# Weak values and weak coupling maximizing the output of weak measurements

## Abstract

In a weak measurement, the average output 〈o〉 of a probe that measures an observable A{sup -hat} of a quantum system undergoing both a preparation in a state ρ{sub i} and a postselection in a state E{sub f} is, to a good approximation, a function of the weak value A{sub w}=Tr[E{sub f}A{sup -hat} ρ{sub i}]/Tr[E{sub f}ρ{sub i}], a complex number. For a fixed coupling λ, when the overlap Tr[E{sub f}ρ{sub i}] is very small, A{sub w} diverges, but 〈o〉 stays finite, often tending to zero for symmetry reasons. This paper answers the questions: what is the weak value that maximizes the output for a fixed coupling? What is the coupling that maximizes the output for a fixed weak value? We derive equations for the optimal values of A{sub w} and λ, and provide the solutions. The results are independent of the dimensionality of the system, and they apply to a probe having a Hilbert space of arbitrary dimension. Using the Schrödinger–Robertson uncertainty relation, we demonstrate that, in an important case, the amplification 〈o〉 cannot exceed the initial uncertainty σ{sub o} in the observable o{sup -hat}, we provide an upper limit for the more general case, and a strategy to obtainmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22314826

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 345; Journal Issue: Complete; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AMPLIFICATION; APPROXIMATIONS; COUPLING; HILBERT SPACE; MATHEMATICAL SOLUTIONS; OPTIMIZATION; QUANTUM MECHANICS; QUANTUM STATES; SCHROEDINGER EQUATION; SYMMETRY; UNCERTAINTY PRINCIPLE

### Citation Formats

```
Di Lorenzo, Antonio, E-mail: dilorenzo.antonio@gmail.com.
```*Weak values and weak coupling maximizing the output of weak measurements*. United States: N. p., 2014.
Web. doi:10.1016/J.AOP.2014.03.007.

```
Di Lorenzo, Antonio, E-mail: dilorenzo.antonio@gmail.com.
```*Weak values and weak coupling maximizing the output of weak measurements*. United States. doi:10.1016/J.AOP.2014.03.007.

```
Di Lorenzo, Antonio, E-mail: dilorenzo.antonio@gmail.com. Sun .
"Weak values and weak coupling maximizing the output of weak measurements". United States. doi:10.1016/J.AOP.2014.03.007.
```

```
@article{osti_22314826,
```

title = {Weak values and weak coupling maximizing the output of weak measurements},

author = {Di Lorenzo, Antonio, E-mail: dilorenzo.antonio@gmail.com},

abstractNote = {In a weak measurement, the average output 〈o〉 of a probe that measures an observable A{sup -hat} of a quantum system undergoing both a preparation in a state ρ{sub i} and a postselection in a state E{sub f} is, to a good approximation, a function of the weak value A{sub w}=Tr[E{sub f}A{sup -hat} ρ{sub i}]/Tr[E{sub f}ρ{sub i}], a complex number. For a fixed coupling λ, when the overlap Tr[E{sub f}ρ{sub i}] is very small, A{sub w} diverges, but 〈o〉 stays finite, often tending to zero for symmetry reasons. This paper answers the questions: what is the weak value that maximizes the output for a fixed coupling? What is the coupling that maximizes the output for a fixed weak value? We derive equations for the optimal values of A{sub w} and λ, and provide the solutions. The results are independent of the dimensionality of the system, and they apply to a probe having a Hilbert space of arbitrary dimension. Using the Schrödinger–Robertson uncertainty relation, we demonstrate that, in an important case, the amplification 〈o〉 cannot exceed the initial uncertainty σ{sub o} in the observable o{sup -hat}, we provide an upper limit for the more general case, and a strategy to obtain 〈o〉≫σ{sub o}. - Highlights: •We have provided a general framework to find the extremal values of a weak measurement. •We have derived the location of the extremal values in terms of preparation and postselection. •We have devised a maximization strategy going beyond the limit of the Schrödinger–Robertson relation.},

doi = {10.1016/J.AOP.2014.03.007},

journal = {Annals of Physics (New York)},

number = Complete,

volume = 345,

place = {United States},

year = {Sun Jun 15 00:00:00 EDT 2014},

month = {Sun Jun 15 00:00:00 EDT 2014}

}