Sum Uncertainty Relations Research Articles

Tighter sum uncertainty relations via (α, β, γ) weighted Wigner–Yanase–Dyson skew information

Abstract We establish tighter uncertainty relations for arbitrary finite observables via (α, β, γ) weighted Wigner–Yanase–Dyson ((α, β, γ) WWYD) skew information. The results are also applicable to the (α, γ) weighted Wigner–Yanase–Dyson ((α, γ) WWYD) skew information and the weighted Wigner–Yanase–Dyson (WWYD) skew information. We also present tighter lower bounds for quantum channels and unitary channels via (α, β, γ) modified weighted Wigner–Yanase–Dyson ((α, β, γ) MWWYD) skew information. Detailed examples are provided to illustrate the tightness of our uncertainty relations.

A tomographic approach to the sum uncertainty relation and quantum entanglement in continuous variable systems

Entropic uncertainty relations (EURs) have been examined in various contexts, primarily in qubit systems, including their links with entanglement, as they subsume the Heisenberg uncertainty principle. With their genesis in the Shannon entropy, EURs find applications in quantum information and quantum optics. EURs are state-dependent, and the state has to be reconstructed from tomograms (which are histograms readily available from experiments). This is a challenge when the Hilbert space is large, as in continuous variable (CV) systems and certain hybrid quantum (HQ) systems. A viable alternative approach therefore is to extract as much information as possible about the unknown quantum state directly from appropriate tomograms. Many variants of EURs can be extracted from tomograms, even for CV systems. In earlier work we have defined many tomographic entanglement indicators (TEIs) that can be readily calculated from tomograms without knowledge of the density matrix, and have reported on their efficacy as entanglement indicators in various contexts in both CV and HQ systems. The specific objectives of the present work are as follows: (i) To use the tomographic approach to investigate the links between EURs and TEIs in CV and HQ systems as they evolve in time. (ii) To identify the TEI that most closely tracks the temporal evolution of EURs. We consider two generic systems. The first is a multilevel atom modeled as a nonlinear oscillator interacting with a quantized radiation field. The second is the Λ-atom interacting with two radiation fields. The former model accomodates investigations on the role of the initial state of the field and the ratio of the strengths of interaction and nonlinearity in the connection between TEIs and EURs. The second model opens up the possibility of examining the connection between mixed state bipartite entanglement and EURs, when the number of atomic levels is finite. Since the tomogram respects the requirements of classical probability theory, this effort also sheds light on the extent to which TEIs reflect the temporal behaviour of those EURs which are rooted in the Shannon entropy.

Tighter sum uncertainty relations via metric-adjusted skew information

In this paper, we first provide three general norm inequalities, which are used to give new uncertainty relations of any finite observables and quantum channels via metric-adjusted skew information. The results are applicable to its special cases as Wigner-Yanase-Dyson skew information. In quantifying the uncertainty of channels, we discuss two types of lower bounds and compare the tightness between them, meanwhile, a tight lower bound is given. The uncertainty relations obtained by us are stronger than the existing ones. To illustrate our results, we give several specific examples.

Tighter uncertainty relations based on modified weighted Wigner–Yanase–Dyson skew information of quantum channels

We use a novel formation to illustrate the () modified weighted Wigner–Yanase–Dyson (() MWWYD) skew information of quantum channels. By using operator norm inequalities, we explore the sum uncertainty relations for arbitrary N quantum channels and for unitary channels. These uncertainty inequalities are shown to be tighter than the existing ones by a detailed example. Our results are also applicable to the MWWYD skew information and the () modified weighted Wigner–Yanase–Dyson (() MWWYD) skew information of quantum channels as special cases.

Sum Uncertainty Relations Based on (α,β,γ) Weighted Wigner-Yanase-Dyson Skew Information

We introduce (α,β,γ) weighted Wigner-Yanase-Dyson ((α,β,γ) WWYD) skew information and (α,β,γ) modified weighted Wigner-Yanase-Dyson ((α,β,γ) MWWYD) skew information. We explore the sum uncertainty relations for arbitrary N mutually noncommutative observables based on (α,β,γ) WWYD skew information. A series of uncertainty inequalities are derived. We show by detailed example that our results cover and improve the previous ones based on the original Wigner-Yanase (WY) skew information. Finally, we establish new sum uncertainty relations in terms of the (α,β,γ) MWWYD skew information for arbitrary N quantum channels.

Correction to: Sum Uncertainty Relations Based on (α,β,γ) Weighted Wigner-Yanase-Dyson Skew Information

Correction to: Sum Uncertainty Relations Based on (α,β,γ) Weighted Wigner-Yanase-Dyson Skew Information

Product and sum uncertainty relations based on metric-adjusted skew information

The metric-adjusted skew information establishes a connection between the geometrical formulation of quantum statistics and the measures of quantum information. We study uncertainty relations in product and summation forms of metric-adjusted skew information. We present lower bounds on product and summation uncertainty inequalities based on metric-adjusted skew information via operator representation of observables. Explicit examples are provided to back our claims.

