Contextual quantum metrology

Precision measurement has played a crucial role in the development of natural science and engineering since measurement is a means for observing nature. As a technology for precision measurement, quantum metrology has recently drawn attention with a wide range of applications such as microscopy [1], imaging [2, 3], patterning [4, 5], gravitational wave detection [6, 7, 8], and time keeping [9, 10]. Quantum metrology enables measurements going beyond precision of the standard quantum limit which can be obtained from the most-classical state in quantum physics. One of the resources for the precision enhancement is entanglement, a nonclassical property of quantum states [11, 12, 13, 14]. However, an entangled state can easily lose its property through interaction with other objects, while the interaction is inevitable in metrology. This makes it challenging to generate and manipulate an entangled state. Due to the limitations, it is difficult in practice to attain the entanglement-enabled enhancement of precision. If easy-to-implement resources for metrology are found, the performance of quantum metrology can be greatly enhanced, as well as its practicality. In this work, we demonstrate that contextuality of measurement selection [15], another nonclassical property, is an easy-to-implement resource for quantum metrology.

Specifically, contextuality here refers to dependency of quantum systems on measurement context [16]. Unlike classical predictions, quantum predictions for a given measurement can change depending on whether another measurement is performed simultaneously or not. Bell-Kochen-Specker theorem first showed that quantum physics is contextual [17, 18], and this has been experimentally proved on various quantum systems [19, 20, 21, 22]. Also, it has been revealed that the contextuality can be a resource for quantum information processing tasks such as quantum key distribution [23, 24], universal quantum computing [25], quantum state discrimination [26], and quantum machine learning [27, 28]. Yet, whether the contextuality can be a resource for quantum metrology remains a question.

To demonstrate the precision enhancement from the contextuality of measurement selection, we propose a method which we call contextual quantum metrology (coQM). Unlike conventional quantum metrology, the coQM utilizes two measurement settings and their contextuality. In our experiment, we adopt an optical polarimetry devised to measure concentration of sucrose solution [29], and modify its scheme for the coQM. Our experiment employs two polarization measurements in mutually unbiased (or complementary) bases, and their selection context is implemented by toggling a polarizing beam splitter ‘in’ and ‘out’ from its optical path. Our setup is scalable in that the size of the experiment does not increase along with the increase of the number of probe photons. Also, the enhanced precision is attainable without error correction or mitigation which requires overhead [30, 31]. We finally show that the precision of coQM can go beyond the precision limit of conventional quantum metrology [32, 33, 34] by a factor of $1.4$ to $6.0$.

Fig. 1 shows an experimental schematic for the coQM. Here, the coQM estimates the concentration of sucrose solution by following four steps: preparing a polarized single photon as a probe light (see Methods), interacting the photon with the sucrose solution, measuring the polarization, and calculating an estimate via maximum likelihood estimator (MLE) using the operational quasiprobability which will be discussed later as in Eq. (1). The photon interacts with the sucrose solution as it propagates through the solution. Afterwards, the photon polarization rotates by angle $\alpha cl$, where $\alpha$, $c$, and $l$ are the specific rotation of the sucrose solution, the concentration of the solution, and traveling length of light in the solution, respectively [29]. We can decide the concentration by measuring the polarization change for a given specific rotation and traveling length. All the procedures here look similar to those of the conventional quantum metrology [35], but the measurement and the estimation are steps that differentiate the coQM from the conventional approach.

