LOCC distinguishable orthogonal product states with least entanglement resource

Haiquan Li; Naihuan Jing; Xilin Tang; Ze Gu

REVIEW 4 major objections 5 minor 29 references

LOCC distinguishable orthogonal product states with least entanglement resource

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper constructs four families of orthogonal product states that are indistinguishable by LOCC without entanglement, and proves that sharing one C^2⊗C^2 maximally entangled state (one ebit) makes every family perfectly distinguishable…

desk verdict A genuinely new set of LOCC-indistinguishable product-state families with one-ebit LOCC protocols, but the C^n⊗C^5 construction is ill-defined as printed and needs a major fix. read the letter →

arxiv 1908.03434 v1 pith:37KHUXO5 submitted 2019-08-09 quant-ph

Haiquan Li , Xilin Tang , Naihuan Jing , Ze Gu This is my paper · ORCID

classification quant-ph PACS 03.67.Mn03.65.Ud

keywords LOCCdistinguishabilityorthogonalproductstateslocalindistinguishabilitynonlocalitywithoutentanglementmaximallyentangledstateoneebitresourcequantumdiscrimination

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 constructs four families of pairwise orthogonal product states that cannot be distinguished by local operations and classical communication (LOCC) when the two parties share no entanglement: 2n−1 states in C^n⊗$C^{4}$ and in C^n⊗$C^{5}$, 2(n+2l)−8 states in C^n⊗$C^{{2l}}$, and 2(n+2k+1)−7 states in C^n⊗$C^{{2k+1}}$. It then proves a converse: in every family, the states become perfectly distinguishable by LOCC once Alice and Bob share a single $C^{2}$⊗$C^{2}$ maximally entangled state, i.e., one ebit. If correct, this shows that these particular instances of "nonlocality without entanglement" do not require higher-dimensional entanglement or extra communication; one shared Bell pair is sufficient. The significance is that a minimal two-qubit entangled resource unlocks local distinguishability for all the constructed product-state sets.

What carries the argument

The constructions are tiling-like lists of product states built around a "stopper" state |φ⟩ = (|1⟩−|2⟩+⋯+(−1)^{n−1}|n⟩)_A ⊗ (|1⟩−|2⟩+⋯+(−1)^{m−1}|m⟩)_B, together with tile states of the form |i+(i+1)⟩_A ⊗ |j⟩_B and |i⟩_A ⊗ |j+(j+1)⟩_B. The load-bearing mechanism is the non-disturbing measurement lemma: to prove LOCC indistinguishability, the paper shows that any measurement operator preserving the orthogonality of the whole set must be a scalar matrix, forcing the measurement to be trivial. The distinguishing protocol uses Cohen's ancilla method: Alice and Bob share the two-qubit maximally entangled state (|00⟩+|11⟩)/√2, Alice performs a controlled two-outcome measurement that correlates the ancilla with the first-party state, and then a sequence of local projections separates the remaining states in each branch.

What would settle it

Compute all inner products among the states in Eq. (1) for n=6 in $C^{6}$⊗$C^{4}$: if any two states are not orthogonal or the number of distinct states is not 11, the central claim collapses. Also run the explicit protocol of Theorem 4 on this instance and check that at every branch the surviving states remain pairwise orthogonal; a branch with overlapping states would refute one-ebit sufficiency.

Watch

Extended reading notes

Core claim

The central claim is that the orthogonal product states listed in Eq. (1) (for C^n⊗$C^{4}$, n>4), Eq. (4) (for C^n⊗$C^{{2l}}$, n≥2l>4), and Eq. (9) (for C^n⊗$C^{{2k+1}}$, n≥2k+1≥5) are LOCC-indistinguishable without assistance, yet can be perfectly distinguished by LOCC when Alice and Bob share a suitable $C^{2}$⊗$C^{2}$ maximally entangled state. The indistinguishability proofs follow the Walgate–Hardy criterion: any first-party measurement that preserves mutual orthogonality of the listed states must have its POVM operator proportional to the identity, so neither Alice nor Bob can extract information by going first. The distinguishability protocols append the entangled ancilla and let Alice or Bob perform a controlled two-outcome measurement that splits the set into branches, after which the states in each branch are separated by further local projective measurements. The paper therefore establishes that one ebit of entanglement is an upper bound on the resource needed to distinguish each of these families.

