The Exercise

A general measurement on a quantum system has a measurement operator $M_i$ for each outcome $i$. It determines a positive operator valued measure (POVM): a family of positive operators $E_i$, one for each outcome, that sum to the identity. The exercise is to show that each measurement operator is a unitary times the positive square root of the corresponding POVM element. This is the polar decomposition of $M_i$.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦

A measurement $\{M_i\}$ on $S$ with a finite set of outcomes is a family of measurement operators $M_i$ on the state space of $S$, one for each outcome $i$, satisfying the completeness relation $$\sum_i M_i^\dagger M_i = I .$$(1)^✦

The POVM of a measurement $\{M_i\}$ on $S$ is the POVM whose element for the outcome $i$ is the operator $$E_i = M_i^\dagger M_i$$(2)^✦ on the state space of $S$. The square root $\sqrt{E}$ of a positive operator $E$ is the positive operator whose square is $E$.

The Claim

What Is Checked

The claim is stated over a measurement on a quantum system and the POVM that the measurement determines through (2). Of this construction the library proves that it yields a POVM: every $E_i$ is a positive operator, and the completeness relation (1) gives $\sum_i E_i = I$.