Local sum uncertainty relations for angular momentum operators of bipartite permutation symmetric systems

We show that violation of the variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi [Phys. Rev. A 68 032103 (2003)], reflects entanglement in the equal bipartitions of an N-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A 364 203 (2007)] of the two-qubit reduced system of a permutation symmetric N-qubit state.

Improving sum uncertainty relations with the quantum Fisher information

We show how preparation uncertainty relations that are formulated as sums of variances may be tightened by using the quantum Fisher information to quantify quantum fluctuations. We apply this to derive stronger angular momentum uncertainty relations, which in the case of spin-$1/2$ turn into equalities involving the purity. Using an analogy between pure-state decompositions in the Bloch sphere and the moment of inertia of rigid bodies, we identify optimal decompositions that achieve the convex- and concave-roof decomposition of the variance. Finally, we illustrate how these results may be used to identify the classical and quantum limits on phase estimation precision with an unknown rotation axis.

Erratum: Tighter sum uncertainty relations based on metric-adjusted skew information [Phys. Rev. A 104 , 052414 (2021)

Received 17 December 2021DOI:https://doi.org/10.1103/PhysRevA.104.069905©2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum channelsQuantum information processingQuantum Information

Tighter sum uncertainty relations via variance and Wigner–Yanase skew information for N incompatible observables

We study the sum uncertainty relations based on variance and skew information for arbitrary finite N quantum mechanical observables. We derive new uncertainty inequalities which improve the exiting results about the related uncertainty relations. Detailed examples are provided to illustrate the advantages of our uncertainty inequalities.

Tighter sum uncertainty relations based on metric-adjusted skew information

Uncertainty principle is one of the most essential features of quantum mechanics, and it reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, we focus on the uncertainty relations based on metric-adjusted skew information. By making full use of the norm property, we establish an uncertainty relation for arbitrary finite $n$ observables based on metric-adjusted skew information, and we give two new uncertainty relations for arbitrary finite $N$ quantum channels based on metric-adjusted skew information. For arbitrary two observables and two channels, the uncertainty relations we give are not only better than the uncertainty relations detailed in [Quantum Inf. Process. 20, 72 (2021)], but also are the equations. The equation of the uncertainty relation is more accurate than the inequation of the usual uncertainty relation, which has important advantages in the application of quantum information technology, such as quantum communication and the quantum precision measurement. Meanwhile, our results are suitable to Wigner-Yanase-Dyson skew information that is a special metric-adjusted skew information and Wigner-Yanase skew information that is a special Wigner-Yanase-Dyson skew information. Some examples about Wigner-Yanase-Dyson skew information are given and show that the new lower bounds are tighter. The results play an important role in quantum information processing in this paper.

A note on uncertainty relations of arbitrary N quantum channels

Uncertainty principle plays a vital role in quantum physics. The Wigner–Yanase skew information characterizes the uncertainty of an observable with respect to the measured state. We generalize the uncertainty relations for two quantum channels to arbitrary N quantum channels based on Wigner–Yanase skew information. We illustrate that these uncertainty inequalities are tighter than the existing ones by detailed examples. Especially, we also discuss the uncertainty relations for N unitary channels, which could be regarded as variance-based sum uncertainty relations with respect to any pure state.

An uncertainty view on complementarity and a complementarity view on uncertainty

Since the uncertainty about an observable of a system prepared in a quantum state is usually described by its variance, when the state is mixed, the variance is a hybrid of quantum and classical uncertainties. Besides that, complementarity relations are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate the complementarity relation and can even reach zero for a maximally mixed state. So, to fully characterize a quanton it is not sufficient to consider its wave-particle aspect; one has also to regard its correlations with other systems. In this paper, we discuss the relation between quantum correlations and local classical uncertainty measures, as well as the relation between quantum coherence and quantum uncertainty quantifiers. We obtain a complete complementarity relation for quantum uncertainty, classical uncertainty, and predictability. The total quantum uncertainty of a d-paths interferometer is shown to be equivalent to the Wigner-Yanase coherence and the corresponding classical uncertainty is shown to be a quantum correlation quantifier. The duality between complementarity and uncertainty is used to derive quantum correlations measures that complete the complementarity relations for $l_1$-norm and $l_2$-norm coherences. Besides, we show that Brukner-Zeilinger's invariant information quantifies both the wave and particle characters of a quanton and we obtain a sum uncertainty relation for the generalized Gell Mann's matrices.

Bound on Efficiency of Heat Engine from Uncertainty Relation Viewpoint.

