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Information Fragility or Robustness Under Quantum Channels

Nicholas Laracuente, Graeme Smith

TL;DR

The paper addresses whether universal lower bounds exist for information preservation under small quantum noise, complementing known universal decay bounds. It shows that such lower bounds can fail: even infinitesimal noise can erase a large portion of mutual information for some states, with depolarizing/dephasing and finite-group channels as key examples, while also identifying structured regimes (via CMLSI and commuting states) where multiplicative lower bounds can be established. The work provides quantitative converse bounds and discusses private capacity implications by constructing channels that leak almost everything to the environment yet retain nonzero private rates. Overall, the results illuminate the fragility and resilience of quantum correlations under noise and offer guidance for designing robust quantum communication and error-correction strategies.

Abstract

Quantum states naturally decay under noise. Many earlier works have quantified and demonstrated lower bounds on the decay rate, showing exponential decay in a wide variety of contexts. Here we study the converse question: are there uniform upper bounds on the ratio of post-noise to initial information quantities when noise is sufficiently weak? In several scenarios, including classical, we find multiplicative converse bounds. However, this is not always the case. Even for simple noise such as qubit dephasing or depolarizing, mutual information may fall by an unbounded factor under arbitrarily weak noise. As an application, we find families of channels with non-zero private capacity despite arbitrarily high probability of transmitting an arbitrarily good copy of the input to the environment.

Information Fragility or Robustness Under Quantum Channels

TL;DR

The paper addresses whether universal lower bounds exist for information preservation under small quantum noise, complementing known universal decay bounds. It shows that such lower bounds can fail: even infinitesimal noise can erase a large portion of mutual information for some states, with depolarizing/dephasing and finite-group channels as key examples, while also identifying structured regimes (via CMLSI and commuting states) where multiplicative lower bounds can be established. The work provides quantitative converse bounds and discusses private capacity implications by constructing channels that leak almost everything to the environment yet retain nonzero private rates. Overall, the results illuminate the fragility and resilience of quantum correlations under noise and offer guidance for designing robust quantum communication and error-correction strategies.

Abstract

Quantum states naturally decay under noise. Many earlier works have quantified and demonstrated lower bounds on the decay rate, showing exponential decay in a wide variety of contexts. Here we study the converse question: are there uniform upper bounds on the ratio of post-noise to initial information quantities when noise is sufficiently weak? In several scenarios, including classical, we find multiplicative converse bounds. However, this is not always the case. Even for simple noise such as qubit dephasing or depolarizing, mutual information may fall by an unbounded factor under arbitrarily weak noise. As an application, we find families of channels with non-zero private capacity despite arbitrarily high probability of transmitting an arbitrarily good copy of the input to the environment.
Paper Structure (10 sections, 17 theorems, 98 equations, 1 figure)

This paper contains 10 sections, 17 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Semigroups
  4. Example(s) of Sudden Information Decay
  5. Proof of Fragility Under Finite Group Channels
  6. Channels with Non-Zero Private Rate Despite Arbitrarily High Loss
  7. Situation-Specific Decay Converse Bounds
  8. CMLSI Converse Bounds
  9. Classical and Semiclassical Converse Bounds
  10. Starting from Already Decayed States

Key Result

Theorem 1.1

Let $\Phi^t = \exp( - t \mathcal{L})$ be any quantum Markov semigroup acting on $B$ within bipartite system $A \otimes B$, generated by Lindbladian $\mathcal{L}$ with fixed point projection $\mathcal{E}$. In each case as above, $f_{...} : \mathbb{R}^+ \rightarrow [0,1)$, and $f_{...}(0) = 0$.

Figures (1)

  • Figure 1: (A) Bloch sphere visualization of $\rho_{\theta, \lambda}$ when $d=2$. (B) Visualization of the X-Z plane in the Bloch sphere.

Theorems & Definitions (35)

  • Theorem 1.1: Decay Converse
  • Theorem 1.2
  • Theorem 2.1: Theorem 5.1 from bluhm_continuity_2022
  • Corollary 2.2
  • Proposition 2.3: Pinsker's Inequality & Refinement
  • Lemma 2.4: Lemma 2.1 from gao_complete_2022
  • Remark 2.5
  • Example 3.1: Depolarizing and Dephasing
  • Lemma 3.2
  • proof
  • ...and 25 more