{"id":"0d10abd4-1ba1-4a4a-bcb2-11ff4eb3736a","arxiv_id":"2607.13223","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Three W-like states achieve maximal single-qubit entropy and one-maximal tangle for one subsystem, and can support perfect teleportation and dense coding, unlike the W state.","lead":"This paper computes four entanglement measures for three specific W-class states and compares them with the W state, claiming advantages in robustness and perfect teleportation/superdense coding. The numerical comparisons are largely correct, but the claimed novelty is overstated because prior work already used these states for the same tasks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's conditional verdict was motivated by a suspicion that the inherited formulas might be wrong and that the uniqueness theorems lack derivations. My stress-test shows the formulas are correct, the tables are consistent, and the protocols work. The remaining issues—novelty overstatement, deferred proofs, typos—are real but do not threaten the central claims. Therefore I do not find a load-bearing concern that would change the verdict.","tokens_in":15880,"tokens_out":52228,"duration_ms":383784,"concrete_test":"Independently compute the reduced density matrices for |ϑ⟩, |η⟩, |ξ⟩, and W by direct partial trace of the state vectors, then evaluate S(ρ), τ_xy, negativity, and τ_x(yz) from the definitions; compare every entry with Tables 1–4. Also simulate the teleportation and superdense-coding protocols for a random single-qubit input state |φ⟩ using the paper's measurement bases and recovery unitaries, checking that the final state has fidelity 1 with |φ⟩.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the central claims and the supporting calculations. The entanglement measures (vNEE, 2-tangles, negativities, tangles) are correctly derived for the W-class states used; I re-derived the reduced density matrices for the three states and the W state and found agreement with Tables 1–4. The maximization claims in Properties 1.1–1.4 and the lemmas are supported by the explicit formulas, and Theorems 1–3 are plausible via qubit permutation symmetry. The teleportation and superdense-coding protocols are valid: the four measurement states are orthogonal, and the total state lies in their span, so a projective measurement extended arbitrarily to the full Hilbert space yields the correct outcomes with unit probability. The reader's weakest assumption—that the formulas from [4,6] might be wrong—does not land; I independently checked them. The incomplete 'complicated calculation' proofs and minor typographical issues are not load-bearing for the correctness of the main results.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies three W-like three-qubit states |ϑ′⟩, |η′⟩, |ξ′⟩ and their Schmidt forms |ϑ⟩, |η⟩, |ξ⟩, comparing them with the W state using four entanglement measures: the 2-tangles τxy, negativities, von Neumann entanglement entropy, and the 1-vs-2 tangles τx(yz). It claims that each of the three states has one single-qubit reduced density matrix with maximal entropy ln 2 and one tangle equal to 1, that tracing out a suitable qubit leaves a two-qubit state more entangled than the corresponding reduced state of the W state, and that these states enable perfect teleportation and superdense coding while the W state does not. Theorems 1–3 assert uniqueness of the maximizers within restricted Schmidt families; Results 1–4 compare W-state averages; Theorems 4–5 give explicit protocols.","tokens_in":16111,"tokens_out":43346,"duration_ms":344443,"significance":"If the results are correct, the paper provides a useful, small catalogue of W-class states with asymmetric entanglement extremality and explicit teleportation/dense-coding circuits. The explicit calculations I checked—the Schmidt forms, Tables 1–4, Lemma 1, Theorem 1, and the protocols for |ξ⟩ and |ϑ⟩—are internally consistent and reproducible from the displayed formulas. The contribution is incremental but publishable in a specialized quantum-information venue, provided the proof gaps identified below are filled. The main weakness is that several central assertions are supported only by 'a calculation yields' or by external citations rather than by derivations in the manuscript.","major_comments":[{"comment":"The proofs of Theorems 2 and 3 consist solely of 'A complicated calculation yields' with no displayed equations or derivative bounds. These theorems are the main uniqueness results for |η⟩ and |ξ⟩. Since |η⟩ and |ξ⟩ are qubit permutations of |ϑ⟩, the derivative analysis used in Theorem 1 and Properties 1.1–1.4 can be adapted; please supply the actual calculation or a symmetrization argument.","section":"Theorems 2 and 3 (Sections 'THE STATES HAVING THE MAXIMAL VNEE S(ρB)' and 'S(ρC)')"},{"comment":"The maximization of A_neg over the W-like states in Eq. (2) is asserted via 'A calculation shows' and 'One can verify'. This extremum is load-bearing for the W-state comparison. Please include an explicit two-variable optimization (or a bounding inequality) showing that the conditional maximum (√5−1)/6 occurs at λ0=λ2=λ3=1/√3.","section":"Result 1 (Section 'A_neg, Aτx(yz), Avnee, AND Aτxy FOR THE W STATE')"},{"comment":"The claim that the W state cannot be used for perfect teleportation or superdense coding is not proved in this manuscript; it rests entirely on references [20,21,24]. Since this negative statement is part of the paper's headline comparison, please state explicitly the no-go criterion used in those references (e.g., the W-class condition for deterministic perfect teleportation) and verify that the standard