⚛️ quantum physics

Reduced State Embedding for Quantum Cryptography

This paper introduces a reduced state embedding technique that optimizes high-dimensional quantum key distribution by selecting a specific subset of symbols tailored to channel errors, thereby resolving the trade-off between noise resilience and detection complexity to achieve higher secure key rates, as validated by experimental data and theoretical analysis.

Original author: Amit Kam, Kfir Sulimany, Shai Tsesses, Uzi Pereg

Published 2026-11-10

📖 7 min read🧠 Deep dive

CC BY 4.0

Original author: Amit Kam, Kfir Sulimany, Shai Tsesses, Uzi Pereg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest for unbreakable communication, scientists have long turned to the strange rules of quantum mechanics. Unlike traditional encryption, which relies on complex math problems that could eventually be solved by powerful computers, quantum key distribution uses the fundamental behavior of light particles to create secret codes. If a spy tries to listen in, the very act of measuring the light changes it, alerting the users to the breach. For decades, researchers have tried to make these systems faster and more robust by packing more information into each particle of light. Instead of using a particle to represent just a single bit of data, like a coin that is either heads or tails, they began using high-dimensional states that can represent many possibilities at once. The logic seemed sound: more possibilities mean more information per photon and a higher tolerance for noise. However, in the messy reality of the physical world, simply adding more options often backfires. As the number of possible states grows, the equipment needed to create and detect them becomes incredibly complex, and the signals begin to interfere with one another, introducing errors that actually weaken the security rather than strengthen it.

A team of researchers has now found a way to navigate this paradox by doing less, not more. In a study published in a leading journal, they demonstrated that the most secure and efficient way to send quantum keys is not to use the full capacity of the available space, but to deliberately choose a smaller, cleaner subset of options. The team, working with a system capable of handling twenty-five different states of light, discovered that using only five of those states produced a significantly higher rate of secure keys than using all twenty-five. This counterintuitive result, which they call "reduced state embedding," suggests that the best strategy for quantum communication is to filter out the noise before it even becomes a problem, rather than trying to correct it after the fact.

The researchers built their experiment on a foundation of entangled photons, particles of light that are linked in such a way that measuring one instantly reveals the state of the other. They generated these pairs and sent them through a complex optical setup involving a series of mirrors and phase plates, known as a multi-plane light converter. This device allowed them to manipulate the spatial shape of the light, effectively creating a twenty-five-dimensional grid where each position represented a different symbol. In a standard high-dimensional approach, the sender would choose any of the twenty-five positions to encode a message, and the receiver would attempt to detect which one was sent. The researchers, however, tested a different strategy. They asked what would happen if, for every transmission, they restricted the sender to a smaller group of choices—say, only five specific positions out of the twenty-five available.

To understand why this works, one must look at how noise behaves in these systems. In the real world, light does not travel perfectly; it scatters, and the equipment introduces small errors. In a twenty-five-dimensional system, if a photon is meant to land in one specific spot, noise might push it slightly off course. If the system is crowded with twenty-five possible targets, a small shift might cause the photon to land in a neighboring spot that is also a valid message, creating confusion. The receiver might think the sender chose a different symbol than they actually did. By restricting the valid messages to a smaller, carefully chosen group of five spots, the researchers created a buffer. If noise pushes a photon out of this small group, the receiver simply discards that attempt as inconclusive, rather than mistaking it for a different valid message. This process acts like a sieve, letting only the cleanest signals through to be counted as part of the secret key.

The team tested this idea by systematically varying the size of the subset they used, moving from two states up to all twenty-five. They measured how often the receiver successfully detected a message and how often errors occurred among those successful detections. The results were clear and decisive. As they increased the number of states used, the rate of errors grew faster than the benefit of having more options. The secure key rate—the amount of secret information they could actually extract—peaked when they used exactly five states. At this point, the system was robust enough to filter out the majority of the noise while still carrying enough information to be efficient. Using more than five states caused the error rate to climb so high that the security of the key was compromised, and using fewer than five states meant they were throwing away too much potential information.

This finding challenges the prevailing assumption that "bigger is better" in quantum communication. For years, the field has focused on expanding the dimensionality of the space to squeeze more data into every photon. The researchers show that this approach hits a wall where the physical imperfections of the hardware and the environment outweigh the theoretical gains. Their method of reduced state embedding does not try to fight the noise with more complex correction algorithms after the message is received. Instead, it designs the transmission itself to be inherently resistant to noise. By choosing a specific, smaller set of states that are well-separated from the noise patterns of the channel, they effectively turn physical errors into discarded attempts rather than corrupted data. This is a crucial distinction because it means the system remains secure even when the noise level is high, a scenario where full-dimensional systems would fail.

