# Heisenberg scaling with weak measurement: a quantum state discrimination point of view

## Abstract

We examine the results of the paper “Precision metrology using weak measurements” (Zhang et al. arXiv:1310.5302, 2013) from a quantum state discrimination point of view. The Heisenberg scaling of the photon number for the precision of the interaction parameter between coherent light and a spin one-half particle (or pseudo-spin) has a simple interpretation in terms of the interaction rotating the quantum state to an orthogonal one. To achieve this scaling, the information must be extracted from the spin rather than from the coherent state of light, limiting the applications of the method to phenomena such as cross-phase modulation. We next investigate the effect of dephasing noise and show a rapid degradation of precision, in agreement with general results in the literature concerning Heisenberg scaling metrology. We also demonstrate that a von Neumann-type measurement interaction can display a similar effect with no system/meter entanglement.

### Keywords

Quantum metrology State discrimination Weak value amplification## 1 Introduction

Aharonov et al. [1] introduced the concept of a weak value as controlling an anomalously large deflection of an atomic beam passing through a Stern–Gerlach apparatus. The deflection size is controlled by pre- and post-selected states, as well as the size of the magnetic field gradient. In the concluding paragraph, they mention that “another striking aspect of this experiment becomes evident when we consider it as a device for measuring a small gradient of the magnetic field...Our choosing (of the post-selection state) yields a tremendous amplification”. The price one pays for this amplification is the loss of a large fraction of events due to the post-selection. Nevertheless, the relevant information about the parameter in question is concentrated into these small number of events [2]. This technique has been adapted to optical metrology and has been successfully implemented in many experiments to precisely estimate various parameters, such as beam deflection, phase or frequency shifts. For recent reviews of this active area of research, see Refs. [3, 4].

While still obeying the standard quantum limit, weak value amplification experiments have been shown to be capable of extracting nearly all of the theoretically available information about the estimated parameter in a relatively simple way. Further, it has been shown that in comparison to a standard experimental technique, and given the presence of certain types of noise sources or technical limitations obscuring the measurement process, the weak value-type experiment can have better precision (even when using optimal statistical estimators), even though the detector only collects a small fraction of the light in the experiment [2]. There have also been a number of recent advances that propose to improve the intrinsic inefficiency of the post-selection. For example, in the optical context, it is possible to recycle the rejected photons, further improving the sensitivity of the technique [5]. This then gathers all the photons in the experiment through repeated cycles of selection, leading to higher power on the detector with the enhanced signal.

Quantum-enhanced metrology is based on using quantum resources, such as entanglement, to estimate a parameter of interest better than an analogous classical technique could do with similar resources—typically photon number. Proposed applications of this field range from precision measurements in optical interferometry to gravity wave detection [6]. Recently, Pang et al. [7] proposed combining the weak value technique with additional entangled quantum degrees of freedom to further increase the weak value at the same post-selection probability, or to keep the same weak value while boosting the post-selection probability. This technique leads to Heisenberg scaling of the parameter estimation precision with the number of auxiliary degrees of freedom, using quantum entanglement as a resource. These advances lead us naturally to consider how other quantum resources manifest in the context of weak measurements, which is the subject of the present article.

An important tool in quantum-enhanced metrology is the Fisher information. Classically, this quantity indicates how much information about the parameter of interest is encoded in the probability distribution of a random variable that is being measured. It is an important quantity because it sets the (Cramér–Rao) bound for the minimum variance of any unbiased estimator for the parameter of interest. Any estimator that achieves that bound is said to be efficient. The quantum mechanical extension of the Fisher information analogously gives the quantum Cramér–Rao bound, which indicates the minimum variance achievable using any measurement strategy. Despite these powerful properties, the formal expressions for the Fisher information do not necessarily provide deeper insight about the physics of the detection method and can even obscure what are essentially simple physical effects. In this paper, we will use both quantum Fisher information and the distinguishability of two quantum states as ways to quantify the smallest measurable parameter. These measures are related to one another: the mean squared distance between a quantum state and that state slightly shifted by a classical parameter is proportional to the quantum Fisher information about the parameter in the quantum state [8]. The usefulness of weak measurements has also been considered in the problem of state distinguishability, which is related to the current problem [9].

A conundrum involving Fisher information was recently presented by Zhang et al. [10], who considered a coherent state of photons interacting with a spin-1/2 particle to estimate a small coupling parameter in the interaction Hamiltonian. Notably, this example is a variation of the original weak value amplification scenario [1], but using a different parameter regime that more commonly appears in cavity and circuit QED [11, 12]. Even though the coherent state used in their example is typically considered to be a classical quantity that does not provide quantum resources, the authors showed the surprising result that the Fisher information about the coupling parameter seemed to scale at the optimal Heisenberg limit as the average number of photons was increased, rather than at the standard quantum limit that one would typically expect. This result raises several immediate questions: is there a simple physical explanation of this apparent Heisenberg scaling, and can this scaling really be used to enhance the estimation of the interaction parameter in an experiment?

