Reduced State Embedding for Error Correction in Quantum Cryptography

TL;DR

This work addresses the fundamental trade-off in high-dimensional QKD between increased information capacity and vulnerability to channel noise by introducing reduced-state embeddings: encoding information in a $k$-dimensional subspace of a $d$-dimensional Hilbert space and treating non-kept outcomes as erasures. The authors develop a theory for three channel models—depolarizing, modulo, and block-biased—and derive closed-form expressions for kept probability, $k$-ary dit error, and secure-key-rate bounds via the Devetak–Winter rate, identifying noise-dependent interior optima with $k<d$. They validate the framework experimentally in a $d=25$ QKD system, finding an optimal $k=5$ that maximizes the key rate under realistic block-biased noise, demonstrating modulation and erasure-based error handling at the physical layer. The results offer a practical path to higher secure key rates in high-dimensional QKD and motivate broader application of quantum error-correction concepts at the transmission stage, with scalability across degrees of freedom and platform architectures.

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 error-correction 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. In the framework of quantum error-correction, our reduced-state embedding realizes an explicit erasure-type error-correction within the quantum channel. We demonstrate the advantage of our scheme in realistic quantum channels, producing a higher secure key rate. We validate our approach using a 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 to error-correction and modulation for quantum cryptography.

Figures (7)

• Figure 1: Conceptual visualizations of the noisy channels. The states are represented by nodes in a graph, where the distance between adjacent nodes indicates the transition probability between the corresponding states. (a) Depolarizing channel (Section \ref{['Subsection:Dep']}). In the depolarization model, each state is equally distant from every other state, as every pair has the same transition probability. The states sit at the vertices of a regular simplex (e.g., triangle for $d=3$, tetrahedron for $d=4$), hence every pairwise distance is identical. (b) Modulo channel (Section \ref{['Subsection:Mod']}). States are arranged at equal spacing on a ring. Transition is only possible between two nearest-neighbors $i\pm 1 \pmod d$. (c) Block-bias channel (Section \ref{['Subsection:block']}). The state space is partitioned into disjoint 5-state blocks. Within each block, all-to-all transitions occur with equal probability, and there is a weak coupling between blocks.
• Figure 2: Physical-noise threshold and secure key rate for the depolarizing channel.(a) Physical-noise threshold $\varepsilon_{\mathfrak{D}}^{\mathrm{th}}$ for different values of $d$. The heatmap shows the threshold of the tolerable depolarization probability $\varepsilon$, for a positive Devetak--Winter key rate, as a function of the signal-set size $k$ and the space dimension $d$. (b) Secure key rate $R$ as a function of signal-set size $k$, for a fixed dimension $d=25$. Each curve corresponds to a different physical noise parameter $\varepsilon$. For every $\varepsilon$, a black dot marks the optimal signal-set size, which maximizes the secret key rate. The plot highlights the trade-off between increasing signal-set size and noise accumulation. Initially, as $k$ increases, the key rate increases as well. For larger $k$, however, noise accumulation may suppress performance. Notably, for $\varepsilon < 0.083$, the optimal performance occurs when the signal-set size is strictly smaller than the space dimension, i.e., $k<25$. This confirms that encoding into a reduced subspace is preferable to using the full Hilbert space dimension.
• Figure 3: Encoding a signal set of size $k$ on the cycle graph $C_6$, for $k=2,\dots,6$. Red nodes represent chosen states in the subset $\mathscr S_b$, corresponding to basis $b$. Blue edges indicate internal adjacencies in $W(\cdot)$ ("confusions") that generate errors within the kept set, and green dashed edges are boundary adjacencies in $B(\cdot)$ that lead to inconclusive outcomes. For $k\leq 3$, the encoding removes all internal adjacencies, i.e., $W=0$. Whereas, for $k>3$, some adjacencies are unavoidable, causing a trade-off between kept probability $\alpha$ and dit error rate $Q$.
• Figure 4: Physical-noise threshold and secure key rate for modulo channel.(a) Heatmap of the physical-noise threshold $\varepsilon_{\mathfrak M}^{\mathrm{th}}$. Results correspond to the evenly spaced encoding strategy. The triangular region $k \leq d$ marks valid encodings, with lighter colors indicating higher tolerance to physical noise. The plateau at $\varepsilon_{\mathfrak M}^{\mathrm{th}}=1/2$ identifies the maximal noise-tolerance regime, occurring whenever adjacent symbols can be completely avoided ($W=0$), so errors vanish and only inconclusive outcomes remain.(b) Secure key rate $R$ as a function of signal-set size $k$, for a fixed dimension $d=25$. Each curve corresponds to a different physical noise parameter $\varepsilon$. For every $\varepsilon$, a black dot marks the signal-set size that maximizes the key rate. Notably, for $\varepsilon < 0.0325$, the optimal performance occurs at $k<d$, indicating that encoding into a reduced subspace is preferable to using the full Hilbert space dimension.
• Figure 5: Physical-noise threshold and secure key rate for the block-bias channel.(a) Heatmap of the physical-noise threshold $\varepsilon_{2}^{\mathrm{th}}$ as a function of the Hilbert space dimension $d$ and signal-set size $k$, with intra-block depolarization fixed at $\varepsilon_{1}=0.07$. The contour lines highlight threshold levels, showing how the tolerance to inter-block noise depends on both $d$ and the chosen signal-set. (b) Secure key rate $R$ as a function of signal-set size $k$, for a space dimension of $d=4,9,16,25$. The optimal signal-set size is $k=\sqrt{d}$, where $\varepsilon_1=0.3$ and $\varepsilon_2=0.07$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4