Fisher information analysis on post-selection involved quantum precision measurements using optical coherent states

The concept

of quantum weak values AAV88 ; AV90 has been found useful in quantum metrology. A novel scheme termed as weak-value amplification (WVA) has been developed for detecting tiny effects such as the spin Hall effect of light Kwi08 and measuring weak signals How09a ; How09b ; How10a ; How10b ; Guo13 , owing to the advantages of improving the precision of weak measurements by post-selection and protecting the weak measurements against technical noises Sim10 ; Ste11 ; Nish12 ; Ked12 ; Jor14 ; Li20 ; Bru15 ; Bru16 ; How17 .

To be more specific, for the measurement of small transverse deflections of an optical beam How09a ; How09b ; How10a ; How10b , the WVA technique can lead to experimental sensitivity beyond the detector’s resolution. In this context, it allowed to use high power lasers with low power detectors while maintaining the optimal signal-to-noise ratio (SNR), and obtained the ultimate limit in deflection measurement with a large beam radius. Actually, theoretical analysis has pointed out that the WVA technique can outperform conventional measurement in the presence of detector saturation Lun17 . This advantage of the WVA technique was recently illustrated by achieving a precision of 6 times higher than that of the conventional measurement ZLJ20 . Other improvements along the line of the WVA technique include such as strengthening the WVA via photon recycling Jor13 ; Sim15 ; Jor21 and applying a technique called dual WVA (DWVA) which allows for a precision metrology with sensitivity several orders of magnitude higher than the standard approach Zen19 . Also, it has been pointed out that the WVA technique can reduce the technical noise in some circumstances Kwi08 ; How09a ; How09b ; Sim10 ; Ste11 ; Nish12 ; Ked12 ; Jor14 ; Li20 , and can even utilize it to outperform standard measurement by several orders of magnitude by means of the imaginary weak-value measurements Sim10 ; Ste11 ; Ked12 ; Jor14 ; Li20 .

The WVA technique involves an essential procedure of post-selection to the state of the measured system. It is true that the meter state conditioned on the successful post-selection contains more information about the parameter to be estimated than the meter state in conventional measurement for parameter estimation. However, since the probability of successful post-selection is low, the practically attainable Fisher information will be greatly reduced. It is largely this balance consideration that has caused controversial debates in literature Nish12 ; Ked12 ; Jor14 ; Li20 ; Tana13 ; FC14 ; Kne14 ; ZLJ15 ; Aha15 . In Ref. Jor14 , using the Fisher-information (FI) metric, it was proved that WVA technique can put all of the Fisher information about the parameter to be estimated into a small portion of the events and show how this fact alone gives technical advantages. Meanwhile, there existed diverse opinions about the WVA advantages Tana13 ; FC14 ; Kne14 ; ZLJ15 ; Aha15 . For instance, in Ref. ZLJ15 , it was concluded that both the weak measurement and post-selection cannot enhance the metrological precision. The analysis was based on a consideration that the Fisher information contained in the WVA technique is only a part of the total Fisher information associated with the post-selection measurement and the subsequent measurement on the meter state, while the reference standard is the quantum Fisher information (QFI) of the joint entangled state of the entire system-plus-meter after coupling interaction (with the strength parameter to be estimated). Obviously, such a reference standard is too stringent, which is not the standard for comparison with the conventional precision measurement, such as considered in Refs. Ked12 ; Jor14 ; Li20 and in many others.

In precision metrology, another important criterion is the scaling of the metrological precision with the resource (e.g. photon numbers), aiming to excess the standard quantum limit (SQL) and to achieve the Heisenberg limit (HL). The general viewpoint is that, in order to achieve the HL, quantum resources such as the NOON states Dow08 ; Mac11 ; Ste14 are typically required. However, it was found in Ref. ZLJ15 and further analyzed in Aha15 , that it is possible to achieve the HL with weak measurement using the optical coherent state, which is a classical resource, as a meter to coupled to a qubit system. In this case, the information encoded in the distribution of post-selection process (but not in the post-selected meter state) can be scaled at the Heisenberg limit with the average photon numbers of the coherent state.

For employing the optical coherent state as a meter, we become aware of an interesting conclusion from an earlier study Lui10 , which states that for any linear coupling interaction, metrological precision better than using the optical coherent state must come from non-classical effect. Noting that this statement was proved in the absence of post-selection procedure, we may thus raise a question: can the post-selection involved in the WVA technique violate the conclusion proved in Ref. Lui10 ? Of great interest is that, as we will see in this work, the answer is YES.

