Inevitability of knowing less than nothing

Gilad Gour; Mark M. Wilde; S. Brandsen; Isabelle Jianing Geng

doi:10.22331/q-2024-11-20-1529

Inevitability of knowing less than nothing

Gilad Gour¹, Mark M. Wilde², S. Brandsen³, and Isabelle Jianing Geng⁴

¹Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa 3200003, Israel
²School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA
³Department of Physics, Duke University, Durham, North Carolina 27708, USA
⁴Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, Alberta, Canada T2N 1N4

Published:	2024-11-20, volume 8, page 1529
Eprint:	arXiv:2208.14424v3
Doi:	https://doi.org/10.22331/q-2024-11-20-1529
Citation:	Quantum 8, 1529 (2024).

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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.

Popular summary

Quantum information science is famously known for having features that are not present in the classical case. Among these, perhaps the feature that distinguishes it the most from the classical case is that our knowledge of the state of two particles can be greater than our knowledge of the individual states of the particles. Indeed, for an Einstein–Podolsky–Rosen entangled state, our knowledge of the joint state of the two particles is complete. However, the state of an individual particle is described by an unbiased probabilistic ensemble, and we thus have incomplete knowledge of the state of an individual particle.

Given this peculiar situation, one of the earliest puzzles in quantum information science was to devise a quantum generalization of the concept of conditional entropy. In the classical case, conditional entropy is meant to capture the uncertainty about the state of one particle given access to a second particle that is potentially statistically dependent on the first.

In quantum information theory, the traditional approach to resolving such puzzles is to devise a physically meaningful task for which an information quantity is an optimal rate for accomplishing the task. For the case of quantum conditional entropy, this approach was successfully applied by Horodecki, Oppenheim, and Winter and led to the introduction of the quantum state merging protocol. Although these works brought great insight to the peculiar situation mentioned above, they mainly focus on a particular definition of the conditional entropy.

It is also well known that there are an infinite number of ways to define quantum conditional entropy, with particular examples reviewed in the paper of Gour et al. Several of these alternate definitions have physical meanings as well. However, all of these definitions are based on particular mathematical formulas for quantum conditional entropy that hitherto have not been derived on the basis of simple physical principles. As such, the aforementioned contributions do not address the question of why any plausible quantum conditional entropy can be negative when defining it in the most general way possible, which is consistent with some simple physical principles.

It is the aim of the paper of Gour et al. to address this pressing foundational question, and it is done so by introducing a physically motivated framework for defining quantum conditional entropy, based on two simple postulates. The two postulates are related to the second law of thermodynamics and the extensivity of entropy, widely accepted in scenarios of physical interest. This novel framework is essentially the simplest way of defining any plausible quantum conditional entropy, and the main conclusion is that any such quantum conditional entropy must take on a negative value when evaluated for the EPR state. This finding can be summarized as the “inevitability of knowing less than nothing” in the quantum world. As such, the approach answers the aforementioned question, i.e., why the quantum conditional entropy can be negative.

► BibTeX data

@article{Gour2024inevitabilityof,
  doi = {10.22331/q-2024-11-20-1529},
  url = {https://doi.org/10.22331/q-2024-11-20-1529},
  title = {Inevitability of knowing less than nothing},
  author = {Gour, Gilad and Wilde, Mark M. and Brandsen, S. and Geng, Isabelle Jianing},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1529},
  month = nov,
  year = {2024}
}

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