⚛️ quantum physics

No-deleting principle for two unitary copies

This paper extends the no-deleting principle by defining "unitary copies" of a quantum state as $u|\psi\rangle$ and $U|\psi\rangle$ for arbitrary unitary operators $u$ and $U$ , and proves that it is impossible to delete one of these two non-identical copies of an unknown quantum state.

Original authors: Dafa Li

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dafa Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Quantum Copy-Paste" Problem

To understand this paper, we first need to look at how the quantum world differs from our normal, everyday world.

In the Classical World (Your Computer): If you have a digital file, you can copy it perfectly. You can also delete one copy while keeping the other. It's like having two identical PDFs; you can trash one, and the other remains untouched.
In the Quantum World: Things are much stricter.
1. No-Cloning: You cannot make a perfect copy of an unknown quantum state. (Imagine trying to photocopy a secret message that changes every time you look at it; the machine breaks).
2. No-Deleting: Even if you do have two identical copies of a quantum state, you cannot delete one of them while keeping the other.

This paper focuses on the second rule: The No-Deleting Principle.

The New Twist: "Unitary Copies"

In previous research, scientists showed you couldn't delete one of two identical copies (like two exact photocopies).

In this paper, the author, Dafa Li, asks a new question: What if the copies aren't exactly identical, but are "transformed" versions of the original?

He introduces the idea of "Unitary Copies."

The Analogy: Imagine you have a secret recipe (the quantum state $|\psi\rangle$ $∣ ψ ⟩$ ).
- Copy A is the recipe written in English.
- Copy B is the same recipe, but translated into French (or perhaps written in a different font, or rotated on the page).
- In the quantum world, these translations are done by "Unitary Operators" (let's call them $U$ and $U'$ ). They change the look or "orientation" of the state, but they don't destroy the information.

Li defines these two transformed versions ( $U|\psi\rangle$ and $U'|\psi\rangle$ ) as "two unitary copies."

The Experiment: Can We Delete One?

The paper tries to build a "Quantum Deletion Machine."

The Goal: Feed the machine two copies of a secret (one original, one transformed). The machine should delete the second copy, turning it into a blank, standard "empty" state, while leaving the first copy alone.
The Setup:
- Input: Copy 1 (Transformed) + Copy 2 (Transformed) + A blank "trash can" (Ancilla).
- Desired Output: Copy 1 (Transformed) + Blank State + Trash Can (now holding the deleted info).

The Result: The Machine Fails

Li proves mathematically that this machine cannot work.

Here is the breakdown of why, using a simple metaphor:

Imagine you have two photos of a cat.

Photo A is the original.
Photo B is the original, but rotated 90 degrees.

You want to use a magic eraser to delete Photo B and turn it into a blank white square, while keeping Photo A exactly as is.

Li shows that because quantum mechanics follows strict rules of linearity (like how mixing paints works: if you mix red and blue to get purple, the machine must handle the "purple" input by mixing the results of the "red" and "blue" inputs), the math breaks down.

If the machine works for a "Red Cat" (State A) and a "Blue Cat" (State B), it must also work for a "Purple Cat" (a mix of both).
When Li runs the math for a "Purple Cat," the machine fails to produce a clean "Blank Square." Instead, the information from the deleted copy leaks into the "trash can" in a way that depends on the original state.
The Conclusion: The machine doesn't actually delete the information; it just moves it. The "deleted" copy isn't gone; it has been transferred to the trash can (the ancilla), but the original copy remains.

The "Magic Mirror" Connection

The paper also explores a relationship between the old "Standard Deletion Machine" (which deletes identical copies) and this new "Unitary Deletion Machine."

The Finding: You can turn a Standard Machine into a Unitary Machine (and vice versa) by adding a "mirror" step.
The Metaphor: Imagine you have a machine that deletes a standard document. If you put a mirror in front of the input (to change the language) and another mirror in front of the output (to change it back), you create a machine that deletes a "translated" document.
The Catch: Even with these mirrors, the machine still doesn't delete. It just moves the information from the document to the trash can. The "deletion" is an illusion; the information is conserved, just relocated.

Summary

The Claim: You cannot delete one of two "transformed" (unitary) copies of an unknown quantum state.
The Reason: Quantum laws (specifically linearity) force the information to be preserved. If you try to delete it, the machine simply shuffles the information into a third container (the ancilla) rather than erasing it.
The Takeaway: In the quantum world, information is indestructible. You can hide it, move it, or transform it, but you cannot simply make it disappear while keeping a copy. This holds true even if the copies look different (are "unitary copies") from each other.

