Complementary Observables Research Articles

Arithmetic, mutually unbiased bases and complementary observables

Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.

COMPLEMENTARITY AND THE ALGEBRAIC STRUCTURE OF FOUR-LEVEL QUANTUM SYSTEMS

The history of complementary observables and mutual unbiased bases is shortly reviewed. A characterization is given in terms of conditional entropy of subalgebras due to Connes and Størmer. The extension of complementarity to noncommutative subalgebras is considered as well. Possible complementary decompositions of a four-level quantum system are described and a characterization of the Bell basis is obtained: The MASA generated by the Bell basis is complementary to both M2(ℂ) tensor factors.

Angular signatures of annihilating dark matter in the cosmic gamma-ray background

The extragalactic cosmic gamma-ray background (CGB) is an interesting channel to look for signatures of dark matter annihilation. In particular, besides the imprint in the energy spectrum, peculiar anisotropy patterns are expected compared to the case of a pure astrophysical origin of the CGB. We take into account the uncertainties in the dark matter clustering properties on subgalactic scales, deriving two possible anisotropy scenarios. A clear dark matter angular signature is achieved when the annihilation signal receives only a moderate contribution from subgalactic clumps and/or cuspy haloes. Experimentally, if galactic foreground systematics are efficiently kept under control, the angular differences are detectable with the forthcoming GLAST observatory, provided that the annihilation signal contributes to the CGB for a fraction $\ensuremath{\gtrsim}10\char21{}20%$. If, instead, subgalactic structures have a more prominent role, the astrophysical and dark matter anisotropies become degenerate, correspondingly diluting the dark matter signature. As complementary observables we also introduce the cross correlation between surveys of galaxies and the CGB and the cross correlation between different energy bands of the CGB, and we find that they provide a further sensitive tool to detect the dark matter angular signatures.

Estimating the expectation values of spin-1/2 observables with finite resources

We examine the problem of estimating the expectation values of two observables when we have a finite number of copies of an unknown qubit state. Specifically we examine whether it is better to measure each of the observables separately on different copies or to perform a separable nonadaptive joint measurement of the observables on each copy. We find that joint measurements can sometimes provide an advantage over separate measurements, but only if we make estimates of an observable based solely on the results of measurements of that observable. If we instead use both sets of results to estimate each observable then we find that individual measurements will be better. Finally we consider estimating the expectation values of three complementary observables for an unknown qubit.

Complementarity in quantum systems

Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A1 and A2 of B(H) are called complementary if the traceless subspaces of A1 and A2 are orthogonal (with respect to the Hilbert-Schmidt inner product). When both subalgebras are maximal Abelian, then the concept reduces to complementary observables or mutually unbiased bases. In the paper several characterizations of complementary subalgebras are given in the general case and several examples are presented. For a 4-level quantum system, the structure of complementary subalgebras can be described very well, the Cartan decomposition of unitaries plays a role. It turns out that a measurement corresponding to the Bell basis is complementary to any local measurement of the two-qubit-system.

Understanding the origin of the solar cyclic activity for an improved earth climate prediction

This review is dedicated to the processes which could explain the origin of the great extrema of the solar activity. We would like to reach a more suitable estimate and prediction of the temporal solar variability and its real impact on the Earth climatic models. The development of this new field is stimulated by the SoHO helioseismic measurements and by some recent solar modelling improvement which aims to describe the dynamical processes from the core to the surface. We first recall assumptions on the potential different solar variabilities. Then, we introduce stellar seismology and summarize the main SOHO results which are relevant for this field. Finally we mention the dynamical processes which are presently introduced in new solar models. We believe that the knowledge of two important elements: (1) the magnetic field interplay between the radiative zone and the convective zone and (2) the role of the gravity waves, would allow to understand the origin of the grand minima and maxima observed during the last millennium. Complementary observables like acoustic and gravity modes, radius and spectral irradiance from far UV to visible in parallel to the development of 1D–2D–3D simulations will improve this field. PICARD, SDO, DynaMICCS are key projects for a prediction of the next century variability. Some helioseismic indicators constitute the first necessary information to properly describe the Sun–Earth climatic connection.

Formulation of the uncertainty relations in terms of the Rényi entropies

Quantum mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the Renyi entropies. The proof of these inequalities requires the use of the exact expression for the (p,q)-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in N-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Renyi entropies.

