Majority-Agreed Key Distribution using Absolutely Maximally Entangled Stabilizer States

Ramij Rahaman; Sourin Das; Sowrabh Sudevan

REVIEW 3 major objections 6 minor 124 references

Majority-Agreed Key Distribution using Absolutely Maximally Entangled Stabilizer States

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Quantum keys from AME states work with any majority's cooperation

desk verdict Theorem 1 is solid and worth citing; the sufficiency proof of Theorem 3 is broken by a concrete attack the authors themselves acknowledge in Section V. read the letter →

arxiv 2411.15545 v1 pith:QFNX6UJK submitted 2024-11-23 quant-ph

Sowrabh Sudevan , Ramij Rahaman , Sourin Das This is my paper · ORCID

classification quant-ph MSC 81P9481P4081P68 PACS 03.67.Dd03.67.Mn

keywords majority-agreedkeydistributionabsolutelymaximallyentangledstatesstabilizergraphquantumconferencekeysself-testingmultipartiteentanglement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An earlier proposal introduced absolutely maximally entangled (AME) states—multipartite states whose every half-sized reduction is maximally mixed—and suggested they could enable majority-agreed key distribution (MAKD), in which any two parties can set up a secret key only with the cooperation of a majority of the parties sharing the state. This paper proves that when the shared AME state also has stabilizer structure, the cooperation count is exactly optimal: any $\lfloor n/2\rfloor-1$ other parties suffice, and no smaller set can. It extends the same guarantee to qudit AME stabilizer states, where examples are more plentiful than for qubits. It then removes the AME requirement entirely for stabilizer states, showing that secure QKD between two parties is possible exactly when those two qubits are not separated by a product structure; for graph states this is simply connectivity. The concrete payoff is a set of protocols—MAKD, conference keys, multiple independent keys per shared state—that use graph states within reach of current preparation and self-testing methods.

What carries the argument

The load-bearing object is the stabilizer group of the shared state—the set of Pauli tensor products that leave the state invariant. For an AME stabilizer state every stabilizer has weight at least $\lfloor n/2\rfloor+1$, and the proof of Theorem 1 counts the independent stabilizers supported on a chosen subset through the reduced-density-matrix expansion $\rho_A = 2^{-|A|}\sum_{\sigma\in S_A}\sigma$, using the AME condition to force the subgroup size. For the graph-state generalization, the machinery is the graph itself: stabilizer generators $S_i = X_i\prod_{j\in N(i)} Z_j$, the path product $\sigma = \prod_{k\in P(i,j)} S_k$ along a shortest path between communicants, and the equivalence between separability and a block-diagonal tableau (a disconnected graph). Fact 1 tells which stabilizers are safe to use: no substring supported on one communicant may itself be a stabilizer, otherwise the announced permission bits leak the key.

What would settle it

For every qubit AME stabilizer state with $n=5$ or $n=6$, and for every subset of $\lfloor n/2\rfloor+1$ qubits, compute the subgroup of stabilizers supported on that subset; a single subset whose supported subgroup is just the identity would refute Theorem 1, and the analogous qudit computation refutes Theorem 2. For Theorem 3, a stabilizer state whose two target qubits lie in different factors of a product decomposition but which nevertheless admits a key stabilizer with no offending substring would be a counterexample.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: in a stabilizer state that is also AME, for any choice of $\lfloor n/2\rfloor+1$ qubits there is a stabilizer element supported only on those qubits—exactly one when $n$ is odd and three when $n$ is even. Consequently, QKD between any two of the $n$ parties can be completed with the cooperation of any $\lfloor n/2\rfloor-1$ other parties, and AME structure makes fewer cooperators impossible. Theorem 2 lifts the statement to $n$-qudit stabilizer AME states, where at least one such stabilizer exists on every chosen subset of $\lfloor n/2\rfloor+1$ qudits. Theorem 3 replaces AME by a weaker condition: a shared stabilizer state permits secure QKD between qubits $i$ and $j$ if and only if the state is not separable with $i$ and $j$ in different factors; for graph states, connectedness of the underlying graph is the exact criterion. The paper also proves Fact 1, the necessary condition that a usable key stabilizer must not contain a substring that is itself a stabilizer on one communicant, and uses products of overlapping graph-state stabilizers along paths to build conference keys, parallel independent keys, and fixed authorizer sets.

