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

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Uncertainty, entropy, and non-Gaussianity for mixed states

Aikaterini Mandilara, Evgueni Karpov, Nicolas J. Cerf

Published Apr 30, 2010• 2 citations
Publisher

Abstract

In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated. Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded uncertainty relation by adding an additional parameter characterizing the state, namely its degree of non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation. The common points and differences between the two extensions of the uncertainty relation help us to draw more general conclusions concerning the bounds on the non-Gaussianity of mixed states.

DOI
10.1117/12.854750

Related papers

Non-Gaussianity and entropy-bounded uncertainty relations: Application to detection of non-Gaussian entangled states

Kyunghyun Baek, Hyunchul Nha

We suggest an improved version of the Robertson--Schr\"odinger uncertainty relation for canonically conjugate variables by taking into account a pair of characteristics of states: non-Gaussianity and mixedness quantified by using fidelity and entropy, respectively. This relation is saturated by both Gaussian and Fock states and provides a strictly improved bound for any non-Gaussian states or mixed states. For the case of Gaussian states, it is reduced to the entropy-bounded uncertainty relation derived by Dodonov. Furthermore, we consider readily computable measures of both characteristics and find a weaker but more readily accessible bound. With its generalization to the case of two-mode states, we show applicability of the relation to detect entanglement of non-Gaussian states.

arXiv:1811.10207•Physical Review A•Oct 10, 2018•14 citations

Purity-bounded uncertainty relations in multidimensional space—generalized purity

Mikalai U. Karelin

Uncertainty relations for mixed quantum states (precisely, purity-bounded position–momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in the general multidimensional case. An expression for a family of mixed states at the lower bound of uncertainty relation is obtained. It is shown that in the case of entropy-bounded uncertainty relations, the lower-bound state is thermal, and a transition from the one-dimensional problem to the multidimensional one is trivial. Results of numerical calculation of the relation of the lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.

arXiv:quant-ph/0411067•Journal of Physics A Mathematical and General•Jun 29, 2005•3 citations

Purity- and Gaussianity-bounded uncertainty relations

Aikaterini Mandilara, Evgueni Karpov, Nicolas J. Cerf

Bounded uncertainty relations provide the minimum value of the uncertainty assuming some additional information on the state. We derive analytically an uncertainty relation bounded by a pair of constraints, those of purity and Gaussianity. In a limiting case this uncertainty relation reproduces the purity-bounded derived by Man'ko and Dodonov and the Gaussianity-bounded one (Mandilara and Cerf 2012 Phys. Rev. A 86 030102R).

arXiv:0910.3473•Journal of Physics A Mathematical and Theoretical•Jan 7, 2014•7 citations

Role of quantum non-Gaussian distance in entropic uncertainty relations

Wonmin Son

A Gaussian distribution of a quantum state with continuous spectra is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{p}$ are featured explicitly. From the construction of the entropic uncertainty relation, it follows in a transparent manner that (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modified form, ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}\ensuremath{\hbar}{e}^{\mathcal{N}}/2$; (ii) the additional factor $\mathcal{N}$ quantifies the quantum non-Gaussianity of the probability distributions of two observables; and (iii) the lower bound of the entropic uncertainty relation for a non-Gaussian continuous-variable (CV) mixed state becomes stronger with purity. The optimality of specific non-Gaussian CV states for the refined uncertainty relation has been investigated and the existence of a new class of CV quantum state is identified.

arXiv:1504.05725•Physical Review A•Jul 17, 2015•14 citations

Purity- and entropy-bounded uncertainty relations for mixed quantum states

V. V. Dodonov

We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr 2 and 37% if the `superpurity' lim r →0 [Tr ( 1+1/ r )] r is chosen as a measure of deviation).

Journal of Optics B Quantum and Semiclassical Optics•Apr 5, 2002•68 citations

Quantum uncertainty of mixed states based on skew information

Shunlong Luo

The uncertainty of a mixed state has two quite different origins: classical mixing and quantum randomness. While the classical aspect (mixedness) is significantly quantified by the von Neumann entropy, it seems that we still do not have a well accepted measure of quantum uncertainty. In terms of the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurements, we will propose an intrinsic measure for synthesizing quantum uncertainty of a mixed state and investigate its fundamental properties. We illustrate how it arises naturally from a naive hidden-variable approach to entanglement and how it exhibits a simple relation to the notion of negativity, which is an entanglement monotone introduced quite recently. We further show that it has a dramatic nonextensive feature resembling the probability law relating operations of two events. This measure of quantum uncertainty provides an alternative quantity complementary to the von Neumann entropy for studying mixedness and quantum correlations.

Physical Review A•Feb 14, 2006•97 citations
In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated. Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded uncertainty relation by adding an additional parameter characterizing the state, namely its degree of non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation. The common points and differences between the two extensions of the uncertainty relation help us to draw more general conclusions concerning the bounds on the non-Gaussianity of mixed states.
We suggest an improved version of the Robertson--Schr\"odinger uncertainty relation for canonically conjugate variables by taking into account a pair of characteristics of states: non-Gaussianity and mixedness quantified by using fidelity and entropy, respectively. This relation is saturated by both Gaussian and Fock states and provides a strictly improved bound for any non-Gaussian states or mixed states. For the case of Gaussian states, it is reduced to the entropy-bounded uncertainty relation derived by Dodonov. Furthermore, we consider readily computable measures of both characteristics and find a weaker but more readily accessible bound. With its generalization to the case of two-mode states, we show applicability of the relation to detect entanglement of non-Gaussian states.
Uncertainty relations for mixed quantum states (precisely, purity-bounded position–momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in the general multidimensional case. An expression for a family of mixed states at the lower bound of uncertainty relation is obtained. It is shown that in the case of entropy-bounded uncertainty relations, the lower-bound state is thermal, and a transition from the one-dimensional problem to the multidimensional one is trivial. Results of numerical calculation of the relation of the lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.
A Gaussian distribution of a quantum state with continuous spectra is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{p}$ are featured explicitly. From the construction of the entropic uncertainty relation, it follows in a transparent manner that (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modified form, ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}\ensuremath{\hbar}{e}^{\mathcal{N}}/2$; (ii) the additional factor N\mathcal{N}N quantifies the quantum non-Gaussianity of the probability distributions of two observables; and (iii) the lower bound of the entropic uncertainty relation for a non-Gaussian continuous-variable (CV) mixed state becomes stronger with purity. The optimality of specific non-Gaussian CV states for the refined uncertainty relation has been investigated and the existence of a new class of CV quantum state is identified.
We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr 2 and 37% if the `superpurity' lim r →0 [Tr ( 1+1/ r )] r is chosen as a measure of deviation).
The uncertainty of a mixed state has two quite different origins: classical mixing and quantum randomness. While the classical aspect (mixedness) is significantly quantified by the von Neumann entropy, it seems that we still do not have a well accepted measure of quantum uncertainty. In terms of the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurements, we will propose an intrinsic measure for synthesizing quantum uncertainty of a mixed state and investigate its fundamental properties. We illustrate how it arises naturally from a naive hidden-variable approach to entanglement and how it exhibits a simple relation to the notion of negativity, which is an entanglement monotone introduced quite recently. We further show that it has a dramatic nonextensive feature resembling the probability law relating operations of two events. This measure of quantum uncertainty provides an alternative quantity complementary to the von Neumann entropy for studying mixedness and quantum correlations.