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arxiv: 2606.07880 · v1 · pith:OIIYSQ5Onew · submitted 2026-06-05 · 🪐 quant-ph

Defining Unique, Redundant, and Synergistic Quantum Information

Sean Ericson , Hailin Wang , S. J. van Enk This is my paper

Pith reviewed 2026-06-27 21:25 UTC · model grok-4.3

classification 🪐 quant-ph
keywords partial information decompositionquantum error correctiondecoherenceunique informationredundant informationsynergistic informationquantum information
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The pith

Unique quantum information must be absent from erasure-correctable qubit subsets

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

The paper extends the classical partial information decomposition to quantum states to define unique, redundant, and synergistic quantum information. It shows that unique information plays a central role in quantum error correction because any erasure-correctable subset of encoding qubits must contain zero unique information. Synergistic information appears when logical operations are supported only on the full set of qubits. Redundant information is crucial to the decoherence mechanism that explains the emergence of the classical world from quantum mechanics. A sympathetic reader cares because these relations link an information decomposition directly to practical quantum technologies and fundamental questions about reality.

Core claim

We extend the classical ideas of the Partial Information Decomposition (PID) to the quantum domain and quantify unique, redundant, and synergistic quantum information. We show that unique information plays the central role in quantum error correction codes: any erasure-correctable subset of encoding qubits must contain zero unique information. Synergistic information between two disjoint subsets of encoding qubits appears when a logical operation is supported on the whole set but not on the subsets separately. In a different application of our PID, we show that redundant quantum information is the crucial feature of the decoherence mechanism proposed by Zurek et al. to explain how the classi

What carries the argument

The quantum partial information decomposition that partitions the information between subsystems into unique, redundant, and synergistic parts, with operational meanings in error correction and decoherence.

Load-bearing premise

That the classical PID can be extended to quantum states while retaining the operational interpretations for quantum error correction and decoherence without being disrupted by non-commutativity or measurement effects.

What would settle it

Finding an erasure-correctable set of qubits in a quantum code that nonetheless has positive unique information under the quantum PID definition would disprove the claim that unique information must be zero in such sets.

Figures

Figures reproduced from arXiv: 2606.07880 by Hailin Wang, Sean Ericson, S. J. van Enk.

Figure 1
Figure 1. Figure 1: FIG. 1. Collective and individual two-bit capacities for the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. C-PID for two instances of the mixed-Bell state. In a) the spectator states are [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Distributions of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Mutual information between system and environ [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. a) Collective and individual superdense capacities for the decoherence model as a function of the number of environment [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. a) Collective and individual superdense capacities for the generalized decoherence model with [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

We extend the classical ideas of the Partial Information Decomposition (PID) to the quantum domain and quantify unique, redundant, and synergistic quantum information. We show that unique information plays the central role in quantum error correction codes: any erasure-correctable subset of encoding qubits must contain zero unique information. Synergistic information between two disjoint subsets of encoding qubits appears when a logical operation is supported on the whole set but not on the subsets separately. In a different application of our PID, we show that redundant quantum information is the crucial feature of the decoherence mechanism proposed by Zurek \textit{et al.} to explain how the classical world emerges out of the quantum world.

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 extends the classical partial information decomposition (PID) to quantum states, introducing definitions for unique, redundant, and synergistic quantum information. It claims that unique information is central to quantum error correction, with any erasure-correctable subset of encoding qubits containing zero unique information; synergistic information arises when a logical operation is supported on the full set but not on disjoint subsets; and redundant information is the key feature in Zurek et al.'s decoherence mechanism for the emergence of classicality from quantum systems.

