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Vestnik Moskovskogo Universiteta Seriya 3 Fizika Astronomiya

Quantum Communication Scheme Described by a Regular Four-Dimensional Complex Polytope

Alexander Yu. Vlasov

Published Jan 1, 2025• 0 citations
Publisher

Abstract

At present, generalisations of quantum communication protocols from qubits to systems with higher- dimensional state spaces (qudits) commonly employ mutually unbiased bases (MUB), the construction of which is known for any dimension equal to a prime power. However, in low dimensions, there also exist formally more symmetric state systems described by regular complex polytopes, which are a generalisation of the concept of regular polytopes to complex spaces. This work considers the application of a model originally proposed by R. Penrose and based on the geometry of the dodecahedron and two entangled particles with spin 3/2. In a more general case, two arbitrary quantum systems with four basis states (ququarts) can be used instead. It was subsequently shown that this system of 40 states is equivalent to the Witting con guration and is related to the four-dimensional complex polytope described by Coxeter. The paper proposes a quantum key distribution protocol based on contextuality, utilising this con guration.

DOI
10.55959/msu0579-9392.80.2530405

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At present, generalisations of quantum communication protocols from qubits to systems with higher- dimensional state spaces (qudits) commonly employ mutually unbiased bases (MUB), the construction of which is known for any dimension equal to a prime power. However, in low dimensions, there also exist formally more symmetric state systems described by regular complex polytopes, which are a generalisation of the concept of regular polytopes to complex spaces. This work considers the application of a model originally proposed by R. Penrose and based on the geometry of the dodecahedron and two entangled particles with spin 3/2. In a more general case, two arbitrary quantum systems with four basis states (ququarts) can be used instead. It was subsequently shown that this system of 40 states is equivalent to the Witting con guration and is related to the four-dimensional complex polytope described by Coxeter. The paper proposes a quantum key distribution protocol based on contextuality, utilising this con guration.
Abstract At present, generalizations of quantum communication protocols from qubits to systems with higher-dimensional state spaces (qudits) commonly employ mutually unbiased bases (MUB), the construction of which is known for any dimension equal to a prime power. However, in low dimensions, there also exist formally more symmetric state systems described by regular complex polytopes, which are a generalization of the concept of regular polytopes to complex spaces. This work considers the application of a model originally proposed by R. Penrose and based on the geometry of the dodecahedron and two entangled particles with spin 3/2. In a more general case, two arbitrary quantum systems with four basis states (ququarts) can be used instead. It was subsequently shown that this system of 40 states is equivalent to the Witting configuration and is related to the four-dimensional complex polytope described by Coxeter. The paper proposes a quantum key distribution protocol based on contextuality, utilizing this configuration.
Quantum key distribution (QKD) protocols most often use two conjugate bases in order to verify the security of the quantum channel. In the majority of protocols, these bases are mutually unbiased to one another, which is to say they are formed from balanced superpositions of the entire set of states in the opposing basis. Here, we introduce a high-dimensional QKD protocol using qubit-like states, referred to as Fourier-qubits (or F-qubits). In our scheme, each F-qubit is a superposition of only two computational basis states with a relative phase that can take d distinct values, where d is the dimension of the computational basis. This non-mutually-unbiased approach allows us to bound the information leaked to an eavesdropper, maintaining security in high-dimensional quantum systems despite the states’ seemingly two-dimensional nature. By simplifying state preparation and measurement, our protocol offers a practical alternative for secure high-dimensional quantum communications. We experimentally demonstrate this protocol for a noisy high-dimensional QKD channel using the orbital angular momentum degree of freedom of light and discuss the potential benefits for encoding in other degrees of freedom. High dimensional quantum key distribution (QKD) systems will allow for higher key generation rate, but with added complexity for creating and detecting high dimensional quantum states. The authors demonstrate a QKD protocol using “qubit-like” qudit states, “F-qubits”, with simpler generation and detection, maintaining the benefits of high dimensional QKD protocols.
The topic of this thesis is Quantum Cryptography. Based on the laws of Quantum Mechanics, this allows two parties to share a secure key, by using quantum states to carry classical information. Namely, the Quantum Key Distribution (QKD) process, completed with the classical algorithm One Time Pad, leads to an unconditionally secure cryptosystem. Most QKD protocols, like the famous BB84, are realized on unidirectional quantum channels and have probabilistic character. Recently, a new protocol, named LM05, has been proposed, which works on bidirectional quantum channels and in a deterministic way. This thesis is mainly concerned with protocols extending LM05 from binary to d-ary alphabets (dmajor2), by using multi-level quantum systems in d-dimensional Hilbert spaces. The construction of such protocols relies on the notion of Mutually Unbiased Bases (MUB). But the total number of MUB in a Hilbert space of dimension d is known only if d is a prime power. Accordingly, the new protocols are realized under this assumption. As a preliminary step, an explicit expression for MUB encompassing powers of both even and odd primes is discussed. The first proposed extension, called EM09, uses shift operators on MUB to encode information and the usual quantum measurement to realize the control procedure. This guarantees maximal security for dimension d=3 against a powerful individual attack. The second extension, named EM11, is characterized by an innovative control strategy based on a suitable unitary transformation rather than on quantum measurement. Such protocol only works for d an odd prime power, due to the particular choice of the control operator, which is proved to be the only possible with the required properties. The EM11 protocol leads to a relevant improvement. In fact, the security against the same attack is much better than in the EM09 and it increases in terms of the dimension d. Some partial results are also obtained about the possible extensions of the probabilistic protocol called SARG04 to higher dimensions. Finally, it is considered the use of the EM09 and EM11 protocols in the Quantum Direct Communication, where the meaningful message is transmitted without any encryption, as allowed by their deterministic character. In this context, the asymptotic security turns out to be optimal in dimension d=2 for EM09 and in dimension d=3 for EM11.
The application of quantum mechanics contributes to the field of cryptography with very important advantage as it offers a mechanism for detecting the eavesdropper. The pioneering work of quantum key distribution uses mutually unbiased bases (MUBs) to prepare and measure qubits (or qudits). Weak mutually unbiased bases (WMUBs) have weaker properties than MUBs properties, however, unlike MUBs, a complete set of WMUBs can be constructed for systems with composite dimensions. In this paper, we study the use of weak mutually unbiased bases (WMUBs) in quantum key distribution for composite dimensional finite systems. We prove that the security analysis of using a complete set of WMUBs to prepare and measure the quantum states in the generalized BB84 protocol, gives better results than using the maximum number of MUBs that can be constructed, when they are analyzed against the intercept and resend attack.
Works in two way deterministic quantum cryptography protocols using qubits and qutrits have seen the nontrivial generalization of the protocol to accommodate the use of all mutually unbiased bases (MUB). Here we consider the use of qunits (quantum systems of n dimensional Hilbert space) in a similar protocol and working with cases where the number of MUB is n +1 we show that n qunits are necessary to generalize the use of all available MUBs. We also provide a recipe that would allow for encoding with unitary efficiency.