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arxiv: 2606.26558 · v1 · pith:LFDXFEX4new · submitted 2026-06-25 · 🪐 quant-ph · math.QA

Graph Structures for Local Distinguishability of Quantum Product States

Sooyeong Kim , David W. Kribs , Michael Nathanson , Rajesh Pereira , Sarah Plosker This is my paper

Pith reviewed 2026-06-26 05:08 UTC · model grok-4.3

classification 🪐 quant-ph math.QA
keywords quantum product statesLOCC distinguishabilitytwo-way protocolsgraph theorybipartite stateslocal operationsfinite communication roundsstate discrimination
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The pith

Sets of bipartite product states are distinguishable by two-way LOCC after finite steps precisely when their graphs satisfy derived closure properties.

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

The paper extends one-way LOCC graph methods to full two-way protocols for distinguishing sets of quantum product states. It provides the first detailed look at the graphs tied to bipartite product states that become distinguishable through two-way communication in a finite number of rounds. The authors extract basic closure properties of this graph set and point to specific graph families that always permit distinction as well as families that block it. This combinatorial lens matters because it replaces direct quantum calculations with checks on graph structure for deciding distinguishability. A reader would care if the correspondence holds, as it turns a quantum information question into a graph-theoretic one.

Core claim

We extend prior graph-based identification of one-way LOCC distinguishable product states to the setting of two-way LOCC. The work supplies the first significant analysis of the graphs that correspond to bipartite product states distinguishable by two-way protocols after finitely many steps. Basic closure properties of the distinguishable graph set are obtained, together with concrete classes of graphs that guarantee local distinguishability and classes that do not.

What carries the argument

The mapping that sends sets of bipartite product states to graphs while preserving the distinction between one-way and two-way LOCC distinguishability.

If this is right

  • If two graphs belong to the distinguishable set, certain combinations or modifications of them remain in the set.
  • Some explicit graph families, such as particular trees or complete graphs, always correspond to distinguishable states under two-way LOCC.
  • Other graph families lie outside the distinguishable set and therefore block finite two-way distinction.
  • The closure properties allow systematic generation of new distinguishable graphs from known ones without returning to the quantum states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same graph lens might be used to bound the number of communication rounds needed for distinction.
  • It could connect local distinguishability questions to existing algorithms for testing graph properties such as planarity or bipartiteness.
  • The approach may generalize to settings with more than two parties once the corresponding hypergraph structures are defined.

Load-bearing premise

The graph representation of product-state sets keeps enough structure that two-way LOCC distinguishability after finite rounds can be read off from graph-theoretic properties alone.

What would settle it

Exhibit a concrete collection of bipartite product states whose associated graph lies inside the claimed distinguishable class yet cannot be perfectly distinguished by any finite two-way LOCC protocol.

Figures

Figures reproduced from arXiv: 2606.26558 by David W. Kribs, Michael Nathanson, Rajesh Pereira, Sarah Plosker, Sooyeong Kim.

Figure 1
Figure 1. Figure 1: Graph representation of local orthogonality relations in Example 2.4. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graph representation of local orthogonality relations in Example 3.6. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Measurement tree for a two–way LOCC protocol in Example 3.6 with Alice going first. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graph clique sum G = H ⊕ H in Example 4.5. We claim that any product state representation with G = GA = GB of G can be distinguished with LOCC by building on an existing protocol associated with the graph H. One such representation of (unnormalized) states is given below. Note that the specifics of Bob’s representation are not needed to reduce the problem from G to H. |ψ1⟩ = |0⟩A ⊗ |1 + 2⟩B |ψ2⟩ = |0 + 1⟩A… view at source ↗
Figure 5
Figure 5. Figure 5: G = GA = GB ∈ G / We indicate the known graph class relationships in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Graph classes relative to G. G1 is the set of chordal graphs, and G0 is the set of split graphs. 5 Conclusion and Outlook This work opens up several new lines of investigation. Most obvious is a direct continuation of the work we have begun on determining the structure of the class of distinguishable graphs for full LOCC. The current work is largely concerned with determining and deriving the basic graph c… view at source ↗
read the original abstract

We consider the problem of distinguishing sets of quantum product states with local operations and classical communication (LOCC). Recent work has used graph theory to identify sets of product states distinguishable with one-way LOCC. We extend these efforts to full two-way LOCC, with the first significant analysis of the set of graphs corresponding to bipartite product states that can be distinguished with two-way protocols after finitely many steps. We derive basic closure properties of the set of distinguishable graphs and identify some classes of graphs that guarantee local distinguishability and some graphs that do not. We also include several examples and forward-looking comments.

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

0 major / 3 minor

Summary. The paper extends graph-theoretic methods previously used for one-way LOCC distinguishability of bipartite quantum product states to the two-way LOCC setting. It analyzes the graphs corresponding to sets of product states that are distinguishable via finite-round two-way protocols, derives basic closure properties of the distinguishable graph set, identifies certain graph classes that are (or are not) locally distinguishable, and provides examples along with forward-looking comments.

Significance. If the mapping from product-state sets to graphs and the derived closure properties are valid, the work supplies a concrete combinatorial handle on two-way LOCC distinguishability, a regime that is strictly more powerful than one-way LOCC yet still lacks a complete characterization. Explicit identification of distinguishable and non-distinguishable graph families, together with the closure results, offers a reusable toolkit that could be applied to concrete state-discrimination tasks and to the design of finite-round LOCC protocols.

minor comments (3)
  1. [Abstract] The abstract states that the work supplies 'the first significant analysis'; this phrasing is subjective and could be replaced by a more neutral claim such as 'the first systematic analysis of closure properties' to avoid inviting reviewer push-back on the word 'significant'.
  2. Notation for the graph-to-state mapping (presumably introduced in §2 or §3) should be stated once in a dedicated paragraph or table so that later closure proofs can refer to it without re-deriving the correspondence each time.
  3. The forward-looking comments at the end would benefit from explicit pointers to open graph-theoretic questions (e.g., decidability of membership in the distinguishable set) rather than remaining at the level of general remarks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on extending graph-theoretic methods to two-way LOCC distinguishability of product states, including the derived closure properties and identification of distinguishable graph classes. The recommendation for minor revision is noted, and we will address any editorial or minor issues accordingly. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends prior graph-theoretic analysis of one-way LOCC distinguishability to two-way protocols by defining a mapping from product states to graphs and deriving closure properties. No step reduces by construction to its own inputs, no fitted parameters are relabeled as predictions, and no load-bearing premise rests solely on self-citation chains. The derivation remains self-contained against the stated definitions of the graph mapping and LOCC semantics, consistent with external prior work rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities can be extracted from the provided text.

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discussion (0)

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Reference graph

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