arxiv: 2604.12577 · v2 · submitted 2026-04-14 · 🪐 quant-ph

Ternary Quantum Eraser Cryptography

Ahmed Halawani , Yahya Meshalwi Khabrani , Abdulaziz Al-Mogheeth , Zheng-Hong Li , M. Al-Amri This is my paper

Pith reviewed 2026-05-10 15:26 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum key distributionquantum eraserternary polarizationeavesdropping attacksfour-dimensional Hilbert spacerandomized orderingindividual attacks

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The pith

A ternary quantum eraser protocol using three 120-degree polarization states in randomized three-photon groups limits an eavesdropper's success probability to 54 percent against individual attacks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a ternary quantum eraser cryptography scheme to address the security limitation of binary quantum eraser protocols. Binary versions allow an eavesdropper to recover the encoded bit with 85 percent probability, but the new approach employs three polarization states separated by 120 degrees and transmits them in three-photon groups whose temporal order is randomized. This design reduces the distinguishability of individual states and increases the combinatorial difficulty of guessing the correct ordering, which together bound the maximum success probability for an individual eavesdropper at 54 percent inside the four-dimensional path-polarization Hilbert space. The protocol retains the core operational advantage of quantum eraser methods by using interference to identify matching encoding choices automatically, without any basis reconciliation step, and delivers a sifted efficiency of 0.30 bits per photon. A reader would care because the scheme offers a concrete route to stronger security in quantum key distribution while preserving the simplicity that makes quantum eraser protocols attractive.

Core claim

The central claim is that the ternary quantum eraser protocol, which encodes information using three polarization states at 120 degree angular separation and transmits them in three-photon groups with randomized temporal ordering, establishes that an eavesdropper's maximum success probability against individual attacks is bounded at 54 percent within the four-dimensional path-polarization Hilbert space, thereby providing substantially improved security over binary quantum eraser implementations while preserving automatic interference-based basis identification and a binary-equivalent efficiency of 0.30 bits per photon.

What carries the argument

The three polarization states with 120 degree separation transmitted in three-photon groups with randomized temporal ordering, operating inside the four-dimensional path-polarization Hilbert space, which simultaneously reduces single-photon distinguishability and imposes combinatorial uncertainty on multi-photon eavesdropping.

If this is right

An individual eavesdropper's maximum success probability is limited to 54 percent.
The protocol achieves a sifted efficiency of 0.30 bits per photon.
Automatic identification of matching and mismatching encoding choices occurs through interference, eliminating basis reconciliation.
Security improvement arises from both reduced state distinguishability and the added complexity of unknown photon ordering.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The randomized ordering technique could be adapted to other multi-photon quantum key distribution protocols to increase uncertainty for eavesdroppers.
Practical implementations would need to verify that real-world noise and detection imperfections do not allow the eavesdropper success rate to exceed the calculated bound.
Extending the state symmetry to four or more angles might produce even tighter bounds on information leakage in similar interference-based schemes.

Load-bearing premise

The 54 percent security bound holds only for individual eavesdropping attacks and assumes that the 120 degree state symmetry together with the randomized three-photon ordering behave exactly as idealized without collective or coherent attacks.

What would settle it

An experiment that lets an eavesdropper attempt to discriminate the three 120-degree polarization states from individual photons or small groups drawn from randomized three-photon sequences and measures whether the actual guessing probability stays at or below 54 percent.

Figures

Figures reproduced from arXiv: 2604.12577 by Abdulaziz Al-Mogheeth, Ahmed Halawani, M. Al-Amri, Yahya Meshalwi Khabrani, Zheng-Hong Li.

**Figure 3.** Figure 3: FIG. 3. Probability of correct identification as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Eve’s correct identification probability [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Probability of correct identification [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 7.** Figure 7: plots Bpole(γA) and Ef f (γA) for the binary eraser, with γA the angle between the two transmitted states (overlap cos γA). The standard configuration sits at γA = π/4, where Bpole reaches the binary discrimination bound (0.85) and Ef f = 0.25. Bpole follows from the overlap of the two transmitted states (a general twostate discrimination result); Ef f reflects the eraser’s sifting mechanism, in which m… view at source ↗

**Figure 6.** Figure 6: FIG. 6. Probability of correct identification [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Polarization state geometric representation for the [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Probability of single detector clicking [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Conditional probability of correct identification [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Conditional probability of correct identification [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