Load-bearing premise

Everything rests on the listed states in Eqs. (1), (4), and (9) being pairwise orthogonal with exactly the stated cardinalities; the paper itself flags an un-integrated indexing correction in Eq. (1), so a reader must verify the lists before relying on the theorems.

Editorial extensions

If this is right

Each of the four constructed families is LOCC-indistinguishable without assistance, so each is a new example of nonlocality without entanglement in dimensions C^n⊗C^m with m≥4.
For each family, a single C^2⊗C^2 maximally entangled state is sufficient for perfect LOCC discrimination, so one ebit is an upper bound on the entanglement resource needed.
The same protocol style works across even and odd Bob-side dimensions, indicating a uniform route from local indistinguishability to one-ebit-assisted distinguishability.
Since the indistinguishable sets consist entirely of product states, the result shows that entanglement is required only as a discrimination resource, not as a property of the states themselves.

Reading between the lines

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

The paper proves sufficiency, not necessity: the title's "least" is relative to Cohen's method, and nothing here shows that a Bell pair is required for these sets; a matching lower bound remains open.
The alternating-sign stopper state appears to be the engine of indistinguishability, which suggests that other stopper-based tilings in higher dimensions should also be unlocked by one ebit.
The indexing slip in Eq. (1) means a numerical implementation should regenerate the list from the corrected |ϕ7⟩=|3+4⟩_A|2⟩_B before testing the protocol.
A natural testable extension is to ask whether the state counts 2n−1, 2(n+2l)−8, and 2(n+2k+1)−7 are minimal among product-state sets in these dimensions that require one ebit.

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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

The paper does real work: it writes down explicit orthogonal product states in C^n⊗C^4, C^n⊗C^{2l}, and C^n⊗C^{2k+1} and gives concrete LOCC protocols showing one shared ebit suffices to distinguish them. The techniques are standard — Walgate–Hardy, Cohen's tiling, Zhang et al.'s one-ebit trick — but the particular state lists and the counts (2n−1, 2(n+2l)−8, etc.) are new and not in the cited papers. Theorem 1's proof is worked out in detail, and the even-dimensional Theorem 2 looks plausible, with the protocol in Theorem 4 spelled out. That is genuine progress in a narrow but active subfield.

The soft spots are more than cosmetic. The big one is Eq. (9), the promised C^n⊗C^5 family. For k=2 (m=5), the states use |0+1> and |5+6>, which do not exist in a 5-dimensional space. So the family that Theorem 3 and Theorem 6 claim to handle is simply not defined as printed. That is a load-bearing flaw, not a typo: local indistinguishability and the distinguishing protocol both operate on an explicit list, and the list fails. The proof of Theorem 3 is also a sketch — it says Bob-first follows by 'obvious symmetry when m=n' without showing the set is actually symmetric, and it compresses the Alice-first case into a few bullets. The note after Theorem 1 admitting that |ϕ_{n+1}> should be |ϕ_7> reinforces the impression that the indexing is not yet reliable. Pairwise orthogonality and cardinality of the large lists are asserted, not verified. Finally, the 'least entanglement resource' in the title and abstract is an overclaim: the paper proves sufficiency of one ebit, not optimality.

None of this kills the core idea for the even-dimensional families. But as it stands, the odd-dimensional family is broken for the headline case, and the proofs are uneven. This is exactly the kind of manuscript that needs a serious referee's time: a referee can require the authors to fix Eq. (9), write out the missing cases, and verify orthogonality. I would not cite it in its current form, but I would send it out.

Referee Report

4 major / 5 minor

Summary. The paper constructs explicit families of orthogonal product states in bipartite systems C^n⊗C^4, C^n⊗C^5, C^n⊗C^{2l}, and C^n⊗C^{2k+1}, proves their LOCC indistinguishability without shared entanglement, and then presents LOCC protocols that perfectly distinguish each family when Alice and Bob share a C^2⊗C^2 maximally entangled state. The indistinguishability proofs follow the Walgate-Hardy criterion: any POVM element that preserves orthogonality of the set must be proportional to the identity. The distinguishing protocols use an auxiliary maximally entangled pair and sequential projective measurements. Theorems 1 and 4 treat the C^n⊗C^4 case in detail; Theorems 2, 3, 5, and 6 generalize to even and odd Bob dimensions.

