Uncertainty relations and approximate quantum error correction

TL;DR

This paper derives new uncertainty relations based on guessing probabilities, linking them to quantum error correction and wave-particle duality, with implications for quantum information theory.

Contribution

It introduces novel uncertainty relations formulated through guessing probabilities, connecting them to approximate quantum error correction and wave-particle duality.

Findings

New uncertainty relations for conjugate observables based on guessing probabilities.

A condition for approximate quantum error correction related to recoverability of information.

A tight relation on guessing probabilities applicable to wave-particle duality.

Abstract

The uncertainty principle can be understood as constraining the probability of winning a game in which Alice measures one of two conjugate observables, such as position or momentum, on a system provided by Bob, and he is to guess the outcome. Two variants are possible: either Alice tells Bob which observable she measured, or he has to furnish guesses for both cases. Here I derive new uncertainty relations for both, formulated directly in terms of Bob's guessing probabilities. For the former these relate to the entanglement that can be recovered by action on Bob's system alone. This gives a condition for approximate quantum error correction in terms of the recoverability of "amplitude" and "phase" information, implicitly used in the recent construction of efficient quantum polar codes. I also find a tight relation on the guessing probabilities for the latter game, which has application to wave-particle duality relations.