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The sum of entanglement and subsystem coherence is invariant under quantum reference frame transformations

Carlo Cepollaro, Ali Akil, Paweł Cieśliński, Anne-Catherine de la Hamette, Časlav Brukner

TL;DR

This work investigates how quantum reference frame (QRF) changes affect coherence and entanglement, showing that for ideal QRFs there exists an invariant decomposition of these resources. By defining ideal QRF transformations $S^{(C)\to (A)}=\int_G dg\, |g^{-1}\rangle_{CA}\langle g| \otimes U^\dagger(g)$, the authors prove two exact conservation laws for selected quantifier pairs: $\mathcal{C}_e^{(C)}+\mathcal{E}_e^{(C)}=\mathcal{C}_e^{(A)}+\mathcal{E}_e^{(A)}$ and $\mathcal{C}_{l^2}^{(C)}+\mathcal{E}_l^{(C)}=\mathcal{C}_{l^2}^{(A)}+\mathcal{E}_l^{(A)}$, implying a trade-off where increasing entanglement reduces subsystem coherence and vice versa. A simple three-qubit example illustrates this exchange, while a broader analysis shows a weaker trade-off holds for any pair of measures when a frame with zero coherence or zero entanglement exists. The paper also discusses Bell-test violations under QRFs, showing that nonlocality persists across frames via redistribution between states and measurements. These results illuminate the quantum information structure of QRFs and point toward future work on multipartite systems and quantum gravity, where diffeomorphism-like transformations may reveal deeper invariants.

Abstract

Recent work on quantum reference frames (QRFs) has demonstrated that superposition and entanglement are properties that change under QRF transformations. Given their utility in quantum information processing, it is important to understand how a mere change of perspective can produce or reduce these resources. Here we find a trade-off between entanglement and subsystem coherence under a QRF transformation, in the form of a conservation theorem for their sum, for two pairs of measures. Moreover, we find a weaker trade-off for any possible pair of measures. Finally, we discuss the implications of this interplay for violations of Bell's inequalities, clarifying that for any choice of QRF, there is a quantum resource responsible for the violation. These findings contribute to a better understanding of the quantum information theoretic aspects of QRFs, offering a foundation for future exploration in both quantum theory and quantum gravity.

The sum of entanglement and subsystem coherence is invariant under quantum reference frame transformations

TL;DR

This work investigates how quantum reference frame (QRF) changes affect coherence and entanglement, showing that for ideal QRFs there exists an invariant decomposition of these resources. By defining ideal QRF transformations , the authors prove two exact conservation laws for selected quantifier pairs: and , implying a trade-off where increasing entanglement reduces subsystem coherence and vice versa. A simple three-qubit example illustrates this exchange, while a broader analysis shows a weaker trade-off holds for any pair of measures when a frame with zero coherence or zero entanglement exists. The paper also discusses Bell-test violations under QRFs, showing that nonlocality persists across frames via redistribution between states and measurements. These results illuminate the quantum information structure of QRFs and point toward future work on multipartite systems and quantum gravity, where diffeomorphism-like transformations may reveal deeper invariants.

Abstract

Recent work on quantum reference frames (QRFs) has demonstrated that superposition and entanglement are properties that change under QRF transformations. Given their utility in quantum information processing, it is important to understand how a mere change of perspective can produce or reduce these resources. Here we find a trade-off between entanglement and subsystem coherence under a QRF transformation, in the form of a conservation theorem for their sum, for two pairs of measures. Moreover, we find a weaker trade-off for any possible pair of measures. Finally, we discuss the implications of this interplay for violations of Bell's inequalities, clarifying that for any choice of QRF, there is a quantum resource responsible for the violation. These findings contribute to a better understanding of the quantum information theoretic aspects of QRFs, offering a foundation for future exploration in both quantum theory and quantum gravity.
Paper Structure (16 sections, 2 theorems, 65 equations, 1 figure)

This paper contains 16 sections, 2 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A simple example
  3. Main results
  4. Bell tests in QRFs
  5. Conclusion
  6. QRF transformations are incoherent operations
  7. Conservation of diagonal elements up to permutation
  8. Continuous case
  9. Discrete case
  10. Proofs of theorems
  11. Proof of Theorem 1: Entanglement entropy and coherence entropy
  12. Proof of Theorem 1: Linear entropy and $l^2$ coherence
  13. Uniqueness proof
  14. Counterexample for a generic inequality
  15. Proof of Theorem 2
  16. ...and 1 more sections

Key Result

Theorem 1

The sum of entanglement and subsystem coherence $\mathcal{E}+\mathcal{C}$ is invariant under QRF transformations of pure bipartite states, for $(\mathcal{E},\mathcal{C})=(\mathcal{E}_e,\mathcal{C}_e)$ or $(\mathcal{E}_l,\mathcal{C}_{l^2})$.

Figures (1)

  • Figure 1: Three qubits $A$, $B$, and $C$ are described relative to one another. On the left, systems $A$ and $B$ are described relative to system $C$. While $C$ is in the state ${|{\uparrow}\rangle}$ relative to itself, $A$ is in a coherent superposition of ${|{\uparrow}\rangle}$ and ${|{\downarrow}\rangle}$ and $B$ is in the state ${|{\uparrow}\rangle}$. Coherently changing reference frame to that of $A$, we find that $B$ and $C$ are in an entangled state. In general, superposition and entanglement depend on the QRF.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2