Testing of quantum phase in matter-wave optics

Řeháček, J.; Hradil, Z.; Zawisky, M.; Pascazio, Saverio; Rauch, H.; Peřina, Jan

doi:10.1103/physreva.60.473

Cited by 23 publications

(15 citation statements)

References 19 publications

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“…To date, the maximum-likelihood approach has been applied to various quantum problems from quantum phase estimation [7] to reconstruction of entangled optical states [8,9].…”

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confidence: 99%

Diluted maximum-likelihood algorithm for quantum tomography

et al. 2007

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We propose a refined iterative likelihood-maximization algorithm for reconstructing a quantum state from a set of tomographic measurements. The algorithm is characterized by a very high convergence rate and features a simple adaptive procedure that ensures likelihood increase in every iteration and convergence to the maximum-likelihood state. We apply the algorithm to homodyne tomography of optical states and quantum tomography of entangled spin states of trapped ions and investigate its convergence properties.

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“…To date, the maximum-likelihood approach has been applied to various quantum problems from quantum phase estimation [7] to reconstruction of entangled optical states [8,9].…”

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confidence: 99%

Diluted maximum-likelihood algorithm for quantum tomography

et al. 2007

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“…By averaging over ∆ it is easy to show that one obtains again (12). By plugging the average operator (12) into (5) one finally gets…”

Section: Fluctuations In Neutron Opticsmentioning

confidence: 96%

“…The trace of ρ O yields the relative frequency of neutrons in the ordinary channel. Suppose now that the phase shift ∆ fluctuates according to a probability law w(∆ − ∆ 0 ), ∆ 0 being the average phase (operationally defined as the phase that is measured-or inferred [12]-in an interferometric experiment). Therefore one has d∆ w(∆) = 1, d∆ w(∆)∆ = 0.…”

Section: Fluctuations In Neutron Opticsmentioning

confidence: 99%

Decoherence, fluctuations and Wigner function in neutron optics

Facchi

Mariano

Pascazio

et al. 2003

J. Opt. B: Quantum Semiclass. Opt.

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Abstract. We analyze the coherence properties of neutron wave packets, after they have interacted with a phase shifter undergoing different kinds of statistical fluctuations. We give a quantitative (and operational) definition of decoherence and compare it to the standard deviation of the distribution of the phase shifts. We find that in some cases the neutron ensemble is more coherent, even though it has interacted with a wider (i.e. more disordered) distribution of shifts. This feature is independent of the particular definition of decoherence: this is shown by proposing and discussing an alternative definition, based on the Wigner function, that displays a similar behavior. We briefly discuss the notion of entropy of the shifts and find that, in general, it does not correspond to that of decoherence of the neutron.

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“…which is, in fact, the completeness relation of a POVM with measured outputs {f i }. Special cases of the solution (10) have been discussed recently for the phase [8], the diagonal elements of the density matrix [9] and the reconstruction of the 1/2 spin state [10]. Quantum interpretation offers a new viewpoint on the maximum likelihood estimation.…”

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confidence: 99%

Quantum theory of incompatible observations

Hradil¹,

Summhammer²

2000

J. Phys. A: Math. Gen.

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Maximum likelihood principle is shown to be the best measure for relating the experimental data with the predictions of quantum theory.Quantum theory describes events on the most fundamental level currently available. The synthesis of information from mutually incompatible quantum measurements plays the key role in testing the structure of the theory. The purpose of this Letter is to show a unique relationship between quantum theory and the mathematical statistics used to obtain optimal information from incompatible observations: Quantum theory prefers the relative entropy (maximum likelihood principle) as the proper measure for evaluation of the distance between measured data and probabilities defined by quantum theory.In the standard textbooks [1], a quantum measurement is represented by a hermitian operatorÂ, whose spectrum determines the possible results of the measurement A|a = a|a .(1)In the following, the Dirac notation will be used and for the sake of simplicity a discrete spectrum will be assumed. Eigenstates are orthogonal a|a ′ = δ aa ′ and the corresponding projectors provide the closure relation a |a a| =1.Projectors predict the probability for detecting a particular value of the q-variable a represented by the operator A as p a = a|ρ|a , provided that the system has been prepared in a quantum state ρ. This mathematical picture corresponds to the experimental reality in the following sense: When the measurement represented by the operatorÂ is repeated N times on identical copies of the system, the number a particular output a is collected N a times. The relative frequencies f a = Na N will sample the true probability as f a → p a fluctuating around them. The exact values are reproduced only in the asymptotical limit N → ∞. Experimentalist's knowledge may be expressed in the form of a diagonal density matrixprovided that error bars of the order 1/N are associated with the sampled relative frequencies. This should be understood as mere rewriting of the experimental data {N, N a }. Similar knowledge may be obtained by observations, which can be parameterized by operators diagonal in the |a basis, i.e. by operators commuting with operatorÂ. But the possible measurement of non-commuting operators yields new information, which cannot be derived from the measurement ofÂ. From this viewpoint it seems to be advantageous to consider the sequential synthesis of various noncommuting observables. In this case several operatorŝ A j , j = 1, 2, . . . will be measured by probing of the system N times together. Now, one expects to gain more than just the knowledge of the diagonal elements of the density matrix in some a priori given basis. This sequential measurement of non-commuting observables should be distinguished from the similar problem of approximate simultaneous measurement of non-commuting observables [2]. As in the case of the measurement of a hermitian operators, the result of sequential measurements of noncommuting operators may be represented by a series of projectors |y i y i |. This should be accompa...

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Testing of quantum phase in matter-wave optics

Cited by 23 publications

References 19 publications

Diluted maximum-likelihood algorithm for quantum tomography

Diluted maximum-likelihood algorithm for quantum tomography

Decoherence, fluctuations and Wigner function in neutron optics

Quantum theory of incompatible observations

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