Impossible to conjugate or reflect an unknown quantum state via a unitary evolution

arXiv:2609.11536 · quant-ph · Submitted 2026-09-10 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Impossible to conjugate or reflect an unknown quantum state via a unitary evolution".

Mira: Here's the gist of the paper. The gist It is impossible to conjugate an unknown arbitrary quantum state via a universal physical device or linear/unitary operation.

Kai: This paper proposes the no-conjugating theorem, which states that it is impossible to conjugate an unknown arbitrary quantum state via a universal unitary operator or linear/unitary operation The authors situate their work within the context of established theorems such as the no-cloning theorem, no-deleting theorem, no-hiding theorem, no-flip theorem, and no-broadcasting.

Mira: These prior theorems form the foundation for understanding limitations in quantum information processing.

Kai: Mathematical Formulation of Conjugation The paper mathematically investigates the existence of a universally unitary operator U that can transform an arbitrary unknown state ψ⟩ = α0⟩ + β1⟩ into a state ψ′⟩ = p0⟩ + q1⟩, where p and q are constrained by the amplitudes α and β.

Paper summary: Kai: So wrapping up the paper "Impossible to conjugate or reflect an unknown quantum state via a unitary evolution," we see that the main message is quite definitive. It’s about establishing a hard limit on what universal unitary operations can do with an unknown quantum state.

Mira: The authors prove it by showing that for any arbitrary input psi> = alpha0> + beta1>, there are specific target states, like the original or its bit-flipped version, that are reachable via a universal unitary operator U two <ref:2609.11536#pg2>.

Lev: The title itself says it all: "Impossible to conjugate or reflect an unknown quantum state via a unitary evolution," and it confirms this impossibility through algebraic methods derived from the general form of the two-by-two unitary operator U two <ref:2609.11536#pg1,an unknown quantum state via a unitary evolution>.

Kai: For someone just listening, the big picture is that you can't have a universal tool that lets you transform any unknown qubit into any other arbitrary configuration using only standard unitary evolution.

Mira: This limitation impacts how we think about quantum gates and how we design systems for quantum cryptography because it shows that perfect state manipulation isn't always possible with a single universal operation two <ref:2609.11536#pg1>.

Conclusion: Kai: So we're wrapping up this look at the paper "Impossible to conjugate or reflect an unknown quantum state via a unitary evolution."

Mira: It basically boils down to showing that you can't use a standard, universal unitary operation U to take any random qubit state and turn it into another arbitrary target state.

Lev: From my side, the real question is how much overhead this imposes on actual hardware trying to implement these operations in a real circuit.

Kai: Yeah, so if you're building something with superconducting qubits that need to perform some kind of state transformation—a conjugation or reflection—you can’t just rely on any universal unitary gate to do it perfectly.

Mira: Exactly, the authors are showing that this isn't just a limitation in theory; it comes from the fundamental math of how those states interact.

Lev: And for error correction, this means you can’t assume some universal operation can clean up or manipulate an unknown corrupted state without violating these constraints.

Kai: So what does this mean practically for someone just listening? It limits the kinds of universal quantum gates we design and use in our experiments.

Mira: It explains why certain kinds of perfect state manipulation you might be hoping for with a single tool just aren't possible under those exact mathematical rules.

Lev: It’s a constraint on the toolbox, meaning any system designed to handle unknown states will have specific limitations baked into its design from the start.

Kai: Right, so it sets a boundary on what we can build and what we can expect from universal quantum operations.

Mira: And that boundary is defined by these no-go theorems, which are just more formal ways of saying the transformation you want isn't physically possible with that kind of evolution.

Tsinghua University

quant-ph

Submitted: 2026-09-10

Updated: 2026-10-21

Comments: 9 pages, no figure

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist It is impossible to conjugate an unknown arbitrary quantum state via a universal physical device or linear/unitary operation.

Key concepts

No-Cloning Theorem: This theorem states that it is impossible to create an exact copy of an unknown quantum state. It is a cornerstone of quantum information, meaning you cannot perfectly duplicate an arbitrary qubit without losing information or violating physical laws.
Conjugation via Unitary Evolution: This refers to trying to transform any unknown quantum state |ψ⟩ into another specific state |ψ′⟩ using only a universal unitary operator (a reversible quantum gate). The paper demonstrates that for an arbitrary input state, this transformation is not universally possible.
No-Go Theorems: These are established mathematical proofs showing that certain operations are fundamentally impossible in quantum mechanics. The paper derives new no-go theorems concerning conjugation, showing that universal unitary evolution cannot perform the desired state transformations for all unknown inputs.

Summary

The gist

It is impossible to conjugate an unknown arbitrary quantum state via a universal physical device or linear/unitary operation.

Introduction and Context

This paper proposes the no-conjugating theorem, which states that it is impossible to conjugate an unknown arbitrary quantum state via a universal unitary operator or linear/unitary operation The authors situate their work within the context of established theorems such as the no-cloning theorem, no-deleting theorem, no-hiding theorem, no-flip theorem, and no-broadcasting. These prior theorems form the foundation for understanding limitations in quantum information processing.

Mathematical Formulation of Conjugation

The paper mathematically investigates the existence of a universally unitary operator U that can transform an arbitrary unknown state ψ⟩ = α0⟩ + β1⟩ into a state ψ′⟩ = p0⟩ + q1⟩, where p and q are constrained by the amplitudes α and β. The general form of a 2 by 2 unitary operator is given as U = u v −e iθv∗e iθu∗. By expanding the transformation equation, the authors derive expressions for u and v in terms of α, β, p, and q.

