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Optimal detection of dissipation in Lindbladian dynamics

Yiyi Cai

Abstract

Experimental implementations of Hamiltonian dynamics are often affected by dissipative noise arising from interactions with the environment. This raises the question of whether one can detect the presence or absence of such dissipation using only access to the observed time evolution of the system. We consider the following decision problem: given black-box access to the time-evolution channels $e^{t\mathcal{L}}$ generated by an unknown time-independent Lindbladian $\mathcal{L}$, determine whether the dynamics are purely Hamiltonian or contain dissipation of magnitude at least $ε$ in normalized Frobenius norm. We give a randomized procedure that solves this task using total evolution time $\mathcal{O}(ε^{-1})$, which is information-theoretically optimal. This guarantee holds under the assumptions that the Lindblad generator has bounded strength and its dissipative part is of constant locality with bounded degree. Our work provides a practical method for detecting dissipative noise in experimentally implemented quantum dynamics.

Optimal detection of dissipation in Lindbladian dynamics

Abstract

Experimental implementations of Hamiltonian dynamics are often affected by dissipative noise arising from interactions with the environment. This raises the question of whether one can detect the presence or absence of such dissipation using only access to the observed time evolution of the system. We consider the following decision problem: given black-box access to the time-evolution channels generated by an unknown time-independent Lindbladian , determine whether the dynamics are purely Hamiltonian or contain dissipation of magnitude at least in normalized Frobenius norm. We give a randomized procedure that solves this task using total evolution time , which is information-theoretically optimal. This guarantee holds under the assumptions that the Lindblad generator has bounded strength and its dissipative part is of constant locality with bounded degree. Our work provides a practical method for detecting dissipative noise in experimentally implemented quantum dynamics.
Paper Structure (34 sections, 21 theorems, 184 equations)

This paper contains 34 sections, 21 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Technical overview
  4. Bell sampling as a decay observable.
  5. Pauli-diagonal setting.
  6. Implementing the twirled dynamics.
  7. Connection to purity testing.
  8. Related work
  9. Testing Hamiltonian dynamics
  10. Certification of Hamiltonian dynamics
  11. Learning open-system generators.
  12. Outlook
  13. Relaxations of structural assumptions.
  14. Dependence on locality.
  15. Alternative norms.
  16. ...and 19 more sections

Key Result

Theorem 1.1

Consider the class of time-independent Lindblad generators $\mathcal{L} = -i[H, \cdot] + \mathcal{D}$ on $n$ qubits such that the jump operators defining $\mathcal{D}$ have constant locality and bounded degree (see Definition def:locality_and_degree), and $\|\mathcal{L}\|_{\diamond} \leq L$. Then th

Theorems & Definitions (39)

  • Theorem 1.1: Informal version of Theorem \ref{['thm:final_certification']}
  • Corollary 1.2
  • Definition 2.1: $k$-local jump operators of degree $\Delta$
  • Lemma 2.2: Bell identity for the superoperator trace
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Stability of the Bell identity probability
  • proof
  • Lemma 2.5: Paley-Zygmund
  • ...and 29 more