### Abstract

This paper starts with a brief review of the topic of strong and weak pre-and post-selected (PPS) quantum measurements, as well as weak values, and afterwards presents original work. In particular, we develop a nonperturbative theory of weak PPS measurements of an arbitrary system with an arbitrary meter, for arbitrary initial states of the system and the meter. New and simple analytical formulas are obtained for the average and the distribution of the meter pointer variable. These formulas hold to all orders in the weak value. In the case of a mixed preselected state, in addition to the standard weak value, an associated weak value is required to describe weak PPS measurements. In the linear regime, the theory provides the generalized Aharonov-Albert-Vaidman formula. Moreover, we reveal two new regimes of weak PPS measurements: the strongly-nonlinear regime and the inverted region (the regime with a very large weak value), where the system-dependent contribution to the pointer deflection decreases with increasing the measurement strength. The optimal conditions for weak PPS measurements are obtained in the strongly-nonlinear regime, where the magnitude of the average pointer deflection is equal or close to the maximum. This maximum is independent of the measurement strength, being typically of the order of the pointer uncertainty. In the optimal regime, the small parameter of the theory is comparable to the overlap of the pre-and post-selected states. We show that the amplification coefficient in the weak PPS measurements is generally a product of two qualitatively different factors. The effects of the free system and meter Hamiltonians are discussed. We also estimate the size of the ensemble required for a measurement and identify optimal and efficient meters for weak measurements. Exact solutions are obtained for a certain class of the measured observables. These solutions are used for numerical calculations, the results of which agree with the theory. Moreover, the theory is extended to allow for a completely general post-selection measurement. We also discuss time-symmetry properties of PPS measurements of any strength and the relation between PPS and standard (not post-selected) measurements.

Original language | English |
---|---|

Pages (from-to) | 43-133 |

Number of pages | 91 |

Journal | Physics Reports |

Volume | 520 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Nov 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Foundations of quantum mechanics
- Measurement theory
- Precision metrology
- Quantum information processing
- Weak values

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physics Reports*,

*520*(2), 43-133. https://doi.org/10.1016/j.physrep.2012.07.001

**Nonperturbative theory of weak pre-and post-selected measurements.** / Kofman, Abraham G.; Ashhab, Sahel; Nori, Franco.

Research output: Contribution to journal › Article

*Physics Reports*, vol. 520, no. 2, pp. 43-133. https://doi.org/10.1016/j.physrep.2012.07.001

}

TY - JOUR

T1 - Nonperturbative theory of weak pre-and post-selected measurements

AU - Kofman, Abraham G.

AU - Ashhab, Sahel

AU - Nori, Franco

PY - 2012/11/1

Y1 - 2012/11/1

N2 - This paper starts with a brief review of the topic of strong and weak pre-and post-selected (PPS) quantum measurements, as well as weak values, and afterwards presents original work. In particular, we develop a nonperturbative theory of weak PPS measurements of an arbitrary system with an arbitrary meter, for arbitrary initial states of the system and the meter. New and simple analytical formulas are obtained for the average and the distribution of the meter pointer variable. These formulas hold to all orders in the weak value. In the case of a mixed preselected state, in addition to the standard weak value, an associated weak value is required to describe weak PPS measurements. In the linear regime, the theory provides the generalized Aharonov-Albert-Vaidman formula. Moreover, we reveal two new regimes of weak PPS measurements: the strongly-nonlinear regime and the inverted region (the regime with a very large weak value), where the system-dependent contribution to the pointer deflection decreases with increasing the measurement strength. The optimal conditions for weak PPS measurements are obtained in the strongly-nonlinear regime, where the magnitude of the average pointer deflection is equal or close to the maximum. This maximum is independent of the measurement strength, being typically of the order of the pointer uncertainty. In the optimal regime, the small parameter of the theory is comparable to the overlap of the pre-and post-selected states. We show that the amplification coefficient in the weak PPS measurements is generally a product of two qualitatively different factors. The effects of the free system and meter Hamiltonians are discussed. We also estimate the size of the ensemble required for a measurement and identify optimal and efficient meters for weak measurements. Exact solutions are obtained for a certain class of the measured observables. These solutions are used for numerical calculations, the results of which agree with the theory. Moreover, the theory is extended to allow for a completely general post-selection measurement. We also discuss time-symmetry properties of PPS measurements of any strength and the relation between PPS and standard (not post-selected) measurements.

AB - This paper starts with a brief review of the topic of strong and weak pre-and post-selected (PPS) quantum measurements, as well as weak values, and afterwards presents original work. In particular, we develop a nonperturbative theory of weak PPS measurements of an arbitrary system with an arbitrary meter, for arbitrary initial states of the system and the meter. New and simple analytical formulas are obtained for the average and the distribution of the meter pointer variable. These formulas hold to all orders in the weak value. In the case of a mixed preselected state, in addition to the standard weak value, an associated weak value is required to describe weak PPS measurements. In the linear regime, the theory provides the generalized Aharonov-Albert-Vaidman formula. Moreover, we reveal two new regimes of weak PPS measurements: the strongly-nonlinear regime and the inverted region (the regime with a very large weak value), where the system-dependent contribution to the pointer deflection decreases with increasing the measurement strength. The optimal conditions for weak PPS measurements are obtained in the strongly-nonlinear regime, where the magnitude of the average pointer deflection is equal or close to the maximum. This maximum is independent of the measurement strength, being typically of the order of the pointer uncertainty. In the optimal regime, the small parameter of the theory is comparable to the overlap of the pre-and post-selected states. We show that the amplification coefficient in the weak PPS measurements is generally a product of two qualitatively different factors. The effects of the free system and meter Hamiltonians are discussed. We also estimate the size of the ensemble required for a measurement and identify optimal and efficient meters for weak measurements. Exact solutions are obtained for a certain class of the measured observables. These solutions are used for numerical calculations, the results of which agree with the theory. Moreover, the theory is extended to allow for a completely general post-selection measurement. We also discuss time-symmetry properties of PPS measurements of any strength and the relation between PPS and standard (not post-selected) measurements.

KW - Foundations of quantum mechanics

KW - Measurement theory

KW - Precision metrology

KW - Quantum information processing

KW - Weak values

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UR - http://www.scopus.com/inward/citedby.url?scp=84867709031&partnerID=8YFLogxK

U2 - 10.1016/j.physrep.2012.07.001

DO - 10.1016/j.physrep.2012.07.001

M3 - Article

VL - 520

SP - 43

EP - 133

JO - Physics Reports

JF - Physics Reports

SN - 0370-1573

IS - 2

ER -