Decoherence assisting a measurement-driven quantum evolution process

We study the problem of driving an unknown initial mixed quantum state onto a known pure state without using unitary transformations. This can be achieved, in an efficient manner, with the help of sequential measurements on at least two unbiased bases. However here we found that, when the system is affected by a decoherence mechanism, only one observable is required in order to achieve the same goal. In this way the decoherence can assist the process. We show that, depending on the sort of decoherence, the process can converge faster or slower than the method implemented by means of two complementary observables.

A simple optical demonstration of quantum cryptography using transverse position and momentum variables

We discuss a simple experiment illustrating the fundamentals of quantum cryptography. Our experiment is performed using novel complementary observables: the transverse position and momentum of photons. In the classical optics regime the experiment serves as an interesting and simple classroom demonstration of the principles of quantum key distribution.

Operational approach to complementarity and duality relations

We present a fully operational and consistent approach to complementarity. In contrast to previous approaches, in this proposal the duality relations emerge exclusively from the outcomes of simultaneous measurements performed on every run of the experiment and under the same experimental conditions. This can be done without assuming any definite relationship between the measurement performed and the complementary observables being studied.

Comprehensive test of entanglement for two-level systems via the indeterminacy relationship.

A 3-setting Bell-type inequality enforced by the indeterminacy relation of complementary local observables is proposed as an experimental test of 2-qubit entanglement. The proposed inequality has the advantage of being a sufficient and necessary criterion of separability. Therefore any entangled 2-qubit state cannot escape the detection by this kind of test. It turns out that the orientation of the local testing observables plays a crucial role in our perfect detection of entanglement.

Uncertainty reconciles complementarity with joint measurability

The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A new joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-independent consideration show that this uncertainty relation is logically connected with the joint measurability of the POVMs in question.

Complementarity and duality relations for finite-dimensional systems

We generalize to systems with arbitrary finite dimension a measure of quantum fluctuations (the certainty) previously introduced for two-dimensional systems. Using this measure, we study the duality relations satisfied by complementary observables looking for states with minimum joint fluctuations (maximum certainty states). We extend the duality relations to encompass several complementary observables simultaneously.

Solution to the King’s Problem in Prime Power Dimensions

It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1 - 5 (2001)].

Complementary observables for the determination of | Vub| in inclusive semileptonic B decays

The determination of | V ub | from inclusive semileptonic B decays is limited by uncertainties in modeling the decay distributions in b → u l ν transitions. The largest uncertainties arise from the limited knowledge of the appropriate b quark mass and Fermi momentum to use in the parameterization of the shape function. In this talk a new method is presented in which these parameters are constrained by the same data used to measure | V ub |. Two complementary observables can be constructed, one for discriminating between b → u l ν transitions and background and the other for constraining the b quark mass and the shape function, thereby reducing the theoretical uncertaintites in | V ub |.

Complementary observables for the determination of | Vub| in inclusive semileptonic B decays

The determination of | V ub | from inclusive semileptonic B decays is limited by uncertainties in modeling the decay distributions in b→ uℓ ν transitions. The largest uncertainties arise from the limited knowledge of the appropriate b quark mass and Fermi momentum to use in the parameterization of the shape function. This Letter presents a new method in which these shape function parameters are constrained by the same data used to measure | V ub |. The method requires measurements of the momenta of both the charged lepton and the neutrino in semileptonic B decays. From these quantities two complementary observables can be constructed, one for discriminating between b→ uℓ ν transitions and background and the other for constraining the shape function. Using this technique the uncertainties in | V ub | from the shape function may be significantly reduced.

ON ACCARDI'S NOTION OF COMPLEMENTARY OBSERVABLES

We discuss some examples of complementary observables, in the sense of Accardi. We show that the trace property is not sufficient to determine such a pair of observables.

Invariant information and quantum state estimation.

The invariant information, introduced by C. Brukner and A. Zeilinger [Phys. Rev. Lett. 83, 3354 (1999)], is reconsidered from the point of view of quantum state estimation. We show that this quantity is directly related to the mean error of the standard reconstruction from the measurement of a complete set of mutually complementary observables. We give its generalization in terms of the Fisher information. Provided that the optimum reconstruction is adopted, the information loses its invariant character.

Complementarity in multiple beam interference

We show that for multipath quantum interferometers the visibility of the interference and `which-path' information are not always complementary observables. This implies that there are path measurements that do not destroy the interference.

Exact uncertainty relations

The Heisenberg inequality \Delta X \Delta P \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The statistics of complementary observables are thus connected by an ``exact'' uncertainty relation.