Quantum theory of measurement and ergodicity conditions

Daneri, A.; Loinger, A.; Prosperi, G. M.

doi:10.1016/0029-5582(62)90528-x

Cited by 274 publications

(109 citation statements)

References 23 publications

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“…This subject is referred to as hybrid (quantum-classical) dynamics. Besides the measurement theory it is supposed to apply in quantum chemistry [95,96,103] (where the full modeling of quantum degrees of freedom is difficult) and in quantum 21 What is allowed for Jupiter, is not allowed for the rind 22 Don't throw away old shoes before you have new ones gravity [110], where the proper quantum dynamics of the gravitational field is not known. There are several versions of the hybrid dynamics.…”

Section: Dutch Proverbmentioning

confidence: 99%

“…Wenn i wieder, wieder komm 29 From the German folk song "Muß i denn" Daneri, Loinger and Prosperi approach quantum measurements via the quantum ergodic theory [21]. Such an approach is anticipated in the late forties by the works of Jordan [186] and Ludwig [187].…”

Section: Ergodic Theory Approachmentioning

confidence: 99%

“…Under the influence of the system to be measured this conservation is broken, and there is a possibility to correlate different values of the measured observable with the pointer values. Daneri, Loinger and Prosperi invoke the basic assumption of ergodic theory and treat the overall density matrix via 28 Who cares about every little feather should not make the bed 29 When I come, come again time-averaging [21]. The time-averaged density matrix satisfies the necessary requirements for an ideal measurement.…”

Section: Ergodic Theory Approachmentioning

confidence: 99%

“…The differences ∆ ± defined by (4.15) satisfy 46 You have to keep your balance 21) and give rise to derivatives with respect to m according to…”

Section: Large N Expansionmentioning

confidence: 99%

“…In many approaches, starting from von Neumann [4,17,21] the word "collapse" or "reduction" is taken in a weak acception, referring to S alone. Such theoretical analyses involve only a proof that, in a basis that diagonalizes the tested observable, the off-diagonal blocks of the marginal density matrixr(t) of S fade out, but not necessarily those of the full density matrixD(t) of S + A.…”

Section: Dutch Expressionmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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