On the Complementarity of the Quadrature Observables

Lahti, Pekka; Pellonpää, Juha-Pekka

doi:10.1007/s10701-009-9373-y

Cited by 3 publications

(4 citation statements)

References 25 publications

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“…As the key ingredient of the T-C procedure is the preservation of Gaussianity, it is therefore a natural step to investigate how this procedure can be generalized and extended to more involved non-Markovian scenarios, still preserving Gaussianity. Indeed our aim is further supported by recent work [19,20,21], in which it has been proved that it is in principle possible to make tomographic measurements of the probability densities associated to every quadrature in phase space (for example in quantum optics it could be realized by means of homodyne detection). As a final remark we note that other methods to measure the covariance matrix of Gaussian states have been discussed in [22].…”

Section: From Tomograms To Cumulantsmentioning

confidence: 68%

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A tomographic approach to non-Markovian master equations

Bellomo¹,

Pasquale²,

Gualdi³

et al. 2010

J. Phys. A: Math. Theor.

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We propose a procedure based on symplectic tomography for reconstructing the unknown parameters of a convolutionless non-Markovian Gaussian noisy evolution. Whenever the time-dependent master equation coefficients are given as a function of some unknown time-independent parameters, we show that these parameters can be reconstructed by means of a finite number of tomograms. Two different approaches towards reconstruction, integral and differential, are presented and applied to a benchmark model made of a harmonic oscillator coupled to a bosonic bath. For this model the number of tomograms needed to retrieve the unknown parameters is explicitly computed.

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Section: From Tomograms To Cumulantsmentioning

confidence: 68%

“…where the ratio Sj (0)/ Sj (t) is the experimentally measurable quantity. Hence, by performing two distinct measurements of this ratio we can evaluate (21) at two different times. We thus obtain a system of two numerically solvable equations, which allows us to retrieve the time-independent parameters α and ω c .…”

Section: Example: Integral Approachmentioning

confidence: 99%

A tomographic approach to non-Markovian master equations

Bellomo¹,

Pasquale²,

Gualdi³

et al. 2010

J. Phys. A: Math. Theor.

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“…The relation between quadratures and Weyl systems is very well known, both in the case F = R [34,35,36,37] and when F is a finite field as in the present paper [5,27,38,39,40]. In the latter case, the use of Weyl systems to construct quadrature systems essentially goes back to [2].…”

Section: -Covariant Quadratures and Their Associated Weyl Systemsmentioning

confidence: 87%

Covariant mutually unbiased bases

2016

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The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension $2^r$ covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance

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“…, π}, is complementary [42]. Moreover, any family of the pairwise complementary observables {Q θ | θ ∈ S}, with a dense set S ⊂ [0, 2π), is informationally complete [43].…”

Section: Examples Of Complementaritymentioning

confidence: 99%

Complementary Observables in Quantum Mechanics

et al. 2019

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To the memory of Paul Busch, our friend and colleague One may view the world with the p-eye and one may view it with the q-eye but if one opens both eyes simultaneously then one gets crazy.Wolfgang Pauli in a letter to Werner Heisenberg, 19 October 1926. We hope to have demonstrated that one can safely open a pair of complementary 'eyes' simultaneously.He who does so may even 'see more' than with one eye only. The means of observation being part of the physical world, Nature Herself protects him from seeing too much and at the same time protects Herself from being questioned too closely: quantum reality, as it emerges under physical observation, is intrinsically unsharp. It can be forced to assume sharp contours -real properties -by performing repeatable measurements. But sometimes unsharp measurements will be both, less invasive and more informative. Paul Busch et co in the Epilogue of [1].Abstract. We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity -some of which new -in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul's work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.

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On the Complementarity of the Quadrature Observables

Cited by 3 publications

References 25 publications

A tomographic approach to non-Markovian master equations

A tomographic approach to non-Markovian master equations

Covariant mutually unbiased bases

Complementary Observables in Quantum Mechanics

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