In this work, following the line of analysis in Refs. Ked12 ; Jor14 ; Li20 , where the Gaussian meter state (i.e. the transverse spatial wavefunction of a light beam) was considered, we further consider the probe meter as an optical coherent state and explore its possible advantages in the WVA metrology. This type of meter state is particularly relevant to the optical cavity-QED or the solid-state circuit-QED set-up realization. Also, as in Ref. Li20 , our analysis will go beyond the Aharonov-Albert-Vaidman (AAV) limit by considering arbitrary measurement-coupling strengths. By means of the Fisher-information metric, the first important result is that using the optical coherent state as a probe meter, which is cheap and not a “quantum resource”, the WVA measurement can definitely outperform the conventional measurement in the absence of the post-selection strategy. This result is very different from the one of using the Gaussian meter state, where the Fisher information contained in the WVA measurement can, at most, reach the same as in the conventional measurement Jor14 ; Li20 . In this context we will show that, for the measurement of coherent state in the absence of post-selection, we must consider quadrature measurement, since the usual photon-number measurement cannot extract the quantum Fisher information (QFI) encoded in the phase-shifted coherent state. The second important result is that, in the presence of the post-selection strategy, the WVA measurement using a mixture of coherent states can work better than using a single pure coherent state. This result is of great interest when making a connection with the claim in Ref. Lui10 , as mentioned above. Since our analysis covers the range of arbitrary coupling strengths, being somewhat surprising, we find that it is possible to achieve a scaling behavior with the average photon numbers even better than the HL, when the measurement-coupling strength is properly increased to violate the AAV limit.

Let us consider

a two-state quantum system (qubit) with states denoted by $|g\rangle$ and $|e\rangle$, which is coupled to an optical meter prepared in coherent state $|\alpha\rangle$, while the meter state will be finally monitored by a classical apparatus via photon number or field quadrature measurement, as schematically shown in Fig. 1. In dispersive regime, the coupling interaction Hamiltonian can be assumed as $H_{\rm int}=\chi\hat{\sigma}_{z}\hat{n}$, where $\hat{\sigma}_{z}$ is the atomic Pauli operator $\hat{\sigma}_{z}=|e\rangle\langle e|-|g\rangle\langle g|$ and $\hat{n}=\hat{a}^{\dagger}\hat{a}$ is the photon number operator (with $\hat{a}^{\dagger}$ and $\hat{a}$ the creation and annihilation operators of a photon). Over some time, the coupling interaction will generate a unitary operation $U=e^{i\lambda\hat{\sigma}_{z}\hat{n}}$ on the system-plus-meter joint state, where $\lambda$ is the integrated interaction strength, which is the right parameter to be metrologically estimated.

Assuming the system initially prepared in a superposition state characterized by $|\psi_{i}\rangle={\rm cos}\frac{\theta_{i}}{2}|g\rangle+{\rm sin}\frac{\theta_{i}}{2}e^{i\phi_{i}}|e\rangle$, the coupling interaction will entangle the joint state of the system-plus-meter as

In the WVA strategy, a post-selection procedure is involved by performing a projective measurement on the system, before finally measuring the meter state for the parameter-$\lambda$ estimation. Let us assume the post-selection with state $|\psi_{f}\rangle$, which has the same form of $|\psi_{i}\rangle$ by replacing only the Bloch-vector polar angles with $\theta_{f}$ and $\phi_{f}$. After post-selection for the system state, the meter state reads as

Here we introduced $\phi_{0}=\phi_{i}-\phi_{f}$. The state $|\tilde{\Phi}_{f}\rangle$ is not normalized. Let us denote the normalized meter state as $|\Phi_{f}\rangle=|\tilde{\Phi}_{f}\rangle/\sqrt{p_{a}}$, where the normalization factor is given by $p_{a}=\langle\tilde{\Phi}_{f}|\tilde{\Phi}_{f}\rangle$, which is also the probability of successful post-selection, while the subscript “a” simply indicates the accepted case of post-selection. This notation will be used hereafter for other quantities. In practice, after post-selection for the system state, subsequent measurement to the meter state $|\Phi_{f}\rangle$ should be done, e.g., performing the photon-number measurement and obtaining the distribution probability $P_{f}(n)$. This probability encodes the information of the parameter $\lambda$; one can thus calculate the Fisher information $F_{a}$, which characterizes the metrological precision for the parameter $\lambda$ via the Cramér-Rao inequality WM09 . In the WVA scheme, the key quantity to be compared with the conventional approach is the reduced Fisher information $p_{a}F_{a}$ (to be termed as WVA-FI in this work).

