{"id":"ca419b76-e958-4835-8946-26e44e5e6ec6","arxiv_id":"2411.15077","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"PME states for any even number of qudits are shown to be generated by applying controlled-phase gates to separable phase states, and a quantum secret sharing protocol is sketched for them.","lead":"This paper builds planar maximally entangled (PME) states by applying controlled-phase gates to simple product phase states, and then sketches a quantum secret sharing protocol using them. The construction is explicit for two, three, four, eight, and any even number of qudits, but much of it reproduces known graph-state PME constructions.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The PME construction is valid, but the QSS protocol in Sec. 6 fails: for K=4 a single player can recover the secret, so the claimed secret-sharing threshold is not realized.","rationale":"The reader's weakest assumption targets the linear-independence criterion (Eq. 28) for composite qudit dimension d. While that is a real caveat, it is not load-bearing for the explicit construction: the Bell-pair product (Eq. 55) is PME for any d by a direct interval argument that does not invoke Eq. (28), and the controlled-phase graph state generated from separable phase states with edges pairing opposite sites is locally equivalent to Eq. (55). The incorrect star/line identifications in Sec. 5.3 are presentational errors, not the crux of the central claim. The crux is the QSS section: the paper promises a secret-sharing protocol, but as written the protocol is inconsistent, and for the minimal instance it reduces to plain teleportation to one player. If the QSS claim were merely suggested future work, the rest of the paper would be a reasonable conditional accept; as it stands, the title and abstract assert a QSS protocol, so this needs to be fixed or removed. I therefore keep the reader's CONDITIONAL verdict: the PME construction deserves publication after corrections, but the QSS section requires major revision or a clear caveat that the proposed scheme is not a threshold protocol.","tokens_in":15903,"tokens_out":17032,"duration_ms":161709,"concrete_test":"Implement the K=4, d=2 case of the Sec. 6 scheme. Take |λ>=α|0>+β|1>, |PME>=|P^+>_{13}|P^+>_{24}, dealer at qudit 1. Project S and qudit 1 onto the Bell state |Φ^+> (or any outcome) and compute the post-measurement state on qudits 2,3,4. If the result is (I⊗U_3⊗I)|λ>_3|P^+>_{24} for a known single-qudit unitary U_3, then player 3 alone can recover the secret from the announced outcome, confirming that the claimed K−K/2 player threshold is not attained. Also check whether qudits 2 and 4 retain any correlation with the secret.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central PME construction (Eq. 55) is sound: the product of Bell pairs |P^+>_{l,l+K/2} is PME for any d, since any adjacent interval of length ≤ K/2 contains at most one member of each pair. The load-bearing problem is the claimed QSS application (Sec. 6). For the smallest nontrivial case K=4, Eq. (45) gives |PME>=|P^+>_{13}|P^+>_{24}. If the dealer holds qudit 1 and prepares |λ>_S⊗|PME>, a Bell measurement on S and qudit 1 teleports |λ> entirely to qudit 3, leaving qudits 2 and 4 in a Bell pair uncorrelated with the secret. After the dealer announces the outcome, player 3 alone applies the local correction from Eq. (64) and recovers |λ>; no collaboration of K−K/2=2 players is required. The protocol therefore does not implement a threshold QSS. More broadly, Eqs. (61)–(65) are not a well-defined protocol: Eq. (63) is dimensionally inconsistent, and the correction operator (64) is not specified in terms of which shares it acts on or how the classical outcome is split among players. Since the secret-sharing claim is part of the title and abstract, this is a load-bearing gap; the Bell-pair PME states may support QSS through known graph-state access structures, but this paper does not provide a working scheme.