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High-Dimensional Deterministic Secure Quantum Communication with Reed-Solomon Erasure Coding
L. F. A. de Sousa Moura, G. L. Zanin, P. H. Souto Ribeiro, C. Becker Westphall
TL;DR
Existing DSQC systems face low bit rates and channel loss, motivating a higher-dimensional resilient design. The paper proposes Reed-Solomon erasure coding with prepare-and-measure quantum states, reporting a scheme that avoids quantum memory and uses commercially available technology. Its scope includes a trade-off: increasing the number of mutually unbiased bases raises Eve's error but makes implementation harder and requires more parity photons.
Problem
DSQC suffers from low bit rate, while quantum communication also loses information through network noise, motivating higher-dimensional and fault-tolerant approaches.
Method
The paper combines classical Reed-Solomon erasure coding with high-dimensional prepare-and-measure qudit exchange using single-photon transverse spatial degrees of freedom, decoy states, and MUB measurements.
Results
The proposed protocol transmits deterministic messages through noisy networks without quantum memory and is described as implementable with commercially available technology.
Takeaways & Limitations
The architecture keeps quantum communication focused on sensitive data while relying largely on classical resources and existing technology.
Takeaways & Limitations
Using more than two MUBs increases Eve's introduced error but makes high-dimensional implementation harder and requires more parity photons.
Abstract
from arXiv · showhide
Deterministic Secure Quantum Communication (DSQC) is a quantum cryptographic technique engineered to transfer a message through a quantum channel, requiring an auxiliary classical channel for eavesdropping verification and decoding, but without prior key distribution. This article presents a theoretical high-dimensional prepare and measure DSQC protocol using the Reed-Solomon erasure coding to ensure data resilience to noise. This protocol offers the following benefits: it eliminates the need for quantum memory or entanglement, it can be built with commercially available technology, and its higher capacity improves the overall transmission rate.
I. INTRODUCTION
DSQC transfers messages through quantum states with classical assistance, but its low bit rate and vulnerability to network noise motivate higher-dimensional, fault-tolerant designs. This article proposes a Reed-Solomon-based prepare-and-measure DSQC protocol for resilient transmission without quantum memory.
- DSQC transfers messages through quantum states while requiring an auxiliary classical channel for successful receiver-side decoding.
- Both QSDC and DSQC suffer from low bit rate, making throughput an immediate concern for these protocols.
- Higher dimensionality and fault-tolerant approaches are proposed to mitigate low transmission rates and information loss caused by network noise.
- The proposed protocol combines classical Reed-Solomon erasure coding with quantum prepare-and-measure qudit exchange to transmit deterministic messages through noisy networks.
- The scheme avoids quantum memory and includes a practical implementation proposal using single photons and transverse spatial degrees of freedom.
II. REED-SOLOMON BASED DSQC SECURED BY BB84 NON CLONING STRATEGY
The protocol encodes message symbols with Reed-Solomon redundancy, transmits them as high-dimensional photonic states, and uses MUB measurements and decoys to test channel security. Bob reconstructs the message after retaining at least k valid photons.
- The sender appends Reed-Solomon parity information so the message remains recoverable when some transmitted information is lost.
- Alice and Bob randomly select between two mutually unbiased bases, while decoy photons support QBER estimation against intercept-resend attacks.
- For the example message DATA, the symbols 3-1-5-1 become the polynomial P(x) = 3 + 1 · x^1 + 5 · x^2 + 1 · x^3.
- Alice evaluates the polynomial at n = 10 positions modulo q, producing the codeword C = {10, 0, 1, 8, 5, 9, 4, 7, 2, 6}.
- Bob can recover the original message when he has at least k valid photons after basis mismatches and photon losses.
- Solving the four-equation example yields c0 = 3, c1 = 1, c2 = 5, and c3 = 1.
- Reed-Solomon redundancy permits transmission of additional information without weakening the protocol's security.
A. Splitting the transmission into blocks
Reed-Solomon limits each transmission block to fewer than q photons, so longer messages must be split into multiple blocks. Each block is encoded by a separate polynomial and evaluated at the same ten positions.
- In a q-dimensional alphabet, Reed-Solomon coding requires n < q photons per transmission block.Evaluating at x = 0 is usually discarded because it gives the trivial value P(0) = c0.
- The 20-character string JADE_STORED_HER_DATA is split into five four-character blocks when q = 11.
- Each four-character block is represented by its own degree-three polynomial P1(x) through P5(x).
- The five polynomials are evaluated at ten x values, producing five codewords corresponding to ten photons per block.
- The resulting codewords can be sent through five communications or concatenated into a single transmission.
B. Preparing a precise number of photons
The protocol sizes each photon block using the message length, statistical margin, and Reed-Solomon alphabet limit. The parameter Z can trade transmission cost against successful reconstruction probability.
- B. Preparing a precise number of photons: Each block’s photon count is chosen using a statistical advantage beyond the message length k.The number of successfully measured photons follows a binomial distribution with mean n/2, so n = 2k is insufficient for reliable reconstruction.