^{1}Here, \({\hat{\sigma }}_z\) is a Pauli operator, and \({\hat{n}}\) is a photon number operator. This results in the entangled state

The authors go on to look at projection of the system state onto a final state \(|\psi _f\rangle \), where this state has the same form as \(|\psi _i\rangle \), with the subscript \(i\) replaced by \(f\) on the parameters [10]. Specifically, a strong measurement will project the system onto \(|\psi _f\rangle \) or onto the state orthogonal to \(|\psi _f\rangle \) (since the system is two dimensional, there are no other options). The scaling of the post-selected parameter estimation is optimized when pre- and post-selected states are parallel \(|\psi _i\rangle = |\psi _f\rangle = (|- \rangle + |+ \rangle )/\sqrt{2}\), so we focus on this case for simplicity of calculation. The orthogonal state is then clearly \(|\psi _f^\perp \rangle = (|-\rangle - |+\rangle )/\sqrt{2}\).

*post-selection*has a Fisher information that scales as \(N^2 = |\alpha |^4\), giving

*Heisenberg*scaling in the photon number for the precision of estimating \(g\).

The main purpose of this paper is to give physical insight into why Heisenberg scaling for the parameter \(g\) can be obtained at all, and further, why it comes mainly from the probability of projecting on the system state, as opposed to mining the meter states for information, as is usually done in weak value amplification experiments [3]. Zhang, Datta, and Walmsley write that “How this conditioning step using a classical measurement apparatus achieves a precision beyond the standard quantum limit is therefore an interesting open question.” We answer this question here and give a simple physical argument showing how this scaling is possible. We are primarily concerned with the large \(N\) limit (Heisenberg scaling), rather than finding general expressions as in Ref. [10].

## 2 Approach

We approach the question by mapping the problem of obtaining a precise estimate for the parameter \(g\) onto a different problem: under what conditions can one distinguish the entangled state \(| \Psi \rangle \) from the separable state \(|\Psi _0\rangle \)? It is well known in quantum physics that two states can only be reliably distinguished if they are orthogonal to one another [13]. Therefore, \(g\) must be large enough to move the initial separable state to an orthogonal state. This sets the scale of the minimum value for \(g\) that can be reliably distinguished when using the two-state system as a probe. Unless the states are distinguishable, no processing techniques will help in the metrological task.

## 3 Spatial shift of independent meter states

## 4 Cavity QED interaction

*coherently*, the scaling with photon number shows a quantum advantage compared to the incoherent photon accumulations that lead to the standard quantum limit. We note that while we initially specialized to projecting the system in its original basis preparation, in this limit of small \(g\), the effect of the state rotation can be seen for any fixed measurement basis, so long as it is in the plane of rotation.

Now we can see why the information about the parameter \(g\) is mainly found in the probability of the selection, \(p_{\pm }\) in Ref. [10]. We notice that up to factors of 2 and shifts by constant factors, the selection probability (5) has the same form as the overlap between entangled and initial state (9). As we see above, the main effect is the coherent phase rotation of the system state, and consequently, this is the effect that will give a large change of the selection probability that can be used to deduce the value of \(g\). With this insight, it makes perfect sense that the post-measurement meter states have relatively little information that can be extracted, and the Heisenberg scaling appears only in the post-selection probability.

## 5 Implementations

There can be, however, other uses of this technique. In cross-phase modulation, a single photon, prepared in a superposition of two polarizations (for example), can interact nonlinearly with a coherent beam with a large average photon number. Depending on the polarization state of the single photon, the phase of the coherent beam is changed by different amounts. This nonlinearity is very difficult to create optically, and single-photon “cross-Kerr” nonlinearities have not been seen yet in the lab. However, Feizpour et al. [16] have shown that in some cases, a single photon can be made to act like many through a weak value amplification process. In this case of cross-Kerr interaction, the difference of rotation angle for the two polarizations is identified with the \(g\) parameter, and the method of measuring the frequency of projection of the polarization back on the original state can estimate that parameter. The standard error on \(g\) will scale as \(1/N\) (the Heisenberg limit in photon number) times \(1/\sqrt{\nu }\), where \(\nu \) is the number of projections on the two-state system (the standard quantum limit in measurement realizations). Thus, for this technique to be useful, we need the number of photons \(N\) per projection to be large and need to be able to repeatedly measure the single-photon polarization. Measuring the changes in the coherent state of the light is irrelevant for the Heisenberg scaling on the precision of \(g\). Unfortunately, linear optics is unable to realize an interaction of the form Eq. (2) and will instead create products of coherent states. Thus, any interferometric set-ups will be unable to create this state. To create the state, nonlinear methods are needed. We note, however, that since we can ignore the state of the light entirely (since we know the average photon number \(N\)), the relevant state is not a Schrödinger cat state, since the coherent state can be traced out entirely, leaving just the phase-shifted qubit state to work with.