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a construction of Planar Maximally Entangled (PME) states by starting from separable phase states of K qudits and applying controlled-phase gates. For K even, the construction is claimed to terminate in the explicit product of Bell pairs in Eq. (55), |PME(K,d)> = ∏_{l=1}^{K/2} |P^+>_{l,l+K/2}, which is indeed a PME state for any d. The paper also presents examples for K=2,3,4,8, discusses star and line graph states, and proposes a quantum secret sharing (QSS) protocol in Section 6. The QSS section is the advertised application in the title and abstract, and it is this part that is not sound: the protocol as written fails even for the smallest nontrivial case K=4, where a single player can recover the secret.","tokens_in":16208,"tokens_out":9142,"duration_ms":90595,"significance":"If the paper's claims were fully correct, an elementary phase-state route to PME states for arbitrary even K and arbitrary qudit dimension would be a useful pedagogical and constructive contribution, and the QSS application would be of practical interest. The Bell-pair form in Eq. (55) is directly checkable, involves no fitting parameters, and the reduced-density verification for those states is straightforward; these are genuine strengths. However, Eq. (55) is exactly Definition II imported from Ref. [40], so the original content resides in the phase-state derivation and in the QSS protocol. The phase-state derivation is essentially a graph-state rewriting, and the QSS protocol is invalid for K=4. The star and line graph examples are also not certified as PME and are in fact counterexamples to such a certification. Thus the advertised contribution is not established as stated.","major_comments":[{"comment":"The QSS protocol does not implement the claimed threshold. For K=4, Eq. (45) gives |PME(4,d)> = |P^+>_{13}|P^+>_{24}. If the dealer holds qudit 1 and performs a Bell measurement on the secret qudit S and qudit 1, the standard teleportation identity transfers the secret entirely to qudit 3, while qudits 2 and 4 are left in a Bell state uncorrelated with the secret. After the dealer announces the measurement outcome, player 3 alone can apply the local correction from Eq. (64) and recover |λ>. This directly contradicts the protocol's closing claim that at least K−K/2 = 2 collaborating players are required. The correction formulas are also not a well-defined protocol: Eq. (63) mixes a projection outcome with a state and leaves |φ_j> undefined, while Eq. (64) does not specify on which share or shares the operators act or how the classical outcome is split among players. Because the QSS claim is in the title and abstract, this is a load-bearing error.","section":"Sec. 6, Eqs. (61)–(65)"},{"comment":"The star and line graph states are presented as PME-related states without the required certification. In fact, the four-qudit star state of Eq. (41) with Q12=Q13=Q14=1 is not PME. For the adjacent subset A={2,3}, the reduced density matrix has the off-diagonal element <0,1|ρ_A|1,0> = 1/d^2 (obtained by tracing out qudits 1 and 4), so ρ_A is not proportional to the identity. The paper never verifies the PME condition for all connected subsets; it only states Eq. (28) for a single bipartition and then jumps to the Bell-pair product in Eq. (45). The relation between the star/line states and the valid Bell-pair state of Eq. (45) needs to be made explicit, or these examples should be removed or relabeled.","section":"Sec. 5.3–5.4, Eqs. (41), (42), (48)–(50)"},{"comment":"The certification criterion in Eq. (28) is quoted from Ref. [53] for one bipartition (A1,A2), but PME requires the reduced state to be maximally mixed for every connected subset of size at most K/2 under periodic boundary conditions. The paper never proves or even states that the criterion applies to all such bipartitions. It also does not justify the criterion for composite qudit dimension d: for d composite, linear independence over the ring Z_d is not equivalent to full column rank over a field, so the quoted 'if and only if' needs a proof or a reference that covers this case. This gap is exactly what allows the non-PME star state of Eq. (41) to pass unexamined.","section":"Sec. 5, Eq. (28)"}],"minor_comments":[{"comment":"Equation (36) is dimensionally inconsistent: the left-hand side is the full bipartite density matrix |PME(2,d)><PME(2,d)|, a d^2×d^2 operator, while the right-hand side is the d×d identity. It should read ϱ_{A2} = Tr_{A1}(|PME(2,d)><PME(2,d)|) = (1/d) I_d.","section":"Sec. 5.1, Eq. (36)"},{"comment":"The text says 'five qubits system (K = 8)'; this should be 'eight-qudit system (K=8)', and the notation in Eqs. (48)–(49) uses both subscripts '5' and '8' for the same density matrix.","section":"Sec. 5.4, paragraph before Eq. (48)"},{"comment":"Equation (57) contains '|µ1, µ2, ........., µK/2>' and