Measure Of Quantum Uncertainty Research Articles

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.

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.

On the Justification and Validity of the Kennard Inequality

In 1927, Earle Hesse Kennard derived an inequality describing Heisenberg’s uncertainty principle. Since then, we have traditionally been using the standard deviation as the measure of uncertainty in quantum mechanics. But Jan Hilgevoord asserts that the standard deviation is neither a natural nor a generally adequate measure of quantum uncertainty. Specifically, he asserts that the standard deviations are inadequate to use as the quantum uncertainties in the single- and double-slit diffraction experiments. He even tells that from these examples it will become clear that the standard deviation is the wrong concept to express the uncertainty principle generally and that the Kennard relation has little to do with the uncertainty principle. We will investigate what are adequate as the measures of quantum uncertainty. And, beyond that, we will investigate the effects of multiplying the two uncertainties; namely, characteristics which is hiding in deep interior of the Kennard inequality. Through investigations we’ll come to naturally realize that his assertions were wrong. All of our discussions will help raise understanding of the Heisenberg uncertainty principle. Our discussions will afford us an opportunity to think about the essence of the Fourier transform. The aim of this paper is to draw conclusions about whether the Kennard inequality is justified or not.

Measurability of the polynomial invariant of degree 2 for even-N qubit states

Experimentally quantification of the entanglement measures due to some unphysical properties in their definition is a difficult and important problem in quantum information theory. In this paper, we show that for even N-partite pure states of qubits, the polynomial invariant of degree 2 as the measure of entanglement has a physical interpretation, since it is experimentally measurable by the local projective measurements. Then, we consider the special cases of even-N qubit rank 2 mixed state and show that the polynomial invariant of degree 2 can be written in terms of Wigner–Yanase information as a measure of quantum uncertainty of mixed states.

Quantum coherence versus quantum uncertainty

The notion of measurement is of both foundational and instrumental significance in quantum mechanics, and coherence destroyed by measurements (decoherence) lies at the very heart of quantum to classical transition. Qualitative aspects of this spirit have been widely recognized and analyzed ever since the inception of quantum theory. However, axiomatic and quantitative investigations of coherence are attracting great interest only recently with several figures of merit for coherence introduced [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. While these resource theoretic approaches have many appealing and intuitive features, they rely crucially on various notions of incoherent operations which are sophisticated, subtle, and not uniquely defined, as have been critically assessed [Chitambar and Gour, Phys. Rev. Lett. 117, 030401 (2016)]. In this paper, we elaborate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate, and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of a measurement (with respect to a state). Consequently, coherence measures may be set into correspondence with measures of quantum uncertainty. In particular, we take average quantum Fisher information as a measure of quantum uncertainty, and introduce the corresponding measure of coherence, which is demonstrated to exhibit desirable properties. Implications for interpreting quantum purity as maximal coherence, and quantum discord as minimal coherence, are illustrated.

Local quantum uncertainty and bounds on quantumness for orthogonally invariant class of states

Local quantum uncertainty (in short LQU) was introduced by Girolami et. al.(Phy. Rev. Lett. \textbf{110}, 240402) as a measure of quantum uncertainty in a quantum state as achievable on single local measurement. However, such quantity do satisfy all necessary criteria to serve as measure of discord like quantum correlation and it has no closed formula except only for $2\otimes n$ system. Here, we consider orthogonal invariant class of states which includes both the Werner and Isotropic class of states and explore the possibility of closed form formula. Further, we extend our quest to the possibility of closed form of geometric discord and measurement induced nonlocality for this class. We also provide a comparative study of the bounds of general quantum correlations with entanglement, as measured by negativity, for an interesting subclass of states.

Collision entropy and optimal uncertainty

We propose an alternative measure of quantum uncertainty for pairs of arbitrary observables in the 2-dimensional case, in terms of collision entropies. We derive the optimal lower bound for this entropic uncertainty relation, which results in an analytic function of the overlap of the corresponding eigenbases. Besides, we obtain the minimum uncertainty states. We compare our relation with other formulations of the uncertainty principle.

Quantum uncertainty of mixed states based on skew information

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.

Quantum versus classical uncertainty

The uncertainty of an observable in a quantum state is usually described by variance. This description is well suited when the states are pure. But when the states are mixed, things become subtle, and the variance is a hybrid of quantum and classical uncertainties. Motivated by the notion of Fisher information in statistical inference, we establish a decomposition of the variance into quantum and classical parts. The key observation is that the Wigner-Yanase skew information (a distinguished version of quantum Fisher information) can be interpreted as a measure of quantum uncertainty. We also establish a decomposition of the conventional covariance into quantum and classical parts. The results provide a new perspective for understanding uncertainty and correlation and are used to quantify entanglement, as well as to establish a new uncertainty relation in purely quantum terms.

On Skew Information

In this paper, we show that skew information introduced by Wigner and Yanase, which is a natural informational extension of variance for pure states, can be interpreted as a measure of quantum uncertainty. By virtue of skew information, we establish a new uncertainty relation in the spirit of Schrodinger, which incorporates both incompatibility (encoded in the commutator) and correlations (encoded in a new correlation measure related to skew information) between observables, and moreover is stronger than the conventional ones.

The standard deviation is not an adequate measure of quantum uncertainty

The standard deviation is not an adequate measure of quantum uncertainty

Generalized entropy as a measure of quantum uncertainty

A generalized entropy is used in order to advance a different form of expressing the Uncertainty Principle of Quantum mechanics. We consider the generalized entropic formulation for different pairs of incompatible observables. In particular, we study the number-phase entropic uncertainty measure for the case of coherent states within the Pegg-Barnett theory. We also tackle the situation of operators with continuous spectra, where a correlation functional is calculated in terms of generalized joint and marginal entropies, for harmonic oscillator wavefunctions.