# Understanding Weak Values Without New Probability Theory

## Abstract

The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable \(\hat {A}\) with post-selection 〈*F*| may be out of the eigenvalue range of \(\hat {A}\). This is because the weak value of \(\hat {A}\) with the post-selection is, in general, not the expectation value of \(\hat {A}\), but the expectation value of \(\hat {A}| F\rangle \langle F|\) boosted by the post-selection.

## Keywords

Weak value Weak measurement Post-selection Anomalous weak value Born ruleNearly three decades have passed since Aharonov et al. [1] introduced weak measurements and values. Nevertheless, they remain a subject of debate. Recently, Vaidman [2, 3] analyzed the nested Mach-Zehnder interferometer experiment with two-state vector formalism and insisted that the past of a quantum particle could be described according to the weak trace. Li et al. [4, 5] challenged Vaidman’s claim and insisted that the weak trace could be understood without any unusual probability theory if the disturbances of the weak measurements are considered. However, they agreed with Vaidman with regard to the physical meaning of the weak values.

Moreover, Ferrie and Combes [6, 7] argued that weak values are classical statistic quantities, which gave rise to a number of rebuttals [8, 9, 10, 11, 12]. In particular, Pusey [13] showed that anomalous (imaginary, negative, and unbounded) weak values are non-classical and proofs of contextuality. However, he did not show how the contextuality is responsible for the anomalous weak values.

*I*〉 and post-state 〈

*F*|; its real and imaginary parts are, which are accompanied with some constant factors, essentially the expectation values of \((1/2)(|F\rangle \langle F|\hat {A}+\hat {A}|F\rangle \langle F|)\) and \((i/2)(|F\rangle \langle F|\hat {A}-\hat {A}|F\rangle \langle F|)\) for |

*I*〉 boosted by 1/|〈

*I*|

*F*〉|

^{2}via the post-selection, respectively. If \(\hat {A}\) and |

*F*〉〈

*F*| do not commute, these values are completely different from the real and imaginary parts of the expectation value of \(\hat {A}\) for |

*I*〉 with the post-selection. Moreover, even if \(\hat {A}\) is a projection operator, \(\hat {A}|F\rangle \langle F|\) is not. Therefore, we have no reason to expect the weak value of \(\hat {A}\) within its eigenvalue range.

*g*

_{ A }is the coupling constant. \(\hat {H}_{A}\) is assumed to be constant and roughly equivalent to the total Hamiltonian \(\hat {H}\) over some interaction time

*t*

_{ A }. The initial wavefunction

*ϕ*

_{ A }(

*x*) of the measuring apparatus is assumed to be

*x*

_{ A }is the position of the pointer of the measuring device. The initial state |Φ

_{ A }(0)〉 = |

*I*〉|

*ϕ*

_{ A }〉, where |

*I*〉 is the initial state of the observed system, of the unified system of the observed system and the measuring device, evolves unitarily obeying the Schrödinger equation:

*g*

_{ A }

*t*

_{ A },

*F*| for the measured system is

*S*of the observed system and the ensemble

*M*of the measuring device after their unitary interaction are both separately obtained by combining all the elements of sub-ensembles, each of which is described by its own ket. Then, each element of

*S*belongs to one of the sub-ensembles

*E*

_{ i },

*i*= 1,2,⋯ described by |

*s*

_{ i }〉 and each element of

*M*belongs to one of the sub-ensembles

*E*

_{ α },

*α*= 1,2,⋯ described by |

*m*

_{ α }〉, such that the sub-ensemble

*ε*

_{i, α}of the unified system, whose elements belong to both

*E*

_{ i }and

*E*

_{ α }, is described by the density matrix

*ε*is the union of all the

*ε*

_{i, α}, the density matrix \(\hat \rho ^{\prime }\) describing

*ε*should be written as the weighted sum of all the \(\hat \rho _{i,\alpha }\):

*P*

_{i, α}are suitable factors. However,

*ε*is defined to be described by (6), such that it should be described by the density matrix (7). \(\hat \rho _{A}(t)\) and \(\hat \rho ^{\prime }\) are necessarily different, except in the case that |Φ

_{ A }(

*t*)〉 is a product of a vector |

*S*〉 in the Hilbert space of the observed system and a vector |

*M*〉 in the Hilbert space of the measuring apparatus, i.e.,

We must say for the above reason that both the observed system and the measuring device do not have separate ensembles of their own. Therefore, we conclude that the operation of 〈*F*| on (6) changes the unified system and (9) is not the state of the measuring device right after their unitary interaction, i.e. right after *t*_{ A }.

*F*| via a projection measurement. Their interaction Hamiltonians are (2) and

*ϕ*

_{ F }〉 is the initial state of the measuring device of \(\hat {F}\) whose wave function is assumed to be

*x*

_{ F }is the position of the pointer of the measuring device of \(\hat {F}\).

