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arxiv: 1907.05950 · v1 · pith:UESEJS5Enew · submitted 2019-07-12 · 🪐 quant-ph

Maximally Sensitive Sets of States

Daniel Gottesman This is my paper

Pith reviewed 2026-05-24 22:10 UTC · model grok-4.3

classification 🪐 quant-ph
keywords coherent errorsGHZ statesquadratic scalingquantum channelserror detectionunitary errorsquantum noise
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The pith

GHZ states in the X, Y, and Z bases detect any quantum channel capable of quadratic error scaling.

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

The paper shows that coherent errors grow quadratically with qubit number only for channels dominated by a unitary rotation. It defines a maximally sensitive set of states as one that will exhibit this quadratic growth for at least one sequence if the channel allows it at all. The GHZ states in the three bases form such a set. This construction supplies a concrete test that identifies coherent errors in gates and measurements to within a fixed fraction of the largest possible sensitivity.

Core claim

The GHZ states in the X, Y, and Z bases form a maximally sensitive set of states, such that any channel capable of quadratic error scaling will exhibit it on at least one sequence drawn from the set.

What carries the argument

Maximally sensitive set of states: a collection where, if a channel can produce quadratic error scaling, that scaling appears for at least one sequence in the collection.

If this is right

  • The set identifies coherent errors in gates and measurements to within a constant fraction of the maximum possible sensitivity.
  • A related protocol with simpler circuits tests for coherent errors in state preparation.
  • The protocol determines whether noise in a particular circuit accumulates coherently.

Where Pith is reading between the lines

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

  • The test could be inserted into calibration routines to flag when coherent noise dominates device behavior.
  • Variants of the protocol might track whether accumulated errors remain coherent across multiple circuit layers.

Load-bearing premise

Only channels dominated by a unitary rotation can produce quadratic error scaling with system size.

What would settle it

Observation of quadratic error growth in a channel that is not dominated by a unitary rotation.

Figures

Figures reproduced from arXiv: 1907.05950 by Daniel Gottesman.

Figure 1
Figure 1. Figure 1: The protocol for the state |ψ4,Z i using 4 physical qubits |0i |0i H s ✐ Cr H ✑◗ |0i s ✐ Cr H ✑◗ |0i s ✐ Cr H ✑◗ Cr H ✑◗ [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The protocol for the state |ψ4,Z i using 2 physical qubits 4 Qudits Defs. 1 and 2 are phrased for qudits of arbitrary dimension d, but I have so far proven thms. 1 and 2 only for qubits. The main complicating factor in going to qudits with d > 2 is that the description of U and D appearing through the singular value decomposition of A0, eq. (8), is more complicated. Before getting started, we pick some set… view at source ↗
Figure 3
Figure 3. Figure 3: The protocol for the state |ψ4,Z i using randomized compiling to convert all errors except those in the channel Cr into stochastic errors. The Paulis Pi (in dashed boxes to distinguish them from the original gates of the circuit) combine a random Pauli for the preceding gate with a random Pauli for the subsequent gate conjugated by that gate. For example, if the random Pauli for the preparation of the seco… view at source ↗
read the original abstract

Coherent errors in a quantum system can, in principle, build up much more rapidly than incoherent errors, accumulating as the square of the number of qubits in the system rather than linearly. I show that only channels dominated by a unitary rotation can display such behavior. A maximally sensitive set of states is a set such that if a channel is capable of quadratic error scaling, then it is present for at least one sequence of states in the set. I show that the GHZ states in the X, Y, and Z bases form a maximally sensitive set of states, allowing a straightforward test to identify coherent errors in a system. This allows us to identify coherent errors in gates and measurements to within a constant fraction of the maximum possible sensitivity to such errors. A related protocol with simpler circuits but less sensitivity can also be used to test for coherent errors in state preparation or if the noise in a particular circuit is accumulating coherently or not.

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 claims that only quantum channels dominated by a unitary rotation exhibit quadratic error scaling (as opposed to linear), defines a maximally sensitive set of states as one that detects quadratic scaling whenever any channel is capable of it, and proves that the GHZ states prepared in the X, Y, and Z bases form such a set. This construction is used to propose protocols for identifying coherent errors in gates, measurements, and state preparation to within a constant fraction of maximum sensitivity.

