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arxiv: 2604.23501 · v1 · submitted 2026-04-26 · 🪐 quant-ph

Recognition: unknown

Quantum average correlation based on average coherence

Xiaoyu Ma , Qing-Hua Zhang , Cong Xu
Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:23 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum correlationaverage coherenceskew informationmutually unbiased basesHaar measurewave-particle dualitycomplementarity relation
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The pith

A new measure of average quantum correlation is the difference between global and local skew information.

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

The paper defines an average correlation for bipartite quantum systems using the difference between global and local skew information, inspired by equivalent definitions of average coherence through mutually unbiased bases or Haar integration. It proves this measure is nonnegative, contractive under local channels, and invariant under local unitaries, while also showing the two definitions are equivalent. A complementarity relation is established linking the correlation to wave-particle duality with the environment. This matters for developing tools to quantify correlations in a way consistent with quantum coherence and duality principles.

Core claim

We define an average correlation for bipartite systems as the difference between global and local skew information. This correlation measure is shown to satisfy essential properties including non negativity, contractivity under local quantum channels, and local unitary invariance. We further prove the equivalence between the average correlation defined via mutually unbiased bases and that defined via unitary groups. Finally, we derive a complementarity relation that connects wave-particle duality with the average correlation between a system and its environment.

What carries the argument

The difference between global and local skew information, serving as the definition of average correlation based on average coherence.

If this is right

  • It is nonnegative for bipartite states.
  • It contracts under local quantum channels.
  • It is invariant under local unitary operations.
  • The MUB-based and unitary-group definitions are equivalent.
  • It obeys a complementarity relation with wave-particle duality.

Where Pith is reading between the lines

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

  • Similar differences could be explored with other coherence quantifiers to define additional correlation measures.
  • The equivalence allows choosing the more convenient averaging method for explicit calculations in specific systems.
  • The complementarity could be checked in experiments that control both coherence and system-environment coupling.

Load-bearing premise

The difference between global and local skew information forms a meaningful correlation measure that inherits the key properties of average coherence.

What would settle it

A counterexample bipartite state where the average correlation is negative, or where the MUB and unitary definitions disagree, would disprove the main results.

read the original abstract

This paper studies the quantification and structural properties of quantum average correlation based on average coherence. Motivated by two mathematically equivalent approaches to define average coherence: one by averaging over complete sets of mutually unbiased bases, and the other by integrating over all orthogonal bases under the Haar measure, we define an average correlation for bipartite systems as the difference between global and local skew information. This correlation measure is shown to satisfy essential properties including non negativity, contractivity under local quantum channels, and local unitary invariance. We further prove the equivalence between the average correlation defined via mutually unbiased bases and that defined via unitary groups. Finally, we derive a complementarity relation that connects wave-particle duality with the average correlation between a system and its environment.

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 paper defines an average correlation measure for bipartite quantum systems as the difference between global and local average skew information, motivated by two equivalent definitions of average coherence (via averaging over mutually unbiased bases or integrating over the Haar measure on unitary groups). It claims to prove that this measure satisfies non-negativity, contractivity under local quantum channels, and invariance under local unitaries; establishes equivalence of the two averaging-based definitions; and derives a complementarity relation linking the correlation to wave-particle duality between a system and its environment.

Significance. If the properties and equivalence are rigorously established, the work offers a coherence-based correlation quantifier with desirable operational features and a link to foundational duality relations, which could aid studies in quantum information and foundations. The explicit equivalence proof between MUB and Haar averaging methods, along with the complementarity derivation, represents a strength by unifying distinct mathematical approaches to averaging.