If we do not specify the concrete scheme of measurement on the meter state, we can talk about and calculate the QFI $Q_{a}$ encoded in the meter state $|\Phi_{f}\rangle$, using the formula

Actually, the reduced QFI (or, WVA-QFI) $p_{a}Q_{a}$ represents the maximum information attainable by the WVA measurement. In Fig. 2 we display the result of $p_{a}Q_{a}$ as a function of the parameter $\lambda$. This $\lambda$-dependence will be used later to discuss an important conclusion, when the WVA scheme is compared with the conventional approach.

In the above analysis,

we employed the tool of Fisher information, which is connected with the parameter estimate precision via the well known Cramér-Rao inequality, to discuss the WVA scheme in comparison with the conventional approach (without post-selection). In contrast to the conclusion based on the Gaussian meter state as typically analyzed in literature Nish12 ; Ked12 ; Jor14 ; Li20 ; Tana13 ; FC14 ; Kne14 , we revealed an important metrological advantage of the optical coherent state if the strategy of post-selection is involved.

In quantum precision metrology, another important criterion is in concern with resource consuming, e.g., by considering the (average) photon-number ($\bar{n}=|\alpha|^{2}$) dependence to be standard-quantum-limit (SQL $\propto\bar{n}$) or Heisenberg-limit (HL $\propto\bar{n}^{2}$) scaling behavior. Usually, the HL scaling can be achieved by exploiting the quantum entanglement of photon pairs Dow08 ; Mac11 ; Ste14 . Therefore, an interesting question is: can the HL precision of the $\bar{n}^{2}$-scaling be achieved by the post-selection strategy, without using any entangled photon pairs?

For optical coherent states, unfortunately, it was found that the FI gained by the post-selected results of measurement scales with $\bar{n}$ linearly ZLJ15 . This means that only the precision of the SQL can be achieved by the WVA technique. However, when applying the procedure of post-selection, it was found that the FI contained in the distribution probability of the post-selection results is possible to achieve the HL of $\bar{n}^{2}$-scaling ZLJ15 . This issue was further highlighted and analyzed in detail by Jordan and Aharonov et al in Ref. Aha15 .

Remarkably, rather than using the single (pure) coherent state as the meter state, it was found that, if using a statistical mixture of coherent states, the HL of $\bar{n}^{2}$-scaling can be achieved by the WVA measurement LCF18 . In the remainder of our present work, we would like to extend the study from the AAV limit of extremely weak measurement, as assumed in Ref. LCF18 , to arbitrary strength of measurement interaction. We will show that, with the increase of the measurement interaction strength, the HL precision (and even better result) can be achieved in the large $\bar{n}$ regime. Moreover, for using classical mixed states, we will make a connection with the contradictory statement proved in Ref. Lui10 , showing that the strategy of post-selection may cause remarkable impact on some existing conclusions and make important addition to wider range of problems Lui08 ; Sil10 .

The statistical mixture of multiple coherent states for the optical meter can be expressed in general as $\rho_{\rm class}=\sum_{j}w_{j}|\alpha_{j}\rangle\langle\alpha_{j}|$, with $|\alpha_{j}\rangle$ the individual coherent states and $w_{j}$ the weighting factors. In parallel to the formulation for the case of the pure state, the measurement interaction between the system and the meter is described by a unitary transformation to the initial product state, $\rho_{T}=U(\rho_{s}\otimes\rho_{\rm class})U^{\dagger}$, where $U=e^{i\lambda\hat{\sigma}_{z}\hat{n}}$. After post-selecting the system state with $|\psi_{f}\rangle$, the meter state reads $\rho_{f}=\langle\psi_{f}|\rho_{T}|\psi_{f}\rangle/\tilde{p}_{a}\equiv\tilde{\rho}_{f}/\tilde{p}_{a}$. Now the probability of successful post-selection is given by $\tilde{p}_{a}={\rm Tr}_{M}[\tilde{\rho}_{f}]$, where ${\rm Tr}_{M}(\cdots)$ means the trace over a complete set of the meter states. Then, the photon-number measurement on the meter state gives the distribution function $P_{f}(n)=\langle n|\rho_{f}|n\rangle$, and the Fisher information can be computed straightforwardly.

