Minimum Uncertainty States Research Articles

Teleportation of a qubit using quasi-Bell states

In this paper, we study the exotic Landau problem at the classical level where two conserved quantities are derived. At the quantum level, the corresponding quantum operators of the conserved quantities provide two oscillator representations from which we derive two Boson Fock spaces. Using the normalized coherent states which are the minimum uncertainty states on noncommutative configuration space isomorphic to each of the boson Fock space, we form entangled coherent states which are Bell- like states labeled quasi-Bell states. The effect of non-maximality of a quasi-Bell state based quantum channel is investigated in the context of a teleportation of a qubit.

Minimum and maximum quantum uncertainty states for qubit systems

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.

Entropic characterization of stabilizer states and magic states

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.

Uncertainty relations revisited

Introductory courses on quantum mechanics usually include lectures on uncertainty relations, typically the inequality derived by Robertson and, perhaps, other statements. For the benefit of the lecturers, we present a unified approach — well suited for undergraduate teaching — for deriving all standard uncertainty relations: those for products of variances by Kennard, Robertson, and Schrödinger, as well as those for sums of variances by Maccone and Pati. We also give a brief review of the early history of this topic and try to answer why the use of variances for quantifying uncertainty is so widespread, while alternatives are available that can be more natural and more fitting. It is common to regard the states that saturate the Robertson inequality as “minimum uncertainty states” although they do not minimize the variance of one observable, given the variance of another, incompatible observable. The states that achieve this objective are different and can be found systematically.

Uncertainty relation for non-Hermitian systems

We construct an uncertainty relation for arbitrary finite-dimensional $\mathcal{PT}$-invariant non-Hermitian quantum systems within a special inner product framework. This construction is led by ``good observables,'' which are a more general class of operators. We show that the cumulative gain in the quantum Fisher information when measuring two good observables for such non-Hermitian systems is much better than their Hermitian counterpart. Minimum uncertainty states being the best candidates for this gain near the exceptional point supports the intelligent or simultaneous non-Hermitian quantum sensors.

Two-dimensional quantum motion of a levitated nanosphere

We report on the two-dimensional (2D) dynamics of a levitated nanoparticle in an optical cavity. The motion of the nanosphere is strongly coupled to the cavity field by coherent scattering and heavily cooled in the plane orthogonal to the tweezer axis. Due to the characteristics of the 2D motion and the strong optomechanical coupling, the motional sideband asymmetry that reveals the quantum nature of the dynamics is not limited to mere scale factors between Stokes and anti-Stokes peaks, as customary in quantum optomechanics, but assumes a peculiar spectral dependence. We introduce and discuss an effective thermal occupancy that quantifies how close the system is to a minimum uncertainty state and allows us to consistently characterize the particle motion. By rotating the polarization angle of the tweezer beam we tune the system from a 1D cooling regime, where we achieve a best thermal occupancy of $0.51\ifmmode\pm\else\textpm\fi{}0.05$, to a regime in which the fully 2D dynamics of the particle exhibit strong nonclassical properties. We achieve a strong 2D confinement with thermal occupancy of $3.4\ifmmode\pm\else\textpm\fi{}0.4$ along the warmest direction and around one in the orthogonal one. These results represent a major improvement over previous experiments considering both the 1D and the 2D motion and pave the way toward the preparation of tripartite optomechanical entangled states and novel applications to directional force and displacement quantum sensing.

Optomechanical parametric oscillation of a quantum light-fluid lattice

Two-photon coherent states are one of the main building pillars of non-linear and quantum optics. It is the basis for the generation of minimum-uncertainty quantum states and entangled photon pairs, applications not obtainable from standard coherent states or one-photon lasers. Here we describe a fully-resonant optomechanical parametric amplifier involving a polariton condensate in a trap lattice quadratically coupled to mechanical modes. The quadratic coupling derives from non-resonant virtual transitions to extended discrete excited states induced by the optomechanical coupling. Non-resonant continuous wave (cw) laser excitation leads to striking experimental consequences, including the emergence of optomechanically induced inter-site parametric oscillations and inter-site tunneling of polaritons at discrete inter-trap detunings corresponding to sums of energies of the two involved mechanical oscillations (20 and 60 GHz confined vibrations). We show that the coherent mechanical oscillations correspond to parametric resonances with threshold condition different to that of standard linear optomechanical self-oscillation. The associated Arnold tongues display a complex scenario of states within the instability region. The observed new phenomena can have applications for the generation of entangled phonon pairs, squeezed mechanical states relevant in sensing and quantum computation, and for the bidirectional frequency conversion of signals in a technologically relevant range.