major comments (2)
  1. [Section 3] Section 3 (properties of the measure): Non-negativity of the average correlation, defined as the difference between global and local averaged skew information, is not automatic from the non-negativity of individual skew information I(ρ, K) = −½ Tr([√ρ, K]²). An explicit inequality relating the global and local averages (e.g., via properties of the partial trace or convexity under the specific averaging) must be provided, as subadditivity does not follow immediately; this is load-bearing for the central definition.
  2. [Section 4] Section 4 (equivalence proof): The claimed equivalence between the MUB-based and Haar-measure (unitary group) versions of the average correlation is load-bearing for the measure's consistency. The proof must detail how the two averaging procedures yield identical values for the global-minus-local difference, without gaps in commuting the averaging with the global/local split.
minor comments (2)
  1. Notation for global versus local skew information and the two averaging procedures should be standardized and introduced with explicit operator definitions early in the manuscript to improve readability.
  2. [Section 5] The complementarity relation derivation would benefit from a brief discussion of its assumptions and potential experimental implications to strengthen the connection to wave-particle duality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment point by point below, providing clarifications and committing to revisions where the proofs require strengthening to meet the required rigor.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (properties of the measure): Non-negativity of the average correlation, defined as the difference between global and local averaged skew information, is not automatic from the non-negativity of individual skew information I(ρ, K) = −½ Tr([√ρ, K]²). An explicit inequality relating the global and local averages (e.g., via properties of the partial trace or convexity under the specific averaging) must be provided, as subadditivity does not follow immediately; this is load-bearing for the central definition.

    Authors: We agree that non-negativity of the defined average correlation (global minus local averaged skew information) does not follow immediately from the non-negativity of the individual skew information and requires an explicit demonstration. While the manuscript invokes the convexity of skew information and the averaging procedures over MUBs or the Haar measure, we acknowledge that a direct inequality relating the global and local averages—leveraging properties of the partial trace and the specific form of the averaging—should be supplied to rigorously establish I_global_avg(ρ, K) ≥ I_local_avg(ρ, K). In the revision, we will insert a dedicated lemma and proof in Section 3 that derives this inequality explicitly, confirming non-negativity without relying on implicit subadditivity assumptions. revision: yes

  2. Referee: [Section 4] Section 4 (equivalence proof): The claimed equivalence between the MUB-based and Haar-measure (unitary group) versions of the average correlation is load-bearing for the measure's consistency. The proof must detail how the two averaging procedures yield identical values for the global-minus-local difference, without gaps in commuting the averaging with the global/local split.

    Authors: We appreciate the referee highlighting the need for greater detail in the equivalence proof. The manuscript establishes equivalence for average coherence via MUB averaging and Haar integration, then defines the correlation as the difference; however, we concur that the proof should explicitly verify that the averaging operations commute with the global-local decomposition to ensure the differences coincide. In the revised Section 4, we will expand the argument with intermediate steps: first showing the equivalence for the global skew information term, then for the local terms (accounting for the partial trace), and finally confirming the subtracted differences are identical under both averaging schemes, closing any potential gaps in the commutation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines average correlation as the difference between global and local averaged skew information (using either MUB averaging or Haar integration), motivated by the known equivalence of those two averaging procedures for coherence but without reducing the new measure to a tautology. Non-negativity, contractivity, and unitary invariance are claimed to be proved explicitly rather than following automatically from the definition of skew information I(ρ,K). The equivalence between the MUB-based and Haar-based versions of the correlation measure is also proved inside the paper as a separate step. No parameters are fitted and then relabeled as predictions, no self-definitional loops exist, and no load-bearing self-citations reduce the central claims to prior unverified assertions by the same authors. The derivation chain is self-contained and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard quantum-information concepts rather than new free parameters or physical entities. Skew information and the two averaging procedures for coherence are taken from prior literature; the new correlation is a derived definition, not an invented particle or force.

axioms (2)
  • domain assumption Skew information is a valid quantifier of non-commutativity that can be subtracted globally versus locally to produce a correlation measure.
    Invoked when the average correlation is defined as global minus local skew information.
  • standard math The two mathematically equivalent definitions of average coherence (MUB averaging and Haar integration) can be used interchangeably for the correlation construction.
    Stated as motivation and used to prove equivalence of the resulting correlation measures.
invented entities (1)
  • Average correlation measure (global minus local skew information) no independent evidence
    purpose: To quantify bipartite quantum correlation via average coherence
    Newly defined quantity whose properties are then proven; no independent experimental signature is supplied.

pith-pipeline@v0.9.0 · 5407 in / 1632 out tokens · 38138 ms · 2026-05-08T06:23:16.444475+00:00 · methodology

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