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Physical Review A

Correlations in open quantum systems and associated uncertainty relations

Sergey A. Ponomarenko, E. Wolf

Published May 9, 2001• 17 citations
Publisher

Abstract

We study how correlations in a state of an open quantum system affect the intrinsic uncertainties of the expectation values of an arbitrary pair of noncommuting observables. We show that for such observables, there exist Heisenberg-type uncertainty relations that take fully into account correlations in the state of the system. If the quantum system is in a pure state, such uncertainty relations reduce to the conventional one. We obtain an equation for the density operator of a general state that minimizes the new uncertainty relations, and demonstrate that in the important case of coordinate and momentum operators, the minimum-uncertainty states are displaced squeezed thermal states.

DOI
10.1103/physreva.63.062106

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We study how correlations in a state of an open quantum system affect the intrinsic uncertainties of the expectation values of an arbitrary pair of noncommuting observables. We show that for such observables, there exist Heisenberg-type uncertainty relations that take fully into account correlations in the state of the system. If the quantum system is in a pure state, such uncertainty relations reduce to the conventional one. We obtain an equation for the density operator of a general state that minimizes the new uncertainty relations, and demonstrate that in the important case of coordinate and momentum operators, the minimum-uncertainty states are displaced squeezed thermal states.
We consider states leading to the equality sign in the uncertainty inequalities associated with correlations in open quantum systems which have been recently derived by Ponomarenko and Wolf [Phys. Rev. A 63, 062106 (2001)]. The new inequalities involve fluctuations defined in terms of the square of the density operator that characterizes mixed states. We find the minimum-correlation states associated with the quadratures of single-mode and two-mode electromagnetic fields in a cavity and for the angular momentum operators which can describe atomic degrees of freedom. We show that while in the case of single-mode quadratures the functional form of the minimum-correlation state is uniquely specified, this is not so for the other pairs of noncommuting operators. In general, the states with the least amount of correlations are mixed and they exhibit squeezing.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.
We study a possible improvement of uncertainty relations. The Heisenberg uncertainty relation employs commutator of a pair of conjugate observables to set the limit of quantum measurement of the observables. The Schrödinger uncertainty relation improves the Heisenberg uncertainty relation by adding the correlation in terms of anti-commutator. However both relations are insensitive whether the state used is pure or mixed. We improve the uncertainty relations by introducing additional terms which measure the mixtureness of the state. For the momentum and position operators as conjugate observables and for the thermal state of quantum harmonic oscillator, it turns out that the equalities in the improved uncertainty relations hold.
We derive several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space. We show that our bounds are tighter in various cases than the ones existing in the current literature. Using the uncertainty relation for the unitary operators, we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anticommuting observables in higher dimensions. With regard to the minimum-uncertainty states, we derive the minimum-uncertainty state equation by the analytic method and relate this to the ground-state problem of the Harper Hamiltonian. Furthermore, the higher-dimensional limit of the uncertainty relations and minimum-uncertainty states are explored. From an operational point of view, we show that the uncertainty in the unitary operator is directly related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. This shows a principle of preparation uncertainty, i.e., for any quantum system, the amount of visibility for two general noncommuting unitary operators is nontrivially upper bounded.
The uncertainty principle constitutes a cornerstone of quantum mechanics. However, its conventional formulations do not adequately describe uncertainty in scenarios involving conditional measurement. In this work, we propose uncertainty relations based on conditional variance and conditional expectation. We define the conditional expectation operator E ( A ̂ | B ̂ ) , its dual operator E ( B ̂ | A ̂ ) , and a corresponding conditional covariance term R corr . We demonstrate that even for noncommuting observables A ̂ and B ̂ , their conditional expectation operators can possess a nonvanishing commutator. This leads to a strengthened uncertainty relation applicable to both pure and mixed states. In addition to the standard term arising from the noncommutativity of the observables themselves, our relation incorporates three distinct contributions: one from the noncommutativity of the conditional expectations, one from quantum coherence, and one from the conditional covariance. These contributions collectively yield a significantly tighter lower bound. Numerical simulations verify the generality and enhanced tightness of the proposed relation. Our results provide a refined theoretical tool for the joint estimation of multiple observables in composite quantum systems, with potential applications in quantum precision metrology and secure quantum key distribution.
Abstract 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 .