Quantum cycles in established heat engines can be modeled with various quantum systems as working substances. For example, a heat engine can be modeled with an infinite potential well as the working substance to determine the efficiency and work done. However, in this method, the relationship between the quantum observables and the physically measurable parameters—i.e., the efficiency and work done—is not well understood from the quantum mechanics approach. A detailed analysis is needed to link the thermodynamic variables (on which the efficiency and work done depends) with the uncertainty principle for better understanding. Here, we present the connection of the sum uncertainty relation of position and momentum operators with thermodynamic variables in the quantum heat engine model. We are able to determine the upper and lower bounds on the efficiency of the heat engine through the uncertainty relation.

Sum Uncertainty Relations: Uncertainty Regions for Qubits and Qutrits

We investigate the notion of uncertainty region using the variance based sum uncertainty relation for qubits and qutrits.We compare uncertainty region of the qubit (a 2-level system) with that of the qutrit (3-level system) by considering sum uncertainty relation for two non-commuting Pauli-like observables, acting on the two dimensional qubit Hilbert space. We identify that physically valid uncertainty region of a qubit is smaller than that of a qutrit. This implies that an enhanced precision can be achieved in the measurement of incompatible Pauli-like observables acting on the 2-dimensional subspace of a qutrit Hilbert space. We discuss the implication of the reduced uncertainties in the steady states of Lambda, V and Cascade types of 3-level atomic systems. Furthermore, we construct a two-qubit permutation symmetric state, corresponding to a 3-level system and show that the reduction in the sum uncertainty value -- or equivalently, increased uncertainty region of a qutrit system { is a consequence of quantum entanglement in the two-qubit system. Our results suggest that uncertainty region can be used as a dimensional witness.

Sum uncertainty relations based on metric-adjusted skew information

We show that the sum uncertainty relations for Wigner–Yanase skew information introduced in Chen et al. (Quantum Inf Process 15:2639–2648, 2016) can hold for an arbitrary metric adjusted skew information. A refinement of one main result in that paper is formulated via a series of lower bounds consisting of the skew information of any prescribed size of the combinations. We also study the metric-adjusted skew information-based uncertainty relations for quantum channels in the spirit of Fu et al. (Quantum Inf Process 18:258, 2019).

Tighter uncertainty relations based on Wigner-Yanase skew information for observables and channels

Uncertainty principle is the basis of quantum mechanics. It reflects the basic law of the movement of microscopic particles. Wigner-Yanase skew information, as a measure of quantum uncertainties, is used to characterize the intrinsic features of the state and the observable. In this paper, we mainly investigate the sum uncertainty relations for both quantum mechanical observables and quantum channels based on skew information. We establish a new uncertainty relation in terms of skew information for n observables, which is saturated (thus it holds as equality) for two incompatible observables. We also present two uncertainty relations for arbitrary finite N quantum channels by using skew information. Our uncertainty relations have tighter lower bounds than the existing ones. Detailed examples are provided.

Entanglement criterion for four-partite systems based on local sum uncertainty relations

We propose an experimentally feasible criterion for the detection of entanglement in an arbitrary four-partite quantum system. It is written as local sum uncertainty relations for suitably chosen observables of the subsystems. The relations are satisfied by all separable states so that their violation implies the presence of the entanglement. This criterion generalizes the tighter criterion for tripartite systems introduced in our previous work (Y. Akbari-Kourbolagh and M. Azhdargalam, Phys. Rev. A 97, 2018) to the four-partite cases. It can be used to detect the entanglement of both discrete- and continuous-variable quantum states without having to know their complete density matrices. Moreover, in contrast to the well-established positive partial transpose criterion, it is also able to detect bound entangled states. Its efficiency is shown by the examples of four-qubit and Gaussian four-mode states.

Quantifying parallelism of vectors is the quantification of distributed n-party entanglement

The three way distributive entanglement is shown to be related to the parallelism of vectors. Using a measurement based approach we form a set of 2-dimensional vectors, representing the post measurement states of one of the parties. These vectors originate at the same point and have an angular distance between them. The area spanned by a pair of such vectors is a measure of the entanglement of formation. This leads to a geometrical manifestation of the 3-tangle in 2-dimensions, from an inequality in area which generalizes for n-qubits to reveal that the n-tangle also has a planar structure. Quantifying the genuine n-party entanglement in every bi-partition, we show that the genuine n-way entanglement does not manifest in n-tangle. A new quantity geometrically similar to 3-tangle is then introduced that represent the genuine n-way entanglement. Extending our formalism to 3-qutrits, we show that the non locality without entanglement arises from a condition under which the post measurement state vectors of a separable state show parallelism. A connection to non trivial sum uncertainty relation analogous to Maccone and Pati uncertainty relation (Maccone and Pati 2014 Phys. Rev. Lett. 113 260401) is then shown using decomposition of post measurement state vectors along parallel and perpendicular direction of the pre-measurement state vectors.