Load-bearing premise

The security analysis assumes the cooperating permission-holding parties are individually honest and do not secretly pool their measurement outcomes; if all cooperators compare notes, the public permission bits suffice to reconstruct the key.

Editorial extensions

If this is right

For any $n$-qubit AME stabilizer state, each pair of parties can establish a secret key with exactly $\lfloor n/2\rfloor-1$ cooperating parties, and this number is both necessary and sufficient.
For $n$-qudit AME stabilizer states, the same majority-agreed key distribution works because each $\lfloor n/2\rfloor+1$-qudit subset carries at least one stabilizer correlation.
For arbitrary shared stabilizer states, QKD between two parties reduces to a single topological condition: the two qubits must not lie in different factors of a product decomposition—for graph states, a path must connect them.
Products of overlapping stabilizers along graph paths yield conference keys shared by more than two parties and allow several independent keys to be extracted from one resource state.
Scalable Bell inequalities for graph states can certify the shared state before MAKD begins, making the protocols compatible with device self-testing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step is tailoring self-testing inequalities to AME structure rather than borrowing generic graph-state Bell inequalities; the paper leaves key-rate analysis open, and an AME-specific test could substantially improve finite-key efficiency.
The security model presumes the cooperating permission holders do not collude; if a majority secretly pooled their raw outcomes, the published permission bits would determine the key, so a robust implementation would need to detect or prevent such collusion.
Because the number of authorizers is set by path length in a graph state, the same formalism could be used as an access-control layer in quantum networks, letting intermediate parties act as mandatory approval nodes for long-distance keys.
If qudit AME stabilizer states are easier to prepare than qubit ones, the qudit version of Theorem 2 makes MAKD a plausible near-term application in higher-dimensional photonic or atomic platforms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on 2411.15545. The result worth stealing is Theorem 1: for qubit AME stabilizer states, any ⌊n/2⌋+1 parties can find a stabilizer on exactly their qubits, making the MAKD cooperation count tight. The proof via reduced stabilizer entropies is clean and correct. The necessary direction of Theorem 3 is also solid: separability forces any candidate stabilizer to have stabilizer substrings, which leaks the key.

But the sufficiency half of Theorem 3 does not hold as written. Lemma 4 builds σ = S1S3...Sr along the path and claims Fact 1 guarantees security. Fact 1 only blocks substrings of σ; it doesn't block a product of two adjacent stabilizers that overlaps σ partially. The 5-qubit path graph is a concrete counterexample: for communicants 1 and 5, σ = X1X3X5, and idle party 4 can use S1S3 = X1X3Z4 to get X5 = Z4 from the public X3. So a non-cooperating neighbor recovers the key. The paper's own Section V concedes exactly this attack for subset-path stabilizers and proposes the full-path product ∏ S_k as a fix, but Lemma 4 and the proof of Theorem 3 never apply that fix. The sufficiency claim is unproven as written. It's probably repairable, but the current text is internally inconsistent.

The qudit extension is also unproven. Appendix C's argument that S_{N\A} is nontrivial because |A| < |N\A| is wrong on its face, and in fact AME implies the complement has no nontrivial stabilizer for these sizes. Theorem 2 needs real work.

The graph-state material on conference keys and independent keys is interesting and mostly plausible, but it inherits the security flaw. The self-testing section is straightforwardly borrowed, while the die-roll partitioning trick is a nice practical touch, and the paper honestly notes that key rates are not computed.

Bottom line: Theorem 1 alone makes this worth a serious referee. The graph-state generalization is a good idea and likely salvageable, but the security proof needs a substantial rewrite and Theorem 2 needs a correct proof. I'd send it to review with major revision, not desk-reject.

