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CLASSICAL ANALOGUE OF WEAK VALUE IN STOCHASTIC PROCESS

H. Tomita

Published Aug 17, 2016• 0 citations
OpenAlex

Abstract

One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements.We emulate it as a two-time conditional expectation in a classical stochastic model.We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real.The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.

Related papers

CLASSICAL ANALOGUE OF WEAK VALUE IN STOCHASTIC PROCESS

H. TOMITA

One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements. We emulate it as a two-time conditional expectation in a classical stochastic model. We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real. The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.

arXiv:1201.1329•Sep 27, 2012•0 citations

Weak value analogue in classical stochastic process

Hiroyuki Tomita

The time evolution of the two-time conditional probability of the classical stochastic process is described in an analogous form of the quantum mechanical wave equations. By using it, we emulate the same strange behaviors as those of the weak value in the quantum mechanics. A negative probability and abnormal expectations of some quantities remarkablely larger than their inherent norms are found in an example of a stochastic Ising spin system.

arXiv:1109.4302•Sep 20, 2011•1 citation

Observing the Weak Value through Quantum Interference

森, 琢也

In quantum mechanics, it is well known that counterfactual arguments sometimes lead to paradoxes, which implies that our classical intuition is not always reliable.However, it has been pointed out by several authors that the weak value, which was introduced by Aharonov et al. in 1988, may be useful to comprehend quantum mechanics intuitively.Another extraordinary aspect of the weak value is its amplification, which has been exploited for precision measurement.Nevertheless, the physical meaning of the weak value is still obscure due to its complexity of the value and the very fact of the amplification.In this thesis, by considering various situations of quantum interference, we attempt to provide a possible interpretation of the weak value.We find, on one hand, that the imaginary part of the weak value is relevant to the wave nature.On the other hand, in the double-slit experiment the particle nature can be seen in the real part of the position weak value.This property for the real part does not hold for more general cases such as the multiple (more than two) slit experiment.The position weak value, nevertheless, can be interpreted as the average of the classical trajectories based on a complex probability.We also investigate the possibility of knowing which path the particle takes in the multiple-slit experiment without destroying the interference fringes.We show that, because of the statistical nature of the weak value, obtaining the which-path information and the interference pattern is compatible with complementarity.Our study suggests that the weak value is a reasonable physical quantity.

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In this thesis, we develop a probabilistic description of the time-symmetric formulation of quantum mechanics.We derive a generalized form of the probability distribution of a quantum process which is determined by performing pre-and post-selections on a system.It gives the well-known value for an observable associated to a quantum process, called weak value, as an expectation value.We show that the generalized probability satisfies the law of total probability and admits the transitive form by means of conditional probabilities.This motivates us to step forward to a classical ensemble picture in quantum mechanics.In addition, we present the description of quantum measurement in terms of the generalized probability.It is suggested that the difference between the projective measurement and the weak measurement corresponds to the choice of selections of the forward and backward propagating processes. Consistency of the Probability in the Forward and Backward Processes 6 Conclusion and Discussion Bibliography

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Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.

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Jeff Tollaksen

The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.

The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting•Jan 1, 2013•0 citations
One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements.We emulate it as a two-time conditional expectation in a classical stochastic model.We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real.The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.
One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements. We emulate it as a two-time conditional expectation in a classical stochastic model. We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real. The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.
In quantum mechanics, it is well known that counterfactual arguments sometimes lead to paradoxes, which implies that our classical intuition is not always reliable.However, it has been pointed out by several authors that the weak value, which was introduced by Aharonov et al. in 1988, may be useful to comprehend quantum mechanics intuitively.Another extraordinary aspect of the weak value is its amplification, which has been exploited for precision measurement.Nevertheless, the physical meaning of the weak value is still obscure due to its complexity of the value and the very fact of the amplification.In this thesis, by considering various situations of quantum interference, we attempt to provide a possible interpretation of the weak value.We find, on one hand, that the imaginary part of the weak value is relevant to the wave nature.On the other hand, in the double-slit experiment the particle nature can be seen in the real part of the position weak value.This property for the real part does not hold for more general cases such as the multiple (more than two) slit experiment.The position weak value, nevertheless, can be interpreted as the average of the classical trajectories based on a complex probability.We also investigate the possibility of knowing which path the particle takes in the multiple-slit experiment without destroying the interference fringes.We show that, because of the statistical nature of the weak value, obtaining the which-path information and the interference pattern is compatible with complementarity.Our study suggests that the weak value is a reasonable physical quantity.
In this thesis, we develop a probabilistic description of the time-symmetric formulation of quantum mechanics.We derive a generalized form of the probability distribution of a quantum process which is determined by performing pre-and post-selections on a system.It gives the well-known value for an observable associated to a quantum process, called weak value, as an expectation value.We show that the generalized probability satisfies the law of total probability and admits the transitive form by means of conditional probabilities.This motivates us to step forward to a classical ensemble picture in quantum mechanics.In addition, we present the description of quantum measurement in terms of the generalized probability.It is suggested that the difference between the projective measurement and the weak measurement corresponds to the choice of selections of the forward and backward propagating processes. Consistency of the Probability in the Forward and Backward Processes 6 Conclusion and Discussion Bibliography
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.
The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.