W state violates it. Without this, the comparative claim is not self-contained.","section":"Theorems 4 and 5 (Sections on teleportation and superdense coding)"}],"minor_comments":[{"comment":"The sentence 'To guarantee λ0 ≥ 0, λ1 in Eq. (19) must vanish' should be 'To guarantee a real solution for λ0'; non-negativity is not the obstruction—reality of λ0 is.","section":"Lemma 1 proof, Eq. (19)"},{"comment":"In the classical communication mapping after the |ϑ⟩ expansion, '00 to |ς+⟩_{a12}' should read '|ς+⟩_{a23}', since the measurement is on qubits a,2,3. Please check all such subscripts in the appendices.","section":"Appendix A, teleportation for |ϑ⟩"},{"comment":"The paper first defines W-like states as γ|001⟩+β|010⟩+α|100⟩, but later also calls the Schmidt forms |ϑ⟩,|η⟩,|ξ⟩ 'W-like states'. These families overlap only after local unitaries; please clarify the terminology to avoid confusion.","section":"Terminology, Preliminary and Summary"},{"comment":"Result 3 is proved by citing [6] without stating the corresponding theorem. Please quote the relevant result or give the argument, so the reader can verify the claim without consulting the earlier paper.","section":"Result 3"},{"comment":"Fractions such as √5−1/6 in the text should be written with parentheses, (√5−1)/6, to match the table layout and avoid ambiguity.","section":"Tables 1–4 and text"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies substantially on the author's own previous work ([4,6,24,25]) for formulas and criteria. This is not disqualifying—I independently spot-checked the central formulas and protocols—but the omitted derivations for Theorems 2–3 and Result 1 should be supplied. The paper is technically sound in its explicit parts but needs revision to meet the standard of a refereed journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a solid but narrow paper. The numbers in Tables 1-4 are right, the teleportation and dense-coding protocols for |ϑ⟩, |η⟩, |ξ⟩ and their LU equivalents work, and the maximization/uniqueness theorems are plausible. But the abstract's claim that 'no one has compared W-like states with the W state' is false on the paper's own references: Agrawal–Pati and Li–Qiu did that for teleportation and dense coding, and the paper acknowledges this in the proofs. The genuinely new piece is the maximization/uniqueness over the one-parameter families with one marginal entropy at ln 2, and even that is a routine optimization.\n\nWhat the paper does well: it gives explicit Schmidt forms, computes four entanglement measures, and constructs explicit measurement bases and Pauli corrections for teleportation and dense coding for all three states. I spot-checked the reduced density matrices, the negativity formulas (9)–(11), and the tangles; they are correct. The stress-test note is right that the concern about the formulas from [4,6] being wrong does not land—I re-derived them and they are fine.\n\nSoft spots: (1) Theorems 2 and 3 are not proven, just asserted via 'a complicated calculation.' That is a real gap in a manuscript that is otherwise transparent. (2) The novelty is oversold. The fact that these states can do perfect teleportation and dense coding was already known; only the |η⟩ variants and the explicit circuits for the SD forms are new, and that is incremental. (3) Minor typographical and notational issues, including a slightly confusing ordering of Pauli operations in Appendix A—though the operations themselves are correct, so it is a presentation issue, not a mathematical error. (4) The paper leans on the author's own previous formulas, but since those formulas are independently verified here, that is not a serious problem.\n\nWho this is for: researchers working on W-class entanglement or looking for explicit resource states for teleportation/dense coding. It would make a reasonable reference for the three states and their properties.\n\nRecommendation: it deserves a serious referee, but the referee should ask for derivations of Theorems 2 and 3 and a rewrite of the abstract to acknowledge prior comparisons. I would not desk-reject it, but I would not call it a major advance.","headline":"A correct but modest paper: the explicit entanglement calculations and protocols check out, but the novelty is overstated and two key proofs are left to 'a complicated calculation.'","tokens_in":16566,"tokens_out":7778,"would_cite":false,"duration_ms":63097,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40","81P45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the three W-like states |ϑ′⟩, |η′⟩, and |ξ′⟩ each possess one maximally mixed qubit and one maximal one-versus-two tangle, leave two-qubit states that are more entangled than the W state's after tracing out a qubit, an","keywords":["W state","W-like states","von Neumann entanglement entropy","tangle","2-tangle","negativity","quantum teleportation","superdense coding"],"falsifier":"Directly compute the 2-tangle and negativity for a W-class state with the phase term present (λ1 > 0), for example |ψ⟩ = (1/2)|000⟩ + (1/10)|100⟩ + λ2|101⟩ + λ3|110⟩, and check whether the claimed maximal values in Lemmas 1–3 still occur only when λ1 = 0; alternatively, numerically maximize the average negativity over the family in Eq. (22) and test whether a state with λ0, λ2 not both equal to 1/2 beats (√2+1)/12.","tokens_in":15785,"feed_emoji":"🔗","tokens_out":4684,"duration_ms":44869,"temperature":0.7,"pith_summary":"The paper compares three specific W-like pure states of three qubits — named |ϑ′⟩, |η′⟩, and |ξ′⟩ — with the standard W state. It claims that each of these states makes one of the three single-qubit reduced density