The "mean king's problem" with continuous variables

TL;DR

This paper solves the mean king's problem for continuous variables, demonstrating that measurement outcomes of linear combinations of position and momentum can be precisely inferred, and extends to joint measurements of multiple combinations.

Contribution

It introduces a solution to the mean king's problem in the continuous variable domain, including a conjunctive version for joint measurements, which was not previously addressed.

Findings

Outcome of any linear combination of x and p can be ascertained with arbitrary precision.

The solution applies to a conjunctive version involving joint measurements of multiple linear combinations.

The approach extends the mean king's problem to continuous variables, broadening its applicability.

Abstract

We present the solution to the "mean king's problem" in the continuous variable setting. We show that in this setting, the outcome of a randomly-selected projective measurement of any linear combination of the canonical variables x and p can be ascertained with arbitrary precision. Moreover, we show that the solution is in turn a solution to an associated "conjunctive" version of the problem, unique to continuous variables, where the inference task is to ascertain all the joint outcomes of a simultaneous measurement of any number of linear combinations of x and p.