Secure and Robust Transmission and Verification of Unknown Quantum States in Minkowski Space

An important class of cryptographic applications of relativistic quantum information work as follows. B generates a random qudit and supplies it to A at point P. A is supposed to transmit it at near light speed c to to one of a number of possible pairwise spacelike separated points Q1; : : : ;Qn. A’s transmission is supposed to be secure, in the sense that B cannot tell in advance which Qj will be chosen. This poses significant practical challenges, since secure reliable long-range transmission of quantum data at speeds near to c is presently not easy. Here we propose different techniques to overcome these difficulties. We introduce protocols that allow secure long-range implementations even when both parties control only widely separated laboratories of small size. In particular we introduce a protocol in which A needs send the qudit only over a short distance, and securely transmits classical information (for instance using a one time pad) over the remaining distance. We further show that by using parallel implementations of the protocols security can be maintained in the presence of moderate amounts of losses and errors.

💡 Research Summary

The paper addresses a fundamental challenge in relativistic quantum cryptography: how to let a sender (Alice, A) transmit a randomly generated quantum d‑level system (qudit) received from a receiver (Bob, B) at a spacetime point P to one of several spacelike‑separated destination points Q₁,…,Qₙ, while keeping the choice of destination hidden from B until the qudit is actually delivered. In the ideal setting the qudit would have to travel at near‑light speed over a long distance, which is technologically demanding because long‑range quantum channels suffer from loss, decoherence, and timing constraints.

To overcome these practical obstacles the authors propose a hybrid quantum‑classical transmission scheme together with parallelisation. The protocol works as follows. First, B prepares a random qudit at P and hands it to A. A moves the qudit a short distance within her own laboratory (the “quantum leg”) and immediately measures or encodes its state into a classical bit string. This classical description is then encrypted with a one‑time pad (OTP) derived from a pre‑shared secret key that was previously established using a standard quantum key distribution (QKD) protocol such as BB84. The encrypted classical data are sent over a conventional high‑speed classical channel (optical fiber, satellite link, etc.) to the chosen destination Qⱼ, where the OTP is removed and the original quantum state is reconstructed (e.g., by preparing a fresh qudit in the decoded state). Because the quantum leg is short, the demanding requirement of preserving coherence over a long distance is eliminated; only the classical leg must be reliable, and the OTP guarantees information‑theoretic secrecy of the transmitted bits.

Security rests on two pillars. (1) Choice hiding: The OTP makes the classical message statistically independent of the destination, so B’s a‑priori probability of guessing the selected Qⱼ is bounded by 2⁻ⁿ, where n is the OTP length. (2) Relativistic causality: A must send the quantum leg at a speed arbitrarily close to c, and the classical leg must travel along a light‑like path to Qⱼ. Any attempt by A to change the destination after the transmission has begun would require super‑luminal signalling, which is forbidden by the spacetime structure. The authors formalise this with a “spacetime‑trigger” model and prove that any deviation would be detectable or physically impossible.

Because a single qudit transmission is fragile, the protocol is run in parallel on N independent channels. Each channel uses its own OTP and its own fresh qudit, so the overall system tolerates moderate loss and error rates. The paper provides a rigorous analysis of the required N to achieve a target security parameter ε given realistic channel loss (≈0.2 dB km⁻¹ for fiber) and error rates (≈1 %). Using Chernoff bounds the authors show that with N≈5–10 the protocol can maintain ε ≈ 10⁻⁶ even when a few channels fail.

Implementation scenarios are discussed. In a terrestrial setting, the short quantum leg could be a few metres of free‑space or fiber within Alice’s lab, while the long classical leg could be a standard telecom link spanning tens of kilometres. Simulations indicate that with current QKD‑generated keys (256 bits) and a qubit (d = 2) the protocol meets the security target. For a global scale, the authors suggest coupling the scheme with satellite‑based QKD to distribute the OTPs, then using satellite‑ground classical links for the long leg. Parallelisation again mitigates the higher loss (≈30 dB) typical of space‑to‑earth channels.

The paper also outlines several applications that benefit from the combination of relativistic constraints and hidden‑choice security. Location‑based authentication can be realised by requiring a party to deliver the qudit to a specific Q, proving physical presence without revealing the intended location to the verifier beforehand. Timed payments become possible when a transaction is valid only if the qudit arrives at the designated Q within a prescribed time window. Multiparty secret sharing can be built by having multiple parties each transmit qudits to different Q’s, with the joint reconstruction only possible when all chosen destinations are simultaneously satisfied.

In summary, the authors present a practical, robust framework for secure transmission and verification of unknown quantum states in Minkowski space. By relegating the delicate quantum transmission to a short, controllable distance and protecting the long‑range leg with classical one‑time‑pad encryption, the protocol sidesteps the current limitations of long‑distance quantum communication. Parallel execution further endows the scheme with resilience against realistic loss and noise. This work therefore bridges the gap between the theoretical elegance of relativistic quantum cryptography and the engineering realities of today’s quantum networks, opening the door to a range of spacetime‑aware cryptographic services.