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REVIEW 3 major objections 4 minor 58 references

Constructing Multipartite Planar Maximally Entangled States from Phase States and Quantum Secret Sharing Protocol

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict The PME construction is a correct repackaging of known results, but the QSS section is broken and the paper overclaims. read the letter →

arxiv 2411.15077 v2 pith:ZSLWSEXN submitted 2024-11-22 quant-ph math-phmath.MP

Lahoucine Bouhouch , Yassine Dakir , Abdallah Slaoui , Rachid Ahl Laamara This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81P4081P4581P94 PACS 03.67.-a03.67.Mn03.67.Dd
keywords planarmaximallyentangledstatesphasecontrolled-phasegatesquantumsecretsharingquditsystemsmultipartiteentanglementWeyl-Heisenbergalgebragraph
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Sec. 6, Eqs. (61)–(65)] 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.
  2. [Sec. 5.3–5.4, Eqs. (41), (42), (48)–(50)] 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.
  3. [Sec. 5, Eq. (28)] 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.
minor comments (4)
  1. [Sec. 5.1, Eq. (36)] 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.
  2. [Sec. 5.4, paragraph before Eq. (48)] 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.
  3. [Sec. 5.5, Eq. (57)] 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}.
  4. [Sec. 3, Eq. (35)] 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).

Circularity Check

1 steps flagged · score 6.0 of 10

General PME construction (Eq. 55) restates Definition II (Eq. 7) from Ref. [40], making the central 'derivation' from phase states self-definitional; no fit-based or load-bearing self-citation circularity found.

  1. self definitional [Sec. 2 Definition II (Eq. 7) and Sec. 5.5 Eq. (55)]
    "Definition II: For K even dimension (K = 2k) of particles, the PME states shared between 2 k parties is given by |PME (K, d)⟩ = ∏_{l=1}^{k} |P^+⟩_{l,l+k} = ∏_{l=1}^{k} (1/√d ∑_{i=0}^{d-1} |i,i⟩)_{l,l+k}. ... PME states in arbitrary dimension for K = 2k parties gives as |PME (K, d)⟩ = ∏_{l=1}^{K/2} |P^+⟩_{l,l+K/2}."

    Equation (55) is exactly Definition II with k = K/2, so the general 'construction' is the input definition restated. The phase-state evolution in Eqs. (22)-(26) and Eq. (53) yields generic graph states with arbitrary exponents Qij, and the paper never shows that the PME condition or the linear-independence criterion (28) forces Qij = 1 only for the edges (l, l + K/2). Instead, for K = 4 and K = 8 the paper explicitly invokes 'the definition in 2' (Eqs. (45) and (51)), then carries the same formula to K parties in Eq. (55). Thus the central derivation of PME states from phase states reduces to restating the imported Bell-pair definition of PME, rather than to a consequence of phase-state dynamics.

full rationale

The paper contains no fitting, no data, and no load-bearing self-citation: Refs. [17], [18], and [58] are contextual or concern teleportation, not the PME construction. The linear-independence criterion (28) is imported from Ref. [53], an external source, and is independent evidence for the entanglement criterion, though the paper gives no proof for general composite dimensions. The main circularity is definitional: Definition II (Eq. (7)) already asserts that PME(K,d) states are the product of Bell pairs |P^+⟩_{l,l+K/2}, and Sec. 5.5 (Eq. (55)) repeats this as the derived multipartite result. The phase-state machinery does provide a valid graph-state rewriting of these Bell-pair states (controlled-phase gates applied to a separable phase state), which is independent content, so the circularity is partial rather than total. Separately, the QSS protocol in Sec. 6 is a correctness risk rather than a circularity finding: the threshold claim that at least K - K/2 players are needed appears unsupported, since for K = 4 a single player can recover the secret after the dealer's Bell measurement.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The construction depends on the choice of connection numbers Q_{ij}; only the opposite-site pairing is shown to be PME. The PME certification imports the linear-independence criterion from Ref. [53]. No new physical entities are postulated.

free parameters (1)
  • connection numbers Q_{ij} = Q_{ij} in {0,1}; 1 only on pairs (l, l+K/2) for PME states, 1 on star/line graphs in Secs. 5.3-5.4
    The entangled phase states depend on the adjacency matrix Q_{ij}; the PME property holds for the opposite-site pairing, while the star/line choices are not shown to be PME.
assumptions (4)
  • standard math Phase operators E_i admit a common eigenbasis with eigenvalues on the unit circle, giving phase states of the form Eq. (17).
    Invoked in Sec. 3 to define phase states; relies on the polar-decomposition approach from Refs. [50-52].
  • domain assumption A graph state's reduced density matrix for subset A2 is maximally mixed iff the connection vectors from A2 to A1 are linearly independent (Eq. 28).
    This is the certification used in Sec. 4 to identify PME states; it is imported from Ref. [53] and not proved in the paper, including its validity for composite qudit dimensions.
  • domain assumption Definition II, Eq. (7), from Ref. [40] defines PME states as products of Bell pairs for even K.
    The paper's central construction in Sec. 5.5 reduces to this imported definition, making the claimed novelty partly a restatement of the prior literature.
  • standard math Local unitary equivalence preserves the PME property, i.e., maximal mixedness of connected subsets.
    Used when identifying controlled-phase graph states with PME states in Secs. 5.2-5.5; this is a standard and correct fact.

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Pith. "Pith review of Constructing Multipartite Planar Maximally Entangled States from Phase States and Quantum Secret Sharing Protocol." pith.science (2026). https://pith.science/paper/ZSLWSEXN

@misc{pith2026241115077,
  author       = {Pith},
  title        = {Pith review of: Constructing Multipartite Planar Maximally Entangled States from Phase States and Quantum Secret Sharing Protocol},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSLWSEXN}},
  note         = {Machine review of arXiv:2411.15077}
}
abstract

In this paper, we explore the construction of Planar Maximally Entangled (PME) states from phase states. PME states form a class of $n$-partite states in which any subset of adjacent particles whose size is less than or equal to half the total number of particles is in a fully entangled state. This property is essential to ensuring the robustness and stability of PME states in various quantum information applications. We introduce phase states for a set of so-called noninteracting $n$ particles and describe their corresponding separable density matrices. These phase states, although individually separable, serve as a starting point for the generation of entangled states when subjected to unitary dynamics. Using this method, we suggest a way to make complex multi-qubit states by watching how unconnected phase states change over time with a certain unitary interaction operator. In addition, we show how to derive PME states from these intricate phase states for two-, three-, four-, and K-qubit systems. This construction method for PME states represents a significant advance over absolutely maximally entangled (AME) states, as it provides a more accessible and versatile resource for quantum information processing. Not only does it enable the creation of a broader class of multipartite entangled states, overcoming the limitations of AME states, notably their restricted availability in low-dimensional systems; for example, the absence of a four-qubit AME state, but it also offers a systematic construction method for any even number of qudits, paving the way for practical applications in key quantum technologies such as teleportation, secret sharing and error correction, where multipartite entanglement plays a central role in protocol efficiency.

Figures

Figures reproduced from arXiv: 2411.15077 by the authors.

Figure 1
Figure 1. Illustration of an eight-particle Planar Maximally Entangled (PME) state [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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