To be specific, let us consider a simple example of mixed state as

Straightforwardly, one can compute the average photon number as $\bar{n}={\rm Tr}_{M}(\hat{n}\rho_{\rm class})=|\alpha|^{2}+\frac{2}{3}|\delta|^{2}$, and the variance of the number fluctuation as

To gain a preliminary insight, let us assume $|\delta|>>|\alpha|$. Then, we have $\bar{n}\simeq\frac{2}{3}|\delta|^{2}$, and $(\delta n)^{2}\simeq\frac{2}{9}|\delta|^{4}\sim\bar{n}^{2}$. In Ref. LCF18 , being restricted in the AAV limit, it was proved that the FI is directly related with the variance of the photon numbers of the meter state initially prepared, which simply reads $F_{a}^{(n)}=4({\rm Im}\sigma^{w}_{z})^{2}(\Delta n)^{2}$, where $\sigma^{w}_{z}=\langle\psi_{f}|\sigma_{z}|\psi_{i}\rangle/\langle\psi_{f}|\psi_{i}\rangle$ is the famous expression of the AAV weak values. One obtains thus a metrological precision of the Heisenberg $\bar{n}^{2}$-scaling.

For an arbitrary strength of measurement interaction beyond the AAV limit, after post-selection with $|\psi_{f}\rangle$ to the system state, the photon-number measurement on the meter state would yield the probability distribution

Here we have introduced the function $p_{n}(y)=e^{-|y|^{2}}|y|^{2n}/n!$ and the pre-factor

The normalization factor is given by $\tilde{p}_{a}=\frac{1}{3}\sum^{3}_{j=1}p_{a}(\bar{n}_{j})$, where $\bar{n}_{1}=|\alpha|^{2}$ and $\bar{n}_{2,3}=|\alpha\pm\delta|^{2}$, while the $p_{a}$ function reads as

Using the probability distribution given by Eq. (17), the FI can be straightforwardly calculated.

In Fig. 5 we compare the WVA-FI between utilizing mixed and pure states. Results for the former are computed by considering the photon-number measurement, while results for the latter are the optimal QFI $p_{a}Q_{a}$ for a pure state with identical average photon numbers $\bar{n}$. We see clearly that the classical mixed state can work better than the quantum pure state after applying the post-selection strategy. To demonstrate the effect of generalization beyond the AAV limit, we compare the exact numerical results with the ones from the approximated expression under the AAV limit, for a few interaction strengths. With the increase of the strength parameter $\lambda$, greater deviation will occur (not shown in the figure).

The important message got here is that, in the large $\bar{n}$ regime, the mixed state can work much better than a pure state with the same average photon numbers. However, in Ref. Lui10 , it was proved that, within the linear-coupling detection schemes (just as considered in this work), using any classical states $\rho_{\rm class}$ (i.e. statistical mixture of coherent states) cannot surpass the metrology precision by using a quantum pure state $|\alpha_{\rho}\rangle$ with the same $\bar{n}$. This significant difference should originate from the procedure of post-selection in the WVA scheme, since in Ref. Lui10 only the standard measurement scheme (i.e. without post-selection) was considered. Therefore, we can newly claim that, by means of involving the post-selection strategy as involved in the WVA scheme, rather than what was claimed in Ref. Lui10 , properly designed classical mixed states can work much better than the quantum pure states with the same resource (i.e. the same average photon numbers).

Making a connection between the result revealed in Fig. 5 and the statement proved in Ref. Lui10 is quite constructive, which is also relevant to a wider range of problems Lui08 ; Sil10 , i.e., achieving high visibility and super-resolution in the absence of quantum resources (e.g. quantum entanglement). It seems that the simple procedure of post-selection involved in the WVA scheme can make an important addition to some conclusions in the field of quantum metrology, and deeper understanding needs further investigations.

In Fig. 6 we reveal the $\bar{n}$-scaling behavior of the Fisher information associated with photon-number measurement within the WVA scheme. Results for the interaction strengths $\lambda=10^{-3}$ and $10^{-2}$ are shown in Fig. 6(a) and (b), and a comparison between the results from the mixed and pure states is plotted. In the large $\bar{n}$ regime, as indicated by the slope $k=2$ or 1 of the fitting lines, the HL scaling (and beyond) can be achieved by using the mixed state, while using a pure state can largely result in the SQL scaling behavior. We emphasize that the HL scaling behavior revealed here for the mixed meter state is also induced by the post-selection procedure in the WVA scheme, since it was proved in Ref. Lui10 that, for linear coupling interaction, any classical mixture of the coherent states cannot work better than a single pure coherent state (under the condition with identical average photon numbers), while the pure state can only result in the SQL scaling (as indicated in Fig. 6 by the slope $k=1$).

