The uncertainty relation, as one of the fundamental principles of quantum physics, captures the incompatibility of noncommuting observables in the preparation of quantum states. In this work, we derive two strong and universal uncertainty relations for N(N ≥ 2) observables with discrete and bounded spectra, one in multiplicative form and the other in additive form. To verify their validity, for illustration, we implement in the spin-1/2 system an experiment with single-photon measurement. The experimental results exhibit the validity and robustness of these uncertainty relations, and indicate the existence of stringent lower bounds.The uncertainty relation, as one of the fundamental principles of quantum physics, captures the incompatibility of noncommuting observables in the preparation of quantum states. In this work, we derive two strong and universal uncertainty relations for N(N ≥ 2) observables with discrete and bounded spectra, one in multiplicative form and the other in additive form. To verify their validity, for illustration, we implement in the spin-1/2 system an experiment with single-photon measurement. The experimental results exhibit the validity and robustness of these uncertainty relations, and indicate the existence of stringent lower bounds.

arXiv:1801.01395Scientific ReportsApr 5, 201926 citations

Operational measure of incompatibility of noncommuting observables

Somshubhro Bandyopadhyay, Prabha Mandayam

Uncertainty relations are often considered to be a measure of incompatibility of noncommuting observables. However, such a consideration is not valid in general, motivating the need for an alternate measure that applies to any set of noncommuting observables. We present an operational approach to quantifying incompatibility without invoking uncertainty relations. Our measure aims to capture the incompatibility of noncommuting observables as manifest in the nonorthogonality of their eigenstates. We prove that this measure has all the desired properties: It is zero when the observables commute, strictly greater than zero when they do not, and is maximum when they are mutually unbiased. We also obtain tight upper bounds on this measure for any $N$ noncommuting observables and compute it exactly when the observables are mutually unbiased.Uncertainty relations are often considered to be a measure of incompatibility of noncommuting observables. However, such a consideration is not valid in general, motivating the need for an alternate measure that applies to any set of noncommuting observables. We present an operational approach to quantifying incompatibility without invoking uncertainty relations. Our measure aims to capture the incompatibility of noncommuting observables as manifest in the nonorthogonality of their eigenstates. We prove that this measure has all the desired properties: It is zero when the observables commute, strictly greater than zero when they do not, and is maximum when they are mutually unbiased. We also obtain tight upper bounds on this measure for any $N$ noncommuting observables and compute it exactly when the observables are mutually unbiased.

arXiv:1301.4762Physical Review AApr 29, 20135 citations

Generalized quantum measurements and approximate simultaneous measurements of noncommuting observables

Horace P. Yuen

Physics Letters AAug 1, 1982101 citations

Optimal Uncertainty Relations for a Single Observable

Haruki Yamashita, Aina Mayumi, Gen Kimura

Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty $Q_ρ(A)$ , with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient $c_{\rm opt}(ρ)$ for which $V_ρ(A)\ge c_{\rm opt}(ρ)Q_ρ(A)$ holds for every observable $A$ , and show that it depends only on the smallest and largest eigenvalues of $ρ$ . The same optimization principle extends to a new power-commutator family $\frac{1}{2}|[ρ^s,A]|^2$ $(s\ge 1/2)$ , providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty $Q_ρ(A)$, with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient $c_{\rm opt}(ρ)$ for which $V_ρ(A)\ge c_{\rm opt}(ρ)Q_ρ(A)$ holds for every observable $A$, and show that it depends only on the smallest and largest eigenvalues of $ρ$. The same optimization principle extends to a new power-commutator family $\frac{1}{2}|[ρ^s,A]|^2$ $(s\ge 1/2)$, providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.

quant-pharXiv:2605.26331May 27, 20260 citations

Continuous Quantum Measurement of Noncommunting Observables

Maicol A. Ochoa, Wolfgang Belzig, Abraham Nitzan

Bulletin of the American Physical SocietyMar 14, 20170 citations

N

noncommuting observables and compute it exactly when the observables are mutually unbiased.

Q_ρ(A)

, with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient

c_{\rm opt}(ρ)

for which

V_ρ(A)\ge c_{\rm opt}(ρ)Q_ρ(A)

holds for every observable

A

, and show that it depends only on the smallest and largest eigenvalues of

ρ

. The same optimization principle extends to a new power-commutator family

\frac{1}{2}|[ρ^s,A]|^2

(s\ge 1/2)

, providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.