**Figure 12.** Figure 12: shows the probability Q3/QT as a function of λ12/λ10, reaching its maximum of 0.37 at λ12/λ10 = 0 [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Geometric interpretation of subspace projection. [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

read the original abstract

Quantum key distribution protocols based on the quantum eraser phenomenon offer an operational advantage: automatic identification of matching and mismatching encoding choices through interference, eliminating basis reconciliation. However, binary quantum eraser implementations permit an eavesdropper to recover Alice's encoded bit with $85\%$ probability. To overcome this constraint, we introduce a ternary quantum eraser protocol employing three polarization states with $120^\circ$ angular separation, transmitted in three-photon groups with randomized temporal ordering. This extension achieves enhanced security through two complementary mechanisms. First, the reduced distinguishability of symmetrically-arranged quantum states limits single-photon discrimination. Second, the combinatorial complexity of unknown photon ordering constrains multi-photon eavesdropping strategies. Security analysis against individual eavesdropping attacks within the four-dimensional path-polarization Hilbert space establishes that an eavesdropper's maximum success probability is bounded at $54\%$, substantially below the binary discrimination bound. The protocol maintains a binary-equivalent efficiency of 0.30 bits per photon, comparable to established binary QKD protocols at the sifted-rate level, while preserving the operational simplicity inherent to quantum eraser cryptography.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Ternary eraser QKD gives a 54% individual-attack bound via 120° states and randomized ordering, but the gain is scoped narrowly and needs the full derivation checked.

read the letter

The main takeaway is that this protocol uses three 120° polarization states sent in randomized three-photon groups to push the eavesdropper success probability down to 54% for individual attacks in the four-dimensional path-polarization space, versus the 85% figure in binary quantum eraser QKD. The randomized ordering is meant to add combinatorial difficulty for multi-photon strategies while the symmetric states reduce single-photon distinguishability. Efficiency stays at 0.30 bits per photon, which lines up with binary protocols at the sifted level, and the eraser advantage of no basis reconciliation is preserved. The construction is straightforward and the stated scope matches the stress-test note: the modeling is internally consistent for individual attacks with no visible contradictions or unjustified steps. The paper earns credit for spelling out a concrete new bound that is not just a restatement of prior binary results. The soft spot is the narrow attack model. The analysis covers only individual eavesdropping; collective or coherent attacks are left aside, which is standard but limits how far the 54% claim can be taken without further work. The abstract states the bound from Hilbert-space analysis without showing the explicit overlaps or calculation steps, so a referee will need to verify those details in the full text. This is for people already working on quantum eraser or multi-state QKD variants who want to see how ternary encoding changes the numbers. A reader looking for incremental security improvements in a well-defined model will find something usable here. It deserves peer review because the protocol is clearly defined, the bound is presented as calculable, and the limitations are stated up front rather than hidden.

Referee Report

1 major / 1 minor

Summary. The paper proposes a ternary quantum eraser QKD protocol that encodes information in three polarization states separated by 120° and transmits them in randomized three-photon groups. It claims that this construction reduces single-photon distinguishability and adds combinatorial protection against multi-photon strategies, yielding a 54% upper bound on an individual eavesdropper’s success probability inside the four-dimensional path-polarization Hilbert space while achieving a sifted efficiency of 0.30 bits per photon—substantially better than the 85% bound reported for binary quantum-eraser schemes.

Significance. If the 54% bound is rigorously derived, the protocol would constitute a concrete improvement in the security-efficiency trade-off for quantum-eraser cryptography, preserving the operational advantage of automatic basis matching while lowering information leakage to individual attacks. The symmetry-based and ordering-based mechanisms are conceptually appealing and could be of interest to the QKD community if the analysis is made fully explicit and reproducible.

major comments (1)

[Security analysis] Security analysis (the section presenting the 54% bound): the manuscript asserts that the eavesdropper’s maximum success probability is bounded at 54% from Hilbert-space distinguishability of the three 120°-separated states, yet neither the explicit overlap integrals, the POVM used for the individual attack, nor the precise incorporation of the randomized temporal ordering appear in the provided text. Because this calculation is the sole quantitative support for the central security claim, its absence prevents verification that the bound is parameter-free and independent of post-hoc choices.

minor comments (1)

The efficiency figure of 0.30 bits per photon is stated as binary-equivalent; a short derivation or table showing how the three-photon grouping and sifting map onto this rate would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and for highlighting the need for greater explicitness in the security analysis. We address the major comment below and will incorporate the requested details in the revised manuscript.

read point-by-point responses

Referee: Security analysis (the section presenting the 54% bound): the manuscript asserts that the eavesdropper’s maximum success probability is bounded at 54% from Hilbert-space distinguishability of the three 120°-separated states, yet neither the explicit overlap integrals, the POVM used for the individual attack, nor the precise incorporation of the randomized temporal ordering appear in the provided text. Because this calculation is the sole quantitative support for the central security claim, its absence prevents verification that the bound is parameter-free and independent of post-hoc choices.

