State extended uncertainty relations

Physical Review Letters, 2013

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabelling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A 84, 052117 ]. Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory. arXiv:1304.6351v2 [quant-ph]

Physical Review A, 2003

Entangled states represent correlations between two separate systems that are too precise to be represented by products of local quantum states. We show that this limit of precision for the local quantum states of a pair of N-level systems can be defined by an appropriate class of uncertainty relations. The violation of such local uncertainty relations may be used as an experimental test of entanglement generation. * Electronic address: h.hofmann@osa.org

Statistical Complexity, 2011

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the Rényi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and their more direct connection to the observed phenomena is emphasized. Several remaining open problems are mentioned.

2008

The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relations which are all formulated in a more or less intuitive and vague way in Heisenberg's seminal paper of 1927 [1]. These relations are expressions and quantifications of three fundamental limitations of the operational possibilities of preparing and measuring quantum mechanical systems which are stated here informally with reference to position and momentum as a paradigmatic example of canonically conjugate pairs of quantities: ... (A) It is impossible to prepare states in ...

We present a new paradigm for capturing the complementarity of two observables. It is based on the entanglement created by the interaction between the system observed and the two measurement devices used to measure the observables sequentially. Our main result is a lower bound on this entanglement and resembles well-known entropic uncertainty relations. Besides its fundamental interest, this result directly bounds the effectiveness of sequential bipartite operations---corresponding to the measurement interactions---for entanglement generation. We further discuss the intimate connection of our result with two primitives of information processing, namely, decoupling and coherent teleportation.

Physical Review A, 2014

Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.

2013

We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α, β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz-Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square 0 , 1 2 2 in the α-β plane, and a semi-analytical expression on the line β = α. It is seen that previous results are included as particular cases. Moreover, we present an analytical but suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.

Analyzing Heisenberg-Robertson (HR) and Schrödinger uncertainty relations 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 non-commuting observables, A and B, is zero. 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 non-commuting observables in the state under consideration.

Open Systems & Information Dynamics, 2015

We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.

The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relations which are all formulated in a more or less intuitive and vague way in Heisenberg's seminal paper of 1927 [1]. These relations are expressions and quantifications of three fundamental limitations of the operational possibilities of preparing and measuring quantum mechanical systems which are stated here informally with reference to position and momentum as a paradigmatic example of canonically conjugate pairs of quantities:

Physics Letters A, 1997

A geometric framework for quantum statistical estimation is used to establish a series of corrections to the Heisenberg uncertainty relations for canonically conjugate variables. These results apply when the true state of the system belongs to a one-parameter family of unitarily related states, and we are required to estimate the value of the parameter. @ 1997 Elsevier Science B.V.

American Journal of Applied Mathematics, 2015

This paper, deals with the uncertainty relation for photons. In [Phys.Rev.Let.108, 140401 (2012)], and 1 the uncertainty relation was obtained as a sharp inequality by using the energy distribution on space. The relation we obtain here is an alternative to the one given in [Phys.Rev.Let.108, 140401 (2012)] by the use of the position of the center of the energy operator. The fact that the components of the center is non commutative affected the right hand side of the Heisenberg inequality. But this resolved by the increase of the photon energy. Furthermore we study the uncertainty of Heisenberg with respect to angular momentum and Foureir. We end the paper by giving some examples.

Physics Letters A, 1980

A generalized Heisenberg-type uncertainty relation is obtained for two arbitrary operators both in the case of pure and of mixed states. As a rule equality is found to hold for pure quantum states only. New minimizing states called correlated coherent states, are constructed in explicit form, and their properties are studied.

Mathematical Structures and Applications

This contribution to the present Workshop Proceedings outlines a general programme for identifying geometric structures-out of which to possibly recover quantum dynamics as well-associated to the manifold in Hilbert space of the quantum states that saturate the Schrödinger-Robertson uncertainty relation associated to a specific set of quantum observables which characterise a given quantum system and its dynamics. The first step in such an exploration is addressed herein in the case of the observables Q and P of the Heisenberg algebra for a single degree of freedom system. The corresponding saturating states are the well known general squeezed states, whose properties are reviewed and discussed in detail together with some original results, in preparation of a study deferred to a separated analysis of their quantum geometry and of the corresponding path integral representation over such states.

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.