For the “super-Heisenberg” scaling behavior observed in Fig. 6(b), i.e., the scaling better than $k>2$, we may provide a simple understanding as follows, by attributing to violating the AAV limit. Let us denote the photon-number distribution function for the mixed state as $\bar{p}(n)=\frac{1}{J}\sum^{J}_{j=1}p_{n}(\alpha_{j})$, with $p_{n}(\alpha_{j})$ the same function as defined in Eq. (17). In the small $\lambda$ limit, the probability function of photon-numbers measured after post-selection can be approximately expressed as

Then, the FI associated with this distribution function can be simply obtained as

Based on this expression, if neglecting the small $\lambda$ term in the denominator, we obtain $F_{a}^{(n)}\simeq 4({\rm Im}\sigma_{z}^{w})^{2}\langle(n-\bar{n})^{2}\rangle$, where the statistical average $\langle\cdots\rangle=\sum_{n}(\cdots)\bar{p}_{n}$ is introduced. This result means that the photon-number scaling behavior is determined by the initial state of the meter field, and the HL scaling will be achieved as pointed out below Eq. (IV). However, if we keep the series expansion of $(1-z)^{-1}=1+z+z^{2}/2!+\cdots$ to higher orders, e.g., keeping the linear term, we will have an additional term of contribution $\propto\langle(n-\bar{n})^{3}\rangle$, which makes the scaling better than the HL as observed in Fig. 6(b). Moreover, we have checked the $\bar{n}$ dependent behavior for the probability $\tilde{p}_{a}$ of successful post-selection, which can be approximately fitted as $\tilde{p}_{a}=d-b\bar{n}$, with the slope $b>0$. We find that $b$ is very small for $\lambda=10^{-3}$, and becomes larger for $\lambda=10^{-2}$. This additional observation also supports the above explanation for the “super-Heisenberg” scaling behavior shown in Fig. 6(b). It is indeed owing to the violation of the AAV limit. Violation of the AAV limit also explains the scaling result better than the SQL (with slope $k=1$), for the pure state shown in Fig. 6(b).

We have considered

the WVA precision metrology in the case of using the optical coherent state as a probe meter. We calculated the WVA-FI associated with two schemes of measuring the meter’s field, after post-selection to the system state. The first scheme is the photon-number measurement as usually considered; and the second scheme is the field-quadrature measurement by means of homodyne detection, which seems rarely discussed in the WVA community. The field-quadrature measurement is necessarily required by considering the conventional scheme (not involving the strategy of post-selection), since the photon-number measurement cannot extract the quantum Fisher information encoded in the phase-shifted coherent state. The main important results can be briefly summarized as: (i) With the increase of the measurement coupling strength to violate the AAV limit, the WVA scheme can definitely outperform the conventional approach; (ii) the WVA scheme can make a mixture of coherent states work better than a pure coherent state with identical average photon numbers; (iii) the WVA scheme can result in the precision of Heisenberg (or even “super-Heisenberg”) scaling with the photon numbers, but without using any expensive quantum resources.

We notice that all the above unexpected results are rooted in both the post-selection procedure in the WVA scheme and the use of optical coherent states as probe meter. For the above result (i), when considering Gaussian meter states, the conclusion was very different, which has caused extensively controversial debates in literature. The Gaussian meter state corresponds to such as the transverse spatial wavefunction in the Stern-Garlach setup, or the one of a light beam in experiments. In the case of optical coherent states, which are quite relevant to the optical cavity-QED system or solid-state circuit-QED architecture, the above concluding result (i) should be much more decisive. For the results (ii) and (iii), making a connection with the statement proved in Ref. Lui10 is quite constructive, which is also relevant to a wider range of problems Lui08 ; Sil10 , i.e., achieving super-resolution in the absence of quantum resources. Noting that the different conclusion in Ref. Lui10 was proved in the absence of post-selection procedure, we may expect that the simple procedure of post-selection involved in the WVA scheme can drastically change some conclusions in the field of quantum metrology, and deeper investigations are needed in future studies.

Acknowledgements.

— This work was supported by the National Key Research and Development Program of China (No. 2017YFA0303304) and the NNSF of China (Nos. 11675016, 11974011 & 61905174).