Referee Report

3 major / 6 minor

Summary. The paper studies majority-agreed key distribution (MAKD) from absolutely maximally entangled (AME) stabilizer states. The central results are: (i) Theorem 1, stating that in any n-qubit AME stabilizer state, for any choice of ⌊n/2⌋+1 qubits there exists a stabilizer element supported only on those qubits (exactly one for odd n, three for even n), so that QKD between any two parties can be performed with the cooperation of any ⌊n/2⌋−1 other parties, and this number is necessary and sufficient; (ii) Theorem 2, a qudit generalization; and (iii) Theorem 3, extending the construction to arbitrary graph states, where QKD between two qubits is claimed to be possible if and only if the two qubits are not separable across a bipartition, i.e., are connected by a path in the underlying graph. The paper also discusses self-testing of the shared states, conference keys, multiple independent keys, and fixed authorizer sets. The main protocol idea is to use stabilizer elements whose support defines the cooperating parties; the paper's Fact 1 and Lemma 4 are meant to establish that the chosen stabilizers do not leak the key to the public or to other parties sharing the state.

Significance. If Theorem 1 and Theorem 2 are correct, they provide a clean structural characterization of AME stabilizer states that is directly useful for multipartite cryptographic tasks, and the explicit counting (one or three stabilizers for a given majority subset) is an elegant result. The extension to graph states in Theorem 3, if proven, would give a practically relevant criterion based on graph connectivity, and the paper's discussion of conference keys and multiple independent keys adds value for near-term quantum network applications. The paper is written in a self-contained manner, with stabilizer-formalism proofs and explicit protocol steps; the self-testing section borrows known Bell inequalities, which is a reasonable choice. However, the security proof of Theorem 3 is not sound as written, and the qudit proof of Theorem 2 is too terse and contains apparent dimensional errors; these issues directly affect the main claims and must be addressed before the results can be relied upon.

major comments (3)

[Appendix D, Lemma 4, Eq. (D1)] The security claim of Lemma 4 is invalid. The odd-product stabilizer σ = S1·S3·S5·...·Sr is checked only against Fact 1, i.e., against substrings of σ that are stabilizer elements. But Fact 1 does not protect against a party whose qubit lies on the path but outside the support of σ, who can measure one extra local Pauli and combine it with the public announcements. Concretely, take the 5-qubit path graph 1-2-3-4-5 and communicants 1 and 5. The Lemma 4 construction gives σ = X1X3X5 (with party 3 as the only cooperator). The stabilizer group element S1S3 = X1X3Z4 is not a substring of σ, yet it implies X1X3Z4 = +1, and together with the relation X1X3X5 = +1 it yields X5 = Z4. Thus party 4, who is not a cooperator, can measure Z4, hear party 3's public X3 outcome, and recover communicant 5's raw key bit exactly. This contradicts the security claim of Lemma 4 and hence invalidates the sufficiency direction of Theorem 3 as written. Section V explicitly acknowledges this type of attack and proposes the full-path product σ = ∏_{k∈P(i,j)} S_k as a fix, but that fix is not what Lemma 4 or the proof of Theorem 3 proves. The theorem is probably repairable by adopting the full-path construction and proving its security, but the current proof does not establish it.
[Appendix C, proof of Theorem 2] The proof of the qudit theorem is not reliable as written. The text states E(I/d^{|A|}) = E(1/d^{n-|A|} ∑_{σ∈S_{N\A}} σ), which incorrectly sets the reduced density matrix of A to the maximally mixed state; the AME property for a majority subset A only gives E(ρ_A) = |N\A|, not maximal mixedness. Furthermore, the concluding line 'S_{N\A} cannot be a trivial group since |A| < |N\A|' is dimensionally wrong for odd n, where |A| = ⌊n/2⌋+1 > |N\A|. The intended counting argument (entropy of ρ_A equals |A|-k, where d^k = |S_A|) may well yield the claimed stabilizer existence, but as written the derivation needs to be rewritten with the correct entropy equality and the correct sizes of A and N\A.
[Section V and Theorem 3] The paper's own Section V observes that the odd-path stabilizers used in Lemma 4 allow other parties to determine the key ('It is possible that the public outcomes allow some other party possessing a qubit of the shared graph state to determine the secret key'), and it proposes the full-path product as a remedy. This admission is in direct tension with the unqualified wording of Theorem 3, which is stated without the full-path restriction. The theorem and the protocol description need to be aligned: either Theorem 3 should state the specific stabilizer construction that is proven secure, or the security proof must be supplied for the construction that is actually proposed. As it stands, a reader following the proof of Theorem 3 cannot infer a secure protocol.