the summation is written as 'X_{µ1,....,µ2}'; the summation indices should be µ1 through µ_{K/2}.","section":"Sec. 5.5, Eq. (57)"},{"comment":"The controlled-phase gate in Eq. (35) appears to have a typographical error: both factors are written on system j, whereas the standard definition would be |µ><µ|_i ⊗ Z^µ_j (control on i, phase on j), consistent with its subsequent use in Eq. (33).","section":"Sec. 3, Eq. (35)"}],"recommendation":"major_revision","confidential_remarks":"The valid core of the paper, the Bell-pair product in Eq. (55), is present already as Definition II from Ref. [40], so the original contribution is concentrated in the phase-state derivation and the QSS protocol. Since the QSS protocol fails for K=4, the advertised application is not correct as written. I would ask the authors to either remove the QSS claim from the title/abstract and revise Section 6 accordingly, or replace it with a correct graph-state access-structure analysis whose thresholds are verified. The star/line examples should also be corrected or deleted. If these changes are made, the remaining paper would be a modest but self-contained exposition of the phase-state/graph-state route to the known PME states; whether that is sufficient novelty for this journal is a judgment the editor should make."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. The PME construction in Eq. (55) is correct, but it is literally Definition II from Doroudiani and Karimipour [40], rewritten as a product of Bell pairs. And the QSS protocol in Sec. 6 is broken: for the smallest case K=4, a single player can recover the secret, so the claimed threshold is not realized.\n\nCredit where it's due: the paper gives a clean, self-contained derivation that phase states plus controlled-phase gates produce the product-of-Bell-pairs PME states, and it explicitly works through the reduced density matrices for K=2,3,4,8. For a reader who has not seen graph-state notation, this is a readable tutorial. The algebra section is coherent.\n\nThe problems are in proportion: first, novelty. Preparing graph states by applying CP gates to |+> states is standard (Markham-Sanders, Keet et al.), and Eq. (55) is imported from [40]. The abstract overclaims a 'significant advance' over AME states. Second, the star and line graph states in Secs. 5.3–5.4 are labeled PME without checking the linear-independence condition of Eq. (28); for qudit dimension d that is composite, that condition needs real proof. Third, the QSS section is not a scheme. Eq. (63) is dimensionally inconsistent, Eq. (64) does not specify which shares the correction acts on, and the claimed collaboration threshold is false. Concretely, with |PME(4,d)> = |P^+>_{13}|P^+>_{24}, if the dealer holds qudit 1 and does a Bell measurement on the secret and qudit 1, the secret teleports entirely to qudit 3. Player 3 alone applies the correction and recovers it. No second player is needed. That is a load-bearing flaw because 'Quantum Secret Sharing Protocol' is in the title.\n\nWho is this for? A reader who wants a self-contained walkthrough of PME-state-as-graph-state preparation, as long as they already know the states exist. It is not a research advance. My recommendation: send it back. The authors should either remove the QSS claim entirely or replace it with a correct protocol, fix the dimensional errors, verify the star/line cases or drop them, and explicitly state the overlap with [40]. As is, I would not send it to a referee; it is not close to publishable in a research journal.","headline":"The PME construction is a correct repackaging of known results, but the QSS section is broken and the paper overclaims.","tokens_in":16766,"tokens_out":3212,"would_cite":false,"duration_ms":28653,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40","81P45","81P94"],"pacs":["03.67.