Authors: We agree that the explicit derivation must be provided for independent verification. The 54% bound follows from the pairwise overlaps of the three 120°-separated polarization states (inner product -1/2) in the four-dimensional path-polarization space, together with the uniform randomization over the six possible temporal orderings of the three-photon group. In the revised manuscript we will add a self-contained subsection that (i) states the three state vectors and their overlap integrals, (ii) specifies the optimal POVM for an individual attack that maximizes the success probability, and (iii) shows the combinatorial averaging over the unknown ordering that yields the final 54% figure. The calculation is parameter-free once the 120° symmetry and the three-photon randomization are fixed; no post-hoc tuning is involved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; security bound follows from standard state discrimination

full rationale

The paper's central result is a 54% upper bound on individual eavesdropper success probability, obtained by analyzing distinguishability of three 120°-separated polarization states in a four-dimensional path-polarization Hilbert space together with the combinatorial effect of randomized three-photon ordering. No equation or step reduces this bound to a fitted parameter, a self-defined quantity, or a load-bearing self-citation. The derivation relies on ordinary quantum state discrimination calculations whose inputs are the protocol's explicitly stated symmetry and ordering rules; these inputs are not themselves outputs of the security analysis. The protocol description and efficiency claim (0.30 bits/photon) are likewise independent of the bound. Within the explicitly scoped domain of individual attacks the chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The protocol rests on standard quantum mechanics in a four-dimensional Hilbert space and the assumption that only individual attacks are considered; no new physical entities or fitted parameters are introduced beyond the design choice of 120° separation.

axioms (2)

standard math Quantum mechanics in four-dimensional path-polarization Hilbert space
The security analysis is performed within this space as stated in the abstract.
domain assumption Security considered only against individual eavesdropping attacks
The 54% bound is explicitly for individual attacks.

pith-pipeline@v0.9.0 · 5508 in / 1275 out tokens · 33567 ms · 2026-05-10T15:26:35.380054+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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diagonal

Efficiency Analysis for Method I In Method I, Alice encodes each key symbol using a group ofm= 2 photons prepared in two of the three available polarization states. A valid key symbol is es- tablished when Bob’s measurement yields|V⟩detections for both photons, which occurs only when Alice and Bob selected different encoding operations. The sifting probab...

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The sifting criterion requires exactly two|V⟩detections, which uniquely identifies Bob’s measurement choice to Alice

Efficiency Analysis for Method II Method II achieves higher efficiency by transmitting all three polarization states in each group ofm= 3 photons. The sifting criterion requires exactly two|V⟩detections, which uniquely identifies Bob’s measurement choice to Alice. For any valid key generation event, exactly one photon in the group has polarization matchin...

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|U⟩ −1 2 |H⟩+ √ 3 2 |V⟩ ! +|L⟩ −1 2 |H⟩ − √ 3 2 |V⟩ !# , |ψ−⟩= 1√ 2

Security Analysis for Method II The security of Method II depends on both the quan- tum indistinguishability of the trine states and the se- crecy ofσ. Even if Eve could perfectly identify each pho- ton’s polarization state, she would still face uncertainty about Alice’s chosen ordering. To illustrate why this ordering information is crucial, consider wha...

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The maximum value of 0.4 occurs atλ 12 = 0

This optimization is illustrated in Figure 10, which shows the probability of correct identification rate as a function ofλ 12/λ10 when two similar detectors click. The maximum value of 0.4 occurs atλ 12 = 0. c. Case 3: Each detector clicks once (Q 3: permuta- tions of(i, j, k))The most informative scenario for Eve occurs when each photon triggers a diffe...

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Role of the Fourth Detector and Completeness A complete measurement in the four-dimensional Hilbert space requires four orthogonal measurement op- erators{|α 1⟩,|α 2⟩,|α 3⟩,|α 4⟩}. Since Alice’s three states lie in a two-dimensional subspace spanned by{|ϕ0⟩,|ϕ 2⟩}, two important questions arise: can Eve gain additional information by using multiple detect...

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This represents a substantial reduction from the binary discrimination bound

Final Security Bound and Comparison Eve’s maximum success probability of 54%, the ternary discrimination bound, cannot be exceeded regardless of computational resources or measurement technology. This represents a substantial reduction from the binary discrimination bound. Appendix C corroborates this bound through an independent POVM/permutation- uncerta...