minor comments (6)

[Section III, Eq. (14)] The Bell operator Ic is written with expectation values that include products of operators, but the local observables A_i, A'_i are never defined. Please define the measurement settings explicitly, so the expression is comprehensible and the claimed quantum expectation value can be verified.
[Section IV, Fact 1 and following text] The term 'substring' is used informally. A formal definition (restriction of a Pauli string to a subset of qubits, with the remaining factors replaced by identity) would make Fact 1 and the proof of Lemma 4 easier to follow.
[Appendix B] The security analysis in Appendix B only considers a single outside party using one local Pauli observable. The security model (passive, non-colluding adversaries, authenticated classical channels, honest-but-curious cooperating parties) is not stated in Section II or Section V. Please state the assumed adversary model explicitly, since this determines the validity of the claimed security guarantees.
[Figures 5-7] The captions refer to 'green' public qubits; if the figures are printed in black and white, this is not visible. Please use a different visual marker (e.g., dashed circles) for public qubits.
[Reference list] Reference [110] reads 'Noisemodel from backend' and [111] is a Matplotlib citation; these appear to be leftover acknowledgements, not scholarly references. Please remove or properly cite the relevant sources.
[Throughout] There are several typographical issues, including 'Absolutely Maxima lly' in the title, misaligned text in Table I, and inconsistent use of ± in equations (B1). A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the theorems are proved from the stabilizer and graph-state formalism, and the authors' self-citations are contextual only.

full rationale

The central derivations are self-contained. Theorem 1 follows from the stabilizer-state projector identity combined with the AME requirement that all reductions of size at most floor(n/2) are maximally mixed; Theorem 2 extends the same entropy argument to qudits; Theorem 3 is proved through Lemma 2 (separability forces every stabilizer touching both communicants to factor into sub-stabilizers) and Lemma 4 (an explicit odd-indexed path product satisfying the paper's stated Fact 1 criterion). No fitted parameter is renamed as a prediction, and no conclusion is assumed as a premise. The authors' prior works [44] and [78] are cited only as background on d-dimensional cluster states and graph-state constructions of k-uniform states; they do not carry any load-bearing step of the proofs. The self-testing material explicitly borrows an external Bell inequality from [65]. The main concern a skeptical reader might raise is about the security model: Fact 1 is an explicitly stated operational criterion, and the proof of Lemma 4 shows only that no substring of the chosen stabilizer is itself a stabilizer. Whether that criterion adequately excludes attacks by non-cooperating parties who hold other qubits of the shared state is a correctness and security-model issue, not a circularity issue, because the derivation does not reduce to its own conclusion. Accordingly, there is no identifiable circular step and the paper warrants a score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters and no new physical entities. It relies on the stabilizer formalism, the assumption that the shared state is a stabilizer or graph state, and an implicit security model that excludes collusion among the cooperating parties. The latter is the most consequential unstated assumption.

assumptions (4)

domain assumption The shared state is a stabilizer state (Theorems 1, 2, 3) or a graph state (Sections IV and V).
The paper explicitly restricts to stabilizer AME states and graph states; non-stabilizer AME states are outside the paper's scope.
domain assumption The communicants and cooperators have access to authenticated classical channels and can perform the announced Pauli measurements.
Standard QKD assumption, stated in Section III (self-testing paragraph) and used throughout the protocol.
domain assumption The adversary is a member of the public or an individual non-cooperating party; a colluding coalition of all cooperators is not considered.
Appendix B proves security only against individual parties; the protocol is insecure against a coalition of cooperators, so this is a load-bearing unstated assumption.
standard math Standard results from stabilizer formalism and graph state theory: reduced density matrix expansion via the stabilizer subgroup, LC equivalence between stabilizer states and graph states, and graph-state separability iff no edges across the cut.
Used in proofs of Theorems 1-3 (Appendices A-D) without proof.