Significance. If the constructions and protocols are correct, the paper would supply new explicit examples of 'nonlocality without entanglement' that can be resolved with a single ebit of entanglement, adding to the literature on entanglement as a resource for LOCC discrimination. The paper's main strengths are its constructive approach and its explicit one-ebit protocols. The C^n⊗C^4 case is worked out in sufficient detail to be checked line by line, and the use of the Walgate-Hardy criterion is standard and sound. However, the advertised C^n⊗C^5 family and the odd-dimension generalization currently suffer from undefined states and omitted proof cases, so the full claimed result is not yet established.

major comments (4)

[II, Eq. (9)] For k=2, i.e. Bob dimension m=5, the states in Eq. (9) are not defined on the stated Hilbert space: |φ1> = |(k−2)+(k−1)>_A|k>_B = |0+1>_A|2>_B and |φ4> = |(k+2)>_A|(k−2)+(k−1)>_B = |4>_A|0+1>_B use a basis vector |0>, which does not exist in the {|i>}_{i=1}^d convention used throughout the paper; moreover |φ3> = |k>_A|(k+3)+(k+4)>_B = |2>_A|5+6>_B uses |6> in a five-dimensional Bob space, and |φ2> = |5+6>_A|4>_B uses |6> on Alice when n=5. Since Theorem 3 explicitly promises d=2(n−m)+9 locally indistinguishable states in C^n⊗C^5 for n≥5, this is not a cosmetic typo: the promised family is never exhibited, and the entanglement-assisted protocol in Theorem 6 cannot be applied to it. The construction for k=2 must be repaired before the main claim can be assessed.
[II, note after Theorem 1] The author-added note that |ϕ_{n+1}> = |3−5>_A|2>_B in (1) 'should be' |ϕ_7> = |3+4>_A|2>_B is not integrated into the theorem. If the replacement is made literally, the Alice-first proof's deduction a33=a55 from ⟨φ|M_A^† M_A ⊗ I|ϕ_{n+1}>=0 no longer follows with the same vector, and the Bob-first proof's use of |ϕ_{n+1}> to obtain b12=b21=0 is likewise affected. As printed, the theorem contains an unresolved self-identified indexing inconsistency, and the proof does not correspond to a single unambiguous state list.
[II, Theorems 2 and 3] The proofs of the even- and odd-dimension generalizations are only sketches. In Theorem 3, the Bob-first case is dismissed by 'the obvious symmetry when m=n' without showing that the set in (9) is invariant under exchanging Alice and Bob; for n>m the symmetry argument is not available at all. In Theorem 2, the Bob-first case is summarized as 'seen similarly' with only a sample computation. Since the Walgate-Hardy criterion requires ruling out a nontrivial first measurement by either party, the local-indistinguishability claim is incomplete without a full argument for the first-measuring party in each case.
[II, state lists (1), (4), (9)] The paper asserts pairwise orthogonality and the stated cardinalities of the sets in (1), (4), and (9) but does not verify them. The indistinguishability proofs assume the Gram matrix of the original set is diagonal; if any two listed states overlapped, the displayed equations setting inner products to zero would not be valid. A concise orthogonality check, for example by listing disjoint-support arguments or a verification table, should be included for each family.

minor comments (5)

[Title and Abstract] The title and abstract claim 'least entanglement resource,' but the paper proves only sufficiency of a C^2⊗C^2 maximally entangled state, not optimality. I suggest rephrasing to 'one-ebit resource' or 'minimal resource among the protocols considered' unless a lower bound is proved.
[II, Eq. (2)] Several inner products in Eq. (2) suppress nonzero Bob-part overlaps, for example ⟨2+3|1+2⟩=1/2 and ⟨2+3|3⟩=1/√2. The conclusions aij=0 are still valid because these factors are nonzero, but the presentation should note this normalization explicitly.
[II, Eq. (4) and Theorem 2 proof] The expression ⟨ψ_{l−l}| appears twice in the block after Eq. (6) and should be ⟨ψ_{l−1}|. Also, the indexing of the |ψ_{i+l}> states is inconsistent with the later use in Eq. (11), where the range begins at i=0; the notation should be harmonized so that every referenced state label is defined.
[III, Theorem 6 proof, Eq. (14)] The notation |φ'_{1,4}> and |φ'_{2,3}> is undefined; it should be spelled out as separate states |φ'_1>, |φ'_4> and |φ'_2>, |φ'_3>, respectively, to match the definitions in (9).
[III, Theorem 4 proof] There is a typographical error in the sentence introducing the projectors A3i: 'A3i = 2⟩a⟨2|⊗| 2 +i⟩A⟨2 +i|' is missing the initial vertical bar and should read 'A3i = |2⟩a⟨2|⊗| 2 +i⟩A⟨2 +i|'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the constructions and LOCC protocols are explicit, with conclusions not assumed as inputs.