Note: The paper is purely theoretical mathematics. It does not discuss building real-world computers, medical applications, or future technologies. It simply proves that a specific type of quantum operation is impossible.

Technical Summary: No-Deleting Principle for Two Unitary Copies

Problem Statement
The paper addresses a gap in the theoretical framework of quantum information regarding the "no-deleting" principle. While the quantum no-cloning theorem (Wootters and Zurek, 1982) and the quantum no-deleting theorem (Pati and Braunstein, 2000) are well-established for identical copies of an unknown quantum state, the scenario involving non-identical copies had not been formally defined or analyzed. Specifically, the paper investigates whether it is possible to delete one of two "unitary copies" of an unknown state $|\psi\rangle$ . The authors define two unitary copies as $U|\psi\rangle$ and $U'|\psi\rangle$ , where $U$ and $U'$ are arbitrary unitary operators, extending the concept beyond the standard case where $U = U' = I$ (the identity).

Methodology
The authors employ a rigorous mathematical approach based on the linearity of quantum mechanics and unitary evolution to test the feasibility of a generalized deleting machine.

Definition of the Machine: The paper proposes a "deleting machine for two unitary copies," defined by the transformation:
$T(U|\psi\rangle_A U'|\psi\rangle_B |A\rangle_C) = U|\psi\rangle_A |\Sigma\rangle_B U'|A\psi\rangle_C$
Here, $|A\rangle_C$ is the initial ancilla, $|\Sigma\rangle_B$ is a standard state intended to replace the deleted copy, and $U'|A\psi\rangle_C$ is the final state of the ancilla, which may depend on $|\psi\rangle$ .
Linearity Analysis: To test the validity of this machine, the authors apply it to an arbitrary unknown state $|\psi\rangle = \alpha|H\rangle + \beta|V\rangle$ , where $|H\rangle$ and $|V\rangle$ are basis states. By expanding the input state and applying the linearity of the operator $T$ , they derive the Left-Hand Side (LHS) of the transformation.
Polynomial Consistency Check: The authors compare the LHS (derived via linearity) with the Right-Hand Side (RHS) (derived from the target definition). They analyze the resulting expressions as homogeneous quadratic polynomials in the complex coefficients $\alpha$ and $\beta$ .
Orthogonality Constraints: By equating the coefficients of the polynomials, the authors deduce necessary conditions for the ancilla states $|A_H\rangle_C$ and $|A_V\rangle_C$ . They demonstrate that for the transformation to hold for all $\alpha$ and $\beta$ , these ancilla states must be orthonormal.

Key Contributions and Results

Formal Definition: The paper provides the first formal definition of "unitary copies" ( $U|\psi\rangle$ and $U'|\psi\rangle$ ) and the corresponding deleting machine for such non-identical copies.
Proof of Impossibility: The central result is the proof that the proposed deleting machine cannot function as intended. The analysis reveals that the transformation does not actually delete the unitary copy $U'|\psi\rangle_B$ and replace it with a standard state $|\Sigma\rangle$ . Instead, the mathematical derivation shows that the term $U'|\psi\rangle_B$ is effectively moved to the ancilla system.
Equivalence to Standard Case: The paper demonstrates that the impossibility of deleting unitary copies is consistent with the standard no-deleting theorem.
- In Section 3.1, the authors show that a standard deleting machine can be used to construct a machine for unitary copies, but the result is merely a transfer of the state to the ancilla, not a deletion.
- In Section 3.2, they show the reverse: a machine for unitary copies can be used to construct a standard deleting machine, which similarly fails to delete the state, instead moving it to the ancilla.

Significance and Claims
The paper claims to extend the fundamental limits of quantum information manipulation. By defining unitary copies and proving that the no-deleting principle holds for them, the authors reinforce the robustness of the no-deleting theorem against variations in the input state's representation (i.e., unitary transformations).

The significance lies in clarifying that the inability to delete a quantum state is not limited to identical copies but extends to any pair of unitary copies of an unknown state. The paper concludes that the "deleting" process in this generalized context is physically indistinguishable from moving the information to an ancillary system, thereby confirming that true deletion of one copy of an unknown quantum state remains impossible, even when the copies are related by arbitrary unitary operators. The work relies on theoretical derivation and does not propose experimental implementations or specific applications beyond the foundational understanding of quantum information constraints.