Normalized Hermitian observable defines a two-outcome POVM

Given: A finite-dimensional nonzero complex inner-product space $H$, a self-adjoint operator $M$ with $\mathrm{\parallel }M\mathrm{\parallel }\le 1$, and a density operator $\rho$ on $H$.

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: a finite-dimensional normal operator has an orthonormal eigenbasis.

The adjoint $T^{\ast }:W\to V$ is characterised by $⟨Tv,w{⟩}_W=⟨v,T^{\ast }w{⟩}_V$: $⟨Mx,y⟩=⟨x,M^{\ast }y⟩$.

Real and complex inner-product spaces and their induced length: $⟨\cdot ,\cdot ⟩$ is conjugate-symmetric, and the convention is linear in its first argument.

Non-negative and positive operators: nonnegative means self-adjoint with nonnegative quadratic form.

Finite positive effect family: an effect has both $E$ and $I-E$ nonnegative, and a finite complete positive family is a nonempty finite family of nonnegative operators summing to $I$.

Finite POVMs and their outcome law: a finite POVM is a finite family of nonnegative operators summing to $I$, with outcome weights $p_{\rho }(a)=\mathrm{tr}(\rho E_a)$ for a density operator $\rho$.

Density operator: $\rho$ is nonnegative and $\mathrm{tr}(\rho )=1$.

Trace is a linear functional on $M_n(F)$: matrix trace is linear.

Trace of a product of positive semidefinite matrices is nonnegative: the trace of a product of positive semidefinite matrices is nonnegative real.

Trace of a product of Hermitian matrices is real: the trace of two Hermitian matrices' product is real.

The basis-independent trace of an endomorphism of a finite-dimensional vector space: endomorphism trace is the matrix trace in any chosen basis.

For $A\in M_{m\times n}(F)$ and $B\in M_{n\times m}(F)$, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$: $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ for compatible finite square products.

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: the operator norm bounds $\mathrm{\parallel }Tv\mathrm{\parallel }$ for each unit vector $v$.