- B. Preparing a precise number of photons: A block may contain at most q − 1 photons, requiring the message length k to be optimized when messages are divided into blocks.The Reed-Solomon construction imposes n < q, with n = q − 1 as the maximum block size.
- B. Preparing a precise number of photons: The variable Z controls the balance between successful communication probability and the total number of transmitted photons.Alice can adjust Z in the photon-count and block-size equations for a given message length.
III. EXPERIMENTAL IMPLEMENTATION
The experimental implementation encodes symbols in the transverse spatial degrees of freedom of single photons and uses randomly chosen mutually unbiased bases. Matching bases recover the symbol, while mismatched bases spread detection across the array and reveal no symbol information.
- III. EXPERIMENTAL IMPLEMENTATION: The protocol encodes message and control parameters in the transverse spatial degrees of freedom of a light beam using a spatial light modulator.The modulator divides the optical field into spatial cells, with each cell representing one symbol of a d-dimensional alphabet.
- III. EXPERIMENTAL IMPLEMENTATION: Alice and Bob use a publicly agreed d-dimensional spatial alphabet, illustrated with d = 11.Alice encodes a symbol into a specific diffracted cell using a hologram, while unused modulator regions are discarded.
- III. EXPERIMENTAL IMPLEMENTATION: When Alice and Bob choose the same basis, Bob recovers the symbol prepared by Alice.Position-basis imaging reproduces the transverse position, while corresponding Fourier and inverse-Fourier operations recover momentum-basis symbols.
- III. EXPERIMENTAL IMPLEMENTATION: When Alice and Bob choose different bases, the detected intensity spreads over the array and provides no information about the transmitted symbol.The same blurred, approximately uniform detection pattern occurs when an eavesdropper measures in the wrong basis.
- III. EXPERIMENTAL IMPLEMENTATION: The detection stage can use single-photon CCD cameras or matrix avalanche photodiodes instead of single-pixel avalanche photodiodes.Matrix detectors can map each hologram region onto a specific detector pixel.
- III. EXPERIMENTAL IMPLEMENTATION: Heralded single photons are generated through SPDC, with one photon providing the herald and its twin traveling through the communication channel.The heralding event supplies a timing reference and correlates detection events.
A. Experimental setup with heralded photons
The proposed setup uses SPDC-generated photon pairs, a telescope link, and a photon-counting matrix gated by the heralding detection. Heralding mitigates photon-loss effects and supplies timing control, supporting implementation with current commercial technology.
- A. Experimental setup with heralded photons: A 405 nm continuous-wave laser and type II BBO SPDC source can produce orthogonally polarized 810 nm photon pairs for the setup.A polarizing beam splitter separates the signal and idler photons, with one photon serving as the herald.
- A. Experimental setup with heralded photons: Bob detects the modulated photon through a telescope link and lens using a photon-counting matrix gated by Alice’s heralding detection.The heralding photon produces the gate pulse for the receiver’s matrix detector.
- A. Experimental setup with heralded photons: Heralding mitigates photon-loss effects by counting only coincident herald-and-detection events and provides a time tag for sequential ten-character packets.Available detector options include matrix APDs, i-CCM cameras, and i-CMOS cameras with built-in time taggers.
- A. Experimental setup with heralded photons: The protocol can be implemented with current commercially available technology.The proposed experimental setup uses existing optical, photon-detection, and time-tagging components.
IV. DISCUSSION AND SECURITY ANALYSIS
The scheme combines classical resources, decoy-state verification, and mutually unbiased bases to support simple, technology-ready DSQC without quantum memory. Decoy states help detect intercept-resend attacks, while increasing the number of bases raises errors but complicates implementation.
- The protocol relies mostly on classical resources, existing technology, and no quantum memory, while restricting quantum communication to sensitive data.
- Decoy states determine whether communication should continue and make the protocol resilient against intercept-resend attacks.
- For b = 2, Eve’s introduced error is limited to 50%.
- Using b > 2 increases Eve’s introduced error but makes suitable high-dimensional systems harder to find and requires more parity photons.
V. CONCLUSION
The conclusion presents a high-dimensional DSQC architecture that combines Reed-Solomon erasure coding with TSDF-encoded single photons. It reports mathematical protection against network noise, resilience to intercept-resend attacks, and adaptability to current laboratory capabilities.
- The protocol integrates classical Reed-Solomon erasure coding with high-dimensional quantum states to address channel loss and low bit-rate.
- The flowchart summarizes the proposed DSQC protocol.
- TSDF-encoded single photons expand channel capacity beyond standard binary quantum key distribution schemes.
- Blockwise polynomial evaluation mathematically protects deterministic-message transmission against network noise, while decoy states and MUBs provide resilience to intercept-resend attacks.
- The proposed implementation uses spatial light modulators, heralded single photons from spontaneous parametric down-conversion, and APD matrix detectors.
- Structured-light multiplexing with classical erasure codes is presented as a pragmatic pathway toward secure, high-capacity communication in noisy environments.