The experiment was conducted in a controlled laboratory setting using a specific type of noise pattern that mimics real-world fiber optic and free-space communication, where errors tend to cluster in groups rather than appearing randomly. The researchers confirmed that their theoretical predictions matched the experimental data perfectly. They derived mathematical formulas that describe exactly how the optimal number of states depends on the type of noise present. For some types of noise, the best subset size might be different, but the principle remains the same: there is always an optimal middle ground where the size of the message set is smaller than the total available space. This optimal point balances the need for information capacity with the need for signal clarity.

The implications of this work extend beyond just making quantum keys faster. It suggests a new way of thinking about how to build quantum networks. Rather than building systems that try to handle every possible state perfectly, engineers might design systems that are tailored to the specific errors of their environment, using a smaller, optimized set of states. This approach could make quantum cryptography more practical for real-world deployment, where noise and imperfections are unavoidable. The researchers also noted that this strategy could be applied to other quantum technologies, such as secure communication between multiple parties or even quantum computing, where protecting information from errors is a constant struggle.

In the end, the study offers a lesson in the value of constraint. In a field often driven by the desire to maximize every variable, the researchers found that limiting the choices available to the sender and receiver actually maximized the security and efficiency of the system. By embracing a smaller, cleaner subset of the available quantum states, they turned a weakness—the noise that plagues all physical systems—into a feature that helps filter out errors. This work provides a concrete path forward for high-dimensional quantum cryptography, showing that the key to unlocking the full potential of quantum communication may lie not in using everything available, but in knowing exactly what to leave out.

Technical Summary: Reduced State Embedding for Quantum Cryptography

Problem Statement

Quantum Key Distribution (QKD) protocols, such as BB84, face a fundamental trade-off between noise resilience and implementation complexity. While encoding information in high-dimensional Hilbert spaces ( $d$ -dimensional) theoretically increases information capacity per photon and raises the tolerable error rate threshold, practical implementations suffer from cross-mode coupling, detection complexities, and mode mismatch. These physical imperfections accumulate with dimension, often negating theoretical advantages and reducing the secure key rate. Standard approaches typically employ the full set of $d$ orthogonal states in two mutually unbiased bases (MUBs), but this "full-dimensional" strategy is experimentally challenging and may inadvertently increase an eavesdropper's potential knowledge due to noise-induced confusion.

The core problem addressed is how to leverage the potential of high-dimensional encoding without succumbing to the noise accumulation and detection inefficiencies associated with using the entire Hilbert space. The authors posit that the practical advantage of large Hilbert spaces is not realized by using all $d$ states, but by encoding within a carefully chosen, smaller subspace tailored to the channel's specific error structure.

Methodology

The paper introduces a Reduced State Embedding framework where information is encoded in a subset of $k$ states ( $k < d$ ) within a $d$ -dimensional Hilbert space. This approach integrates state reduction directly into the physical-layer modulation of the quantum signal, rather than treating error correction solely as a post-processing step.

Theoretical Framework

Signal Sets and Protocol: The protocol adapts the BB84 scheme. Alice selects a basis ( $Z$ or $X$ ) and a symbol $x \in \{0, \dots, k-1\}$ uniformly at random, sending the corresponding state. Bob performs a $(k+1)$ -outcome Positive Operator-Valued Measure (POVM). Outcomes $0 \dots k-1$ are "conclusive" (kept), while the $(k+1)$ -th outcome is "inconclusive" (discarded).
Channel Models: The authors analyze the performance of this reduced-state embedding under three distinct noise models to derive closed-form expressions for the secure key rate, error thresholds, and sifting efficiency:
- Depolarizing Channel: A symmetric noise model where the state is randomized across the full Hilbert space with probability $\epsilon$ .
- Modulo Channel: A structured noise model representing nearest-neighbor cyclic flips (relevant for multicore fibers), where errors correspond to transitions between adjacent states in a cycle graph.
- Block-Biased Channel: A model reflecting realistic experimental imperfections where noise concentrates within contiguous subsets (blocks) of modes, common in multimode fibers and free-space channels.
Optimization: For each channel model, the authors derive the optimal subset size $k$ and the specific geometry of the signal set (e.g., truncation vs. spaced selection) that maximizes the Devetak-Winter secure key rate. They define the "kept probability" ( $\alpha$ ) and the conditional "dit error rate" ( $Q$ ) as functions of the channel parameters and the embedding dimension $k$ .

