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Uncertainty principle reveals the intrinsic differences between the classical and quantum worlds, which plays a significant role in quantum information theory. By using ρ -absolute variance, we introduce the uncertainty of quantum channels and explore its properties. By using Cauchy–Schwarz inequality and the parallelogram law, we establish the product and summation forms of the uncertainty relations for arbitrary two quantum channels, respectively. The summation form of the uncertainty inequalities based on the ρ -absolute variance for arbitrary N quantum channels is also investigated, and the optimal lower bounds are presented. We illustrate our results by several typical examples. [ABSTRACT FROM AUTHOR]

In this paper, we construct stronger lower bounds for the variance-based uncertainty relations of two unitary operators, via the power-mean inequalities. It is proved that these bounds are tighter in any interval, compared to the bounds given by Bong et al. (Phys Rev Lett 120:230402, 2018) and by Yu et al. (Phys Rev A 100:022116, 2019). Moreover, we build descending sequences of lower bounds for the uncertainty relations of two or three unitary operators. For illustration, some explicit examples are given to show the tightness of our bounds, in both pure and mixed state cases. [ABSTRACT FROM AUTHOR]

Uncertainty principle is one of the most fundamental features in quantum mechanics and plays a significant role in quantum information processing. We establish tighter summation form of the uncertainty relations based on metric-adjusted skew information via operator representation of observables, which improves the existing results. By employing the methodologies of sampling coordinates of observables, we also present tighter product form of the uncertainty relations. Detailed examples are given to illustrate the advantages of our uncertainty relations. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

Heisenberg's uncertainty principle is originally represented by the tradeoff relations between the error of measuring an observable A and the consequential disturbance to the value of another observable B with respect to a particular quantum state. Many different error-disturbance tradeoff relations have been proposed and experimentally tested in various scenarios. The error-disturbance uncertainty relations is incorporated into multiparameter estimation theory recently, but they are all the tradeoff relations between the errors of measuring two observables. Here, we derive the error-disturbance tradeoff relations for three observables A, B and C and demonstrate that the new relation is violated, while the relations derived from triple of two observables A and B, A and C and B and C directly are valid by some examples. [ABSTRACT FROM AUTHOR]

The uncertainty relation is considered to be one of key features of quantum mechanics which distinguishes quantum and classical systems. Recently, we developed a new formulation of the uncertainty relation based on the generalized scheme of variational principle, the stochastic variational method (SVM). In this method, the uncertainty relation is related to the nondifferentiability of observables and thus can be obtained even in classical stochastic systems. This new formulation resolves the famous paradox in quantum mechanics, the angular uncertainty relation without introducing artificial assumptions. In this paper, we show that the fluctuations of position and momentum for a nonrelativistic viscous fluid element satisfies the uncertainty relation analogous to the corresponding quantum mechanical one. Such a fluctuation is sensitive to the temperature gradient at the freezeout surface and can affect the collective flow anisotropy in relativistic heavy‐ion collisions. [ABSTRACT FROM AUTHOR]

In this paper, we use the geometric-arithmetic mean inequality to build stronger uncertainty relations based on the Wigner-Yanase skew information of N observables. It is proved that, in comparison to the lower bounds given by Zhang et al. (Phys. Lett. A 387, 127029 2021) and by Zhang and Fei (Quantum Inf. Process. 20(12), 384 2021), our bounds are stronger in any interval. In addition, we construct two descending sequences of lower bounds for the uncertainty relations of N observables. Detailed examples are provided to show the advantages of our bounds. [ABSTRACT FROM AUTHOR]

The variance of quantum channels involving a mixed state gives a hybrid of classical and quantum uncertainties. We seek certain decomposition of variance into classical and quantum parts in terms of the Wigner–Yanase skew information. Generalizing the uncertainty relations for quantum observables to quantum channels, we introduce a new quantity with better quantum mechanical nature to describe the uncertainty relations for quantum channels. We derive several uncertainty relations for quantum channels via variances and the Wigner–Yanase skew information. [ABSTRACT FROM AUTHOR]

Owing to the devise applications of molecules in industries, the bound state solution of the non-relativistic wave equation with a molecular potential function has been obtained in a closed-form using the Nikiforov-Uvarov method. The solutions of the bound state are then applied to study the information-theoretic measures such as the one-dimensional Shannon and Renyi entropic densities. The expectation values for the position and momentum spaces were obtained to verify the Heisenberg's uncertainty principle. Utilizing the energy spectrum equation, the thermodynamic vibrational partition function is obtained via the Poisson summation. Other thermodynamic function variations with absolute temperature have been obtained numerically for four diatomic molecules (H2, N2, O2, and HF) using Maple 18 software. The Shannon global entropic sum inequality has also been verified. The Renyi sum for constrained index parameters satisfies the global entropic inequality. The thermodynamic properties of the four molecules are similar and conform to works reported in the existing literature. The obtained vibrational energies are in fair agreement with the ones obtained using other forms of potential energy. The result further indicates that the lowest bounds for the Shannon, Renyi, and Heisenberg inequalities are ground states phenomena. [ABSTRACT FROM AUTHOR]

