Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Minimum and maximum quantum uncertainty states for qubit systems

Huihui Li, Shunlong Luo, Yue Zhang

Published Mar 11, 2024• 5 citations
Publisher

Abstract

Abstract We introduce the notion of (renormalized) quantum uncertainty and reveal its basic features. In terms of this quantity, we completely characterize the minimum and maximum quantum uncertainty states for qubit systems involving Pauli matrices. It turns out that the minimum quantum uncertainty states consist of both certain pure states and certain mixed states, in sharp contrast to the case of conventional Heisenberg uncertainty relation. The maximum quantum uncertainty states are H-type magic states arising from the stabilizer formalism of quantum computation, and can be obtained from minimum quantum uncertainty states via the T-gate.

DOI
10.1088/1751-8121/ad32a0

Related papers

Characterization of stabilizier states and magic states in terms of Tsallis and Rényi entropies for qubit systems

Jiayu He, Bowen Wang, Shuangshuang Fu

Abstract Uncertainty relations are fundamental in quantum mechanics, distinguishing it from classical physics by setting limits on the precision of incompatible measurements. For qubit systems, considering the three incompatible Pauli observables, we propose two entropic quantifiers of uncertainty exploiting the Tsallis entropies and Rényi entropies. We analyze the minimum and maximum uncertainty states and uncover their close connections with some important families of states in the stabilizer formalism. Explicitly, our findings show that for the two entropic quantifiers of uncertainties with suitable parameters, the stabilizer states correspond to the minimum uncertainty states, while the T-type magic states correspond to the maximum uncertainty states. Additionally, we identify that the H-type magic states serve as the saddle points in the Tsallis entropy framework. These results expand the recent proposed characterizations of stabilizer and magic states via the Shannon entropic uncertainty relation [Li et al 2024 Phys. Scr. 99, 035117].

Physica Scripta•Dec 10, 2024•1 citation

Spin minimum uncertainty states for refined uncertainty relations

Hao Dai, Yue Zhang

Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.

quant-pharXiv:2512.17307•Journal of Physics A Mathematical and Theoretical•Dec 22, 2025•0 citations

Entropic characterization of stabilizer states and magic states

Huihui Li, Shunlong Luo, Yue Zhang

Abstract Quantum states with minimum or maximum uncertainty are of special significance due to their extreme properties. Celebrated examples are coherent states induced from certain Lie groups and intelligent states for various uncertainty relations. In this work, by virtue of the Maassen-Uffink entropic uncertainty relation, we introduce an entropic quantifier of uncertainty and use it to characterize several important families of states in the stabilizer formalism of quantum computation. More specifically, we show that the stabilizer states and T-type magic states stand at the two extremes of the entropic quantifier of uncertainty: The former are precisely the minimum entropic uncertainty states, while the latter are precisely the maximum entropic uncertainty states. Moreover, interpolating between the above two extremes, the H-type magic states are the saddle points of the entropic quantifier of uncertainty. These entropic characterizations reveal some intrinsic features of stabilizer states, H- and T-type magic states, and cast novel light on the resource-theoretic viewpoint of regarding the stabilizer states as free states and the T-type magic states as the most precious source states in the stabilizer quantum theory.

Physica Scripta•Feb 12, 2024•6 citations

Zero uncertainty states in the presence of quantum memory

Huangjun Zhu

Abstract The uncertainty principle imposes a fundamental limit on predicting the measurement outcomes of incompatible observables even if complete classical information of the system state is known. The situation is different if one can build a quantum memory entangled with the system. Zero uncertainty states (in contrast with minimum uncertainty states) are peculiar quantum states that can eliminate uncertainties of incompatible von Neumann observables once assisted by suitable measurements on the memory. Here we determine all zero uncertainty states of any given set of nondegenerate observables and determine the minimum entanglement required. It turns out all zero uncertainty states are maximally entangled in a generic case, and vice versa, even if these observables are only weakly incompatible. Our work establishes a simple and precise connection between zero uncertainty and maximum entanglement, which is of interest to foundational studies and practical applications, including quantum certification and verification.

arXiv:2003.02103•npj Quantum Information•Mar 8, 2021•9 citations

Minimum uncertainty states in the presence of quantum memory

Huangjun Zhu

The uncertainty principle imposes a fundamental limit on predicting the measurement outcomes of incompatible observables even if complete classical information of the system state is known. The situation is different if one can build a quantum memory entangled with the system. Minimum uncertainty states are peculiar quantum states that can eliminate uncertainties of incompatible von Neumann observables once assisted by suitable measurements on the memory. Here we determine all minimum uncertainty states of any given set of observables and determine the minimum entanglement required. It turns out all minimum uncertainty states are maximally entangled in a generic case, and vice versa, even if these observables are only weakly incompatible. Our work establishes a precise connection between minimum uncertainty and maximum entanglement, which is of interest to foundational studies and practical applications, including quantum certification and verification.

