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arxiv: 2512.00484 · v2 · submitted 2025-11-29 · 🪐 quant-ph

Local distinguishability of five orthogonal product states on bipartite and tripartite quantum systems

Guang-Bao Xu , Zi-Yan Hao , Hua-Kun Wang , Yu-Guang Yang , Dong-Huan Jiang This is my paper

Pith reviewed 2026-05-17 03:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords orthogonal product stateslocal distinguishabilityLOCCbipartite systemstripartite systemsquantum nonlocalityvector of orthogonal relations
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The pith

Five orthogonal product states on bipartite systems fall into six structural categories defined by their orthogonal relations, with five categories allowing perfect distinction by LOCC alone.

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

The paper examines the local distinguishability of five orthogonal product states in both bipartite and tripartite quantum systems. It defines a vector of orthogonal relations to capture the pairwise orthogonality patterns that determine the structure of any such set. Using this vector, five-state sets in bipartite systems are sorted into six categories, and the authors show that five of the six categories permit perfect distinction using only local operations and classical communication. The remaining category receives separate case-by-case treatment. The same vector approach is applied to tripartite systems, yielding eight categories whose distinguishability properties are determined individually.

Core claim

By introducing the vector of orthogonal relations, the structures of any five bipartite orthogonal product states are partitioned into six categories; five of these categories admit perfect local distinguishability via LOCC, while the sixth is analyzed on a case-by-case basis. For five tripartite orthogonal product states the same vector yields eight categories, each of which is assigned an explicit local-distinguishability status.

What carries the argument

The vector of orthogonal relations, which records the specific pattern of orthogonality between each pair of the five states and thereby groups the sets into structurally equivalent classes.

Load-bearing premise

The vector of orthogonal relations captures every structural feature that affects whether five OPSs can be distinguished locally, so that the six (or eight) categories exhaust all possibilities.

What would settle it

An explicit example of five bipartite OPSs whose orthogonality pattern matches one of the first five categories yet cannot be perfectly distinguished by any LOCC protocol.

Figures

Figures reproduced from arXiv: 2512.00484 by Dong-Huan Jiang, Guang-Bao Xu, Hua-Kun Wang, Yu-Guang Yang, Zi-Yan Hao.

Figure 1
Figure 1. Figure 1: FIG. 1: The possible graphs of five bipartite OPSs with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The possible graphs of five bipartite OPSs with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The possible graphs of five bipartite OPSs with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The possible graphs of five bipartite OPSs with [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The possible graphs of five OPSs with the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The possible graphs of five OPSs with the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The possible graphs of five tripartite OPSs with [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The possible graphs of five OPSs with the [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The possible graphs of five tripartite OPSs [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The possible graphs of five OPSs with the [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The possible graphs of five tripartite OPSs [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
read the original abstract

Local distinguishability of orthogonal quantum states can effectively reduce the consumption of quantum resources and lower economic costs in quantum protocols. Although numerous achievements have been made regarding local distinguishability of orthogonal quantum states, some fundamental issues have not been effectively addressed. For example, the local distinguishability of five orthogonal product states (OPSs) is still unknown up to now. In this paper, we give the properties of local distinguishability of five OPSs on bipartite and tripartite quantum systems. Firstly, to characterize the structure of a set of bipartite OPSs, we propose the concept of the vector of orthogonal relations for a set of bipartite OPSs. Secondly, we classify the structures of five bipartite OPSs into six categories by this concept and prove that five of these six categories can be perfectly distinguished by local operations and classical communication (LOCC). Thirdly we show that the local distinguishability of each case of the sixth category singly. On the other hand, we first divide the structures of five tripartite OPSs into eight categories by the vectors of orthogonal relations of five tripartite OPSs. Then we give the local distinguishability of each category. Our work enriches the research results of quantum nonlocality and will provide a clear understanding of the local distinguishability of five OPSs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the vector of orthogonal relations to classify structures of five orthogonal product states (OPSs). For bipartite systems it partitions them into six categories, proves LOCC distinguishability for five categories, and treats the sixth via explicit case analysis. For tripartite systems it partitions into eight categories and determines distinguishability for each.

