Abstract We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the support of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state-independent uncertainty relations. Our results may be generalized to multiple observable cases in higher dimensional spaces.Abstract We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the support of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state-independent uncertainty relations. Our results may be generalized to multiple observable cases in higher dimensional spaces.

arXiv:2205.03193Communications in Theoretical PhysicsApr 29, 20221 citation

Tight State-Independent Uncertainty Relations for Qubits

Alastair A. Abbott, Pierre-Louis Alzieu, Michael J. W. Hall +1 more

The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.

arXiv:1512.02383MathematicsFeb 24, 201644 citations

Variance-based uncertainty relations for incompatible observables

Bin Chen, Ning-Ping Cao, Shao-Ming Fei +1 more

arXiv:1608.06075Quantum Information ProcessingJun 15, 201639 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

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.