Viscous control of minimum uncertainty state in hydrodynamics

A minimum uncertainty state for position and momentum of a fluid element is obtained. We consider a general fluid described by the Navier–Stokes–Korteweg (NSK) equation, which reproduces the behaviors of a standard viscous fluid, a fluid with the capillary action and a quantum fluid, with the proper choice of parameters. When the parameters of the NSK equation is adjusted to reproduce Madelung’s hydrodynamic representation of the Schrödinger equation, the uncertainty relation of a fluid element reproduces the Kennard and the Robertson–Schrödinger inequalities in quantum mechanics. The derived minimum uncertainty state is the generalization of the coherent state and its uncertainty is given by a function of the shear viscosity. The viscous uncertainty can be smaller than the inviscid minimum value when the shear viscosity is smaller than a critical value which is similar in magnitude to the Kovtun–Son–Starinets bound. This uncertainty reflects the information of the fluctuating microscopic degrees of freedom in the fluid and will modify the standard hydrodynamic scenario, for example, in heavy-ion collisions.

Quantum Wigner entropy

We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $ln\ensuremath{\pi}+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-R\'enyi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.

Making squeezed-coherent states concrete by determining their wavefunction

With the successes of the laser interferometer gravitational-wave observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based approach; (ii) an approach that uses an expansion in terms of the simple-harmonic oscillator wavefunctions; and (iii) a fully operator-based approach. We do this to illustrate that the concept of the wavefunction can be introduced no matter what methodology an instructor wishes to use. We hope that working with the wavefunction will help demystify the concept of a squeezed-coherent state.

The Persico equation for minimum uncertainty states

We present an important and forgotten result of fundamental quantum mechanics obtained in 1930 by Enrico Persico, consisting of an eigenvalue equation for minimum uncertainty states.

Zero uncertainty states in the presence of quantum memory

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.

Information-theoretic aspects of Werner states

In a seminal study of quantum states with Einstein–Podolsky–Rosen correlations (entanglement) admitting a hidden-variable model (Werner, 1989), Werner introduced the dichotomy of entanglement/separability and devised a family of highly symmetric states, now termed the Werner states, some of which exhibit entanglement but no Bell nonlocality. It turns out that the Werner states have a rich structure of correlations and constitute a paradigm which has played an innovative role in both theoretical and experimental explorations of quantum information. Given the theoretical significance and wide applications of the Werner states, here we first give a concise review of information contents of the Werner states, and then present an information-theoretic characterization of them in terms of the Wigner–Yanase skew information: The Werner states are identified as the states with the minimum quantum uncertainty with respect to a natural family of observables (i.e., the generators of the diagonal unitary group). For this purpose, we introduce a measure of quantum uncertainty which is of independent interest in studying asymmetry, coherence, and uncertainty, and reveal its fundamental properties. We further identify the Bell triplet states as the opposite states of the Werner states in the sense that they have the maximal amount of quantum uncertainty. Analogously, we provide a similar characterization of the isotropic states as the minimum quantum uncertainty states with respect to a closely related family of operators.

New family of parasupercoherent states: entanglement, uncertainties, and statistical properties

The general parasupersymmetric annihilation operator of arbitrary order does not reduce to the Kornbluth–Zypman general supersymmetric annihilation operator for the first order. In this paper, we introduce an annihilation operator for a parasupersymmetric harmonic oscillator that in the first order matches with the Kornblouth–Zypman results. Then, using the latter operator, we obtain the parasupercoherent states and calculate their entanglement, uncertainties, and statistics. We observe that these states are entangled for any arbitrary order of parasupersymmetry and their entanglement goes to zero for the large values of the coherency parameter. In addition, we find that the maximum of the entanglement of parasupercoherent states is a decreasing function of the parasupersymmetry order. Moreover, these states are minimum uncertainty states for large and also small values of the coherency parameter. Furthermore, these states show squeezing in one of the quadrature operators for a wide range of the coherency parameter, while no squeezing in the other quadrature operator is observed at all. In addition, using the Mandel parameter, we find that the statistics of these new states are subPoissonian for small values of the coherency parameter.

