In an usual quantum measurement the measuring device interacts strongly with a single quantum system. The state of the measured system is significantly altered, according to the measurement postulate. However, having a large ensemble of identically prepared quantum systems it is possible to extract information about the quantum state with arbitrarily small disturbance of the quantum state. This process is called “weak measurements” [1–11].
In an usual quantum measurement in order to extract information from a single system the interaction strengt of the quantum system with the detector should be large. If the interaction strenght is small, to obtain meaningful information about the quantum system the process of interaction has to be performed on an ensemble of identical quantum systems. Each system with its own detector is prepared in the same initial state. After time \(\tau\) the readings of the detectors are collected and averaged.
We will describe the microscopic component of the measuring device using quantum mechanics and the rest of the device classically. Let us consider the interaction of the quantum system with the microscopic part of the detector described by similar Hamiltonian as in a simple von Neumann measurement, \[ H=\gamma p_q A\,, \] where \(A\) is the observable of the quantum system, \(p_q\) is the momentum of the detector conjugate to the coordinate of the detector \(q\); the parameter \(\gamma\) describes the strength of the interaction. In contrast to a usual quantum measurement, the parameter \(\gamma\) is small. The interaction Hamiltonian results in a small shift of the detector's position \(q\). If the initial state of the detector has a width much larger than this shift, the state of the quantum system is only slightly changed.
The operator \(A\) has the eigenstates \(|a\rangle\) with eigenvalues \(a\). If the state of the quantum system before the measurement has coefficients \(\psi_a=\langle a|\psi\rangle\) in the basis of eigenstates of operator \(A\), the detector is in the pure state \(\phi(q)\), then the initial state of system and the detector is \[ \Psi_a (q)=\psi_a \phi(q)\,. \] Assuming that the duration of the interaction is \(\tau\), after the interaction the state becomes \[ \Psi_a^{\prime}(q)=\exp\left(-\frac{\mathrm{i}}{\hbar}\gamma\tau p_q a\right)\Psi_a (q) =\psi_a \phi(q-\gamma\tau a)\approx\psi_a \phi(q)-\gamma\tau a\psi_a \frac{\mathrm{d}\phi(q)}{\mathrm{d}q}\,. \] Applying the projective measurement of the postion \(q\) by the rest of the detector with the result \(q'\) yields the post-measurement state of the quantum system \[ \psi_a (q')=\frac{1}{\sqrt{P(q')}}\psi_a \phi(q'-\gamma\tau a)\,, \] where \[ P(q')=\sum_a |\psi_a|^2|\phi(q'-\gamma\tau a)|^2 \] is the probability to obtain the outcome \(q'\). We see that the post-measurement state differes little from the pre-measurement state if for all values \(a\) for which \(\psi_a\neq 0\) we have \(\gamma\tau a\ll\epsilon\), where \(\epsilon\) is the width of function \(\phi\). Since the width \(\epsilon\) of the detector's wave function \(\phi(q)\) is large, \(\epsilon\gg\gamma\tau a\), interaction of one quantum system with the measuring device will give almost no information about \(A\). However, the same measurement process can be performed on each member of an ensemble of \(N\) systems prepared in the same state.
Expanding the probability \(P(q')\) into the power series in the powers of the interaction strength \(\gamma\) and keeping only the first-order terms we get \[ P(q')\approx|\phi(q')|^2 - \gamma\tau\langle A\rangle\frac{\mathrm{d}|\phi(q')|^2}{\mathrm{d}q'}\,, \] where \(\langle A\rangle=\sum_a a|\psi_a|^2\) is the average of the observable \(A\). The average of the post-measurement detector's coordinate \(q'\) is \[ \langle q'\rangle=\int q' P(q')\,\mathrm{d}q'\approx\int q'|\phi(q')|^2\,\mathrm{d}q' - \gamma\tau\langle A\rangle\int q'\frac{\mathrm{d}|\phi(q')|^2}{\mathrm{d}q'}\,\mathrm{d}q' =\langle q\rangle + \gamma\tau\langle A\rangle\,. \] Here \(\langle q\rangle\) is the average pre-measurement coordinate of the detector. If the size of the ensemble of identically prepared systems is \(N\), the error in determining the average coordinate of the detector is of the order of \(\frac{\epsilon}{\sqrt{N}}\). The weak measurement allows the determination of the average \[ \langle A\rangle=\frac{1}{\gamma\tau}(\langle q'\rangle-\langle q\rangle) \] with the error in the average \(\langle A\rangle\) of the order of \(\frac{\epsilon}{\gamma\tau\sqrt{N}}\). The error can be made arbitrarily small by taking a large enough size of ensemble \(N\).
Note, that since the interaction Hamiltonian commutes with the detector's momentum \(p_q\), the average detector's momentum does not change and is the same before and after the interaction; \(\langle p_q^{\prime}\rangle=\langle p_q\rangle\).
Since the state of the quantum system does not significantly change during the interaction with the detector in the weak measurement process, after the interaction we can perform the usual projective measurement of another observable \(B\). From the ensemble of systems we can select a subensemble where the measurement of \(B\) yields a value \(b\). Averaging readings of the detectors in this subensemble we can obtain a quantity called “weak value” \(\langle A\rangle_b\) of the observable \(A\). This value can be interpreted as the average of \(A\) given the condition that the final state of quantum system is \(|b\rangle\).
After the measuring interaction is over, there is no further interaction between the systems of the ensemble and the corresponding measuring devices. Consequently, measurement of observable \(B\) will not affect the results of measurement of \(q\). Thus, changing the time ordering between the measurements of the detector's positon \(q\) and the postselection measurements of \(B\) will not affect the outcomes.
