We describe a setup for obtain!ing uncertainty relations for arbitrary pairs of observables related by a Fourier transform. The physical examples discussed here are the standard position and momentum, number and angle...We describe a setup for obtain!ing uncertainty relations for arbitrary pairs of observables related by a Fourier transform. The physical examples discussed here are the standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow for an arbitrary choice of metric for the outcome distance, and the choice of an exponent distinguishing, e.g., absolute and root mean square deviations. The emphasis of this article is on developing a unified treatment, in which one observable takes on values in an arbitrary locally compact Abelian group and the other in the dual group. In all cases, the phase space symmetry implies the eciuality of measurement and preparation uncertainty bounds. There is also a straightforward method for determining the optimal bounds.展开更多
Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables.In terms of the coherence measure based on the Wigner-Yanase skew information,we establish s...Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables.In terms of the coherence measure based on the Wigner-Yanase skew information,we establish several uncertainty relations for coherence with respect to von Neumann measurements,mutually unbiased bases(MUBs),and general symmetric informationally complete positive operator valued measurements(SIC-POVMs),respectively.Since coherence is intimately connected with quantum uncertainties,the obtained uncertainty relations are of intrinsically quantum nature,in contrast to the conventional uncertainty relations expressed in terms of variance,which are of hybrid nature(mixing both classical and quantum uncertainties).From a dual viewpoint,we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement.In particular,it is shown that if the density operators representing the quantum states do not commute,then there is no measurement(reference basis)such that the coherence of these states can be simultaneously small.展开更多
Modifications of the Weyl-Heisenberg algebra are proposed where the classical limit corresponds to a metric in (curved) momentum spaces. In the simplest scenario, the 2D de Sitter metric of constant curvature in momen...Modifications of the Weyl-Heisenberg algebra are proposed where the classical limit corresponds to a metric in (curved) momentum spaces. In the simplest scenario, the 2D de Sitter metric of constant curvature in momentum space furnishes a hierarchy of modified uncertainty relations leading to a minimum value for the position uncertainty . The first uncertainty relation of this hierarchy has the same functional form as the stringy modified uncertainty relation with a Planck scale minimum value for at . We proceed with a discussion of the most general curved phase space scenario (cotangent bundle of spacetime) and provide the noncommuting phase space coordinates algebra in terms of the symmetric and nonsymmetric metric components of a Hermitian complex metric , such . Yang’s noncommuting phase-space coordinates algebra, combined with the Schrodinger-Robertson inequalities involving angular momentum eigenstates, reveals how a quantized area operator in units of emerges like it occurs in Loop Quantum Gravity (LQG). Some final comments are made about Fedosov deformation quantization, Noncommutative and Nonassociative gravity.展开更多
In a recent paper, Sacchi （Phys. Rev. Lett. 96 （2006） 220502） studied the information-disturbance tradeoff in estimating an unknown two-qubit maximally entangled state. In this study, we explore the tradeoff in es...In a recent paper, Sacchi （Phys. Rev. Lett. 96 （2006） 220502） studied the information-disturbance tradeoff in estimating an unknown two-qubit maximally entangled state. In this study, we explore the tradeoff in estimating 13 an unknown three-qubit GHZ state. The optimal estimation process supplies a fidelity of 13/54 and the tradeoff interpolates smoothly between non-informative measurement and optimal estimation process.展开更多
This research work proceeds from the assumption, which was still considered by Einstein, that the quantization of gravity does not require additional external procedures: quantum phenomena can be a consequence of the ...This research work proceeds from the assumption, which was still considered by Einstein, that the quantization of gravity does not require additional external procedures: quantum phenomena can be a consequence of the properties of the universal gravitational interaction, which maps any physical field upon the space-time geometry. Therefore, an attempt is made in this research work to reduce the quantization of physical fields in GRT to the space-time quantization. Three reasons for quantum phenomena are considered: Partition of space-time into a set of unconnected Novikov’s R- and T-domains impenetrable for light paths;the set is generated by the invariance of Einstein’s equations with respect to dual mappings;The existence of electric charge quanta of wormholes, which geometrically describe elementary particles in GRT. This gives rise to a discrete spectrum of their physical and geometric parameters governed by Diophantine equations. It is shown that the fundamental constants (electric charge, rest masses of an electron and a proton) are interconnected arithmetically;The existence of the so-called Diophantine catastrophe, when fluctuations in the values of physical constants tending to zero lead to fluctuations in the number of electric charges and the number of nucleons at the wormhole throats, which tend to infinity, so that the product of the increments of these numbers by the increment of physical constants forms a relation equivalent to the uncertainty relation in quantum mechanics. This suggests that space-time cannot but fluctuate, and, moreover, its fluctuations are bounded from below, so that all processes become chaotic, and the observables become averaged over this chaos.展开更多
It is found that the field of the combined mode of the probe wave and the phase conjugate wave in the process of non-degenerate four-wave mixing exhibits higher-order squeezing to all even orders. The higher-order squ...It is found that the field of the combined mode of the probe wave and the phase conjugate wave in the process of non-degenerate four-wave mixing exhibits higher-order squeezing to all even orders. The higher-order squeezed parameter and squeezed limit due to the modulation frequency are investigated. The smaller the modulation frequency is, the stronger the degree of higher-order squeezing becomes. Furthermore, the hlgher-order uncertainty relations in the process of non-degenerate four-wave mixing are presented for the first time. The product of higher-order noise moments is related to even order number N and the length L of the medium.展开更多
A general uncertainty relation between the change of weighted value which represents learning ability and the discrimination error of unlearning sample sets which represents generalization ability is revealed in the m...A general uncertainty relation between the change of weighted value which represents learning ability and the discrimination error of unlearning sample sets which represents generalization ability is revealed in the modeling of back propagation (BP) neural network. Tests of numerical simulation for multitype of complicated functions are carried out to determine the value distribution (1×10?5~5×10?4) of overfitting parameter in the uncertainty relation. Based on the uncertainty relation, the overfitting in the training process of given sample sets using BP neural network can be judged.展开更多
文摘We describe a setup for obtain!ing uncertainty relations for arbitrary pairs of observables related by a Fourier transform. The physical examples discussed here are the standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow for an arbitrary choice of metric for the outcome distance, and the choice of an exponent distinguishing, e.g., absolute and root mean square deviations. The emphasis of this article is on developing a unified treatment, in which one observable takes on values in an arbitrary locally compact Abelian group and the other in the dual group. In all cases, the phase space symmetry implies the eciuality of measurement and preparation uncertainty bounds. There is also a straightforward method for determining the optimal bounds.