Significance. If the vector is shown to be a complete invariant and the case enumerations exhaustive, the work supplies a systematic classification for an open problem in local distinguishability, thereby enriching the literature on quantum nonlocality and resource-efficient quantum protocols.

major comments (2)
  1. [§3] §3 (definition of the vector of orthogonal relations): the claim that this vector induces precisely six categories for bipartite five-tuples is load-bearing for the 'five of six' result, yet the manuscript provides no explicit argument that the vector is a complete invariant separating all inequivalent structures under local unitaries.
  2. [§4.2] §4.2 (sixth-category case analysis): the enumeration of subcases must be shown to exhaust every possible vector of orthogonal relations falling into the sixth class; any missed configuration would leave the central claim that the remaining category has been fully resolved incomplete.
minor comments (2)
  1. [Abstract] The abstract states that proofs exist for five of six bipartite categories; adding one sentence clarifying that the sixth category is settled by exhaustive case analysis would improve immediate readability.
  2. [Notation] Notation for the components of the orthogonal-relation vector is introduced without an early concrete example; inserting a small illustrative five-tuple with its vector would aid comprehension of the subsequent classification tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive comments. We address each of the major comments in detail below.

read point-by-point responses

  1. Referee: [§3] §3 (definition of the vector of orthogonal relations): the claim that this vector induces precisely six categories for bipartite five-tuples is load-bearing for the 'five of six' result, yet the manuscript provides no explicit argument that the vector is a complete invariant separating all inequivalent structures under local unitaries.

    Authors: We recognize the importance of establishing that the vector of orthogonal relations serves as a complete invariant for the classification under local unitaries. While the manuscript defines the vector and uses it to partition into six categories, an explicit proof of completeness was not included. In the revised manuscript, we will add a detailed argument or lemma proving that two five-tuples of OPSs are locally unitarily equivalent if and only if they have the same vector of orthogonal relations. This will rigorously justify the six categories. revision: yes

  2. Referee: [§4.2] §4.2 (sixth-category case analysis): the enumeration of subcases must be shown to exhaust every possible vector of orthogonal relations falling into the sixth class; any missed configuration would leave the central claim that the remaining category has been fully resolved incomplete.

    Authors: We agree that it is necessary to demonstrate that the case analysis in §4.2 covers all possible configurations within the sixth category. In the revision, we will include an explicit enumeration of all admissible vectors of orthogonal relations that belong to the sixth class, based on the constraints imposed by the orthogonality conditions for five states. We will then map each to the corresponding subcase analyzed, thereby confirming exhaustiveness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; classification and proofs are self-contained

full rationale

The paper defines a new 'vector of orthogonal relations' as a structural descriptor for sets of five OPSs, uses it to partition bipartite cases into six categories and tripartite into eight, then derives LOCC distinguishability results via explicit proofs for five bipartite categories and case-by-case analysis for the sixth (plus full coverage for tripartite). None of these steps reduce by construction to the inputs: the vector is not defined in terms of the final distinguishability claims, no parameters are fitted and relabeled as predictions, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation. The derivation chain consists of independent definitions followed by direct case analysis and protocol constructions, making the results self-contained rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard postulates of quantum mechanics (Hilbert-space tensor products, orthogonality of product states) plus the newly introduced vector of orthogonal relations; no free parameters or invented physical entities appear.

axioms (2)
  • standard math Quantum states live in finite-dimensional Hilbert spaces and product states are tensors of local vectors.
    Invoked throughout the classification of OPSs on bipartite and tripartite systems.
  • domain assumption LOCC protocols consist of local measurements followed by classical communication.
    Standard definition used to define perfect distinguishability.
invented entities (1)
  • vector of orthogonal relations no independent evidence
    purpose: To encode the pattern of which pairs of states are orthogonal in which subsystems and thereby classify structural types.
    New conceptual object proposed in the paper to organize the six bipartite and eight tripartite categories.

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