full rationale

The paper does not contain fitted parameters, inverse definitions, or a load-bearing self-citation chain. Its central claims are explicit mathematical constructions: the state lists in Eqs. (1), (4), and (9) are displayed, and the LOCC-indistinguishability proofs proceed by direct matrix-element computations (e.g., Eqs. (2), (3), (5), (6), and the bullet list in Theorem 3) using the standard Walgate-Hardy criterion, not by assuming the target result. The distinguishability protocols in Theorems 4-6 give explicit Alice/Bob projectors after adding a C^2⊗C^2 maximally entangled state; the entangled state is an input resource, not a renamed version of the claimed conclusion. Cited works such as Cohen [27] and Zhang et al. [28] provide methods, but the present proofs do not reduce to those citations alone. The phrase 'least entanglement resource' is an unsupported optimality overclaim rather than a circular derivation. There are genuine correctness and completeness concerns: Eq. (9) appears ill-defined for k=2 (e.g., basis labels |5+6> outside C^5), and Theorem 3's Bob-first case is only justified by 'obvious symmetry when m=n' rather than shown; however, these are errors or gaps in proof, not instances of circularity. No step in the paper equates a prediction with an input by construction, and no fitted value is relabeled as a prediction. Accordingly, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No data fitting or invented physical entities. The main input assumptions are the Walgate-Hardy LOCC characterization, the claimed orthogonality and cardinality of the constructed lists, and a symmetry reduction in Theorem 3. The dimension variables n, m, l, k are theorem parameters, not fitted constants.

assumptions (3)

standard math Walgate-Hardy characterization: a set of orthogonal pure states is LOCC-indistinguishable if every possible first nontrivial measurement must be trivial because all non-disturbing measurement operators satisfy M^†M proportional to the identity.
This theorem is the bridge used in Sec. II to turn matrix computations into local-indistinguishability conclusions. It is cited as [21] and not proven in the paper.
domain assumption The state lists in Eqs. (1), (4) and (9) are pairwise orthogonal product states with the claimed cardinalities.
Orthogonality and exact size are load-bearing for the indistinguishability proofs but are not checked exhaustively, and the note after Theorem 1 signals index fragility.
ad hoc to paper For m=n in Theorem 3, checking Alice-first is sufficient by the 'obvious symmetry' of the construction.
The proof uses this symmetry reduction without demonstrating that the specific state list is invariant under swapping the two parties.

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Pith. "Pith review of LOCC distinguishable orthogonal product states with least entanglement resource." pith.science (2026). https://pith.science/paper/37KHUXO5

@misc{pith2026190803434,
  author       = {Pith},
  title        = {Pith review of: LOCC distinguishable orthogonal product states with least entanglement resource},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37KHUXO5}},
  note         = {Machine review of arXiv:1908.03434}
}

abstract

In this paper, we construct $2n-1$ locally indistinguishable orthogonal product states in $\mathbb{C}^n\otimes\mathbb{C}^{4}~(n>4)$ and $\mathbb{C}^n\otimes\mathbb{C}^{5}~(n\geq 5)$ respectively. Moreover, a set of locally indistinguishable orthogonal product states with $2(n+2l)-8$ elements in $\mathbb{C}^n\otimes\mathbb{C}^{2l}~(n\geq 2l>4)$ and a class of locally indistinguishable orthogonal product states with $2(n+2k+1)-7$ elements in $\mathbb{C}^n\otimes\mathbb{C}^{2k+1}~(n\geq 2k+1>5)$ are also constructed respectively. These classes of quantum states are then shown to be distinguishable by local operation and classical communication (LOCC) using a suitable $\mathbb{C}^2\otimes\mathbb{C}^2$ maximally entangled state respectively.

Figures

Figures reproduced from arXiv: 1908.03434 by the authors.

**Figure 3.** FIG. 3: Tiling structure of orthogonal product states in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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