As usual, to get the unitary development (2), the two systems are coupled impulsively in time. The shift of the qubit frequency causes a precession in the \(x\)–\(y\) plane of the Bloch sphere that can be read out via projective qubit measurements, following unitary rotations. In fact, this qubit frequency measurement is used routinely to determine the average photon number in the cavity, since the parameter \(g = \hbar \chi \) can be independently determined spectroscopically as the shift in resonance frequency of the coupled cavity. In our case, we are interested in the reverse procedure: a large known photon number \(N\) in the cavity creates a finite phase shift on the qubit for an unknown small coupling parameter \(g\) of the order \(1/N\). Hence, measuring the qubit would allow one to independently determine \(g\) without using the spectroscopy of the cavity.

The difficulty in implementing the scheme outlined above is the fact that, usually, the way to measure the qubit state is with the cavity field itself, which we already showed becomes uncorrelated with the qubit precisely when the qubit is most sensitive to \(g\). Consequently, the implementation requires that we first have a weak unknown interaction of the qubit with one cavity, followed by a strong interaction with another cavity to do the projective measurement. This could be accomplished perhaps with two strip-line resonators, both of which are coupled to a single transmon qubit.

In the second approach, we can find the quantum Fisher information (7) about \(g\) in the light state \(| \alpha e^{\pm i g}\rangle \). We find the result \(F= 4N\), so the quantum Cramér–Rao bound is \(g_{\min } = F^{-1/2} = 1/\sqrt{4 N}\), which is consistent with the approach above, as well as the standard quantum limit scaling expected from a coherent state.

## 6 Effects of dephasing

One outstanding challenge to quantum metrology is the fact that in the presence of small amounts of dephasing noise, the Heisenberg scaling rapidly changes to standard quantum limit scaling. Here, there is some advantage in the sense that photon loss will not have much effect on the coherent state (other than simply lessening the overall magnitude, \(N = |\alpha |^2\)). However, as we already showed, the Heisenberg scaling occurs by measuring the two-state system, so this is not really helpful. The important effect is how the scaling depends on fluctuations on the phase shift that is being measured, \(g\). Phase fluctuations (or other dephasing mechanisms) will then be the most serious detriment to this method, rendering it useless for estimating \(g\) better than the standard quantum limit. Similar difficulties can be seen with N00N states [17].

## 7 Von Neumann measurement revisited

Before concluding, we point out that the effects described with the cavity QED-type interaction (2) can also be seen more easily with the von Neumann interaction (6). Rather than using a meter wavefunction that is Gaussian as is usually considered to extract information about the qubit state, we consider a meter wavefunction of the form of a plane wave of wavelength \(\lambda \), that is \(\langle x|\psi \rangle = \psi (x) = \exp (2\pi i x/\lambda )\). Of course, this plane wave should as usual be normalized, either by putting on a slowly decaying envelop function or with box normalization. Nevertheless, for this discussion, it is simpler to keep it unnormalized, and this will not change the main conclusions. We can equivalently write this state in momentum space as \(\langle p | \phi \rangle = \phi (p) = \delta (p-p_0)\), where the momentum \(p_0 = 2 \pi \hbar /\lambda \).

Given this insight, could this scheme could be implemented in the optical experiments demonstrating weak value amplification by simply monitoring the post-selection probability? A von Neumann-type interaction has been shown using both polarization [18] or which-path [19] degrees of freedom. The answer is that those experiments use a single photon as both the meter (transverse deflection) and system (polarization or which-path), so the number of meter photons per system projection is \(1\).

## 8 Conclusions

We have considered the weak measurement metrology model proposed by Zhang et al. [10] and have given a simple interpretation of the Heisenberg scaling of the Fisher information with photon number shown there: the coherent state interacting nonlinearly with a spin 1/2 particle imparts a coherent phase shift to the spin that can rotate the spin to an orthogonal state for \(g \sim 1/N\). The coherent state carries only information about \(g\) that scales with the standard quantum limit, and the spin (or pseudo-spin) must be measured directly to obtain Heisenberg scaling precision. We have further investigated dephasing effects on the scheme and shown a rapid degradation to the measurement precision, emphasizing that it is the fragile quantum coherence of the spin that leads to the enhanced scaling. This behavior has been argued to be generic to all Heisenberg scaling schemes (see Refs. [20, 21, 22, 23]), so it is not surprising that we also find this behavior here as well. We also showed that the von Neumann measurement interaction also has the phase accumulation effect, provided we prepared the meter states in momentum eigenstates. In that case, there is no entanglement, and therefore, strictly no measurement of the spin by the meter states.

## Footnotes

## Notes

### Acknowledgments

We thank L. Zhang, A. Datta, I. A. Walmsley, and A. N. Korotkov for discussions. Support from the US Army Research office Grant No. W911NF-09-0-01417 and Chapman University is gratefully acknowledged. J.D. acknowledges support from the Advanced Research Projects Activity (IARPA), through the Army Research Office (ARO) Grant No. W911NF-10-1-0334, and also ARO MURI Grant No. W911NF-11- 1-0268.

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