Multi-Operator Quantum Uncertainty Relations from New Cauchy-Schwarz Inequalities

TL;DR

This work presents a comprehensive generalization of the Cauchy–Schwarz inequality to multiple vectors and applies it to quantum mechanics to derive multi-operator uncertainty relations. By formulating balanced and unbalanced multi-vector CS inequalities and introducing multivariance, the authors obtain a suite of results including multi-operator uncertainty relations for Hermitian and non-Hermitian operators, and for normalized and unnormalized states, as well as their mixed-state extensions. A central contribution is the introduction of multi-operator squeezing, defined via balanced M-operator uncertainty relations and categorized into $q/M$-squeezing, with explicit discussion of thresholds and feasibility. The framework emphasizes conceptual simplicity and the practicality of computing the actual uncertainty product, while also connecting to existing notions of squeezing and potential metrological applications. Overall, the paper provides a unified, operator-centric approach to multi-observable quantum correlations and squeezing, with clear pathways to finite-dimensional and mixed-state generalizations.

Abstract

We present new generalizations of Cauchy-Schwarz (CS) inequalities to multiple vectors and use them to derive multi-operator quantum uncertainty relations and propose multi-operator squeezing.

Figures (18)

• Figure 1: Example of the balanced multi-operator uncertainty relation from \ref{['eq:14']} for $M=4$ arbitrary Hermitian operators, for one-qubit mixed states $\rho\equiv \epsilon\Lambda\epsilon^{\dag}$ where $\Lambda\equiv\text{diag}\{\lambda_{1},\lambda_{2}\}$ for $\lambda_1 =c_{\pi/8}^2$, $\!\lambda_2 \,=\,s_{\pi/8}^2$ where $c_x\equiv\cos(x)$ and $s_x\equiv\sin(x)$, and $\epsilon\!\equiv\!(\!\space_{s_{\theta}e^{i\phi}}^{c_\theta}\!\!\space_{\space c_\theta}^{-s_{\theta}e^{-i\phi}})$. Color for illustration only.
• Figure 2: Demonstration of the unbalanced multi-operator uncertainty relation of \ref{['eq:20']} for the example in \ref{['eq:21']} of three arbitrary Hermitian operators and a family of rank-2 two-qubit states parameterized by spectrum $(\lambda_{1},\lambda_{2})$ and concurrence $C$HiWoWoot as in HeXU as $(\lambda_{1},\lambda_{2})\equiv(c_{\vartheta}^2,s_{\vartheta}^2)$, and $C=\eta c_{\vartheta}^2$ where $\eta\in[0,1]$ and $\vartheta\in[0,\frac{\pi}{4}]$. Color for illustration only.
• Figure 3: Demonstration of the balanced multi-operator uncertainty relation of \ref{['eq:23']} for four arbitrary nonHermitian operators and the same family of one-qubit states used in \ref{['fig:1']}. Color for illustration only.
• Figure 4: Examples of the unbalanced multi-operator uncertainty relation of \ref{['eq:30']} for three arbitrary nonHermitian operators and the same family of two-qubit states used in \ref{['fig:2']}. Color for illustration only.
• Figure 5: Demonstration of the multivariance relation of \ref{['eq:13']} for the same example states and operators as \ref{['fig:1']}, for line 3 of \ref{['eq:32']}. Color for illustration only.