No-cloning with unitary scaling

Dafa Li

arxiv: 2604.22960 · v2 · pith:R77H3U5Ynew · submitted 2026-04-24 · 🪐 quant-ph

No-cloning with unitary scaling

Dafa Li This is my paper

Pith reviewed 2026-05-19 16:31 UTC · model grok-4.3

classification 🪐 quant-ph

keywords no-cloning theoremunitary evolutionquantum pure statesquantum informationU-copy

0 comments

The pith

It is impossible to produce a U-copy of an arbitrary unknown quantum pure state using unitary evolution.

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

The paper extends the 1982 no-cloning theorem. Rather than requiring an exact duplicate, it asks whether a fixed unitary operator U can be applied in advance to yield the transformed state U|ψ> for every possible unknown input |ψ>. The authors prove that no unitary evolution acting on the unknown state together with an ancillary system can accomplish this task. A reader would care because the result shows quantum information stays protected against any predetermined linear transformation when the state itself remains unknown.

Core claim

We call U |ψ>, where U is any unitary operator, a U-copy of the state |ψ> and show it is impossible to make a U-copy of an arbitrary unknown quantum pure state by using unitary evolution.

What carries the argument

The U-copy, defined as the state obtained by applying a fixed unitary operator U independent of |ψ> to the unknown pure state |ψ>.

If this is right

The impossibility holds for every choice of the fixed unitary U.
The original no-cloning theorem follows immediately by taking U to be the identity.
No predetermined linear transformation of an unknown pure state can be realized by unitary evolution alone.

Where Pith is reading between the lines

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

The result may reinforce security proofs in quantum cryptography that assume unknown states cannot be manipulated predictably.
Approximate or probabilistic versions of the U-copy task could be examined as a natural next question.
Similar no-go statements might apply to other operations such as adding a fixed phase or rotating an unknown qubit.

Load-bearing premise

The unitary operator U is chosen once and for all without depending on the unknown state |ψ>.

What would settle it

An explicit unitary operator on an enlarged Hilbert space that maps |ψ> tensored with a fixed ancilla state to U|ψ> tensored with some output state for every possible |ψ>.

read the original abstract

It is known that the classical information like strings of bits can be copied. In 1982, Wootters and Zurek proposed the quantum no-cloning principle. No-cloning principle says that it is impossible to make an identical copy of an arbitrary unknown quantum pure state by using unitary evolution. In this paper, we call U $|\psi>$, where U is any unitary operator, a U-copy of the state $|{\psi}>$ and show it is impossible to make a U-copy of an arbitrary unknown quantum pure state by using unitary evolution.

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.

Desk Editor's Note private letter to a colleague

The paper tries to generalize no-cloning to U-copies but the central claim fails because applying U itself produces U|ψ> for any unknown state.

read the letter

The key point is that this paper claims an impossibility result for producing U|ψ> from an unknown |ψ> via unitaries, but the claim doesn't hold because you can just use U as the evolution operator on the system plus ancilla. The direct construction works for every state without needing to know |ψ> in advance. This undercuts the no-go result as stated in the abstract. The paper does introduce the concept of a U-copy for any unitary U and positions it as an extension of the standard no-cloning theorem. It correctly references the 1982 Wootters-Zurek result and tries to generalize the idea that you can't copy unknown quantum states in certain ways. What it does reasonably is keep the focus on unitary evolution and unknown pure states, which aligns with the original theorem's spirit. The definition is straightforward and the intent to broaden the no-go seems honest. However, the soft spot is significant. The stress-test concern applies here: tensoring U with identity on an ancilla gives exactly the U-copy without any state-dependent operations or knowledge of |ψ>. The paper's abstract doesn't specify additional constraints like preserving the original state untouched while creating the scaled version separately. This leaves the impossibility unproven as stated and suggests the argument may not add a real new limit beyond what's already known. The math and setup appear to have this gap, and without a detailed proof that addresses why the direct U doesn't work, it's hard to see the result standing. This paper would mainly appeal to specialists in quantum foundations looking for small extensions of no-cloning ideas. A reader familiar with the original theorem might find the variant interesting in principle but would likely spot the issue quickly. I recommend against sending it for peer review until the authors fix the definition or add the missing conditions that would make the claim nontrivial.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to generalize the quantum no-cloning theorem. It defines a U-copy of an arbitrary unknown pure state |ψ⟩ as the state U|ψ⟩ for any fixed unitary operator U and asserts that it is impossible to produce such a U-copy using unitary evolution on the composite system.

Significance. If the claimed impossibility result were correct, it would represent a broad extension of the no-cloning principle to arbitrary unitary transformations of unknown states. The result as stated does not hold, however, because the definition permits a direct construction via unitary evolution.

major comments (1)

[Abstract] Abstract: the claimed impossibility is contradicted by the explicit construction V = U ⊗ I. Applying V to the initial state |ψ⟩|0⟩ produces U|ψ⟩|0⟩, which contains the U-copy in the first register for every unknown |ψ⟩. This uses only unitary evolution, requires no state-dependent operations, and matches the definition of a U-copy given in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for identifying the flaw in our claimed result. We agree that the impossibility statement as presented does not hold, and we will revise the manuscript to correct this error.

read point-by-point responses

Referee: [Abstract] Abstract: the claimed impossibility is contradicted by the explicit construction V = U ⊗ I. Applying V to the initial state |ψ⟩|0⟩ produces U|ψ⟩|0⟩, which contains the U-copy in the first register for every unknown |ψ⟩. This uses only unitary evolution, requires no state-dependent operations, and matches the definition of a U-copy given in the abstract.