Finite tight frames are interesting in various topics including questions of quantum information. Each complex tight frame leads to a resolution of the identity in the Hilbert space. Symmetric informationally complete measurements are a special class of equiangular tight frames. Applications of such frames in quantum physics deserve more attention than they have obtained. We derive uncertainty relations for a quantum measurement assigned to an equiangular tight frame. Main results follow from estimation of the corresponding index of coincidence. State-dependent and state-independent formulations are both addressed. Also, we discuss applications of considered measurements to detect entanglement and other correlations. [ABSTRACT FROM AUTHOR]

Quantum theory provides us with the best account of microscopic components of matter as well as of radiation. It was introduced in the twentieth century and has experienced a wide range of success. Although the theory's probabilistic predictions of final experimental outcomes is found to be correct with high precision, there is no general consensus regarding what is actually going on with a quantum system "en route", or rather the perceivable intermediate behavior of a quantum system, e.g., the particle's behavior in the double-slit experiment. Neutron interferometry using single silicon perfect crystals is established as a versatile tool to test fundamental phenomena in quantum mechanics, where an incident neutron beam is coherently split in two or three beam paths with macroscopic separation of several centimeters. Here, we present quantum optical experiments with these matter-wave interferometers, studying the effect of the quantum Cheshire Cat in some variants, the neutron's presence in the paths of the interferometer as well as the direct test of a commutation relation. To reduce disturbances induced by the measurement, the interaction strength is lessened and so-called weak interactions are exploited by employing pre- and post-selection procedures. All results of the experiments confirm the predictions of quantum theory; the observed behaviors of the neutron between the pre- and post-selection in space and time emphasize striking and counter-intuitive aspects of quantum theory. [ABSTRACT FROM AUTHOR]

The uncertainty principle is one of the fundamental features of quantum mechanics and plays a vital role in quantum information processing. We study uncertainty relations based on metric-adjusted skew information for finite quantum observables. Motivated by the paper [Physical Review A 104, 052414 (2021)], we establish tighter uncertainty relations in terms of different norm inequalities. Naturally, we generalize the method to uncertainty relations of metric-adjusted skew information for quantum channels and unitary operators. As both the Wigner–Yanase–Dyson skew information and the quantum Fisher information are the special cases of the metric-adjusted skew information corresponding to different Morozova–Chentsov functions, our results generalize some existing uncertainty relations. Detailed examples are given to illustrate the advantages of our methods. [ABSTRACT FROM AUTHOR]

In a recent paper, we have shown how in Madelung's hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment. [ABSTRACT FROM AUTHOR]

In this work, we study the entropic uncertainty and quantum discord in two double-quantum-dot (DQD) system coupled via a transmission line resonator (TLR). Explicitly, the dynamics of the systemic quantum correlation and measured uncertainty are analysed with respect to a general X-type state as the initial state. Interestingly, it is found that the different parameters, including the eigenvalue α of the coherent state, detuning amount δ, frequency ω and the coupling constant g, have subtle effects on the dynamics of the entropic uncertainty, such as the oscillation period of the uncertainty. It is clear to reveal that the quantum discord and the lower bound of the entropic uncertainty are anti-correlated when the initial state of the system is the Werner-type state, while quantum discord and the lower bound of the entropic uncertainty are not anti-correlated when the initial state of the system is the Bell-diagonal state. Thereby, we claim that the current investigation would provide an insight into the entropic uncertainty and quantum correlation in DQDs system, and are basically of importance to quantum precision measurement in practical quantum information processing. [ABSTRACT FROM AUTHOR]

We assess the degree of quantumness of the P, Q, and W quantum optics' approximations in a thermal context governed by the canonical ensemble treatment. First, we remint the reader of the bridge connecting quantum optics with statistical mechanics using the abovementioned approximations at the temperature T. With the ensuing materials, we explore with some detail some features of the above bridge, related to the entropy and to thermal uncertainties. Some new relations concerning the degree of quantumness of the P, Q, and W are obtained by comparison between them and the exact and classical treatments. [ABSTRACT FROM AUTHOR]