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Correct Derivation forT A andT B With Non-Ideal Angles a. Cancellation of the which-path tags in the ideal case In the ideal version of the protocol, Alice and Bob use polarization rotatorsT A andT B that apply opposite ro- tations of equal magnitude, TA =R(θ), T B =R(−θ), (A1) 25 hence that the combined action on the two interferometer arms is: (TB ⊗T A)...

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ϕ+ 0 ” and|α 2⟩as “ϕ + 1

Revised No-Cloning Derivation a. No-cloning constraint for nonorthogonal channel states Let{|ϕ i⟩}be the four channel states of the binary quantum eraser protocol. If a perfect cloning machine existed, it would implement a unitaryUsuch that: U(|ϕ i⟩ ⊗ |0⟩) =|ϕ i⟩ ⊗ |ϕi⟩, (A14) for alli. Taking the inner product between the outputs for two different states...

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60), forming a trine ensemble

Trine-state structure of the ternary encoding In the ternary quantum eraser protocol, Alice encodes each logical symbol using one of the three symmetric po- larization states: |A1⟩,|A 2⟩,|A 3⟩, (C1) separated by 120 ◦ in the equatorial plane of the Bloch sphere, with pairwise overlap⟨A i|Aj⟩=− 1 2 fori̸=j (Eq. 60), forming a trine ensemble. Because the in...

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For three trine states|ψ i⟩with prior probabilitiesp i = 1/3, the Helstrom bound gives: P (1) correct = 1 3 3X i=1 ⟨ψi|Πi|ψi⟩, (C3) where{Π i}is the optimal POVM

Optimal POVM for three symmetric states (Helstrom solution) ForKequally likely symmetric pure states forming a representation of a cyclic group, the optimal measure- ment is known to be the square-root measurement (SRM) or equivalently the Helstrom measurement, which pre- serves the symmetry of the ensemble. For three trine states|ψ i⟩with prior probabili...

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Instead, she sends a group of three photons, each prepared in one of the three non-orthogonal states|A 1⟩,|A 2⟩,|A 3⟩, arranged in a secret temporal ordering

Eve’s access to three photons and the role of temporal ordering In the ternary protocol, Alice does not send a single photon corresponding to a single trine state. Instead, she sends a group of three photons, each prepared in one of the three non-orthogonal states|A 1⟩,|A 2⟩,|A 3⟩, arranged in a secret temporal ordering. Let the temporal ordering for logi...

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C5) with the permutation uncertainty aris- ing from Alice’s randomized ordering (Eqs

Final bound Combining the POVM discrimination limit for each photon (Eq. C5) with the permutation uncertainty aris- ing from Alice’s randomized ordering (Eqs. C8–C11), we obtain the security bound stated in the main text: PEvemax ≈0.54.Thus the ternary quantum eraser pro- tocol reduces Eve’s maximum information gain from the binary-eraser ceiling of 0.85 ...

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Beam-splitter The ideal beam splitter transformation, characterized by the mixing angleθ=π/4 for a 50:50 splitting ra- tio, becomes in practice subject to small deviations. We parameterize the imperfection of the first beam splitter BS1 through a deviationδ 1 such thatθ 1 = π 4 +δ 1.Simi- larly, for the second beam splitter BS2 with deviationδ 2: θ2 = π 4...

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We model this effect by allowing the photon in each path to decay into orthogonal states that no longer participate in interference

Decoherence Decoherence represents a more fundamental imperfec- tion arising from environmental interactions that destroy the coherent superposition between the interferometer paths. We model this effect by allowing the photon in each path to decay into orthogonal states that no longer participate in interference. After the first beam split- ter, the quan...

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We characterize these deviations through four independent parameters:β A andµ A for Alice’s rotators in the upper and lower paths, andβ B andµ B for Bob’s corresponding rotators

Polarization rotators The polarization rotators employed by Alice and Bob also exhibit imperfections, deviating from the ideal 45◦ rotation angles. We characterize these deviations through four independent parameters:β A andµ A for Alice’s rotators in the upper and lower paths, andβ B andµ B for Bob’s corresponding rotators. The imperfect rotator operator...

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Halawaniet al., Information-theoretic security bounds for the ternary quantum eraser protocol (2026), manuscript in preparation

A. Halawaniet al., Information-theoretic security bounds for the ternary quantum eraser protocol (2026), manuscript in preparation

work page 2026