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.     

Wigner-Yanase skew information and uncertainty relations

The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.     

An Informational Characterization of Schrödinger's Uncertainty Relations

Heisenberg's uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement. The information concerning incompatibility (non-commutativity) of observables is well included but that concerning correlation is missing. Schrödinger's uncertainty relations remedy this defect by supplementing the correlation in terms of anti-commutators. However, both Heisenberg's uncertainty relations and Schrödinger's uncertainty relations are expressed in terms of variances, which are not good measures of uncertainty in general situations (e.g., when mixed states are involved). By virtue of the Wigner–Yanase skew information, we will establish an uncertainty relation along the spirit of Schrödinger from a statistical inference perspective and propose a conjecture. The result may be interpreted as a quantification of certain aspect of the celebrated Wigner–Araki–Yanase theorem for quantum measurement, which states that observables not commuting with a conserved quantity cannot be measured exactly.     

A rigorous uncertainty relation (Cauchy Inequality) for an electromagnetic field in quantum superpositions of coherent states and in a two-photon coherent state

It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrödinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.     

Operationally invariant measure of the distance between quantum states by complementary measurements

We propose an operational measure of distance of two quantum states, which conversely tells us their closeness. This is defined as a sum of differences in partial knowledge over a complete set of mutually complementary measurements for the two states. It is shown that the measure is operationally invariant and it is equivalent to the Hilbert-Schmidt distance. The operational measure of distance provides a remarkable interpretation of the information distance between quantum states.     

Fisher information contains all HO-quantum-statistics already at the semiclassical level

We study here the difference between quantum statistical treatments and semiclassical ones, using as the main research tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.     

Complementary relation between quantum entanglement and entropic uncertainty

Quantum entanglement is regarded as one of the core concepts,which is used to describe the nonclassical correlation between subsystems,and entropic uncertainty relation plays a vital role in quantum precision measurement.It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states.An interesting question is naturally raised:is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement?Or if the relation can be applied to estimate the entanglement.In this work,we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation.The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system,and specifically the larger uncertainty will induce the weaker entanglement of the probed system,and vice versa.Besides,we use randomly generated states as illustrations to verify our results.Therefore,we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.     

Averaged Wigner-Yanase-Dyson information as a quantum uncertainty measure

The average of the skew information over the observables was proposed by Luo as a quantum uncertainty measure. In this paper, we investigate the interesting properties of Wigner-Yanase-Dyson (WYD) information, which is a generalization of skew information. Then, by averaging WYD information over the observables we propose a general quantum uncertainty measure of mixed states, and study the properties of the measure. Note that the general quantum uncertainty measure depends on the parameter α and reduces to Luo’s measure when α is equal to 1/2. To get rid of the parameter α, we propose the average of the general measure over the parameter α as a quantum uncertainty measure of mixed states and discuss its properties. The two measures can be considered as the intrinsic properties of mixed state. The construction is reminiscent of the generalized entropies that have shown to be useful in many applications.     

Entropy of the rotating and charged black string to all orders in the Planck length

<正>By using the entanglement entropy method,this paper calculates the statistical entropy of the Bose and Fermi fields in thin films,and derives the Bekenstein-Hawking entropy and its correction term on the background of a rotating and charged black string.Here,the quantum field is entangled with quantum states in the black string and thin film to the event horizon from outside the rotating and charged black string.Taking into account the effect of the generalized uncertainty principle on quantum state density,it removes the difficulty of the divergence of state density near the event horizon in the brick-wall model.These calculations and discussions imply that high density quantum states near the event horizon of a black string are strongly correlated with the quantum states in a black string and that black string entropy is a quantum effect.The ultraviolet cut-off in the brick-wall model is not reasonable.The generalized uncertainty principle should be considered in the high energy quantum field near the event horizon.From the viewpoint of quantum statistical mechanics,the correction value of Bekenstein-Hawking entropy is obtained.This allows the fundamental recognition of the correction value of black string entropy at nonspherical coordinates.     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

Derivation of quantum Chernoff metric with perturbation expansion method

We investigate a measure of distinguishability defined by the quantum Chernoff bound, which naturally induces the quantum Chernoff metric over a manifold of quantum states. Based on a quantum statistical model, we alternatively derive this metric by means of perturbation expansion. Moreover, we show that the quantum Chernoff metric coincides with the infinitesimal form of the quantum Hellinger distance, and reduces to the variant version of the quantum Fisher information for the single-parameter case. We also give the exact form of the quantum Chernoff metric for a qubit system containing a single parameter.     

Discriminating States: the quantum Chernoff bound

We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.     

Uncertainty relations expressed by means of information energies

In this paper, we consider an alternative form of the uncertainty relation for quantum observables of discrete spectrum which makes use of so-called “information energy” as the measure of uncertainty. Then we apply this form of uncertainty relation to the particular case of two mutually orthogonal components of the spin-1/2 operator. We show that the use of this non-standard uncertainty measure simplifies the expression for spreads of observables and for the construction of the corresponding uncertainty relation.     

Quantum Information and Entropy

Thermodynamic entropy is not an entirely satisfactory measure of information of a quantum state. This entropy for an unknown pure state is zero, although repeated measurements on copies of such a pure state do communicate information. In view of this, we propose a new measure for the informational entropy of a quantum state that includes information in the pure states and the thermodynamic entropy. The origin of information is explained in terms of an interplay between unitary and non-unitary evolution. Such complementarity is also at the basis of the so-called interaction-free measurement.     

Monotonically increasing functions of any quantum correlation can make all multiparty states monogamous

Monogamy of quantum correlation measures puts restrictions on the sharability of quantum correlations in multiparty quantum states. Multiparty quantum states can satisfy or violate monogamy relations with respect to given quantum correlations. We show that all multiparty quantum states can be made monogamous with respect to all measures. More precisely, given any quantum correlation measure that is non-monogamic for a multiparty quantum state, it is always possible to find a monotonically increasing function of the measure that is monogamous for the same state. The statement holds for all quantum states, whether pure or mixed, in all finite dimensions and for an arbitrary number of parties. The monotonically increasing function of the quantum correlation measure satisfies all the properties that are expected for quantum correlations to follow. We illustrate the concepts by considering a thermodynamic measure of quantum correlation, called the quantum work deficit.     

Asymptotic Error Rates in Quantum Hypothesis Testing

We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing. The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support. The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.     

Quantum Probability’s Algebraic Origin

Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.     

Tavis-Cummings模型中的几何量子失协特性

采用几何量子失协的计算方法,通过改变两原子初始状态、腔内光子数和偶极-偶极相互作用强度,研究了Tavis-Cummings模型中的几何量子失协特性.结果表明:几何量子失协都是随时间周期性振荡的,选取适当的初态可以使两原子一直保持失协状态,增加腔内光子数和偶极相互作用对几何量子失协有积极的影响.