The problem of a particle moving freely inside a box with rigid, perfectly reflecting walls is a standard exercise in basic quantum mechanics (QM), the box being taken to be a line interval, a square, and a cube, respectively, in the 1, 2, and 3 dimensional cases. The problem illustrates several aspects of QM, such as stationary states, uncertainty relations, eigenvalues, eigenfunctions, orthonormality, completeness, and so on. The purpose of this two-part article is to provide some insights into how the particle-in-a-box problem acts as a paradigm for the understanding of these and other aspects, such as the roles played by dimensionality, discrete and continuous symmetry, degeneracy, etc. Further, we shall also see (in Part 2) how the same problem leads directly to some non-trivial matters such as quantum chaos and deep results in mathematical physics. Finally, it must be emphasized that the particle-in-a-box problem is no longer just a theoretical exercise. With modern technology, it is physically realized in different kinds of nanostructures such as nanowires, atomic corrals, quantum dots, conjugated polymers, among others. This makes it all the more important to analyze various problems of this class and their solutions. [ABSTRACT FROM AUTHOR]

Physical consequences of the Heisenberg uncertainty principle generalized by H.P. Robertson and E. Schrödinger are analyzed. It is shown within the linear quantum oscillator model that the use of frequency-variable external classical action on the oscillator is crucial for new physical and chemical applications. It is also shown that the efficiency of the action is controlled by the mutual correlation coefficient between the coordinate and the momentum of the quantum system. A calculation procedure for choosing the frequency dependence of the action is established such as to maximize this coefficient. Directions are outlined for generalizing and sophisticating the model suitable for solving problems of chemical kinetics and hydrodynamics of liquid media. [ABSTRACT FROM AUTHOR]

In a recent paper, we have shown how in Madelung’s hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment.

We study the possible interpretation of the "universal constants" by the classification of J.-M. Lévy-Leblond. The Planck constant and the speed of light in vacuum are the most common examples of constants of this type. Using Fock’s principle of the relativity w.r.t. observation means, we show that these two constants can be viewed as manifestations of certain relativity. We also show that there is a possibility to interpret the Boltzmann constant in a similar way, and make some comments about the relativistic interpretation of the constant spacetime curvature and gravitational constant G. [ABSTRACT FROM AUTHOR]

We derive strong variance-based uncertainty relations for arbitrary two and more unitary operators by re-examining the mathematical foundation of the uncertainty relation. This is achieved by strengthening the celebrated Cauchy–Schwarz inequality using a method of brackets and convex functions. The unitary uncertainty relations outperform several strong unitary uncertainty relations, notably better than some recent best lower bounds such as [Phys. Rev. Lett. 120, 230402 (2018)] and [Phys. Rev. A. 100, 022116 (2019)]. [ABSTRACT FROM AUTHOR]

The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.

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. [ABSTRACT FROM AUTHOR]

An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form RMSE≥f(α)/(⟨N⟩+12), bounding the root mean square error of any estimate of a random optical phase shift in terms of average photon number, where f(α) is maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase uncertainty to the photon number distribution, such as ΔΦ≥maxnpn, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rényi mutual information of quantum communication channels, related to asymmetry and convolution, and applied to the estimation (with prior information) of unitary shift parameters such as rotation angle and time, and to obtain strong bounds on measures of coherence. Sharper Rényi entropic uncertainty relations are also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra. In the latter case almost-periodic Rényi entropies are introduced for nonperiodic systems.

Uncertainties in flavor and mass eigenstates of neutrinos are considered within the majorization approach. Nontrivial bounds reflect the fact that neutrinos cannot be simultaneously in flavor and mass eigenstates. As quantitative measures of uncertainties, both the Rényi and Tsallis entropies are utilized. Within the current amount of experience concerning the mixing matrix, majorization uncertainty relations need to put values of only two parameters, viz. 1 2 and 1 3 . That is, the majorization approach is applicable within the same framework as the Maassen–Uffink relation recently utilized in this context. We also consider the case of detection inefficiencies, since it can naturally be incorporated into the entropic framework. Short comments on applications of entropic uncertainty relations with quantum memory are given. [ABSTRACT FROM AUTHOR]

We assess the degree of quantumness of the P, Q, and W quantum optics’ approximations in a thermal context governed by the canonical ensemble treatment. First, we remint the reader of the bridge connecting quantum optics with statistical mechanics using the abovementioned approximations at the temperature T. With the ensuing materials, we explore with some detail some features of the above bridge, related to the entropy and to thermal uncertainties. Some new relations concerning the degree of quantumness of the P, Q, and W are obtained by comparison between them and the exact and classical treatments.

