Localization of quantum states within subspaces

TL;DR

This work formalizes a rigorous notion of localization of quantum states by decomposing any non-negative operator $A\ge0$ along a subspace ${\mathcal V}$ into a maximally localized component $B$ (with $\mathrm{ran}(B)\subseteq{\mathcal V}$) and a remainder $C$ with $\mathrm{ran}(C)\cap{\mathcal V}=0$, i.e., $A=B+C$ with $B,C\ge0$. The main result proves existence and uniqueness of such a decomposition, with $B$ given by a Schur-complement construction and several equivalent forms (projection-based or via $A^{-1}$). The authors then develop rich properties: concavity of the localization map $A\mapsto B(A|{\mathcal V})$, trace inequalities, and dual forms; and they translate these results to quantum information, interpreting $\lambda=\mathrm{TB}(\rho_A|{\mathcal V})$ as an inclusion probability, explore its relation to the usual overlap, and discuss entropy-like behavior, measurement challenges, and cryptographic uses. Overall, the paper provides a comprehensive mathematical framework linking operator decomposition to quantum-information localization, with explicit formulas and multiple representations that facilitate applications to density matrices and beyond. The results offer new insights into how quantum states can be probabilistically and structurally localized within subspaces, with potential implications for state discrimination, information masking, and subspace-based quantum protocols.

Abstract

A precise definition is proposed for the localization probability of a quantum state within a given subspace of the full Hilbert space of a quantum system. The corresponding localized component of the state is explicitly identified, and several mathematical properties are established. Applications and interpretations in the context of quantum information are also discussed.

Figures (4)

• Figure 1: Geometric decomposition of a qubit state $\rho$ in the Bloch sphere (graphically simplified to a disk). The state is expressed as a mixture (convex combination or interpolation) of two pure states: $\rho= \lambda | \psi \rangle \langle \psi | + (1-\lambda) | \phi \rangle \langle \phi |$. For any given state $\rho$ and given pure state $\psi$, the weight $\lambda$ is univocally determined, as is the pure state $\phi$ (unless $\lambda=1$). The distances from $\rho$ to $\psi$ and $\phi$ are proportional to $1-\lambda$ and $\lambda$, respectively.
• Figure 2: Schematic representation of the decompositions $\rho_A = \lambda \rho_B + (1-\lambda) \rho_C$ along ${\mathcal{V} }$, and $\rho_A = \lambda^\perp \rho_{B^\perp} + (1-\lambda^\perp) \rho_{C^\perp}$ along ${\mathcal{V} }^\perp$. The states of type $B$ and $B^\perp$ are fully inside ${\mathcal{V} }$ and ${\mathcal{V} }^\perp$ respectively. The sectors $i)$ and $ii)$ have sizes $\lambda$ and $\lambda^\perp$, respectively.
• Figure 3: Illustration of the decompositions of a qubit state $\rho_A$ along one-dimensional subspaces ${\mathcal{V} }$ and ${\mathcal{V} }^\perp$ in the Bloch sphere (eqs. \ref{['eq:3.22']}--\ref{['eq:3.24']}). Since $\rho_A>0$, the operator $\rho_D$ is outside the sphere and does not define a true quantum state. That the straight-line joining $\rho_A$ and $\rho_D$ is a horizontal one illustrates eq. \ref{['eq:3.30']}, which is only granted for qubit states.
• Figure 4: Illustration of the concavity property for a qubit in the Bloch sphere. The states $\rho_1$ and $\rho_2$, and their mixture $\rho$, are decomposed along a one-dimensional subspace ${\mathcal{V} }$ with orthogonal projector $\rho_B$. The weight $\lambda$ of $\rho$ is larger than the average of the weights $\lambda_1$ and $\lambda_2$ of $\rho_1$ and $\rho_2$.