Loss of Information in Quantum Guessing Game

TL;DR

This paper investigates the fundamental differences between qubits and higher-dimensional quantum systems regarding the predictability of measurement outcomes, revealing that qubits allow deterministic guessing under certain conditions, unlike higher dimensions.

Contribution

It extends previous work by demonstrating that for qubits, measurement outcomes can be deterministically guessed regardless of the measurement set, highlighting a key difference from higher-dimensional systems.

Findings

Qubits allow deterministic guessing of measurement outcomes with suitable strategies.

Higher-dimensional systems generally do not permit deterministic guessing, except in specific cases.

The unpredictability of measurement results is not solely due to coherence loss, especially in qubits.

Abstract

Incompatibility of certain measurements -- impossibility of obtaining deterministic outcomes simultaneously -- is a well known property of quantum mechanics. This feature can be utilized in many contexts, ranging from Bell inequalities to device dependent QKD protocols. Typically, in these applications the measurements are chosen from a predetermined set based on a classical random variable. One can naturally ask, whether the non-determinism of the outcomes is due to intrinsic hiding property of quantum mechanics, or rather by the fact that classical, incoherent information entered the system via the choice of the measurement. Authors of [NJP 19, 023038 (2017)] examined this question for a specific case of two mutually unbiased measurements on systems of different dimensions. They have somewhat surprisingly shown that in case of qubits, if the measurements are chosen coherently with the use of a controlled unitary, outcomes of both measurements can be guessed deterministically. Here we extend their analysis and show that specifically for qubits, measurement result for any set of measurements with any a-priori probability distribution can be faithfully guessed by a suitable state preparation and measurement. We also show that up to a small set of specific cases, this is not possible for higher dimensions. This result manifests a deep difference in properties of qubits and higher dimensional systems and suggests that these systems might offer higher security in specific cryptographic protocols. More fundamentally, the results show that the impossibility of predicting a result of a measurement is not caused solely by a loss of coherence between the choice of the measurement and the guessing procedure.