A multipartite state that is not the convex sum of bipartite product states is said to be a genuine multipartite entangled (GME) state, which offers more significant advantages in quantum information compared with entanglement. We propose a sufficient criterion for the detection of GME based on uncertainty relations for chosen observables of subsystems. We apply the criterion to detect the GME properties of noisy n-partite W state when n = 3 , 4 , 5 and 6 and find that the criterion can detect more noisy W states when n ranges from 4 to 6. Moreover, the criterion is also used to detect the genuine entanglement of 3-qutrit states. The result is stronger than that based on GME concurrence and Fisher information. [ABSTRACT FROM AUTHOR]

The article takes under consideration three versions of the ensemble (statistical) interpretation of quantum mechanics and discusses the interconnection of these interpretations with the philosophy of science. To emphasize the specifics of the problem of interpretation of quantum mechanics in the USSR, the Marxist ideology is taken into account. The present paper continues the author's previous analysis of ensemble interpretations which emerged in the USA and USSR in the first half of the 20th century. The author emphasizes that the ensemble approach turned out to be a dead end for the development of the interpretation of quantum mechanics in Russia. The article also argues that in Soviet Russia, the classical Copenhagen (standard) approach to quantum mechanics was used. The Copenhagen approach was developed by Lev Landau in 1919-1931 and became the basis of the Landau-Lifshitz famous course on quantum mechanics, one of the classics of twentieth-century physics literature (the first edition was published in 1947). Although Vladimir A. Fock's approach to the interpretation of quantum mechanics differs from the standard presentation by Lev Landau and Evgeny Lifshitz, Fock put forward a very important principle that complementarity is a "firmly established law of nature". The fundamental writings of Lev Landau, Vladimir Fock and Igor Tamm, the authors of the mid-twentieth century, did a lot to defend the standard point of view such as the popular interpretations by Landau and Lifshitz. This approach can be traced back to Landau's early writings and to Fock's criticism of the ensemble approach. [ABSTRACT FROM AUTHOR]

The complementarity (upper bound) and uncertainty relations (lower bound) of the coherence via skew information under mutually unbiased bases (MUBs) are studied. The complementarity relation for the geometric measure of coherence is also obtained based on the relation between the coherence via skew information and the geometric measure of coherence. As applications, two tighter upper bounds are presented on the minimum error probabilities that discriminate a set of pure states with the least square measurement, which improve the results of Xiong et al. (Phys Rev A 98:032324, 2018). [ABSTRACT FROM AUTHOR]

In this paper, we provide an overview on the foundation and first results of a very recent quantum theory of color perception, together with novel results about uncertainty relations for chromatic opposition. The major inspiration for this model is the 1974 remarkable work by H.L. Resnikoff, who had the idea to give up the analysis of the space of perceived colors through metameric classes of spectra in favor of the study of its algebraic properties. This strategy permitted to reveal the importance of hyperbolic geometry in colorimetry. Starting from these premises, we show how Resnikoff's construction can be extended to a geometrically rich quantum framework, where the concepts of achromatic color, hue and saturation can be rigorously defined. Moreover, the analysis of pure and mixed quantum chromatic states leads to a deep understanding of chromatic opposition and its role in the encoding of visual signals. We complete our paper by proving the existence of uncertainty relations for the degree of chromatic opposition, thus providing a theoretical confirmation of the quantum nature of color perception. [ABSTRACT FROM AUTHOR]

Based on the results of quantum field theory and quantum electrodynamics on virtual photons as quanta of electromagnetic interaction, we discuss the interpretation of quantum indeterminacy, uncertainty relations, and physics, which determine the probabilistic statistical character of quantum phenomena. We discuss the fundamental reasons for the appearance of the Planck constant in the uncertainty relations. The results of this work may be of interest not only for the quantum field theory and elementary particle physics but for other areas of physics as well. [ABSTRACT FROM AUTHOR]

We analyze general uncertainty relations and we show that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations Δ A and Δ B calculated for these vectors is zero: Δ A ⋅ Δ B ≥ 0. We also show that for some pairs of non-commuting observables the sets of vectors for which Δ A ⋅ Δ B ≥ 0 can be complete (total). The Heisenberg, Δ t ⋅ Δ E ≥ ℏ / 2 , and Mandelstam–Tamm (MT), τ A ⋅ Δ E ≥ ℏ / 2 , time–energy uncertainty relations ( τ A is the characteristic time for the observable A) are analyzed too. We show that the interpretation τ A = ∞ for eigenvectors of a Hamiltonian H does not follow from the rigorous analysis of MT relation. We show also that contrary to the position–momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of A and H. [ABSTRACT FROM AUTHOR]