Applying the projective measurement of the postion \(q\) by the rest of the detector with the result \(q'\) and the projective measurement of the observable \(B\) with the result \(b\) yields the post-measurement probability to obtain outcomes \(q'\) and \(b\) \[ P(q',b)=\left|\sum_a \langle b|a\rangle\psi_a \phi(q'-\gamma\tau a)\right|^2\,, \] where \(|b\rangle\) are the eigenstates of operator \(B\). The average coordinate of the detector in the post-selected subensemble is \[ \langle q'\rangle_b =\int q'P(q'|b)\,\mathrm{d}q'\,, \] where \[ P(q'|b)=\frac{P(q',b)}{P(b)} \] is the coditional probability to obtain \(q'\) given that \(B\) has the value \(b\) and \[ P(b)=\int P(q',b)\,\mathrm{d}q' \] is the probability to obtain the measurement outcome \(b\). Expanding the probability \(P(q',b)\) into the power series in the powers of the interaction strength \(\gamma\) and keeping only the first-order terms we get \[ P(q',b)\approx|\phi(q')|^2|\langle b|\psi\rangle|^2 - 2\gamma\tau\mathrm{Re}\left\{\langle\psi|b\rangle\langle b|A|\psi\rangle\phi^{*}(q')\frac{\mathrm{d}\phi(q')}{\mathrm{d}q'}\right\} \] and \[ P(b)\approx|\langle b|\psi\rangle|^2 - \frac{\mathrm{i}}{\hbar}\gamma\tau\langle\psi|[|b\rangle\langle b|,A]|\psi\rangle\langle p_q\rangle\,. \] Here \(\langle p_q\rangle\) is the average pre-measurement momentum of the detector. Thus, the average coordinate of the detector in the subensemble can be approximated as \[ \langle q'\rangle_b\approx\langle q\rangle + \frac{1}{2}\gamma\tau\frac{\langle\psi|\{|b\rangle\langle b|,A\}|\psi\rangle}{|\langle b|\psi\rangle|^2} + \frac{\mathrm{i}}{\hbar}\gamma\tau\frac{\langle\psi|[|b\rangle\langle b|,A]|\psi\rangle}{|\langle b|\psi\rangle|^2} \left(\langle q\rangle\langle p_{q}\rangle-\mathrm{Re}\langle qp_{q}\rangle\right)\,. \] Introducing a complex-valued number, a “weak value” of the observable \(A\) \[ \langle A\rangle_b = \frac{\langle b|A|\psi\rangle}{\langle b|\psi\rangle} \] we can write the change of detector's coordinate as \[ \frac{1}{\gamma\tau}(\langle q'\rangle_{b}-\langle q\rangle) = \mathrm{Re}\langle A\rangle_{b} +\frac{2}{\hbar}\left(\mathrm{Re}\langle qp_{q}\rangle -\langle q\rangle\langle p_q\rangle\right)\mathrm{Im}\langle A\rangle_b\,. \] This expression is the same as the equation relating the change of the average coordinate to the average of the observable \(A\) in a weak measurement without post-selection. However, now the change of detector's coordinate consists of two parts: the first detector-independent part and the second part that depends on the detector's initial state \(\phi(q)\). The latter part is nonzero only when the operators \(A\) and \(B\) do not commute. Thus, in contrast to the average \(\langle A\rangle\) of the observable \(A\) which does not depend on the details of the interaction with the measuring device, for post-selected ensemble even in the limit of the vanishingly small interaction strength \(\gamma\) the change of detector's coordinate depends the measurement details. By changing the detector's initial state the detector-independent and detector-dependent terms can be separated and combined into one complex-valued number \(\langle A\rangle_b\).
Note, that in contrast to the weak measurement without post-selection, the average momentum of the detector in the post-selected ensemble is different from the initial momentum when the operators \(A\) and \(B\) do not commute. The change in the detector's average momentum can be estimated as follows: In momentum representation the state of the quantum system and the detector after the interaction is \[ \tilde{\Psi}_a^{\prime}(p_q )=\psi_a \mathrm{e}^{-\frac{\mathrm{i}}{\hbar}\gamma\tau p_q a}\tilde{\phi}(p_q )\,, \] where \[ \tilde{\phi}(p_q )=\frac{1}{\sqrt{2\pi\hbar}}\int\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}p_q q}\phi(q)\,\mathrm{d}q \] is the initial wave function of the detector in momentum representation. Applying the projective measurement of the observable \(B\) with the result \(b\) yields the post-measurement probability to obtain the momentum \(p_q\) and the value \(b\) \[ P(p_q, b)=|\tilde{\phi}(p_q)|^2\left| \sum_a \langle b|a\rangle\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}\gamma\tau p_q a}\psi_a \right|^2\,. \] The average momentum of the detector in the subensemble is given by the equation \[ \langle p_q^{\prime}\rangle_b = \int p_q \frac{P(p_q, b)}{P(b)}\,\mathrm{d}p_q\,. \] Keeping only terms up to first-order in interaction strength \(\gamma\) we get \[ P(p_q, b)\approx|\tilde{\phi}(p_q)|^2\left(|\langle b|\psi\rangle|^2 -\frac{\mathrm{i}}{\hbar}\gamma\tau p_q \langle\psi|[|b\rangle,\langle b|A]|\psi\rangle\right) \] and \[ \langle p_q^{\prime}\rangle_b \approx \langle p_q\rangle +\frac{2}{\hbar}\gamma\tau\mathrm{Im}\langle A\rangle_b \left(\langle p_q^2\rangle-\langle p_q\rangle^2\right)\,. \] Thus for non-commuting operators \(A\) and \(B\) the average momentum of the detector in the subensemble is different from the initial average momentum.