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Cite this review

Pith. "Pith review of Majority-Agreed Key Distribution using Absolutely Maximally Entangled Stabilizer States." pith.science (2026). https://pith.science/paper/QFNX6UJK

@misc{pith2026241115545,
  author       = {Pith},
  title        = {Pith review of: Majority-Agreed Key Distribution using Absolutely Maximally Entangled Stabilizer States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QFNX6UJK}},
  note         = {Machine review of arXiv:2411.15545}
}

abstract

In [Phys. Rev. A 77, 060304(R),(2008)], Facchi et al. introduced absolutely maximally entangled (AME) states and also suggested ``majority-agreed key distribution"(MAKD) as a possible application for such states. In MAKD, the qubits of an AME state are distributed one each to many spatially separated parties. AME property makes it necessary that quantum key distribution(QKD) between any two parties can only be performed with the cooperation of a majority of parties. Our contributions to MAKD are, $(1)$ We recognize that stabilizer structure of the shared state is a useful addition to MAKD and prove that the cooperation of any majority of parties(including the two communicants) is necessary and sufficient for QKD between any two parties sharing AME stabilizer states. Considering the rarity of qubit AME states, we extended this result to the qudit case. $(2)$ We generalize to shared graph states that are not necessarily AME. We show that the stabilizer structure of graph states allows for QKD between any inseparable bipartition of qubits. Inseparability in graph states is visually apparent in the connectivity of its underlying mathematical graph. We exploit this connectivity to demonstrate conference keys and multiple independent keys per shared state. Recent experimental and theoretical progress in graph state preparation and self-testing make these protocols feasible in the near future.

Figures

Figures reproduced from arXiv: 2411.15545 by the authors.

**Figure 1.** The blue lines show the underlying mathematical graph for the graph state and the red double headed arrows point to distinct pairs of communicants. Next to each intent we give a stabilizer element that can be used.(a) 1 ↔ 2 : X1Z2. (b) 1 ↔ 2, 3 : Z1X2Z3. (c) 1 ↔ 2, 4 : Z4X1Z2Z3 and 1 ↔ 5 : S1S5 = X1Z3X5Z6. of each transmitter is known with certainty. See figure 1 for examples of MAKD using n = 2, 5, 6 graph, AME sta… view at source ↗

**Figure 4.** For 1 ↔ 11, we choose the path, p = {1, 3, 6, 9, 10, 11} and use the stabilizer element S1S6S10 = X1Z2X6Z7Z8X10Z11. V. MORE THAN ONE KEY PER STATE Note that QKD between parties that are far apart with respect to the underlying graph for the shared graph state will require many measurement outcomes to be made public. Using a product of overlapping stabilizers along a path connecting the two parties only ensures that … view at source ↗

**Figure 3.** Table of stabilizer correlations for all [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: The red double headed arrows represent communicants with shared keys. Qubits whose measurement results are public are green. and X8 measurement outcomes being made public. Since parties 3, 5, 7 were not required to participate in QKD between 1 and 9, they remain free t…

**Figure 6.** Figure 6: Performing 1 ↔ 9 using S2 · S4 · S6 · S8 results in enough public measurement outcomes to allow a conference key setup between all the qubits in {1, 3, 5, 7, 9}. Fact 3. If multiple non-overlapping pairs of communicants (the pairs of communicants do not form the verti…

**Figure 7.** Figure 7: (a) 1 ↔ 2 using Z1X2Z3, 4 ↔ 5 using Z5X4Z3. (b) 1 ↔ 3 using Z1X2Z3, 4 ↔ 5 using S2 · S4 · S5 = X2Y4Y5. Note that a common public measurement outcome allows both independent QKD events. requires the use of “overlapping stabilizers” lends importance to the topology of t…

