Schroedinger uncertainty relation and its minimization states

Pramana, 1997

A definition of coherent states is proposed as the minimum uncertainty states with equal variance in two hermitian non-commuting generators of the Lie algebra of the hamiltonian. That approach classifies the coherent states into distinct classes. The coherent states of a harmonic oscillator, according to the proposed approach, are shown to fall in two classes. One is the familiar class of Glauber states whereas the other is a new class. The coherent states of spin constitute only one class. The squeezed states are similarly defined on the physical basis as the states that give better precision than the coherent states in a process of measurement of a force coupled to the given system. The condition of squeezing based on that criterion is derived for a system of spins.

Current science

The notion of uncertainty in the description of a physical system has assumed prodigious importance in the development of quantum theory. Overcoming the early misunderstanding and confusion, the concept grew continuously and still remains an active and fertile research field. Curious new insights and correlations are gained and developed in the process with the introduction of new `measures' of uncertainty or indeterminacy and the development of quantum measurement theory. In this article we intend to reach a fairly up to date status report of this yet unfurling concept and its interrelation with some distinctive quantum features like nonlocality, steering and entanglement/ inseparability. Some recent controversies are discussed and the grey areas are mentioned.

2016

We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the EPR paradox. Entropic uncertainty relations are used to reveal quantum steering for non-Gaussian continuous variable states. Entropic uncertainty relations for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tipartite systems. The Robertson-Schrodinger uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.

2012

We describe a six-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions. The corresponding oscillating photons statistics are discussed and an application to quantum optics and cavity quantum electrodynamics is mentioned.

Journal of Optics B Quantum and Semiclassical Optics, 2002

We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the 'most classical' thermal states is used as a 'measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose 'purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the 'standard purity' Trρ 2 and 37% if the 'superpurity' lim r→0 [Tr(ρ 1+1/r)] r is chosen as a measure of deviation).

2008

The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schrödinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard SU(1, 1) and SU(2) coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schrödinger inequality for the Hermitian components of the suq(1,1) ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form. 1

We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the EPR paradox. Entropic uncertainty relations are used to reveal quantum steering for non-Gaussian continuous variable states. Entropic uncertainty relations for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tripartite systems. The Robertson-Schrodinger uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.

Journal of Mathematical Physics, 2002

States which minimize the Schrödinger-Robertson uncertainty relation are constructed as eigenstates of an operator which is a element of the h(1) ⊕ su(2) algebra. The relations with supercoherent and supersqueezed states of the supersymmetric harmonic oscillator are given. Moreover, we are able to compute gneneral Hamiltonians which behave like the harmonic oscillator Hamiltonian or are related to the Jaynes-Cummings Hamiltonian.

Concepts of physics old and new

It is demonstrated that uncertainty relations (UR), being the general property of functions, do not prevent particles in quantum mechanics (QM) to have precisely and simultaneously defined position and momentum. Classical interpretation of QM, different from Bohmian mechanics is proposed.

Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson-Schrödinger uncertainty relation and Kraus-Maassen Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schrödinger cat states. Again, improvement over both the Robertson-Schrödinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.