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arXiv:2510.19989 · Newton

Reduced-state embedding for quantum cryptography

Amit Kam, Kfir Sulimany, Shai Tsesses, Uzi Pereg

Published Jul 1, 2026• 0 citations
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Abstract

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.

DOI
10.1016/j.newton.2026.100597

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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.
In this thesis, Quantum Key Distribution and quantum communication methods based on high-dimensional spatial coding of light are developed. Their security is based on the fundamental quantum nature of light and profits from the highdimensional Hilbert spaces offered intrinsically by imaging optics. An overview of classical information theory and quantum information is given to introduce the field of quantum cryptography. The advantages of larger dimensional alphabets and the security of the standard two-bases BB84 protocol is shown. The spatial states used for encoding are introduced. An analytic expression for the upper bound on the mutual information for these states is derived, including multiphoton states, detector noise and beam broadening. High-dimensional encoding of single photons is experimentally realized, reaching 10.5 bit per received photon. The dependence of the mutual information on the number of detector pixels is discussed and the experimental values are compared with the theoretical upper bound. It was shown that a standard error-correcting LDPC code is sufficient to achieve practically error-free communication. By adding a second mutually unbiased basis, a large-alphabet QKD system is experimentally realized and characterized. The security of this BB84-like protocol is analyzed in terms of intercept-resend and collective attacks. The key rate after postprocessing is analyzed under realistic circumstances, including finite key length. Finally, a new quantum communication method is demonstrated, which is based on encoding information into wavefronts decomposed over guided modes of a multimode fiber. At the end a step back is made and the similarities and differences of several quantum authentication and quantum cryptography schemes as well as Quantum Data Locking are discussed. This involves comparison of the characteristics of the classical channel, the quantum channel and the necessary dimension of the Hilbert space.
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 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 $n\ensuremath{\ge}2$ 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.
We present the scheme of compatible quantum information analysis of the quantum key distribution (QKD) protocols, which give answers to the following questions: is it possible to improve the quantum bit error rate (QBER) of the 6-state protocol by employing more states, up to infinity, and can we essentially improve the QBER if the multidimensional Hilbert space with dimensionality more than 3 is used? Also, a novel quantum key distribution (QKD) protocol, based on all unselected states of a quantum system, which set the alphabet with continuous set of letters, is proposed. Employing all states of the Hilbert space leads to the maximal quantum uncertainty of transmitted states and therefore an eavesdropper receives the minimal amount of information. For the case of two-dimensional Hilbert space, our protocol allows secure transmission at the error rate higher than that one for the BB84-protocol and comparable with the characteristics of the best known QKD-protocols. However, with increasing the dimensionality of the Hilbert space the critical error rate for our protocol increases and in the limit of infinite-dimensional space the protocol becomes non-threshold.
High-dimensional quantum key distribution (HD-QKD) provides notable advantages over qubit-based QKD, including higher key rates and greater noise resilience. Leveraging the orbital angular momentum states of structured photons to encode information has become a prominent method for implementing HD-QKD. However, practical challenges such as state-dependent diffraction, limited receiving aperture, and atmospheric turbulence impose constraints on the attainable performance and encoding dimension of QKD in free-space links. In this work, we address these challenges with focused, structured photons. Numerical simulations indicate that this method markedly enhances transmission efficiency and improves the robustness against turbulence, resulting in higher key rates and extended transmission distances in free-space HD-QKD. Additionally, focused, structured photons allow us to implement higher-dimensional QKD and enable QKD systems to achieve superior performance with smaller receiving apertures in a turbulent free-space link. This makes our approach particularly advantageous for applications of unmanned aerial vehicles and satellite platforms. Our work contributes to exploring the robust QKD schemes employing structured photons and may promote its application in quantum networks.
Quantum key distribution (QKD) exploits the quantum nature of light to share provably secure keys, allowing secure communication in the presence of an eavesdropper. The first QKD schemes used photons encoded in two states, such as polarization. Recently, much effort has turned to large-alphabet QKD schemes, which encode photons in high-dimensional basis states. Compared to binary-encoded QKD, large-alphabet schemes can encode more secure information per detected photon, boosting secure communication rates, and also provide increased resilience to noise and loss. High-dimensional encoding may also improve the efficiency of other quantum information processing tasks, such as performing Bell tests and implementing quantum gates. Here, we demonstrate a large-alphabet QKD protocol based on high-dimensional temporal encoding. We achieve record secret-key rates and perform the first field demonstration of large-alphabet QKD. This demonstrates a new, practical way to optimize secret-key rates and marks an important step towards transmission of high-dimensional quantum states in deployed networks.