-a","03.67.Mn","03.67.Dd"],"model":"deepseek-v4-flash","headline":"The paper claims that PME states for any even number of qudits can be generated by applying controlled-phase gates to separable phase states, pairing each qudit with its antipodal partner.","keywords":["planar maximally entangled states","phase states","controlled-phase gates","quantum secret sharing","qudit systems","multipartite entanglement","Weyl-Heisenberg algebra","graph states"],"falsifier":"Compute the reduced density matrix of a connected half-block for any state the paper labels PME; for the star-shaped four-qudit state with $Q_{12}=Q_{13}=Q_{14}=1$ and block $\\{2,3\\}$, the result is not maximally mixed, showing that label is applied too broadly, while the same check on the product-of-Bell-pairs formula always yields the maximally mixed state $I/d^{|S|}$ because each connected block contains at most one member of each antipodal pair.","tokens_in":15713,"feed_emoji":"🔗","tokens_out":13754,"duration_ms":122623,"temperature":0.7,"pith_summary":"The paper claims a systematic way to build planar maximally entangled (PME) states—states where every connected block of at most half the particles is maximally entangled—for any even number K of qudits of any dimension d. It starts from fully separable phase states and applies controlled-phase gates that pair each qudit with its antipodal partner, obtaining $|\\mathrm{PME}(K,d)\\rangle = \\prod_{l=1}^{K/2} |P^+\\rangle_{l,l+K/2}$. Because such multipartite entangled states exist where absolutely maximally entangled (AME) states do not (e.g., four qubits), the construction widens the available entanglement resources for quantum information. The paper further shows that these PME states can run a quantum secret sharing scheme in which any connected subset of more than half the players can recover the secret.","feed_headline":"Antipodal Bell pairs yield PME states for any even qudit number","feed_subtitle":"A simple product of Bell pairs on opposite qudits gives planar maximally entangled states and a quantum secret sharing scheme.","key_machinery":"The machinery has three pieces. (1) Phase states: using the Weyl–Heisenberg algebra of qudit raising and lowering operators, the polar decomposition $B^- = E\\sqrt{B^+B^-}$ yields phase operators whose eigenstates are the separable states $|\\varphi_{n_1\\ldots n_K}\\rangle \\propto \\sum_{\\mu} \\Omega^{\\sum_i n_i \\mu_i}|\\mu_1\\ldots\\mu_K\\rangle$. (2) Controlled-phase evolution: applying $U = \\prod_{i<j}(C_{ij})^{Q_{ij}}$ with $C_{ij}|\\mu_i\\mu_j\\rangle = \\Omega^{\\mu_i\\mu_j}|\\mu_i\\mu_j\\rangle$ to a product of such phase states produces the entangled states of Eq. (26). (3) PME certification: following the criterion imported from the graph-state literature, the reduced density matrix of a half-system is maximally mixed iff the rows of the connection matrix $C_{\\mu}$ are linearly independent. The paper's explicit PME construction is the matching state $\\prod_l |P^+\\rangle_{l,l+K/2}$, whose connection matrix is a permutation between antipodal qudits.","core_discovery":"The central claim is that the family of PME states is far larger and more accessible than the AME family, and that its members can be generated directly from product phase states by unitary dynamics. Concretely, the paper argues that for $K=2k$ qudits, the state $|\\mathrm{PME}(K,d)\\rangle = \\prod_{l=1}^{k} |P^+\\rangle_{l,l+k}$, with $|P^+\\rangle = d^{-1/2}\\sum_i |i,i\\rangle$, satisfies the PME condition: tracing out any connected block of at most $k$ adjacent qudits leaves the maximally mixed state $I/d^{|\\mathrm{block}|}$. The paper derives this by expressing controlled-phase gates as operators that act on separable phase states, and it certifies the maximal mixedness through a linear-independence criterion on the connection matrix of the underlying graph. It then uses the PME state to construct a quantum secret sharing protocol in which the dealer performs a Bell measurement and a set of at least $K-\\lfloor K/2\\rfloor$ adjacent players can reconstruct the secret.","pith_inferences":["The antipodal-pair construction works for a combinatorial reason the paper leaves implicit: on a cycle of $2k$ vertices, any connected set of at most $k$ consecutive vertices contains at most one endpoint of each diametrically opposite pair, so the reduced density matrix factorizes into maximally mixed single-qudit states; this suggests the idea transfers to any graph with a pairing such that ever","The star-shaped states displayed in Sections 5.3 and 5.4 differ from the product-of-Bell-pairs formula also called PME there; read charitably, the paper's PME claim concerns the antipodal-pair formula, not every evolved phase state.","The QSS scheme's threshold is geometric rather than algebraic: the recoverable sets are connected blocks of players, which makes the protocol a natural fit for nearest-neighbour quantum networks where interactions are local by construction.","Since each Bell pair $|P^+\\rangle$ is