Derivation of No-Go Theorems

The algebraic solution yields six specific no-go theorems based on the possible forms of ψ′⟩. Theorem 1 states that for an arbitrary unknown state ψ⟩ = α0⟩+β1⟩, there exists a universally unitary operator U such that Uψ⟩ = ψ′ >, where ψ′ > is of the form in Eq. (3), if and only if ψ′ > = α0⟩ + β1⟩ or β0⟩ + α1⟩. Theorem 2 asserts that there is no universally unitary operator U such that Uψ⟩ = ψ′ >, where ψ′ > is one of Φi⟩, i = 1, 2, to 6 in Table 2. These six no-go theorems are summarized in Table 3.

Implications for Quantum Information

The findings of the no-conjugating theorem have significant implications for quantum technology and security.

(1). No conjugate gate for quantum circuit.

This theorem establishes a fundamental limitation on what quantum gates can do, specifically stating that A gate cannot universally conjugate an arbitrary unknown qubit.

(2). Quantum cryptography.

The no-conjugating theorem helps explain why an eavesdropper cannot simply take an unknown quantum state and convert it into its conjugate to obtain information or distinguish quantum states perfectly.

Related No-Go Theorems

The paper also derives related results, such as Corollary 1, which shows that for an arbitrary unknown state ψ⟩ = α0⟩ + β1⟩, there is a universally unitary operator U such that Uψ⟩ = V ψ′ >, where V is any universally unitary operator and ψ′ > is one of Ψ1⟩ and Ψ2⟩ in Table 1. Corollary 2 extends this to show that there is no universally unitary operator U such that Uψ⟩ = V ψ′ >, where V is any universally unitary operator and ψ′ > is one of Φi⟩, i = 1, to 6 in Table 2. These results lead to the general no-conjugating theorems from specific states like α∗e iθ0⟩ + β∗e iω1⟩ and β∗e iθ0⟩ + α∗e iω1〉. The paper also obtains a different proof of the no-flip theorem by considering ψ ⊥⟩, which is called the flip state of ψ⟩.

Conclusion

The no-conjugating theorem provides a definitive statement regarding the impossibility of universal conjugation via unitary evolution. This limitation impacts the design of quantum gates and offers insights into quantum cryptography.

Statements and declarations: No financial interests, no competing interests, no financial supports.

Data Availability Statement: No Data associated in the manuscript.

References

[1] W. K. Wootters and W. H. Zurek., A Single quantum cannot be cloned

[2] Yuan Yao, Phase spaces that cannot be cloned in classical mechanics, J. Math. Phys 64, 102901 (2023)

[3] Koji Yamaguchi and Achim Kempf, Encrypted Qubits can be Cloned, Phys. Rev. Lett. 136, 010801 (2026)

[4] A.K. Pati and S.L. Braunstein, Impossibility of deleting an unknown quantum state

[5] D. Qiu, Some analogies between quantum cloning and quantum deleting, Phys. Rev. A 65, 052303 (2002)

[6] S. Sazim et al., Complementarity of Quantum Correlations in Cloning and Deleting of Quantum State, Phys. Rev. A 91, 062311 (2015)

[7] I. Chakrabarty et al., Deletion, Bell’s Inequality, Teleportation, Quant. Inf. Process, 10, 27 (2011)

[8] Y. Chen et al., Quantum deleting and cloning in a pseudo-unitary system, Front. Phys 16, 53601 (2021)

[9] Braunstein, Samuel L. and Pati, Arun K., Quantum Information Cannot Be Completely Hidden in Correlations: Implications for the Black-Hole Information Paradox, Phys. Rev. Lett. 98 (8), 080502 (2007)

[10] Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin. Noncommuting Mixed States Cannot Be Broadcast. Phys. Rev. Lett 76 (15), 2818–2821 (1996). arXiv:quantph/9511010

[11] Matthew Girling, Cristina Cˆırstoiu, and David Jennings, A simple formulation of no-cloning and no-hiding that admits efficient and robust verification, Phys. Rev. Research 6, 023090 (2024)

[12] Buˇzek, M. Hillery, and R.F. Werner, Optimal Manipulations with Qubits: Universal NOT Gate, Physical Review A 60, R2626–R2629 (1999), e-print

[13] V. Buˇzek, M. Hillery, and R. F. Werner, Universal-NOT gate, Journal of Modern Optics 47, 211–232 (2000)

[14] A. K. Pati and B. C. Sanders, No partial erasure of quantum information, Physics Letters A 359, 31-36 (2006)

[15] Dafa Li, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, CHINA email: lidafa@tsinghua.edu.cn Corresponding author:

Improvements for AI systems

No-conjugating quantum circuit design: AI systems can be constrained to only implement unitary operations that cannot conjugate an arbitrary unknown qubit, preventing universal state transformation via a single gate.
Quantum cryptography enhancement: AI can exploit the no-conjugating theorem to ensure eavesdroppers cannot perfectly distinguish or convert an unknown quantum state into its conjugate for information gain.
No-flip theorem verification: AI can verify the impossibility of flipping an arbitrary unknown quantum state by testing if any universal unitary operator can achieve this transformation, using proofs derived from the paper's results.

Abstract

The no-cloning theorem, the no-deleting theorem, and other no-go theorems play a crucial role in quantum mechanics and quantum information. In this paper, we show that it is impossible to conjugate or reflect an unknown quantum state or conjugate or reflect the bit-flip (the phase-flip) of an unknown state via a universally unitary evolution. We also indicate that the partial conjugating operations are not anti-unitary and it is impossible to universally partially conjugate an unknown state via a unitary evolution. We compute the fidelities of the optimal physical approximations to conjugating and conjugating the bit-flip and phase-flip operations.

Sources

Noncommuting mixed states cannot be broadcast