**Figure 8.** Figure 8: QKD between any party in graph G1 and a party in graph G2 requires the cooperation of at least ⌊r/2⌋ parties in {α1, α2, α3, . . . , αr} such that their corresponding stabilizers overlap. VI. SUMMARY AND OUTLOOK We have proven how shared n-qudit AME stabilizer states a…

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Reference graph

Works this paper leans on

124 extracted references · 67 canonical work pages

[1]

Note that, ⟨p|Z1Z2 |p⟩ = ⟨φ +|Z1Z2 |φ +⟩ = +1

Consider the two states, |p⟩ = |00⟩ ; |φ +⟩ = (1/ √ 2)(|00⟩ + |11⟩). Note that, ⟨p|Z1Z2 |p⟩ = ⟨φ +|Z1Z2 |φ +⟩ = +1. We clarify that the public is always aware of the identity of the shared state and the measurements performed by each party. Despite this knowledge, |φ +⟩ can be used for QKD and |p⟩ cannot. This diﬀerence arises from how, ⟨p|Z1 or 2 |p⟩ = +...
[2]

Any reduction in the degree of violation of this Bell inequality quantiﬁes the extend to which the state has decohered or an eavesdropper has attempted to extracted information

In fact, this violation can be used for self-testing, i.e., to conﬁrm that the qubit distribution was performed faithfully. Any reduction in the degree of violation of this Bell inequality quantiﬁes the extend to which the state has decohered or an eavesdropper has attempted to extracted information. One method to implement E-91 is to have party 1 and par...
[3]

12 we can construct the corresponding Bell inequality for |ψ 5⟩ by a simple basis change Iψ 5 =UIcU †, where U = F1X1G2H3H4Z4

Using Ic and equation. 12 we can construct the corresponding Bell inequality for |ψ 5⟩ by a simple basis change Iψ 5 =UIcU †, where U = F1X1G2H3H4Z4. Now, we must randomly partition the shared states to a set for self-testing and a set for MAKD. Due to the larger number of observables in 14, each party making a random choice between some set of observable...
[4]

To per- 1 2 3 4 Figure 2: |L4⟩ form 1 ↔ 4, use of σ =S1 ·S4 =X1Z2Z3X4 is not secure

Consider the graph state |L4⟩ in ﬁgure 2. To per- 1 2 3 4 Figure 2: |L4⟩ form 1 ↔ 4, use of σ =S1 ·S4 =X1Z2Z3X4 is not secure. When the measurement outcomes Z2 and Z3 are made public, a public party can consider the stabilizer elements S1 = X1Z2 and S4 = Z3X4 to compute the key. After considering these two scenarios, we can state the following necessary c...
[5]

A proof of this can be found within the proof for Theorem 3, Lemma 4

Note that in scenario 2 we see that a stabilizer element constructed from a product of overlapping Si’s for the graph state is a secure choice. A proof of this can be found within the proof for Theorem 3, Lemma 4. It turns out that a lack of separability is not just nec- essary but also suﬃcient for QKD between two qubits of a shared stabilizer state. Qui...
[6]

Now we can state precisely what we refer to as conference key distribution. F act 2. In a shared graph state, let a set of communicant pairs form the edges of a connected graph where the com- municants belonging to these pairs are the vertices and we use stabilizer elements satisfying constraints 1 and 2 for each QKD pair, then all the communicants belong...
[7]

authorizers

Let 1 ↔ 2 be accomplished with S1 = Z5X1Z2 and 1 ↔ 3 with S1 · S3 = X1Z5Z4X3. The public outcome of Z4 results in 2 ↔ 3 also being possible using S3 = Z2X3Z4. We list all distinct three party conference key scenarios using this 5-qubit state in ﬁgure 5. In ﬁgure 6, let parties 1 and 9 attempt QKD using the stabilizer elementS2 ·S4 ·S6 ·S8. This requires X...
[8]

authorizers

In fact, increasing the number of parties along this chain increases the number of permission bits needed for QKD between the parties in G1 and G2. G1 G2α 1 α 2 α 3 α r Figure 8: QKD between any party in graph G1 and a party in graph G2 requires the cooperation of at least ⌊r/ 2⌋ parties in {α 1,α 2,α 3,...,α r} such that their corresponding stabilizers o...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Note that, ⟨p|Z1Z2 |p⟩ = ⟨φ +|Z1Z2 |φ +⟩ = +1