matrices maximally mixed (von Neumann entropy ln 2) and gives the corresponding one-versus-two tangle its maximal value 1, whereas the W state achieves neither. It further claims that tracing out any one qubit leaves a two-qubit state that is more entangled, by both 2-tangle and negativity, than the corresponding leftover of the W state. Because of this combination, the paper argues, these three states are suitable for perfect teleportation and superdense coding while the W state is not, and they offer higher robustness against particle loss.","feed_headline":"Three W-like states beat the W state in four entanglement measures","feed_subtitle":"Each has a maximally mixed qubit, more entangled two-qubit leftovers, and works for perfect teleportation and superdense coding.","key_machinery":"The machinery is the Schmidt decomposition of W-SLOCC states, |ψ⟩ = λ0|000⟩ + λ1 e^{iϕ}|100⟩ + λ2|101⟩ + λ3|110⟩, together with closed-form expressions for four LU-invariant entanglement measures (von Neumann entanglement entropy, 2-tangles, negativities, and one-versus-two tangles) in terms of the λ coefficients. The paper uses these formulas to identify, for each choice of distinguished qubit, the family of states with maximal marginal entropy, solves for the unique member of that family that also maximizes all average measures, and verifies the resource protocols by exhibiting orthogonal four-qubit bases for teleportation and superdense coding.","core_discovery":"The central discovery is that within the SLOCC W class there exist locally-unitarily inequivalent states whose single-qubit marginal can be maximally mixed and whose bipartition entanglement can be maximal — properties previously associated mainly with the GHZ class. Concretely, |ϑ⟩ = (1/√2)|000⟩ + (1/2)|101⟩ + (1/2)|110⟩ (and two cyclic variants |η⟩, |ξ⟩) realize S(ρ_A) = ln 2 and τ_{A(BC)} = 1. The tables show that after tracing out any one of the other two qubits, the 2-tangle and negativity of the remaining pair exceed those of the W state. The author also proves, by explicit construction of four orthogonal measurement bases, that all six states (the three original forms and their Schmid","pith_inferences":["The results suggest a trade-off principle: as one qubit approaches maximal mixedness, the other two qubits become more entangled, so the three states may be optimal for distributed tasks where one party holds a distinguished qubit.","One could experimentally test the predicted robustness by preparing |ϑ⟩, sending it through a lossy or noisy channel, and comparing the 2-tangle of the surviving pair against the W state prepared under the same conditions.","The same optimization approach could be extended to search W-class states that maximize other entanglement monotones, or generalized to four-qubit W-like states, though the uniqueness proofs would need to be checked numerically."],"forward_implications":["If the results are correct, these six states provide a concrete advantage over the W state for any task sensitive to single-particle loss: dropping one qubit still leaves a more entangled pair.","The states join the GHZ state as resources for perfect teleportation and superdense coding, but they belong to the W SLOCC class, which may be easier to prepare or more robust in practical settings.","The pattern shows that maximal mixedness of one qubit is not exclusive to GHZ; a one-parameter family within the W class can achieve it while retaining full three-partite entanglement.","The explicit protocols in the appendices give ready-to-use measurement and correction sets for implementing two-qubit teleportation and dense coding with these states."],"fun_headline_variants":["W-like states achieve maximal single-qubit entanglement, unlike W state","These W-like states beat W state in teleportation and coding","Maximally entangled qubit in W-like states, not W state","W-like states give max entanglement per qubit, beat W state","Superior W-like states: max entanglement and perfect coding"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper assumes without independent derivation that its formulas for von Neumann entropy, 2-tangles, negativities, and one-versus-two tangles, which it imports from earlier work, correctly describe every state in the W SLOCC class; if any formula is only conditionally valid, the maxima, the comparison tables, and the protocol conclusions would lose their support.","fun_headline_variants_meta":{"raw":{"variants":["W-like states achieve maximal single-qubit entanglement, unlike W state","These W-like states beat W state in teleportation and coding","Maximally entangled qubit in W-like states, not W state","W-like states give max entanglement per qubit, beat W state","Superior W-like states: max entanglement and perfect coding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000587,"raw_usage":{"total_tokens":2682,"prompt_tokens":917,"completion_tokens":1765,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":1675}},"tokens_in":661,"tokens_out":1765,"duration_ms":12726,"temperature":1.0,"reasoning_tokens":1675,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T05:49:56.822898+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly compute the 2-tangle and negativity for a W-class state with the phase term present (λ1 > 0), for example |ψ⟩ = (1/2)|000⟩ + (1/10)|100⟩ + λ2|101⟩ + λ3|110⟩, and check whether the claimed maximal values in Lemmas 1–3 still occur only when λ1 = 0; alternatively, numerically maximize the average negativity over the family in Eq. (22) and test whether a state with λ0, λ2 not both equal to 1/2 beats (√2+1)/12.","supporting_citations":[],"review_version":1}