---
author:
- |
    Thomas Decker[^1], Dominik Janzing, and Thomas Beth\
    Institut f[ü]{}r Algorithmen und Kognitive Systeme, Universit[ä]{}t Karlsruhe,\
    Am Fasanengarten 5, D-76131 Karlsruhe, Germany
date: 'August 19, 2003'
title: 'Quantum circuits for single-qubit measurements corresponding to platonic solids'
---

Introduction
============

A key postulate of textbook quantum mechanics is the assumption that measurements correspond to self-adjoint operators $A$ in such a way that the probability of each possible measurement outcome or set of possible outcomes can be computed from the spectral projections of $A$. If the corresponding system Hilbert space is finite dimensional $A$ can be written as $A=\sum_j \lambda_j P_j$ where $P_j$ is the projection on the eigenspace with eigenvalue $\lambda_j$. The probability of the outcome $\lambda_j$ is ${\rm tr}(\rho P_j)$ if the system is in a state with density matrix $\rho$. This type of measurement is called [*von-Neumann measurement*]{}, [*orthogonal measurement*]{} or [*projector-valued measurement*]{}.

Within the standard model of a quantum computer one can easily show that it is in principle possible to implement measurements for all self-adjoint operators $A$ acting on the Hilbert space $(\C^2)^{\otimes l}$, i.e., the state space of a quantum register with $l$ qubits. Since a universal quantum computer allows the implementation of 