Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of preassigned accuracy can be obtained straightforwardly. Our method also works for postive-operator-valued measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional spin squeezing criteria fail because of detector noise.Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of preassigned accuracy can be obtained straightforwardly. Our method also works for postive-operator-valued measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional spin squeezing criteria fail because of detector noise.

arXiv:1705.10679Physical Review LettersOct 27, 201762 citations

State-independent uncertainty relations on multipartite spin $1/2$ -systemsState-independent uncertainty relations on multipartite spin $1/2$-systems

Yiling Wang, Kailash Misra, Naihuan Jing

Uncertainty relations quantify fundamental limits on simultaneous measurement of quantum observables. While conventional formulations are state-dependent, state-independent uncertainty relations (SIURs) impose universal bounds determined solely by the algebraic structure of the operators, with applications across metrology, quantum cryptography, and entanglement detection. Despite extensive study, exact analytical variance-based SIURs have so far been established primarily for one- and bi-partite systems, while entropic SIURs have been extended to multipartite and memory-assisted settings through information-theoretic constructions. In contrast, exact variance-based SIURs beyond the bipartite level have remained analytically unresolved.} Here we develop a representation-theoretic framework for multipartite SIURs in collective spin- $\tfrac{1}{2}$ systems. Using the Clebsch--Gordan decomposition and extremal analysis of total spin variance, we derive exact state-independent bounds up to quintipartite systems. A clear structural dichotomy emerges: odd $n$ systems exhibit strictly positive universal bounds (e.g., $Δ^2(\mathfrak{su}_2)\!\ge\!4/11$ for $n=3$ ), whereas even $n$ admit vanishing variance on trivial sectors but retain positive reduced-space bounds (e.g., $Δ^2(\mathfrak{su}_2)\!\ge\!1/8$ for $n=4$ ). These results establish the first unified, algebraic framework for multipartite variance-based SIURs in qubit ensembles.Uncertainty relations quantify fundamental limits on simultaneous measurement of quantum observables. While conventional formulations are state-dependent, state-independent uncertainty relations (SIURs) impose universal bounds determined solely by the algebraic structure of the operators, with applications across metrology, quantum cryptography, and entanglement detection. Despite extensive study, exact analytical variance-based SIURs have so far been established primarily for one- and bi-partite systems, while entropic SIURs have been extended to multipartite and memory-assisted settings through information-theoretic constructions. In contrast, exact variance-based SIURs beyond the bipartite level have remained analytically unresolved.} Here we develop a representation-theoretic framework for multipartite SIURs in collective spin-$\tfrac{1}{2}$ systems. Using the Clebsch--Gordan decomposition and extremal analysis of total spin variance, we derive exact state-independent bounds up to quintipartite systems. A clear structural dichotomy emerges: odd $n$ systems exhibit strictly positive universal bounds (e.g., $Δ^2(\mathfrak{su}_2)\!\ge\!4/11$ for $n=3$), whereas even $n$ admit vanishing variance on trivial sectors but retain positive reduced-space bounds (e.g., $Δ^2(\mathfrak{su}_2)\!\ge\!1/8$ for $n=4$). These results establish the first unified, algebraic framework for multipartite variance-based SIURs in qubit ensembles.

quant-pharXiv:2607.14376Journal of Physics A Mathematical and TheoreticalJul 17, 20260 citations

Detecting quantum entanglement

Barbara M. Terhal

arXiv:quant-ph/0101032Theoretical Computer ScienceSep 1, 2002273 citations

Entanglement in Quantum Information Processing

Samuel L. Braunstein

NATO science series. Series II, Mathematics, physics and chemistryJan 1, 20051 citation

Geometric and algebraic origins of additive uncertainty relations

Konrad Szymański, Karol Życzkowski

Abstract Constructive techniques to establish state-independent uncertainty relations for the sum of variances of arbitrary two observables are presented. We investigate the range of simultaneously attainable pairs of variances, which can be applied to a wide variety of problems including finding exact bound for the sum of variances of two components of angular momentum operator for any total angular momentum quantum number j and detection of quantum entanglement. Resulting uncertainty relations are state-independent, semianalytical, bounded-error and can be made arbitrarily tight. The advocated approach, based on the notion of joint numerical range of a number of observables and uncertainty range, allows us to improve earlier numerical works and to derive semianalytical tight bounds for the uncertainty relation for the sum of variances expressed as roots of a polynomial of a single real variable.Abstract Constructive techniques to establish state-independent uncertainty relations for the sum of variances of arbitrary two observables are presented. We investigate the range of simultaneously attainable pairs of variances, which can be applied to a wide variety of problems including finding exact bound for the sum of variances of two components of angular momentum operator for any total angular momentum quantum number j and detection of quantum entanglement. Resulting uncertainty relations are state-independent, semianalytical, bounded-error and can be made arbitrarily tight. The advocated approach, based on the notion of joint numerical range of a number of observables and uncertainty range, allows us to improve earlier numerical works and to derive semianalytical tight bounds for the uncertainty relation for the sum of variances expressed as roots of a polynomial of a single real variable.

arXiv:1804.06191Journal of Physics A Mathematical and TheoreticalSep 18, 201924 citations

State-independent uncertainty relations on multipartite spin

1/2

-systems

\tfrac{1}{2}

systems. Using the Clebsch--Gordan decomposition and extremal analysis of total spin variance, we derive exact state-independent bounds up to quintipartite systems. A clear structural dichotomy emerges: odd

n

systems exhibit strictly positive universal bounds (e.g.,

Δ^2(\mathfrak{su}_2)\!\ge\!4/11

for

n=3

), whereas even

n

admit vanishing variance on trivial sectors but retain positive reduced-space bounds (e.g.,

Δ^2(\mathfrak{su}_2)\!\ge\!1/8

for

n=4

). These results establish the first unified, algebraic framework for multipartite variance-based SIURs in qubit ensembles.