基金Supported by the National Natural Science Foundation of China under Grant No.11875317the National Center for Mathematics and Interdisciplinary Sciences,and Chinese Academy of Sciences under Grant No.Y029152K51
文摘Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables.In terms of the coherence measure based on the Wigner-Yanase skew information,we establish several uncertainty relations for coherence with respect to von Neumann measurements,mutually unbiased bases(MUBs),and general symmetric informationally complete positive operator valued measurements(SIC-POVMs),respectively.Since coherence is intimately connected with quantum uncertainties,the obtained uncertainty relations are of intrinsically quantum nature,in contrast to the conventional uncertainty relations expressed in terms of variance,which are of hybrid nature(mixing both classical and quantum uncertainties).From a dual viewpoint,we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement.In particular,it is shown that if the density operators representing the quantum states do not commute,then there is no measurement(reference basis)such that the coherence of these states can be simultaneously small.
文摘Modifications of the Weyl-Heisenberg algebra are proposed where the classical limit corresponds to a metric in (curved) momentum spaces. In the simplest scenario, the 2D de Sitter metric of constant curvature in momentum space furnishes a hierarchy of modified uncertainty relations leading to a minimum value for the position uncertainty . The first uncertainty relation of this hierarchy has the same functional form as the stringy modified uncertainty relation with a Planck scale minimum value for at . We proceed with a discussion of the most general curved phase space scenario (cotangent bundle of spacetime) and provide the noncommuting phase space coordinates algebra in terms of the symmetric and nonsymmetric metric components of a Hermitian complex metric , such . Yang’s noncommuting phase-space coordinates algebra, combined with the Schrodinger-Robertson inequalities involving angular momentum eigenstates, reveals how a quantized area operator in units of emerges like it occurs in Loop Quantum Gravity (LQG). Some final comments are made about Fedosov deformation quantization, Noncommutative and Nonassociative gravity.
基金Supported by the National Basic Research Programme of China, the National Natural Science Foundation of China under Grant Nos 10674128 and 60121503, the Knowledge Innovation Project and the Hundreds of Talents Programme of Chinese Academy of Sciences, and the Doctoral Foundation of the Education Ministry of China under Grant No 20060358043.
文摘In a recent paper, Sacchi （Phys. Rev. Lett. 96 （2006） 220502） studied the information-disturbance tradeoff in estimating an unknown two-qubit maximally entangled state. In this study, we explore the tradeoff in estimating 13 an unknown three-qubit GHZ state. The optimal estimation process supplies a fidelity of 13/54 and the tradeoff interpolates smoothly between non-informative measurement and optimal estimation process.
文摘This research work proceeds from the assumption, which was still considered by Einstein, that the quantization of gravity does not require additional external procedures: quantum phenomena can be a consequence of the properties of the universal gravitational interaction, which maps any physical field upon the space-time geometry. Therefore, an attempt is made in this research work to reduce the quantization of physical fields in GRT to the space-time quantization. Three reasons for quantum phenomena are considered: Partition of space-time into a set of unconnected Novikov’s R- and T-domains impenetrable for light paths;the set is generated by the invariance of Einstein’s equations with respect to dual mappings;The existence of electric charge quanta of wormholes, which geometrically describe elementary particles in GRT. This gives rise to a discrete spectrum of their physical and geometric parameters governed by Diophantine equations. It is shown that the fundamental constants (electric charge, rest masses of an electron and a proton) are interconnected arithmetically;The existence of the so-called Diophantine catastrophe, when fluctuations in the values of physical constants tending to zero lead to fluctuations in the number of electric charges and the number of nucleons at the wormhole throats, which tend to infinity, so that the product of the increments of these numbers by the increment of physical constants forms a relation equivalent to the uncertainty relation in quantum mechanics. This suggests that space-time cannot but fluctuate, and, moreover, its fluctuations are bounded from below, so that all processes become chaotic, and the observables become averaged over this chaos.
文摘It is found that the field of the combined mode of the probe wave and the phase conjugate wave in the process of non-degenerate four-wave mixing exhibits higher-order squeezing to all even orders. The higher-order squeezed parameter and squeezed limit due to the modulation frequency are investigated. The smaller the modulation frequency is, the stronger the degree of higher-order squeezing becomes. Furthermore, the hlgher-order uncertainty relations in the process of non-degenerate four-wave mixing are presented for the first time. The product of higher-order noise moments is related to even order number N and the length L of the medium.
基金Supported by the the Nation Natural Science Foundation of China (No.40271024)
文摘A general uncertainty relation between the change of weighted value which represents learning ability and the discrimination error of unlearning sample sets which represents generalization ability is revealed in the modeling of back propagation (BP) neural network. Tests of numerical simulation for multitype of complicated functions are carried out to determine the value distribution (1×10?5~5×10?4) of overfitting parameter in the uncertainty relation. Based on the uncertainty relation, the overfitting in the training process of given sample sets using BP neural network can be judged.