Consider the two states, |p⟩ = |00⟩ ; |φ +⟩ = (1/ √ 2)(|00⟩ + |11⟩). Note that, ⟨p|Z1Z2 |p⟩ = ⟨φ +|Z1Z2 |φ +⟩ = +1. We clarify that the public is always aware of the identity of the shared state and the measurements performed by each party. Despite this knowledge, |φ +⟩ can be used for QKD and |p⟩ cannot. This diﬀerence arises from how, ⟨p|Z1 or 2 |p⟩ = +...

[2] [2]

Any reduction in the degree of violation of this Bell inequality quantiﬁes the extend to which the state has decohered or an eavesdropper has attempted to extracted information

In fact, this violation can be used for self-testing, i.e., to conﬁrm that the qubit distribution was performed faithfully. Any reduction in the degree of violation of this Bell inequality quantiﬁes the extend to which the state has decohered or an eavesdropper has attempted to extracted information. One method to implement E-91 is to have party 1 and par...

[3] [3]

12 we can construct the corresponding Bell inequality for |ψ 5⟩ by a simple basis change Iψ 5 =UIcU †, where U = F1X1G2H3H4Z4

Using Ic and equation. 12 we can construct the corresponding Bell inequality for |ψ 5⟩ by a simple basis change Iψ 5 =UIcU †, where U = F1X1G2H3H4Z4. Now, we must randomly partition the shared states to a set for self-testing and a set for MAKD. Due to the larger number of observables in 14, each party making a random choice between some set of observable...

[4] [4]

To per- 1 2 3 4 Figure 2: |L4⟩ form 1 ↔ 4, use of σ =S1 ·S4 =X1Z2Z3X4 is not secure

Consider the graph state |L4⟩ in ﬁgure 2. To per- 1 2 3 4 Figure 2: |L4⟩ form 1 ↔ 4, use of σ =S1 ·S4 =X1Z2Z3X4 is not secure. When the measurement outcomes Z2 and Z3 are made public, a public party can consider the stabilizer elements S1 = X1Z2 and S4 = Z3X4 to compute the key. After considering these two scenarios, we can state the following necessary c...

[5] [5]

A proof of this can be found within the proof for Theorem 3, Lemma 4

Note that in scenario 2 we see that a stabilizer element constructed from a product of overlapping Si’s for the graph state is a secure choice. A proof of this can be found within the proof for Theorem 3, Lemma 4. It turns out that a lack of separability is not just nec- essary but also suﬃcient for QKD between two qubits of a shared stabilizer state. Qui...

[6] [6]

Now we can state precisely what we refer to as conference key distribution. F act 2. In a shared graph state, let a set of communicant pairs form the edges of a connected graph where the com- municants belonging to these pairs are the vertices and we use stabilizer elements satisfying constraints 1 and 2 for each QKD pair, then all the communicants belong...

[7] [7]

authorizers

Let 1 ↔ 2 be accomplished with S1 = Z5X1Z2 and 1 ↔ 3 with S1 · S3 = X1Z5Z4X3. The public outcome of Z4 results in 2 ↔ 3 also being possible using S3 = Z2X3Z4. We list all distinct three party conference key scenarios using this 5-qubit state in ﬁgure 5. In ﬁgure 6, let parties 1 and 9 attempt QKD using the stabilizer elementS2 ·S4 ·S6 ·S8. This requires X...

[8] [8]

authorizers

In fact, increasing the number of parties along this chain increases the number of permission bits needed for QKD between the parties in G1 and G2. G1 G2α 1 α 2 α 3 α r Figure 8: QKD between any party in graph G1 and a party in graph G2 requires the cooperation of at least ⌊r/ 2⌋ parties in {α 1,α 2,α 3,...,α r} such that their corresponding stabilizers o...