In the measurement step, the coQM employs two different polarization measurement settings $A$ and $B$ to utilize the contextuality of measurement selection therefrom [15]. $A$ measures the polarization in a specific basis $\{|{H}\rangle,|{V}\rangle\}$ and $B$ does in a basis $\{|{D}\rangle,|{A}\rangle\}$ tilted $45^{\circ}$ from the basis of $A$ by half-wave plate. Our basic setup is the measurement by $B$, and we consider a context of whether measurement $A$ is selected to be performed or not, prior to performing $B$. If $A$ is not selected, the measurement $B$ only is performed, and probability is given by $p(b|B)$ for a binary value $b$. If $A$ is selected, the experimental setup runs the consecutive measurement performing $A$ first and $B$ later. In this case, probability is given by $p(a,b|AB)$ for an outcome pair $(a,b)$, where $a$ is a binary outcome of $A$. In our experiment (Fig. 1), the context of selecting $A$ is implemented by toggling ‘in’ and ’out’ the state of polarizing beam splitter PBS$1$. Depending on the context, the quantum prediction of measurement $B$ changes, so the two probabilities $p(b|B)$ by $B$ only and $p(b|AB)=\sum_{a}p(a,b|AB)$ by the consecutive differ in general, i.e., $p(b|B)\neq p(b|AB)$ [36]. (We will discuss that this difference stems from the incompatibility of measurements [37].) Our metrology utilizes this effect to enhance precision of the polarimetry.

In the estimation step, we employ a noncontextual (or context-free) model, so-called operational quasiprobability [38, 39], which is given by

The context-free condition in our measurement setup is to assume that the prediction of measurement $B$ is invariant under the context of selecting the measurement $A$. This is called the condition of no-signaling in time represented by $p(b|B)=p(b|AB)$, $\forall b$ [36]. The crucial property of the operational quasiprobability is that, for the context-free condition, $w$ is reduced to the probability by the consecutive measurement, $w(a,b)=p(a,b|AB)$, $\forall a,b$. To the contrary, the quantum predictions violate the condition in general, $w(a,b)\neq p(a,b|AB)$, and $w(a,b)$ can even be negative-valued [38, 39].

The conventional quantum metrology estimates a physical parameter $\theta$ with an estimator based on a conditional probability $p(x|\theta)$ for a data set $\{x_{i}\}_{i=1}^{N_{s}}$ (see Supplementary Information). In the coQM, the operational quasiprobability plays a role of the conditional probability for the two data sets ${\bf x}_{B}=\{b_{i}\}_{i=1}^{N_{s}}$ and ${\bf x}_{AB}=\{(a_{j},b_{j})\}_{j=1}^{N_{s}}$. In other words, the coQM calculates an estimate of polarization $\check{\theta}$ with a maximum likelihood estimator given by

where $l_{W}(\theta|{\bf x}_{B},{\bf x}_{AB})$ is a log-likelihood function for $w$ (see Methods). The possible problem caused by this replacement is that $w$ can be negative unlike the conditional probability, so that the log-likelihood function diverges. However, we find that $w$ remains positive for some range of parameters. We focus on the case for $w$ to be applicable to the log-likelihood function without the divergence problem. Finally, for an initial polarization $\theta_{0}$, we derive the estimate of concentration $\check{c}$ from the polarization change as $\check{c}=(\check{\theta}-\theta_{0})/\alpha l$.

Our goal is to demonstrate outperformance of the coQM over the conventional quantum metrology. We employ error of estimate $\Delta\theta$, the standard deviation that the estimate differs from the actual value, to quantify the performance of estimation. The smaller the error is, the more precise the estimate is.

As a baseline of performance, we take the conventional quantum metrology and its error given by quantum Fisher information (QFI) $F_{q}$, $\Delta\theta_{q}={1}/{\sqrt{N_{s}F_{q}}}$, where $N_{s}$ is the number of samples. This is known as the lower bound of error in the conventional method. In the coQM, we propose contextual Fisher information (coFI) to quantify the performance of the coQM,

In the asymptotic limit of $N_{s}\rightarrow\infty$, the error of the coQM $\Delta\theta_{\mathrm{co}}$ approaches to $1/\sqrt{N_{s}F_{\mathrm{co}}}$ (see Supplementary Information for the asymptotic property and estimator of coFI).

Contextuality-enabled enhancement The coQM gains precision enhancement over the conventional quantum metrology if

Our method uses the two data sets ${\bf x}_{B}$ and ${\bf x}_{AB}$. If each data set collects $N_{s}$ samples, the total number of samples is $2N_{s}$ in our method. Reduction factor $\sqrt{2}$ in the error by the conventional is introduced, assuming the conventional takes $2N_{s}$ samples (which is equivalent to comparing $F_{\mathrm{co}}$ to $2F_{q}$).