We construct various systems of coherent states (SCS) on the O(D)-equivariant fuzzy spheres S Λ d ( d = 1 , 2 , D = d + 1 ) constructed in Fiore and Pisacane (J Geom Phys 132:423–451, 2018) and study their localizations in configuration space as well as angular momentum space. These localizations are best expressed through the O(D)-invariant square space and angular momentum uncertainties (Δ x) 2 , (Δ L) 2 in the ambient Euclidean space R D . We also determine general bounds (e.g., uncertainty relations from commutation relations) for (Δ x) 2 , (Δ L) 2 , and partly investigate which SCS may saturate these bounds. In particular, we determine O(D)-equivariant systems of optimally localized coherent states, which are the closest quantum states to the classical states (i.e., points) of S d . We compare the results with their analogs on commutative S d . We also show that on S Λ 2 our optimally localized states are better localized than those on the Madore–Hoppe fuzzy sphere with the same cutoff Λ . [ABSTRACT FROM AUTHOR]

Even though irreversibility is one of the major hallmarks of any real-life process, an actual understanding of irreversible processes remains still mostly semi-empirical. In this paper, we formulate a thermodynamic uncertainty principle for irreversible heat engines operating with an ideal gas as a working medium. In particular, we show that the time needed to run through such an irreversible cycle multiplied by the irreversible work lost in the cycle is bounded from below by an irreducible and process-dependent constant that has the dimension of an action. The constant in question depends on a typical scale of the process and becomes comparable to Planck's constant at the length scale of the order Bohr radius, i.e. the scale that corresponds to the smallest distance on which the ideal gas paradigm realistically applies. [ABSTRACT FROM AUTHOR]

An arbitrarily dense discretization of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized spheres) entanglement. Unlike Meyer's earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken–Specker theorem is not nullified. A physical interpretation of points on the discretized Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorization, where these conditions are defined solely from states on the invariant set. In this finite representation, there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences. [ABSTRACT FROM AUTHOR]

We examine various uncertainty relations: (electromagnetic-field phase (the so-called "Pegg–Barnett operator"))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared. [ABSTRACT FROM AUTHOR]

The paper deals with uncertainty relations for time and energy operators, and the aftermath of the Chernobyl catastrophe is considered as an example. The mathematical approach developed by Holevo is analyzed, which allows us to assign the corresponding observables to non-self-adjoint operators and to establish uncertainty relations for nonstandard canonical conjugate pairs. Relations for calculating the minimal time interval in which the energy jump can be discovered are given. Based on the intensity parameter introduced by the author, which is related to a special statistics called Gentile statistics and to the polylogarithm function, properties of stable chemical elements, such as time fluctuations and the jump of specific energy in the transition from the Bose—Einstein distribution to the Fermi—Dirac distribution, are mathematically described with regard to experimental data. The obtained data are arranged in a table for 255 stable chemical elements. The mathematical approach developed by the author of the present paper allows one to describe the "antipode" (in a certain sense) of the standard thermodynamics, i.e., the thermodynamics of nuclear matter. This field of nuclear physics is very important for the study of properties of radioactive elements and, accordingly, from the standpoint of ensuring nuclear safety. [ABSTRACT FROM AUTHOR]

The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Rényi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state’s angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies–Thakkar, Lieb–Thirring, Redheffer–Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.

In this paper, we provide an overview on the foundation and first results of a very recent quantum theory of color perception, together with novel results about uncertainty relations for chromatic opposition. The major inspiration for this model is the 1974 remarkable work by H.L. Resnikoff, who had the idea to give up the analysis of the space of perceived colors through metameric classes of spectra in favor of the study of its algebraic properties. This strategy permitted to reveal the importance of hyperbolic geometry in colorimetry. Starting from these premises, we show how Resnikoff’s construction can be extended to a geometrically rich quantum framework, where the concepts of achromatic color, hue and saturation can be rigorously defined. Moreover, the analysis of pure and mixed quantum chromatic states leads to a deep understanding of chromatic opposition and its role in the encoding of visual signals. We complete our paper by proving the existence of uncertainty relations for the degree of chromatic opposition, thus providing a theoretical confirmation of the quantum nature of color perception.

The coherence of an ensemble of quantum states is considered — a quantity having a similarity with the uncertainty of the quantum observable that arises from the entropic uncertainty relations. We give estimates for the coherence of the ensemble through the uncertainty of the dual observable and the weak uncertainty of the observable through the coherence of the ensemble. It is shown that this numerical characteristic is closely connected with the relative entropy of discord for the corresponding classical-quantum state. [ABSTRACT FROM AUTHOR]