*g*

_{ A }

*t*

_{ A },

^{(s)}is the partial trace of the observed system. By calculating the expectation value of either \(\hat {x}_{A}\) or \(\hat {x}_{F}\), we can obtain the expectation value of either \(\hat {A}\) or \(\hat {F}\) accurately as follows:

*X*

_{ A }and

*X*

_{ B }simultaneously. However, we cannot know the expectation values of both \(\hat {A}\) and \(\hat {F}\) simultaneously [17]. Its reason is almost the same as the previous discussion: If the ensembles

*M*

_{ A }and

*M*

_{ F }of the two measuring devices after their unitary interaction with the measured system are both separately obtained by combining all the elements of the sub-ensembles, each of them can be described by its own ket. Each element of

*M*

_{ A }belongs to one of the sub-ensembles

*E*

_{ α },

*α*= 1,2,⋯, described by |

*a*

_{ α }〉 and each element of

*M*

_{ F }belongs to one of the sub-ensembles

*E*

_{ β },

*β*= 1,2,⋯, described by |

*f*

_{ β }〉 such that the sub-ensemble

*ε*

_{α, β}of the combined measuring device, whose elements belong to both

*E*

_{ α }and

*E*

_{ β }, is described by the density matrix

*P*

_{α, β}are suitable factors. However, (16) does not take the form of (19) if \(\hat {F}\) and \(\hat {A}\) do not commute. Therefore, \(\hat {x}_{A}\) and \(\hat {x}_{F}\) are entangled, i.e., the position operators of both measuring devices after the unitary interaction with the measured system do not have their own separate ensembles. We should regard the measurement of \(\hat {x}_{A}\) and \(\hat {x}_{F}\) as

*one*manipulation.

Then, we reconsider the process to know what outcome we obtain, i.e., what observable of the *unified* measuring device we read in this manipulation and what observable of the observed system corresponds to the outcome of the unified measuring device.

*X*

_{ A }and

*X*

_{ F }should not be treated separately, as shown above. Because \(\hat {x}_{F}\) is a projection operator,

*X*

_{ F }is 1 or 0 and \({X_{F}^{n}}=X_{F}\ (n\ne 0)\). Here and hereafter, we put

*g*

_{ F }

*t*

_{ F }= 1. On the other hand, we can know only the

*sum*of post-selected (and

*not*selected)

*X*

_{ A }’s, so that the outcome must be regarded as linear of

*X*

_{ A }. Therefore the outcome of the weak measurement with the post-selection is

*X*

_{ A }

*X*

_{ F }and the measured observable is \(\hat {x}_{A}\hat {x}_{F}\). Its expectation value is

*X*

_{ A }

*X*

_{ F }〉, the average of

*X*

_{ A }

*X*

_{ F }. Because \(\hat {x}_{F}\) and \(\hat {x}_{A}\) are entangled and \(\overline {x_{A}x_{F}}\ne \overline x_{A}\cdot \overline x_{F}\), we cannot obtain the expectation value of \(\hat {A}\) if it does not commute with \(\hat {F}\). (We can approximately obtain the expectation value of \(\hat {F}\) because the first measurement is weak.) Instead, we can obtain the expectation value of \((1/2)(\hat {F}\hat {A}+\hat {A}\hat {F})\) via the weak measurement.

*X*

_{ F }= 1, which is approximate selection of the final state 〈

*F*|. Because the post-selection

*X*

_{ F }= 1 (i.e.,

*X*

_{ F }≠ 0) implies

*X*

_{ A }

*X*

_{ F }≠ 0 (if

*X*

_{ A }≠ 0), the average of

*X*

_{ A }

*X*

_{ F }after the post-selection

*X*

_{ F }= 1 is equal to the average of

*X*

_{ A }after the post-selection:

^{(p)}stands for the average after post-selection. Moreover, because 〈

*X*

_{ A }

*X*

_{ F }〉

^{(p)}is the quotient of the sum of post-selected

*X*

_{ A }

*X*

_{ F }’s, which is equal to the sum of all

*X*

_{ A }

*X*

_{ F }’s without any post-selection, divided by the number of the post-selected data, it is boosted by 1/〈

*X*

_{ F }〉:

*X*

_{ F }〉 is nearly equal to \(\overline x_{F}\), because the first measurement is weak. For example, if the measured values are then, 〈

*X*

_{ A }

*X*

_{ F }〉 = 0.4, 〈

*X*

_{ F }〉 = 0.2, 〈

*X*

_{ A }

*X*

_{ F }〉

^{(p)}= 2.

*I*〉〈

*I*| commute, it becomes \(\langle I|\hat {A}|I\rangle \) independently of the post-selection. If \(\left [\hat {A},[\hat {F},|I\rangle \langle I|]\right ]= 0\), it is in proportion to \(\langle I|\hat {A}|I\rangle \). Otherwise, it is not an expectation value of \(\hat {A}\) in any sense, less to be the expectation value of \(\hat {A}\) after the post-selection 〈

*F*|. In contrast, it is the expectation value of \((1/2)(\hat {F}\hat {A}+\hat {A}\hat {F})\) boosted by the post-selection. This is the reason why the weak value of \(\hat {A}\) may be out of the eigenvalue range of \(\hat {A}\).

In summary, our main result comes down to (24), which clarifies that weak values can be completely understood within the framework of conventional quantum mechanics, that is, with Born rule and the general probability theory. Weak measurement with post-selection should be considered as a method to measure an observable which are product of two observables, one of which is a projection operator, and to boost its measured value.

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