Discrete phase-space mappings, tomographic condition and permutation invariance

TL;DR

This paper investigates discrete phase-space mappings for N-qubit systems, revealing a fundamental incompatibility between tomographic conditions and permutation invariance, and provides explicit forms of symmetric mappings.

Contribution

It proves the incompatibility between tomographic conditions and permutation invariance in discrete Wigner functions and offers explicit symmetric mapping forms.

Findings

Tomographic condition conflicts with permutation invariance in Wigner functions.

Explicit forms of permutation-invariant self-dual mappings are provided.

Symmetric projection analysis demonstrates limitations of Wigner function representations.

Abstract

We analyze different families of discrete maps\ in the N-qubit systems in the context of the permutation invariance. We prove that the tomographic condition imposed on the self-dual (Wigner) map is incompatible with the requirement of the invariance under particle permutations, which makes it impossible to project the Wootters-like Wigner function into the space of symmetric measurements. We also provide several \textit{explicit} forms of the self-dual mappings: a) tomographic and b) permutation invariant \ and analyze the symmetric projection in the latter case.