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Quantum geometric tensor determines the pure-state i.i.d. conversion rate in the resource theory of asymmetry for any compact Lie group

Koji Yamaguchi, Yosuke Mitsuhashi, Tomohiro Shitara, Hiroyasu Tajima

Abstract

Quantifying physical concepts in terms of the ultimate performance of a given task has been central to theoretical progress, as illustrated by thermodynamic entropy and entanglement entropy, which respectively quantify irreversibility and quantum correlations. Symmetry breaking is equally universal, yet lacks such an operational quantification. While an operational characterization of symmetry breaking through asymptotic state-conversion efficiency is a central goal of the resource theory of asymmetry (RTA), such a characterization has so far been completed only for the $U(1)$ group among continuous symmetries. Here, we identify the complete measure of symmetry breaking for a general continuous symmetry described by any compact Lie group. Specifically, we show that the asymptotic conversion rate between many copies of pure states in RTA is determined by the quantum geometric tensor, thereby establishing it as the complete measure of symmetry breaking. As an immediate consequence of our conversion rate formula, we also resolve the Marvian-Spekkens conjecture on conditions for reversible conversion in RTA, which has remained unproven for over a decade. Leveraging the connection between symmetry breaking and the theory of quantum reference frames, we also systematically introduce a standardized reference state for frameness based on our asymptotic conversion theory. In addition, by applying our analysis to a standard quantum-thermodynamic scenario, we show that asymptotic state conversion in contact with heat baths generally requires macroscopic coherence in the thermodynamic limit.

Quantum geometric tensor determines the pure-state i.i.d. conversion rate in the resource theory of asymmetry for any compact Lie group

Abstract

Quantifying physical concepts in terms of the ultimate performance of a given task has been central to theoretical progress, as illustrated by thermodynamic entropy and entanglement entropy, which respectively quantify irreversibility and quantum correlations. Symmetry breaking is equally universal, yet lacks such an operational quantification. While an operational characterization of symmetry breaking through asymptotic state-conversion efficiency is a central goal of the resource theory of asymmetry (RTA), such a characterization has so far been completed only for the group among continuous symmetries. Here, we identify the complete measure of symmetry breaking for a general continuous symmetry described by any compact Lie group. Specifically, we show that the asymptotic conversion rate between many copies of pure states in RTA is determined by the quantum geometric tensor, thereby establishing it as the complete measure of symmetry breaking. As an immediate consequence of our conversion rate formula, we also resolve the Marvian-Spekkens conjecture on conditions for reversible conversion in RTA, which has remained unproven for over a decade. Leveraging the connection between symmetry breaking and the theory of quantum reference frames, we also systematically introduce a standardized reference state for frameness based on our asymptotic conversion theory. In addition, by applying our analysis to a standard quantum-thermodynamic scenario, we show that asymptotic state conversion in contact with heat baths generally requires macroscopic coherence in the thermodynamic limit.

Paper Structure

This paper contains 58 sections, 49 theorems, 427 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Resource theory of asymmetry
  4. Quantum geometric tensor
  5. Definition and basic properties
  6. QGT in the RTA
  7. Main results
  8. U(1) group
  9. Reversible asymptotic conversion: Proof of the Marvian-Spekkens conjecture
  10. Irreversibility in asymptotic conversion
  11. Finite groups
  12. Proof sketch of Theorem \ref{['thm:conversion_rate_projective_finite_number']}
  13. QGT as asymmetry monotone
  14. Optimality
  15. Mixed-state conversion
  16. ...and 43 more sections

Key Result

Theorem 1

Let $U$ and $U'$ be projective unitary representations of a compact Lie group $G$ on finite-dimensional Hilbert spaces $\mathcal{H}$ and $\mathcal{H}'$. The conversion rate from a pure state $\psi\in\mathcal{P}(\mathcal{H})$ to another pure state $\phi\in\mathcal{P}(\mathcal{H}')$ is given by if $\mathrm{Sym}_{G}(\psi)\subset \mathrm{Sym}_G(\phi)$, and $R(\psi\to \phi)=0$ otherwise.

Figures (6)

  • Figure 1: Schematic picture of the setup of Theorem \ref{['thm:conversion_rate_projective_finite_number']}, where i.i.d. copies of a pure state $\psi$ are converted into i.i.d. copies of another pure state $\phi$ with an error that vanishes asymptotically.
  • Figure 2: Schematic picture of distillation and dilution setups, in which i.i.d. copies of a mixed state $\rho$ are converted to and from i.i.d. copies of a pure reference state $\phi$.
  • Figure 3: Geometric description of states on the Bloch sphere. Since the group transformation $U(e^{\mathrm{i}\theta})$ corresponds to a rotation about the $z$-axis, intuitively, a quantum state positioned farther from the $z$-axis exhibits greater asymmetry than one located closer to it. This behavior is illustrated by the relation $A_G(\rho_{a_2,\epsilon}) >A_G(\psi_{a_1})$ for $a_1>a_2>\frac{1}{2}$ and sufficiently small $\epsilon$.
  • Figure 4: A schematic figure of the relations among Lemma \ref{['lem:QLAN_asymptotic_conversion']}, Lemma \ref{['lem:rate_changing_part']} and Lemma \ref{['lem:covariance_decreasing_part']}. The direction of the arrow indicates the convertibility with vanishing error. In Lemma \ref{['lem:rate_changing_part']}, we find the growth rate of i.i.d. copies can be set to 1 by adjusting the scaling of the parameters $u$ when analyzing the asymptotic convertibility. Lemma \ref{['lem:covariance_decreasing_part']} shows that the asymptotic conversion between two pure-state models are possible if QGT is reduced during the conversion in the sense of matrix inequality. Combining these lemmas, Lemma \ref{['lem:QLAN_asymptotic_conversion']} is proven.
  • Figure 5: Schematic picture of the synchronization scenario. When $\mathrm{Sym}_G(\psi)\not\subset \mathrm{Sym}_G(\phi)$, the conversion rate from $\psi$ to $\phi$ vanishes, as shown by the dashed arrow in the figure. However, this restriction is circumvented when i.i.d. copies of $\psi\otimes \phi$ are converted into i.i.d. copies of $\phi$, as indicated by the solid arrow in the figure.
  • ...and 1 more figures

Theorems & Definitions (88)

  • Theorem 1
  • Corollary 2
  • proof : Proof of Corollary \ref{['cor:reversible_conversion']}
  • Lemma 3
  • Lemma 4
  • Theorem 5: Upper bound on distillable asymmetry
  • proof
  • Corollary 6: Sufficient condition for vanishing distillable asymmetry
  • Proposition 7: Upper bound on the asymmetry cost
  • Theorem 8
  • ...and 78 more