maximally entangled in any dimension, the validity of the final formula for composite $d$ does not hinge on the linear-independence criterion the paper imports; a direct reduced-density calculation would prove it more simply."],"forward_implications":["For any even $K$ and any dimension $d$, the explicit formula $|\\mathrm{PME}(K,d)\\rangle = \\prod_{l=1}^{K/2}|P^+\\rangle_{l,l+K/2}$ directly yields a valid PME state, including cases like four qubits where no AME state exists.","The construction needs only controlled-phase gates acting on a product input, so it can be realized with pairwise interactions between antipodal qudits rather than a global multipartite gate.","The proposed quantum secret sharing protocol allows a dealer to share a secret so that any connected subset of at least $K-\\lfloor K/2\\rfloor$ players can recover it, while smaller connected subsets cannot, by the no-cloning theorem.","Because PME states exist for parameter ranges where AME states do not, the paper's construction enlarges the pool of usable multipartite entangled resources for teleportation, secret sharing, and error correction."],"supporting_citations":[{"why":"defines PME states and the isometry condition that the paper's construction must satisfy.","marker":"[40]"},{"why":"supplies the linear-independence criterion for maximal mixedness of half-system reduced states.","marker":"[53]"},{"why":"gives the polar-decomposition method used to define phase operators and phase states.","marker":"[50]"},{"why":"introduces the GHZ-based quantum secret sharing scheme that the PME protocol generalizes.","marker":"[41]"},{"why":"provides the AME-based secret sharing framework the paper adapts to PME states.","marker":"[43]"},{"why":"supplies the graph-state quantum secret sharing formalism used for the correction operators.","marker":"[54]"}],"fun_headline_variants":["PME states from Bell pairs: any even qudit count","Even-qudit PME states via antipodal Bell pairs","Quantum secret sharing with planar maximally entangled states","PME states beat AME limits for even qudits","Bell pairs craft PME states for any even qudit number"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction is PME only if every connected block of up to K/2 adjacent qudits indeed ends up maximally mixed; the paper verifies this for selected small cases and asserts it for the general formula, without a proof that the criterion applies to all bipartitions in all dimensions d.","fun_headline_variants_meta":{"raw":{"variants":["PME states from Bell pairs: any even qudit count","Even-qudit PME states via antipodal Bell pairs","Quantum secret sharing with planar maximally entangled states","PME states beat AME limits for even qudits","Bell pairs craft PME states for any even qudit number"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1426,"prompt_tokens":1091,"completion_tokens":335,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":253}},"tokens_in":707,"tokens_out":335,"duration_ms":3801,"temperature":1.0,"reasoning_tokens":253,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:32:42.925521+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the reduced density matrix of a connected half-block for any state the paper labels PME; for the star-shaped four-qudit state with $Q_{12}=Q_{13}=Q_{14}=1$ and block $\\{2,3\\}$, the result is not maximally mixed, showing that label is applied too broadly, while the same check on the product-of-Bell-pairs formula always yields the maximally mixed state $I/d^{|S|}$ because each connected block contains at most one member of each antipodal pair.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines PME states and the isometry condition that the paper's construction must satisfy."},{"cited_title":"Helwig, W","cited_arxiv_id":null,"evidence_quote":"supplies the linear-independence criterion for maximal mixedness of half-system reduced states."},{"cited_title":"Vourdas, SU (2) and SU (1, 1) phase states","cited_arxiv_id":null,"evidence_quote":"gives the polar-decomposition method used to define phase operators and phase states."},{"cited_title":"Hillery, V","cited_arxiv_id":null,"evidence_quote":"introduces the GHZ-based quantum secret sharing scheme that the PME protocol generalizes."},{"cited_title":"Markham, B.C","cited_arxiv_id":null,"evidence_quote":"supplies the graph-state quantum secret sharing formalism used for the correction operators."}],"review_version":1}