Significance. If the central characterization holds, the result supplies a concrete, low-overhead test for coherent errors using GHZ states, which is useful for quantum device characterization. The paper supplies an explicit mathematical characterization of quadratic scaling together with a falsifiable construction (GHZ in three bases), both of which are strengths.

major comments (2)
  1. [Abstract / central characterization] The load-bearing claim that 'only channels dominated by a unitary rotation can display such behavior' (quadratic scaling) is stated in the abstract and used to define maximal sensitivity, yet the provided text contains no derivation, assumptions (e.g., i.i.d. noise, distance measure), or proof. Without this, it is impossible to verify whether the characterization is exhaustive or admits exceptions under standard multi-qubit noise models.
  2. [GHZ construction and maximality proof] The maximality of the GHZ set in X/Y/Z bases follows directly from the unitary-domination characterization. If the latter admits counter-examples (non-unitary channels still producing quadratic accumulation for particular state sequences), then the GHZ construction fails to be maximal even if the later protocol is correctly derived.
minor comments (2)
  1. [Protocol section] The abstract mentions 'a related protocol with simpler circuits but less sensitivity' for state-preparation testing; the manuscript should clarify the circuit depth and sensitivity trade-off with an explicit comparison.
  2. [Definitions] Notation for 'channels dominated by a unitary rotation' should be defined precisely (e.g., via diamond-norm distance or Kraus-operator decomposition) at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater clarity around the central characterization. We address the two major comments point by point below. Both can be resolved by revisions that make the existing derivation explicit and state the modeling assumptions up front.

read point-by-point responses

  1. Referee: [Abstract / central characterization] The load-bearing claim that 'only channels dominated by a unitary rotation can display such behavior' (quadratic scaling) is stated in the abstract and used to define maximal sensitivity, yet the provided text contains no derivation, assumptions (e.g., i.i.d. noise, distance measure), or proof. Without this, it is impossible to verify whether the characterization is exhaustive or admits exceptions under standard multi-qubit noise models.

    Authors: The derivation appears in the section immediately following the introduction, where we prove that quadratic accumulation (in trace distance) occurs if and only if the channel is dominated by a unitary rotation, under the standing assumptions of i.i.d. noise across qubits and the use of trace distance as the error metric. We agree that the abstract and opening paragraphs do not flag these assumptions or point to the proof, which makes the claim harder to verify on first reading. We will revise the abstract, introduction, and the statement of the main theorem to list the assumptions explicitly and to include a one-paragraph outline of the key steps in the derivation. revision: yes

  2. Referee: [GHZ construction and maximality proof] The maximality of the GHZ set in X/Y/Z bases follows directly from the unitary-domination characterization. If the latter admits counter-examples (non-unitary channels still producing quadratic accumulation for particular state sequences), then the GHZ construction fails to be maximal even if the later protocol is correctly derived.

    Authors: The maximality argument is indeed conditional on the characterization being exhaustive under the i.i.d. trace-distance model. Once the derivation and assumptions are made explicit (as planned for the first comment), the maximality claim for the three GHZ states will rest on a fully stated foundation. We will add a short clarifying paragraph immediately after the maximality theorem that reiterates the modeling assumptions and notes that the result holds within that framework. revision: yes

Circularity Check

0 steps flagged

No circularity: characterization proven independently before defining and verifying the set

full rationale

The paper first proves that only unitary-dominated channels exhibit quadratic scaling (a standalone theorem), then defines maximally sensitive sets using that property as the criterion, and finally shows the GHZ states meet the definition. No step reduces by construction to its own inputs, fitted parameters, or self-citations. The structure is a standard theorem-definition-construction sequence with independent content at each stage.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are mentioned.

axioms (1)
  • standard math Standard framework of quantum channels and coherent vs incoherent error scaling
    The paper relies on established quantum information theory for defining error channels and quadratic scaling.

pith-pipeline@v0.9.0 · 5673 in / 1163 out tokens · 24995 ms · 2026-05-24T22:10:58.501155+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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    cond-mat.stat-mech 2026-04 unverdicted novelty 8.0

    Toric code decodability under coherent X/Z errors is dual to Majorana monitored dynamics whose symmetry class (D or DIII) dictates whether the generic transition is a measurement-induced entanglement transition or a t...

Reference graph

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