Authors: We acknowledge the validity of this observation. The operator V = U ⊗ I is indeed a fixed unitary on the composite system. When applied to |ψ⟩|0⟩ it produces U|ψ⟩|0⟩, so that the first register holds exactly the state U|ψ⟩ for any unknown |ψ⟩. This construction satisfies the definition of a U-copy given in the manuscript and uses only unitary evolution. Consequently the impossibility claim in the abstract (and throughout the paper) is incorrect as stated. We will revise the manuscript to remove the erroneous claim. If a related but distinct result was intended—for example, the impossibility of simultaneously retaining |ψ⟩ while producing U|ψ⟩ in an ancillary register—we will reformulate and prove that statement instead. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard no-go argument with independent content

full rationale

The paper defines a U-copy as U|ψ> for fixed unitary U and asserts impossibility under unitary evolution on the composite system. This is a direct generalization of the Wootters-Zurek no-cloning theorem using the standard postulates of quantum mechanics. No parameters are fitted, no self-citations form the load-bearing step, and the central claim does not reduce to a tautology or renaming of its own inputs. The derivation remains self-contained against external benchmarks such as the original no-cloning proof and the definition of unitary operators.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard assumptions of quantum mechanics for closed systems and the definition of U-copy introduced in the paper.

axioms (2)

standard math Quantum evolution of closed systems is unitary
Invoked implicitly when stating that copying must be done by unitary evolution, as in the abstract's reference to the 1982 no-cloning principle.
domain assumption The input state is an arbitrary unknown pure quantum state
Stated directly in the abstract as the target of the impossibility result.

pith-pipeline@v0.9.0 · 5599 in / 1254 out tokens · 46972 ms · 2026-05-19T16:31:54.776867+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Assume that no-cloning with unitary scaling machine works for pure states |ψ⟩ and |ϕ⟩. Then ... ⟨ψ|ϕ⟩=(⟨ψ|ϕ⟩)²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages · 1 internal anchor

[1]

W. K. Wootters and W. H. Zurek. A Single quantum cannot be cloned. Nature 299, 802-803 (1982)

work page 1982
[2]

Nielsen, Chuang, I.L.:Quantum Computation and Quantum Informa- tion(Cambridge Univ

M.A. Nielsen, Chuang, I.L.:Quantum Computation and Quantum Informa- tion(Cambridge Univ. Press, Cambridge, 2000)

work page 2000
[3]

LU-Ming Duan and Guang-Can Guo, A probabilistic cloning machine for replicating two non-orthogonal states, Phys. Lett. A. 243, 261-264 (1998). e-print: arxiv. quant-ph/9704020

work page internal anchor Pith review Pith/arXiv arXiv 1998
[4]

Bennett et al., Phys.Rev.A

C.H. Bennett et al., Phys.Rev.A. 54, 3824-3851,1996

work page 1996
[5]

: A. R. Calderbank, Peter W. Shor, Phys. Rev. A, Vol. 54, No. 2, pp. 1098- 1106, 1996

work page 1996
[6]

Physical Review Letters

Braunstein, Samuel L.; Pati, Arun K., Quantum Information Cannot Be Completely Hidden in Correlations: Implications for the Black-Hole Infor- mation Paradox”. Physical Review Letters. 98 (8) 080502 (2007)

work page 2007
[7]

Noncommuting Mixed States Cannot Be Broad- cast

Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin. Noncommuting Mixed States Cannot Be Broad- cast. Physical Review Letters. 76 (15): 2818–2821 (1996). arXiv:quant- ph/9511010

work page arXiv 1996
[8]

Pati and S.L

A.K. Pati and S.L. Braunstein; Impossibility of deleting an unknown quan- tum state, Nature 404 (2000) p.164. 4

work page 2000

[1] [1]

W. K. Wootters and W. H. Zurek. A Single quantum cannot be cloned. Nature 299, 802-803 (1982)

work page 1982

[2] [2]

Nielsen, Chuang, I.L.:Quantum Computation and Quantum Informa- tion(Cambridge Univ

M.A. Nielsen, Chuang, I.L.:Quantum Computation and Quantum Informa- tion(Cambridge Univ. Press, Cambridge, 2000)

work page 2000

[3] [3]

LU-Ming Duan and Guang-Can Guo, A probabilistic cloning machine for replicating two non-orthogonal states, Phys. Lett. A. 243, 261-264 (1998). e-print: arxiv. quant-ph/9704020

work page internal anchor Pith review Pith/arXiv arXiv 1998

[4] [4]

Bennett et al., Phys.Rev.A

C.H. Bennett et al., Phys.Rev.A. 54, 3824-3851,1996

work page 1996

[5] [5]

: A. R. Calderbank, Peter W. Shor, Phys. Rev. A, Vol. 54, No. 2, pp. 1098- 1106, 1996

work page 1996

[6] [6]

Physical Review Letters

Braunstein, Samuel L.; Pati, Arun K., Quantum Information Cannot Be Completely Hidden in Correlations: Implications for the Black-Hole Infor- mation Paradox”. Physical Review Letters. 98 (8) 080502 (2007)

work page 2007

[7] [7]

Noncommuting Mixed States Cannot Be Broad- cast

Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin. Noncommuting Mixed States Cannot Be Broad- cast. Physical Review Letters. 76 (15): 2818–2821 (1996). arXiv:quant- ph/9511010

work page arXiv 1996

[8] [8]

Pati and S.L

A.K. Pati and S.L. Braunstein; Impossibility of deleting an unknown quan- tum state, Nature 404 (2000) p.164. 4

work page 2000