We here suggest specific cases satisfying the criterion (4). Instead of a theoretical proof, we briefly summarize the theory behind the enhancement of precision by following arguments: For the noncontextual model, the operational quasiprobability $w(a,b|\theta)$ becomes the joint probability of the consecutive measurement $p(a,b|AB)$. In this case, the coQM is reduced and equivalent to the conventional quantum metrology using the consecutive measurement so that the $\Delta\theta_{\mathrm{co}}$ equals or larger than $\Delta\theta_{q}$ (see Supplementary Information). For the contextual model, conversely, $\Delta\theta_{\mathrm{co}}$ can be smaller than $\Delta\theta_{q}$ (see Ref. [40] for rigorous discussions). Fig. 2 a shows simulation results of the contextuality-enabled enhancement on the Bloch sphere.

We perform polarization estimation of $\theta$ with probe states prepared in $|{\psi}\rangle=\cos(\theta/2)|{H}\rangle+e^{i\phi}\sin(\theta/2)|{V}\rangle$ for $0.46\pi\leq\theta\leq 0.55\pi$ and $\phi=0.15\pi$. For the probe states, the operational quasiprobability is given by $w(a,b|\theta)=\left(1+(-1)^{a}\cos\theta+(-1)^{b}\sin\theta\cos\phi\right)/4$. We draw $N_{s}=10^{5}$ samples for each data set to construct the operational quasiprobability, and calculate an estimate with the estimator (2). For polarization estimation of $\theta$, QFI $F_{q}=1$, so the coQM gains the contextuality-enabled enhancement if $\Delta\theta_{\mathrm{co}}<\Delta\theta_{q}/\sqrt{2}=1/\sqrt{2N_{s}}\approx 2.24\times 10^{-3}$. The errors of the coQM are smaller than the error limit of the conventional quantum metrology for the whole selected range of $\theta$ (Fig. 2 $\bf b$). The worst case in our results has $\Delta\theta_{\mathrm{co}}\approx 1.53\times 10^{-3}$ around $\theta=\pi/2$, and the best case has $\Delta\theta_{\mathrm{co}}\approx 3.7\times 10^{-4}$ around each end of the range of $\theta$. This demonstrates that our method elevates the precision of polarimetry by factor of $1.4$ to $6.0$ from the limit of conventional quantum metrology.

We estimate sucrose solutions of three different concentrations $c=0.1$, $0.3$ and $0.5$ g ml^-1. We prepare the probe state with initial parameters $\theta_{0}=0.5\pi$ and $\phi=0.15\pi$. For each concentration, we repeat the estimation $10$ times. The results (Fig. 2 d) show that the errors of estimates by the coQM, $\Delta c_{\mathrm{co}}$, are smaller than the minimum error by the conventional quantum metrology ($\approx 5.9\times 10^{-2}$); For $c=0.1$, $0.3$ and $0.5$ g ml^-1, the mean errors are $\approx 3.7\times 10^{-2}$, $\approx 3.3\times 10^{-2}$, and $\approx 2.8\times 10^{-2}$, respectively. This illustrates that the coQM exceeds the conventional quantum metrology by a wide margin.

The contextuality of measurement selection stems from the incompatibility of quantum measurements [37]. In the scenario of the consecutive measurement, if the two measurements $A$ and $B$ commute, the consecutive measurement is de facto a single measurement; the probabilities $p(b|B)$ and $p(b|AB)$ are equal and the prediction for $B$ is noncontextual. Otherwise, the prediction for the measurement $B$ depends on whether performing the first measurement $A$ and it is contextual except for a case when the initial state is prepared in an eigenstate of the $A$. Thus, the measurement incompatibility is necessary for the contextuality of measurement selection.

