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No Universal Purification in Quantum Mechanics

Zhenhuan Liu, Zhenyu Du, Zhenyu Cai, Zi-Wen Liu

TL;DR

Problem: universal purification of finite copies into input-dependent pure outputs is forbidden by quantum mechanics' linearity and positivity. Approach: establish a general no-go theorem, extend to quantum channels via the Choi representation, and analyze approximate purification with dimension-independent sample-complexity bounds plus an exponential lower bound for pure-dilation preparation. Contributions: a universal no-purification result, robustness bounds for near-pure outputs, and a fundamental exponential lower bound for pure-dilation tasks. Significance: clarifies intrinsic limits on quantum information processing with implications for cooling, error mitigation, and resource-conversion protocols, and highlights open questions about circumventing these limits with virtual processing or explicit protocols.

Abstract

We prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental limitation of quantum information processing. In particular, no quantum operation can transform a finite number of copies of an unknown quantum state or channel into a pure state or channel that depends on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Relaxing the requirement of exact pure output, we further extend our result to establish quantitative sample complexity bounds for approximate purification, independent of any task details or operational constraints. To illustrate the practical consequences of this principle, we examine the task of approximately preparing pure dilation and, for the first time, prove an exponential lower bound on the required sample complexity.

No Universal Purification in Quantum Mechanics

TL;DR

Problem: universal purification of finite copies into input-dependent pure outputs is forbidden by quantum mechanics' linearity and positivity. Approach: establish a general no-go theorem, extend to quantum channels via the Choi representation, and analyze approximate purification with dimension-independent sample-complexity bounds plus an exponential lower bound for pure-dilation preparation. Contributions: a universal no-purification result, robustness bounds for near-pure outputs, and a fundamental exponential lower bound for pure-dilation tasks. Significance: clarifies intrinsic limits on quantum information processing with implications for cooling, error mitigation, and resource-conversion protocols, and highlights open questions about circumventing these limits with virtual processing or explicit protocols.

Abstract

We prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental limitation of quantum information processing. In particular, no quantum operation can transform a finite number of copies of an unknown quantum state or channel into a pure state or channel that depends on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Relaxing the requirement of exact pure output, we further extend our result to establish quantitative sample complexity bounds for approximate purification, independent of any task details or operational constraints. To illustrate the practical consequences of this principle, we examine the task of approximately preparing pure dilation and, for the first time, prove an exponential lower bound on the required sample complexity.

Paper Structure

This paper contains 7 sections, 6 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. No Universal Purification
  3. Approximate purification
  4. Approximate dilation preparation
  5. Discussion and Outlook
  6. Approximate purification (proof of Theorem \ref{['thm:approx_no-purification_distance_between_states']})
  7. Approximate pure dilation (proof of Theorem \ref{['thm:no_approx_purification']})

Key Result

Theorem 1

Let $t$ be any finite integer, and let $\mathcal{M}$ be a nonzero positive trace-non-increasing map. Suppose that for any state $\rho$ supported on a subspace $\mathcal{H}_S \subseteq \mathcal{H}_d$ with $\dim(\mathcal{H}_S)\ge 2$, the output $\mathcal{M}(\rho^{\otimes t})$ is proportional to a pure

Figures (1)

  • Figure 1: A universal quantum purification process $\mathcal{M}$ takes many copies of a mixed state $\rho$ as input and outputs a pure state $\Psi_\rho=\ketbra{\Psi_\rho}{\Psi_{\rho}}$ depending on $\rho$.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Minimax estimation of distributions
  • proof : Proof of Theorem \ref{['thm:no_approx_purification']}