a distinct correlation between them in case of intercept-resend eavesdropping. As a result, Alice and Bob can tolerate lower eavesdropping and higher system error rates without compromising their privacy.
We describe a four-dimensional (D=4\mathcal{D} = 4D=4) single-photon quantum cryptography protocol with up to twenty (D×(22+1)\mathcal{D} \times (2^2 + 1)D×(22+1)) possible states generated by a polarization-, phase- and time-encoding transmitter. This protocol can be experimentally realized with existing technology, drawing from time- and polarization-encoded systems. The protocol is error tolerant and has a maximum raw bit rate of two raw bits per detection, which when combined with state detection efficiency yields a qubit rate of up to one per transmission under ideal assumptions, or up to twice the raw bit rate of two-dimensional protocols such as the well-known BB84 protocol.
Encoding in a high-dimensional Hilbert space improves noise resilience in quantum information processing. This approach, however, may result in cross-mode coupling and detection complexities, thereby reducing quantum cryptography performance. This fundamental trade-off between correctness and secrecy motivates the search for quantum high-dimensional encoding approaches for cryptography. Here, we introduce state embeddings that use a k -symbol subset within a d -dimensional Hilbert space, tailored to the channel's error structure. We demonstrate the advantage of our scheme in realistic quantum channels, producing a higher secure key rate. We validate our approach using d = 25 quantum key distribution (QKD) experimental data, derive closed-form expressions for the key rate and threshold, and determine the optimum at k = 5 . These findings advance high-dimensional QKD and pave the way for error correction and modulation for quantum cryptography.
We present a high-dimensional measurement device-independent (MDI) quantum key distribution (QKD) protocol employing biphotons to encode information. We exploit the biphotons as qutrits to improve the tolerance to error rate. Qutrits have a larger quantum system; hence they carry more bits of classical information and have improved robustness against eavesdropping compared to qubits. Notably, our proposed protocol is independent of measurement devices, thus eliminating the possibility of side-channel attacks. Also, we employ the finite key analysis approach to study the performance of our proposed protocol under realistic conditions where finite resources are used. Furthermore, we simulated the secret key rate for the proposed protocol in terms of the transmission distance for different fixed amounts of signals. The results prove that this protocol achieves a considerable secret key rate for a moderate transmission distance of 90 km by using [Formula: see text] signals. Moreover, the expected secret key rate was simulated to examine our protocol's performance at various intrinsic error rate values, [Formula: see text] caused by misalignment and instability due to the optical system. These results show that reasonable key rates are achieved with a minimum data size of about [Formula: see text] signals which are realizable with the current technology. Thus, implementing MDI-QKD using finite resources while allowing intrinsic errors due to the optical system makes a giant step forward toward realizing practical QKD implementations.