Experimental Validation

The theoretical framework is validated using a $d=25$ QKD system based on spatially entangled photon pairs.

Setup: The system utilizes a Multi-Plane Light Converter (MPLC) to implement programmable unitary mode sorting on spatial modes, realizing two MUBs via 5-point Discrete Fourier Transforms (DFTs) along rows and columns of a $5 \times 5$ grid.
Procedure: For each $k \in \{2, \dots, 25\}$ , the system implements an embedded $k$ -dimensional signal set. Bob applies the corresponding $(k+1)$ -outcome filter.
Measurement: The experiment measures the kept-event probability ( $\alpha$ ) and the conditional dit error rate ( $Q$ ) to compute the secure key rate per signal.

Key Contributions

Reduced State Embedding Strategy: The paper proposes a novel strategy where the modulation scheme itself encodes information in a reduced subspace ( $k < d$ ). This acts as a physical-layer filter, actively discarding noise events that would otherwise contribute to errors in a full-dimensional protocol.
Closed-Form Analytical Results: The authors provide closed-form expressions for the secure key rate and error thresholds for depolarizing, modulo, and block-biased channels. These expressions reveal that the optimal $k$ is an "interior optimum" (strictly less than $d$ ) dependent on the noise level and structure.
Noise-Dependent Optima: The analysis demonstrates that the optimal encoding strategy is fundamentally different for different noise models:
- For depolarizing noise, an intermediate $k$ balances information gain against noise accumulation.
- For modulo noise, optimal sets can be chosen to have zero internal adjacencies ( $W=0$ ), effectively filtering all nearest-neighbor errors, though at the cost of throughput.
- For block-biased noise, the optimal strategy involves distributing the $k$ states evenly across the noise blocks to minimize intra-block confusion.
Experimental Demonstration: The study provides the first experimental validation of reduced-state embedding in a high-dimensional ( $d=25$ ) QKD system, confirming that a reduced subspace ( $k=5$ ) outperforms the full-dimensional protocol ( $k=25$ ) under realistic block-biased noise conditions.

Results

Theoretical Findings: The analysis shows that reducing the state space from $d$ $d$ to $k$ $k$ increases robustness to physical noise (raising the tolerable physical error threshold $\epsilon_{th}$ $ϵ_{t h}$ ) but reduces the sifted throughput (kept probability $\alpha$ $α$ ). The net result is an interior optimum $k_{opt} < d$ $k_{o pt} < d$ where the secure key rate is maximized.
- In the depolarizing channel, $k_{opt}$ is strictly less than $d$ for moderate noise levels.
- In the modulo channel, $k_{opt} = \lfloor d/2 \rfloor$ allows for the suppression of all neighbor-flip errors.
- In the block-biased channel, $k_{opt}$ decreases as noise increases, with the optimal strategy being to spread states across blocks.
Experimental Findings: In the $d=25$ $d = 25$ experimental setup, the secure key rate exhibited a pronounced maximum at $k=5$ .
- As $k$ increased from 2 to 25, the kept probability $\alpha$ and the error rate $Q$ both increased.
- However, the secure key rate peaked at $k=5$ and subsequently declined, confirming the theoretical prediction that using the full Hilbert space ( $k=25$ ) is suboptimal under the specific block-biased noise of the MPLC platform.
- The experimental data aligns with the theoretical prediction for the block-biased channel, validating that operating in an embedded subspace yields higher performance than using all available states.

Significance and Claims

The paper claims to advance high-dimensional QKD by shifting the paradigm from "maximizing dimension" to "optimizing dimension." The significance lies in:

Practicality: It offers a method to improve secure key rates in realistic, noisy channels without requiring complex post-processing error correction or perfect hardware.
Physical-Layer Integration: The approach integrates aspects of state representation and error protection directly into the modulation stage, effectively converting physical errors into erasures (inconclusive outcomes) before post-processing.
Scalability: The method is agnostic to the physical realization of the Hilbert space (spatial, temporal, frequency) and is inherently scalable, provided two MUBs and compatible projective filters are available.
Conceptual Bridge: The authors position reduced-state embedding as a bridge between encoding, modulation, and error correction. It functions similarly to error detection (converting errors to erasures) but is implemented physically rather than via classical post-processing.

The authors conclude that this framework paves the way for error-correction and modulation schemes in quantum cryptography that balance capacity, security, robustness, and practicality, moving beyond the limitations of full-dimensional encoding in noisy environments.