•Mar 4, 2020•1 citation

Minimum-cost quantum measurements for quantum information

Petros Wallden, Vedran Dunjko, Erika Andersson

Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.

arXiv:1312.5205•Journal of Physics A Mathematical and Theoretical•Mar 10, 2014•18 citations
Abstract We introduce the notion of (renormalized) quantum uncertainty and reveal its basic features. In terms of this quantity, we completely characterize the minimum and maximum quantum uncertainty states for qubit systems involving Pauli matrices. It turns out that the minimum quantum uncertainty states consist of both certain pure states and certain mixed states, in sharp contrast to the case of conventional Heisenberg uncertainty relation. The maximum quantum uncertainty states are H-type magic states arising from the stabilizer formalism of quantum computation, and can be obtained from minimum quantum uncertainty states via the T-gate.
Abstract Uncertainty relations are fundamental in quantum mechanics, distinguishing it from classical physics by setting limits on the precision of incompatible measurements. For qubit systems, considering the three incompatible Pauli observables, we propose two entropic quantifiers of uncertainty exploiting the Tsallis entropies and Rényi entropies. We analyze the minimum and maximum uncertainty states and uncover their close connections with some important families of states in the stabilizer formalism. Explicitly, our findings show that for the two entropic quantifiers of uncertainties with suitable parameters, the stabilizer states correspond to the minimum uncertainty states, while the T-type magic states correspond to the maximum uncertainty states. Additionally, we identify that the H-type magic states serve as the saddle points in the Tsallis entropy framework. These results expand the recent proposed characterizations of stabilizer and magic states via the Shannon entropic uncertainty relation [Li et al 2024 Phys. Scr. 99, 035117].
Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.
Abstract Quantum states with minimum or maximum uncertainty are of special significance due to their extreme properties. Celebrated examples are coherent states induced from certain Lie groups and intelligent states for various uncertainty relations. In this work, by virtue of the Maassen-Uffink entropic uncertainty relation, we introduce an entropic quantifier of uncertainty and use it to characterize several important families of states in the stabilizer formalism of quantum computation. More specifically, we show that the stabilizer states and T-type magic states stand at the two extremes of the entropic quantifier of uncertainty: The former are precisely the minimum entropic uncertainty states, while the latter are precisely the maximum entropic uncertainty states. Moreover, interpolating between the above two extremes, the H-type magic states are the saddle points of the entropic quantifier of uncertainty. These entropic characterizations reveal some intrinsic features of stabilizer states, H- and T-type magic states, and cast novel light on the resource-theoretic viewpoint of regarding the stabilizer states as free states and the T-type magic states as the most precious source states in the stabilizer quantum theory.
Abstract The uncertainty principle imposes a fundamental limit on predicting the measurement outcomes of incompatible observables even if complete classical information of the system state is known. The situation is different if one can build a quantum memory entangled with the system. Zero uncertainty states (in contrast with minimum uncertainty states) are peculiar quantum states that can eliminate uncertainties of incompatible von Neumann observables once assisted by suitable measurements on the memory. Here we determine all zero uncertainty states of any given set of nondegenerate observables and determine the minimum entanglement required. It turns out all zero uncertainty states are maximally entangled in a generic case, and vice versa, even if these observables are only weakly incompatible. Our work establishes a simple and precise connection between zero uncertainty and maximum entanglement, which is of interest to foundational studies and practical applications, including quantum certification and verification.
The uncertainty principle imposes a fundamental limit on predicting the measurement outcomes of incompatible observables even if complete classical information of the system state is known. The situation is different if one can build a quantum memory entangled with the system. Minimum uncertainty states are peculiar quantum states that can eliminate uncertainties of incompatible von Neumann observables once assisted by suitable measurements on the memory. Here we determine all minimum uncertainty states of any given set of observables and determine the minimum entanglement required. It turns out all minimum uncertainty states are maximally entangled in a generic case, and vice versa, even if these observables are only weakly incompatible. Our work establishes a precise connection between minimum uncertainty and maximum entanglement, which is of interest to foundational studies and practical applications, including quantum certification and verification.
Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.