Quantifying non-Gaussianity of bosonic fields via an uncertainty relation

While Gaussian states and associated Gaussian operations are basic ingredients and convenient objects for continuous-variable quantum information, it is also realized that non-Gaussianity is an important resource for quantum information processing. The characterization and quantification of non-Gaussianity have been widely studied in the past decade, with several significant measures for non-Gaussianity introduced. In this work, by exploiting an information-theoretic refinement of the conventional Heisenberg uncertainty relation and a physical characterization of Gaussian states as minimum uncertainty states, we introduce an easily computable measure for non-Gaussianity of bosonic field states in terms of the Wigner-Yanase skew information. Fundamental properties, as well as intuitive meaning, of this measure are unveiled. The concept is illustrated by prototypical non-Gaussian states, and compared with various existent measures for non-Gaussianity. Its merit and physical significance are elucidated.

Gaussian states as minimum uncertainty states

In bosonic fields, Gaussian states, which consist of a rather wide family of states including coherent states, squeezed states, thermal states, etc., have many classical-like features, and are usually defined from the mathematical perspective in terms of characteristic functions. It is well known that some special Gaussian states, such as coherent states, are minimum uncertainty states for the conventional Heisenberg uncertainty relation involving canonical pair of position and momentum observables. A natural question arises as whether all Gaussian states can be characterized as minimum uncertainty states. In this work, we show that indeed Gaussian states coincide with minimum uncertainty states for an information-theoretic refinement of the conventional uncertainty relation established in Luo (2005) [40]. This characterization puts Gaussian states on a novel basis of physical significance.

Untangling entanglement and chaos

We present a method to calculate an upper bound on the generation of entanglement in any spin system using the Fannes-Audenaert inequality for the von Neumann entropy. Our method not only is useful for efficiently estimating entanglement, but also shows that entanglement generation depends on the distance of the quantum states of the system from corresponding minimum-uncertainty spin coherent states (SCSs). We illustrate our method using a quantum kicked top model, and show that our upper bound is a very good estimator for entanglement generated in both regular and chaotic regions. In a deep quantum regime, the upper bound on entanglement can be high in both regular and chaotic regions, while in the semiclassical regime, the bound is higher in chaotic regions where the quantum states diverge from the corresponding SCSs. Our analysis thus explains previous studies and clarifies the relationship between chaos and entanglement.

Universal quantum uncertainty relations: Minimum-uncertainty wave packet depends on measure of spread

Based on the statistical concept of the median, we propose a quantum uncertainty relation between semi-interquartile ranges of the position and momentum distributions of arbitrary quantum states. The relation is universal, unlike that based on the mean and standard deviation, as the latter may become non-existent or ineffective in certain cases. We show that the median-based one is not saturated for Gaussian distributions in position. Instead, the Cauchy-Lorentz distributions in position turn out to be the one with the minimal uncertainty, among the states inspected, implying that the minimum-uncertainty state is not unique but depends on the measure of spread used. Even the ordering of the states with respect to the distance from the minimum uncertainty state is altered by a change in the measure. We invoke the completeness of Hermite polynomials in the space of all quantum states to probe the median-based relation. The results have potential applications in a variety of studies including those on the quantum-to-classical boundary and on quantum cryptography.

Minimum-uncertainty states and completeness of non-negative quasiprobability of finite-dimensional quantum systems

We construct minimum-uncertainty states and a non-negative quasi probability distribution for quantum systems on a finite-dimensional space. We reexamine the theorem of Massar and Spindel for the uncertainty relationof the two unitary operators related by the discrete Fourier transformation. It is shown that some assumptions in their proof can be justified by the use of the Perron-Frobenius theorem. The minimum-uncertainty states are the ones that saturate this uncertainty inequality. The continuum limit is closely analyzed by introducing a scale factor in the limiting scheme. Using the minimum-uncertainty states, we construct a non-negative quasi probability distribution. Its marginal distributions are smeared out. However, we show that this quasi probability is optimal in the sense that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance in the phase space is assumed. Generally, it is desirable that the quasi probability is complete, i.e., it contains full information of the state. We show that the obtained quasi probability is indeed complete if the dimension of the state space is odd, whereas it is not if the dimension is even.

Strong Unitary and Overlap Uncertainty Relations: Theory and Experiment.

We derive and experimentally investigate a strong uncertainty relation valid for any n unitary operators, which implies the standard uncertainty relation and others as special cases, and which can be written in terms of geometric phases. It is saturated by every pure state of any n-dimensional quantum system, generates a tight overlap uncertainty relation for the transition probabilities of any n+1 pure states, and gives an upper bound for the out-of-time-order correlation function. We test these uncertainty relations experimentally for photonic polarization qubits, including the minimum uncertainty states of the overlap uncertainty relation, via interferometric measurements of generalized geometric phases.