The noncommutation of observable operators defines the incompatibility among measurements, represented by projection-valued measures (PVM), which we assume in the present work. The notion of incompatibility needs to be generalized if the representation of measurement is generalized to positive operator-valued measure (POVM). This generalization is required, for example, if one considers an open quantum system in a noisy environment. Non-joint measurability (non-JM) is one of the generalizations [37]. The non-JM plays an important role in a contextuality [41, 42], as does the noncommutativity [43]. In fact, non-JM and the contextuality of measurement selection are also closely related as the negativity of operational quasiprobability is the necessary and sufficient condition for non-JM [44, 40].

Recently, there were studies in a similar vein to the present work [45, 46], showing that the noncommutativity can be a resource for quantum metrology. However, their schemes employ a post-selection to discard unwanted measurement outcomes, so there is a tradeoff between success probability and Fisher information; success probability becomes small if Fisher information is large [47]. Quantum post-selected metrology such as weak value amplification methods share this matter [48, 49]. On the contrary, our method utilizes all of measurement outcomes for the estimation [38, 39], implying that the coQM is free from such tradeoff.

This work demonstrates that utilizing the contextuality of measurement selection can enhance the precision measurement. The experiment attains precision beyond the limit of conventional quantum metrology [33, 34]. The coQM has advantages over the conventional method (see Supplementary Information): it can enhance the precision without optimizing the measurements if they are incompatible, and it works even without any entangled state of probe that has been regarded difficult to generate and manipulate. We use the heralded single-photon source to clearly show the performance of coQM per a unit particle of probe. We expect that a multi-photon source can also be adopted for the coQM with the similar settings of measurements [15]. Our method is expected to be applicable to a quantum sensor [50] if the context of measurement selection can be implemented within the sensor’s system. In addition, the approaches employed to demonstrate the contextuality-enabled enhancements can be utilized to characterize quantum devices (see Supplementary Information), which is a fundamental task required to implement quantum technologies.

Estimation or metrology theory employs two models. One is a conditional probability function $p({\mathbf{x}}|\theta)$ of an observing data set $\mathbf{x}$, conditioned on parameter $\theta$ to estimate. The other is an estimator $T({\mathbf{x}})$, which estimates the value of parameter $\theta$, based on the observation. Both types of models are verified whether appropriate for experiments. To find the two types of models with small estimation errors is at the heart of estimation. Once a conditional probability function is assumed, the estimator determines the estimation error.

The estimation error $\Delta\theta$ is known to have a lower bound over all estimators,

This inequality holds when samples are drawn from i.i.d. The lower bound is called Cramér-Rao bound (CRB). CRB contains Fisher information $F$, which is defined by the conditional probability and its dependence of parameter $\theta$:

Note that Fisher information is positive semidefinite. If the conditional probability is parameter-independent, $F$ has the minimum of zero, $F=0$. If it is highly dependent, $F$ is large. As $F$ increases, CRB decreases and results in more precise estimation. An estimator attaining CRB is said optimal. It is known that a maximum likelihood estimator is (approximately) an optimal estimator in limit of large samples [53].

In quantum theory, the conditional probability $p(x|\theta)$ is decomposed into a quantum state $\hat{\varrho}$ and a measurement of positive operator-valued measure (POVM) $\{\hat{m}(x)\}$. Assuming the quantum state is a functional of parameter $\theta$, the conditional probability is given by $p(x|\theta)=\text{Tr}\left(\hat{m}(x)\hat{\varrho}(\theta)\right)$. In this case, CRB can be further optimized over measurements to quantum CRB (QCRB). QCRB contains quantum Fisher information (QFI) defined by

where $\hat{L}:=({\hat{L}\hat{\varrho}(\theta)+\hat{\varrho}(\theta)\hat{L}})/{2}$ is the symmetric logarithmic operator to define derivative of a quantum state $\hat{\varrho}(\theta)$ [34]. For a given probe state, QFI is the maximum value of Fisher information over all measurements, so it is never less than Fisher information $F$, i.e., $F_{q}\geq F$. Equivalently, the error given by quantum Fisher information is never larger than the error given by Fisher information. In that sense, the error given by QFI is the minimum error of quantum metrology using measurements which can be represented by a POVM.