Projective operator closeness bounds output distributions

Let $\mathcal{H}$ be a nonzero finite-dimensional complex inner-product space (Real and complex inner product spaces, with the inner product linear in the first argument) with a fixed orthonormal computational basis $B$ (Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases), and let $U,V:\mathcal{H}\to \mathcal{H}$ be unitary operators, meaning invertible linear isometries (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces). For a unit vector $\psi$, write $U\psi ={\sum }_{b\in B}{\alpha }_b\mathrm{\mid }b⟩$ and $V\psi ={\sum }_{b\in B}{\beta }_b\mathrm{\mid }b⟩$, and define their output laws by $p_U^{\psi }(b)=\mathrm{\mid }{\alpha }_b{\mathrm{\mid }}^2$ and $p_V^{\psi }(b)=\mathrm{\mid }{\beta }_b{\mathrm{\mid }}^2$. Unitarity and orthonormality make these finite probability laws (Finite probability spaces, outcome weights, events, and event probabilities). For laws $p,q$ on $B$, define total variation by $d_{\mathrm{T}\mathrm{V}}(p,q):=\frac{1}{2}{\sum }_{b\in B}\mathrm{\mid }p(b)-q(b)\mathrm{\mid }$. Define projective operator distance by $d_{\mathrm{p}\mathrm{r}}(U,V):={\inf }_{\mathrm{\mid }\lambda \mathrm{\mid }=1}\mathrm{\parallel }U-\lambda V\mathrm{\parallel }$, where $\mathrm{\parallel }\cdot \mathrm{\parallel }$ is the operator norm. If $d_{\mathrm{p}\mathrm{r}}(U,V)\le \epsilon$ for $\epsilon \ge 0$, then

$d_{\mathrm{T}\mathrm{V}}(p_U^{\psi },p_V^{\psi })\le \epsilon \mathrm{.}$

The same bound therefore holds for every computational-basis output law of the two operators. The proof verifies directly that multiplying a vector by a unit scalar leaves these coordinate probabilities unchanged.

The proof uses no axiom of choice: the inequality holds for each phase separately, then for their infimum. Standalone phase invariance follows directly from modulus multiplicativity, so no quantum-specific global-phase theorem is required.