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Inevitability of knowing less than nothing

Gilad Gour, Mark M. Wilde, Sarah Brandsen, Isabelle Jianing Geng

TL;DR

It is proved that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world.

Abstract

A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.

Inevitability of knowing less than nothing

TL;DR

It is proved that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world.

Abstract

A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.
Paper Structure (25 sections, 12 theorems, 146 equations, 2 figures)

This paper contains 25 sections, 12 theorems, 146 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Notation
  4. Uncertainty and entropy
  5. Uncertainty measures
  6. Entropy
  7. Conditional uncertainty and entropy
  8. Conditional uncertainty measures
  9. Conditional entropy
  10. Main result: Inevitability of negative quantum conditional entropy
  11. Conditional entropy from relative entropy
  12. Conclusion
  13. Proofs of properties of entropy
  14. Choi matrix formulations
  15. Choi matrix formulation of conditional unitality
  16. ...and 10 more sections

Key Result

Theorem 1

Let $\mathbf{H}$ be a quantum conditional entropy, as defined in eq:cond-ent-gen-def-1st--eq:QCE-2nd-postulate-add. Then, for every state $\rho_{AB}$, Equality is attained in eq:main-thm if $\rho_{AB}$ is equal to a maximally entangled state $\Phi^{(k)}_{A'B'}$ by the action of local isometric channels. Thus, for every conditional entropy $\mathbf{H}$.

Figures (2)

  • Figure 1: Depiction of the definition of a semi-causal channel $\mathcal{N}$, where the black square represents the discarding channel.
  • Figure 2: Eq. \ref{['eq:semi-causal-breakdown']} asserts that a semi-causal bipartite channel $\mathcal{N}_{AB \rightarrow AB'}$ can be implemented in the depicted fashion.

Theorems & Definitions (20)

  • Definition 1: Majorization
  • Definition 2: Entropy
  • Definition 3: Conditional majorization
  • Definition 4: Conditional entropy
  • Theorem 1
  • Corollary 1
  • Definition 5: Relative entropy
  • Theorem 2
  • Lemma 1
  • Lemma 2: Piani_2006Gour5
  • ...and 10 more