Semantic Security with Infinite Dimensional Quantum Eavesdropping Channel

TL;DR

This work addresses semantic security for wiretap channels where the eavesdropper observes quantum states in an infinite-dimensional setting. It introduces a novel, nonasymptotic direct coding method based on channel resolvability with symmetrization, yielding exponential decay of both average decoding error and eavesdropper advantage under additive cost constraints, and establishing quantum semantic security guarantees. The results unify semantic security with traditional secrecy notions, provide finite-blocklength bounds, and extend to Gaussian cq channels with energy constraints, including practical numerical evaluations. The Gaussian specialization demonstrates the framework’s applicability to optical and wireless scenarios, while the overall approach broadens the applicable scope of quantum secrecy theorems to infinite-dimensional eavesdropper channels and nonasymptotic regimes.

Abstract

We propose a new proof method for direct coding theorems for wiretap channels where the eavesdropper has access to a quantum version of the transmitted signal on an infinite-dimensional Hilbert space and the legitimate parties communicate through a classical channel or a classical input, quantum output (cq) channel. The transmitter input can be subject to an additive cost constraint, which specializes to the case of an average energy constraint. This method yields errors that decay exponentially with increasing block lengths. Moreover, it provides a guarantee of a quantum version of semantic security, which is an established concept in classical cryptography and physical layer security. Therefore, it complements existing works which either do not prove the exponential error decay or use weaker notions of security. The main part of this proof method is a direct coding result on channel resolvability which states that there is only a doubly exponentially small probability that a standard random codebook does not solve the channel resolvability problem for the cq channel. Semantic security has strong operational implications meaning essentially that the eavesdropper cannot use its quantum observation to gather any meaningful information about the transmitted signal. We also discuss the connections between semantic security and various other established notions of secrecy.

Figures (6)

• Figure 1: Quantum wiretap channel models.
• Figure 2: Point-to-point channel models considered in Section \ref{['sec:res-code']}.
• Figure 3: The cascaded channel model consisting of two beam splitters followed by additive thermal noise channels acting on the signals that are arriving at the legitimate receiver and eavesdropper respectively which is used for the numerical evaluation of the theoretical bounds. Both beam splitters are pure-loss channels which is reflected by the presence of the vacuum state $\left\lvert {0} \right\rangle\left\langle {0} \right\rvert$ at the second input of both devices. The third output is the state $\rho_{\textup{env}}^{\textup{out}}$ of photons detected neither by the legitimate receiver nor the eavesdropper.
• Figure 4: Decay of semantic security level $\delta$ depending on the block length of the codebook for various rates $R$. The decoding error is fixed at $\varepsilon=0.01$.
• Figure 5: Decay of decoding error probability $\varepsilon$ depending on the block length of the codebook for various rates. The semantic security level is fixed at $\delta=0.01$.

Theorems & Definitions (69)

• Remark 1
• Definition 1
• Theorem 1
• Theorem 2
• Remark 2
• Theorem \ref{theorem:wiretap-ccq}'
• Theorem \ref{theorem:wiretap-cq}'
• Definition 2
• Definition 3
• Remark 3