Significance. If the proposed quantum PID definitions preserve the operational interpretations of the classical case, the framework could offer a structured way to analyze information partitioning in quantum error correction and decoherence processes. The explicit links to QEC (erasure correction) and pointer-state decoherence provide concrete test cases. The work supplies formal definitions and applications rather than parameter-free derivations or machine-checked proofs.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (QEC application): the claim that 'any erasure-correctable subset of encoding qubits must contain zero unique information' is load-bearing for the central QEC result. The manuscript must demonstrate that the quantum definition of unique information inherits the classical operational semantics (information accessible from one source but not the other, without disturbance) under the non-commuting observables and measurement disturbance inherent to quantum erasure channels; otherwise the zero-unique-information statement does not follow from the definitions alone.
  2. [Abstract and §5] Abstract and §5 (decoherence application): the assertion that 'redundant quantum information is the crucial feature' of Zurek's decoherence mechanism requires an explicit mapping showing that the quantum PID redundant term corresponds to the environment-induced superselection and pointer-basis stability, rather than merely re-labeling classical redundancy. Non-commutativity between system and environment observables risks altering the redundancy interpretation.
minor comments (2)
  1. [§3] Notation for the quantum PID quantities should be introduced with explicit comparison to the classical PID axioms (e.g., non-negativity, monotonicity) to clarify which properties are retained.
  2. [Figures] Figure captions for any illustrative circuits or state diagrams should specify the basis in which the unique/redundant quantities are evaluated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address each major comment below and will incorporate clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (QEC application): the claim that 'any erasure-correctable subset of encoding qubits must contain zero unique information' is load-bearing for the central QEC result. The manuscript must demonstrate that the quantum definition of unique information inherits the classical operational semantics (information accessible from one source but not the other, without disturbance) under the non-commuting observables and measurement disturbance inherent to quantum erasure channels; otherwise the zero-unique-information statement does not follow from the definitions alone.

    Authors: We agree that an explicit demonstration is needed to connect the quantum unique-information definition to the classical operational semantics under measurement disturbance. In the revision we will add a dedicated paragraph in §4 deriving that, for any erasure-correctable subset, the unique term vanishes identically because the quantum mutual information between the logical qubit and that subset equals the mutual information with the complementary subset; this equality follows directly from the definition of unique information and the no-disturbance property of the erasure channel on correctable subspaces, without additional assumptions. revision: yes

  2. Referee: [Abstract and §5] Abstract and §5 (decoherence application): the assertion that 'redundant quantum information is the crucial feature' of Zurek's decoherence mechanism requires an explicit mapping showing that the quantum PID redundant term corresponds to the environment-induced superselection and pointer-basis stability, rather than merely re-labeling classical redundancy. Non-commutativity between system and environment observables risks altering the redundancy interpretation.

    Authors: We accept that a more explicit mapping is required. The revised §5 will contain a step-by-step correspondence: the redundant quantum information is defined via the intersection of quantum mutual informations with distinct environment fragments; this intersection is nonzero precisely when the fragments share the same pointer-basis projectors, thereby enforcing the environment-induced superselection rule. We will also note that the definition is evaluated after decoherence has occurred, so non-commutativity is already accounted for by the pointer-basis restriction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitions and applications are independent of inputs

full rationale

The paper introduces a quantum extension of PID and then applies the resulting quantities to QEC (unique information zero on correctable subsets) and Zurek decoherence (redundant information as key feature). These are presented as consequences of the new definitions rather than tautological reductions; no equations or steps are shown to equate a claimed prediction directly to a fitted parameter or self-referential definition by construction. No load-bearing self-citations or uniqueness theorems imported from the authors' prior work appear in the provided text. The derivation chain remains self-contained against external benchmarks, with the QEC and decoherence statements functioning as interpretive applications rather than forced outputs of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities are identifiable from the provided text.

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Reference graph

Works this paper leans on

56 extracted references · 6 canonical work pages

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    The GHZ and W States Two famously inequivalent [30] tripartite entangled states are the GHZ and W states, given by |GHZ⟩= 1√ 2 (|000⟩+|111⟩),(26) 5 TABLE I. The four cases of the C-PID. Depending on the relative values of the difference between the individual superdense capacities (CA −C B) and the uncorrected one-bit unique capacities (Γ A,Γ B) the PID q...

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