Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unspeakable Coherence Concentration

Benjamin Stratton, Chung-Yun Hsieh, Paul Skrzypczyk

TL;DR

This work formalizes unspeakable coherence as a resource constrained by symmetry and derives how coherence can be concentrated from two copies into a subsystem. It delivers a complete qubit solution, including the optimal two-copy unitary and a constructive multi-qubit protocol that can unboundedly amplify the coherence ratio for certain inputs, while bounding amplification for general states. Beyond qubits, the authors establish fundamental dimension-dependent limits using Ky-Fan norms and LR-D decompositions, and prove no-go results showing that certain global correlations cannot be converted into local coherence. The results illuminate the interplay between global and local symmetry, provide a rigorous framework for coherence concentration under translational covariant dynamics, and suggest practical, scalable methods for generating highly coherent states under symmetry constraints with applications to metrology and thermodynamics.

Abstract

Unspeakable coherence is a key feature separating quantum and classical physics. Modelled as asymmetry with respect to a continuous transformation generated by a physically relevant observable, such as the Hamiltonian or angular moment, unspeakable coherence has been shown to be the relevant notion of coherence for achieving quantum advantage in the tasks of metrology, reference frame alignment and work extraction, among others. A question of both practical and foundational value is: Given some copies of a state with low coherence, can we prepare a more coherent state via coherence non-increasing operations? Here, we study this question in the minimal limiting case: Given two uncorrelated copies of a coherent state, can one, via globally coherence non-increasing unitaries, increase the coherence in a subsystem? We fully solve this problem for qubits, identifying the optimal unitaries and revealing the existence of bound coherence. This is then used to create a completely constructive multi-qubit coherence enhancement protocol, where only effective-qubit unitaries are used. Unexpectedly, in this protocol, we show that there exists states for which the ratio of the input-output coherence can be amplified unboundedly. Extending beyond qubits, we derive two fundamental upper bounds on the amount of local coherence that can be increased and prove a no-go theorem showing that certain global correlations cannot be converted to local coherence.

Unspeakable Coherence Concentration

TL;DR

This work formalizes unspeakable coherence as a resource constrained by symmetry and derives how coherence can be concentrated from two copies into a subsystem. It delivers a complete qubit solution, including the optimal two-copy unitary and a constructive multi-qubit protocol that can unboundedly amplify the coherence ratio for certain inputs, while bounding amplification for general states. Beyond qubits, the authors establish fundamental dimension-dependent limits using Ky-Fan norms and LR-D decompositions, and prove no-go results showing that certain global correlations cannot be converted into local coherence. The results illuminate the interplay between global and local symmetry, provide a rigorous framework for coherence concentration under translational covariant dynamics, and suggest practical, scalable methods for generating highly coherent states under symmetry constraints with applications to metrology and thermodynamics.

Abstract

Unspeakable coherence is a key feature separating quantum and classical physics. Modelled as asymmetry with respect to a continuous transformation generated by a physically relevant observable, such as the Hamiltonian or angular moment, unspeakable coherence has been shown to be the relevant notion of coherence for achieving quantum advantage in the tasks of metrology, reference frame alignment and work extraction, among others. A question of both practical and foundational value is: Given some copies of a state with low coherence, can we prepare a more coherent state via coherence non-increasing operations? Here, we study this question in the minimal limiting case: Given two uncorrelated copies of a coherent state, can one, via globally coherence non-increasing unitaries, increase the coherence in a subsystem? We fully solve this problem for qubits, identifying the optimal unitaries and revealing the existence of bound coherence. This is then used to create a completely constructive multi-qubit coherence enhancement protocol, where only effective-qubit unitaries are used. Unexpectedly, in this protocol, we show that there exists states for which the ratio of the input-output coherence can be amplified unboundedly. Extending beyond qubits, we derive two fundamental upper bounds on the amount of local coherence that can be increased and prove a no-go theorem showing that certain global correlations cannot be converted to local coherence.

Paper Structure

This paper contains 29 sections, 4 theorems, 88 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Unspeakable Coherence
  3. Modes of Coherence
  4. Coherence Concentration Problems
  5. Optimal Qubit Coherence Concentration
  6. Qubit Concatenation
  7. Coherence Concentration Beyond Qubits: Fundamental Limitations
  8. The Role of Initial Correlations and No-Go Results for Coherence Concentration
  9. Discussion
  10. Acknowledgement
  11. Appendix A: Proof of Lemma \ref{['global_Modes_to_local_modes']}
  12. Appendix B: Proof Sketch of Result \ref{['result_3:maxIncreaseQubits']}
  13. Appendix C: Details of Concatenation Protocol
  14. Appendix D: Proof Sketch of Result \ref{['lemma:infinte_concat']}
  15. Appendix E: Proof Sketch of Result \ref{['result:bound_1']}
  16. ...and 14 more sections

Key Result

Lemma 1

Let $\rho_A \coloneqq {\rm tr}_B(\rho_{AB})$ be the reduced state of $\rho_{AB}$ in $A$. Then $\rho_{A}^{(j)} = {\rm tr}_{B}( \rho_{AB}^{(j)} ),$ meaning the $j$th local mode depends only on the $j$th global mode.

Figures (4)

  • Figure 1: Qubit concatenation paths through the Bloch sphere. (A) Paths for $n^0_z=0.7$ and various $n_x^0$. (B) Paths for $n^0_x=0.001$ and various $n_z^0$. Inserts show the $\log_2(\# {\rm ~input~ states})$---the number of steps---needed to converge to within $0.001$ of $n_z=0$, which dictates the maximum reachable coherence.
  • Figure 2: Vector field for the recurrence relations in Eq. \ref{['eq:recurrence relations']}. The length of the vector and its colour detail how much a given point will change under the recurrence relation. For example, a short blue arrow means a small change will occur, a long red arrow means a large change will occur.
  • Figure 3: The value of Result \ref{['result:bound_1']} (bound 1) and Result \ref{['result:bound_2']} (bound 2) plotted for a sample of states in $d=3$ for all possible ranks. The insert shows the number of sampled states for which a given bound is greater than the other.
  • Figure 4: The value of Result \ref{['result:bound_1']} (bound 1) and Result \ref{['result:bound_2']} (bound 2) plotted for a sample of states in $d=4$ for all possible ranks. The insert shows the number of sampled states for which a given bound is greater than the other.

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2: Global Symmetry Implies Local Symmetry
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • Lemma 3
  • proof
  • ...and 4 more