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Optimal Distillation of Qubit Clocks

Sujay Kazi, Iman Marvian

TL;DR

This work addresses the problem of distilling coherence from many noisy qubit clocks under time-translation-invariant (TI) operations. It shows that the minimal output infidelity scales as $\mathcal{I}(\mathcal{E}_N)=\frac{1-\lambda^2}{4\lambda^2}\frac{\sin^2\Theta_{\text{out}}}{\sin^2\Theta_{\text{in}}}\frac{1}{N}+O\left(\frac{1}{N^2}\right)$, tying the leading behavior to the purity of coherence, i.e., the right logarithmic derivative (RLD) Fisher information, and providing an operational meaning to this metric. The authors derive a first-order optimal TI protocol saturating this bound via a Kraus representation after Schur sampling, formulate a boundary-value problem to characterize the exact optimal channel, and extend the analysis to second- and third-order corrections in special cases, including an equatorial setup. They also connect the purity-of-coherence bound to the Morozova–Chentsov–Petz quantum Fisher information framework and discuss the implications for quantum thermodynamics and the resource theory of asymmetry, with potential numerical and experimental relevance for non-asymptotic regimes.

Abstract

We study coherence distillation under time-translation-invariant operations: given many copies of a quantum state containing coherence in the energy eigenbasis, the aim is to produce a purer coherent state while respecting the time-translation symmetry. This symmetry ensures that the output remains synchronized with the input and that the process can be realized by energy-conserving unitaries coupling the system to a reservoir initially in an energy eigenstate, thereby modeling thermal operations supplemented by a work reservoir or battery. For qubit systems, we determine the optimal asymptotic fidelity and show that it is governed by the purity of coherence, a measure of asymmetry derived from the right logarithmic derivative (RLD) Fisher information. In particular, we find that the lowest achievable infidelity (one minus fidelity) scales as $1/N$ times the reciprocal of the purity of coherence of each input qubit, where $N$ is the number of copies, giving this quantity a clear operational meaning. We additionally study many other interesting aspects of the coherence distillation problem for qubits, including computing higher-order corrections to the lowest achievable infidelity up to $O(1/N^3)$, and expressing the optimal channel as a boundary value problem that can be solved numerically.

Optimal Distillation of Qubit Clocks

TL;DR

This work addresses the problem of distilling coherence from many noisy qubit clocks under time-translation-invariant (TI) operations. It shows that the minimal output infidelity scales as , tying the leading behavior to the purity of coherence, i.e., the right logarithmic derivative (RLD) Fisher information, and providing an operational meaning to this metric. The authors derive a first-order optimal TI protocol saturating this bound via a Kraus representation after Schur sampling, formulate a boundary-value problem to characterize the exact optimal channel, and extend the analysis to second- and third-order corrections in special cases, including an equatorial setup. They also connect the purity-of-coherence bound to the Morozova–Chentsov–Petz quantum Fisher information framework and discuss the implications for quantum thermodynamics and the resource theory of asymmetry, with potential numerical and experimental relevance for non-asymptotic regimes.

Abstract

We study coherence distillation under time-translation-invariant operations: given many copies of a quantum state containing coherence in the energy eigenbasis, the aim is to produce a purer coherent state while respecting the time-translation symmetry. This symmetry ensures that the output remains synchronized with the input and that the process can be realized by energy-conserving unitaries coupling the system to a reservoir initially in an energy eigenstate, thereby modeling thermal operations supplemented by a work reservoir or battery. For qubit systems, we determine the optimal asymptotic fidelity and show that it is governed by the purity of coherence, a measure of asymmetry derived from the right logarithmic derivative (RLD) Fisher information. In particular, we find that the lowest achievable infidelity (one minus fidelity) scales as times the reciprocal of the purity of coherence of each input qubit, where is the number of copies, giving this quantity a clear operational meaning. We additionally study many other interesting aspects of the coherence distillation problem for qubits, including computing higher-order corrections to the lowest achievable infidelity up to , and expressing the optimal channel as a boundary value problem that can be solved numerically.

Paper Structure

This paper contains 55 sections, 17 theorems, 508 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Setup: Coherence Distillation
  4. Orders of Optimality
  5. Purity of Coherence
  6. Main Results
  7. Optimal Distillation Protocol
  8. Preprocessing via the Schur Transform
  9. The First-Order Optimal Protocol
  10. Boundary Value Problem
  11. Concentration Argument and Power Series Expansion
  12. Connection to RLD Fisher Information
  13. Discussion
  14. Lower Bound on Infidelity Factor of Single-Shot Coherence Distillation
  15. Deriving the Kraus Representation of the Optimal Distillation Protocol
  16. ...and 40 more sections

Key Result

Theorem 10

Suppose a TI channel $\mathcal{E}_{\text{TI}}$ maps a system with Hamiltonian $H_{\text{in}}$ to a system with Hamiltonian $H_{\text{out}}$. Then for all possible input states $\rho$,

Figures (2)

  • Figure 1: The single-shot qubit coherence distillation problem. Given $N$ copies of a noisy qubit coherent state (orange), how can we create a state (cyan) that maximizes the fidelity with a pure coherent qubit state with the same time parameter $\phi$ (blue), all without any knowledge of that time parameter?
  • Figure 2: A comparison of the first-order infidelity of three different single-shot qubit coherence distillation protocols. The horizontal axis is the input infidelity $\frac{1-\lambda}{2}$, and the vertical axis is the infidelity factor $\delta_1(\mathcal{E})$, meaning that the output infidelity is $\mathcal{I}(\mathcal{E}_N) = \frac{\delta_1(\mathcal{E})}{N} + o(N^{-1})$. The blue curve $\mathcolor{blue}{\delta_1(\mathcal{E}) = \frac{1-\lambda}{2\lambda^2}}$ corresponds to the qubit distillation protocol with full $SU(2)$ covariance devised by Cirac et al., which is based on the Schur transform Cirac1999. The red curve $\mathcolor{red}{\delta_1(\mathcal{E}) = \frac{1}{4\lambda^2}}$ corresponds to a TI measure-and-prepare protocol devised in Marvian2020. Finally, the violet curve $\mathcolor{violet}{\delta_1(\mathcal{E}) = \frac{1-\lambda^2}{4\lambda^2}}$ corresponds to our first-order optimal protocol. Anything below the purple curve is forbidden by the monotonicity of purity of coherence Marvian2020. (This figure is taken from Marvian2020.)

Theorems & Definitions (33)

  • Definition 1: Group-covariant channel
  • Definition 2: Time-translation invariant (TI) channel
  • Definition 3: Single-shot coherence distillation protocol
  • Definition 4: Single-shot qubit coherence distillation protocol
  • Definition 5: Single-shot equatorial qubit coherence distillation protocol
  • Definition 6: Infidelity of a single-shot distillation protocol
  • Definition 7: Absolute optimality, $p^{\text{th}}$-order optimality
  • Definition 8: Infidelity coefficients, infidelity factor
  • Definition 9: Purity of coherence Marvian2020
  • Theorem 10: Monotonicity of purity of coherence under TI channels Marvian2020
  • ...and 23 more