Measuring energy by measuring any other observable

Expectation values are ubiquitous in quantum physics, characterizing different types of behaviors of quantum systems. They are used both as descriptive and predictive tools. To name several: mean values of generic local observables classify many-body systems according to how well they thermalize D’Alessio et al. (2016); Deutsch (2018). Vanishing total magnetization identifies a quantum phase transition Vojta (2003); Sun et al. (2014); Tian et al. (2020). The mean value of homodyne measurement Yuen and Chan (1983); Tyc and Sanders (2004); Shaked et al. (2018); Raffaelli et al. (2018) is evaluated in magnetic resonance imaging Noll et al. (1991) and quantum cryptography protocols Voss (2009), while its variance is used to prove squeezing Davidovich (1996); Takeno et al. (2007) — an essential resource for quantum sensors Lawrie et al. (2019). Variances also appear in Heisenberg’s uncertainty principle Heisenberg (1985); Robertson (1929); Busch et al. (2007). Expectation values are the object of interest in quantum field theory Wightman (1956); Srednicki (2007) and in nuclear physics Bunge et al. (1993); Ikot et al. (2019).

Moments of energy are somewhat special due to their wide range of applications. The mean energy determines the thermodynamic entropy of the system Deutsch (2010); Swendsen (2015); Santos et al. (2011); Šafránek et al. (2021) and its temperature Hovhannisyan and Correa (2018); Mukherjee et al. (2019); Cenni et al. (2021). Its change may represent heat and work Engel and Nolte (2007); Alipour et al. (2016); Modak and Rigol (2017); Goold et al. (2018); De Chiara et al. (2018); Varizi et al. (2020) and its difference defines a measure of extractable work called ergotropy Allahverdyan et al. (2004); Alicki and Fannes (2013); Šafránek et al. (2022). Variance in energy determines the precision in estimating both the time Paris (2008) and temperature Correa et al. (2015) parameters. Both moments, when combined, provide a tight bound on the characteristic time scale of a quantum system Mandelstam and Tamm (1991); Margolus and Levitin (1998); Deffner and Campbell (2017).

Given the breadth of applications, it is clear that measuring and estimating expectation values is of essential importance. This may be however challenging. For example, in quantum many-body systems, the mean energy is considerably difficult to measure, with only a few architecture-specific proposals Villa and De Chiara (2017) and experiments Jiménez et al. (2021) known. This is because energy eigenstates are typically highly entangled. In quantum simulators, measuring in an entangled basis is performed by combining several elementary gates. Each gate has a fixed fidelity, and when many of such gates are combined, the fidelity diminishes making such measurements unreliable Nielsen and Chuang (2002); Reich et al. (2013); Harper and Flammia (2017); Huang et al. (2019). Additionally, experimental setups may allow measurement only of a close but not the exact observable we are interested in. This is the case, for example, in the aforementioned homodyne detection with a finite, instead of infinite, oscillator field strength Tyc and Sanders (2004); Combes and Lund (2022).

In this paper, we show that performing any measurement bounds the probabilities of outcomes, the mean value, and higher moments of any other observable. This means that, quite unintuitively, measurements carry more information than previously known. Any observable yields some information on any other observable. The method uses correlations between the measured, the estimated observable, and the state of the system. It is precisely this interplay that allows us to bound the probability of outcomes of the estimated observable and, from those, its mean value and higher moments.

These results immediately ameliorate the issue mentioned above: even in experimental systems in which we have only a limited ability to measure, we can perform the best possible measurement, and this is enough to estimate the probability distribution of outcomes and the mean value of an observable that we are truly interested in measuring.

The derived bounds are further tightened by measuring in different bases and, in the case of estimating the mean energy, by measuring at different times. After some preliminaries, we show how measurement in any basis bounds the probabilities of the system to have a certain mean energy. From this, we derive two bounds on the mean value of energy: one analytic which is easy to compute, and one tighter which leads to an optimization problem. We discuss situations in which the analytic bound becomes relatively tight. Then we describe a few differences when bounding the mean values of observables other than energy. We illustrate this method on several experimentally relevant models. Finally, we discuss the advantages and drawbacks of this method, possible applications, and future directions.

Next, we consider a Hamiltonian of the system, with spectral decomposition in terms of its eigenvalues and eigenvectors being ${\hat{H}}=\sum_{E}E\outerproduct{E}{E}.$ The probability of finding the system having energy $E$ is given by $p_{E}=\absolutevalue{\innerproduct{E}{\psi_{t}}}^{2}.$ We assume to know the Hamiltonian and its spectral decomposition. However, we presume that we are unable to measure it experimentally. In other words, we cannot perform the measurement in the energy eigenbasis. As we will show next, this does not stop us from estimating its probability of outcomes, and from those also the mean value of energy. Proofs for the following bounds can be found in Appendix A.

If the system is isolated, it evolves unitarily with the time-independent Hamiltonian ${\hat{H}}$, and energy probabilities $p_{E}$ are also time-independent. In contrast, probabilities $p_{i}$ are time-dependent if the measurement basis does not commute with the Hamiltonian, and so are the bounds $a_{E}$ and $b_{E}$. This leads to an interesting observation that measuring at different times can make the bound tighter. Quantitatively, we have

$$a_{E}^{\max}\leq p_{E}\leq b_{E}^{\min},$$

(3)

where $a_{E}^{\max}=\max_{t\in[0,T]}a_{E}$ and $b_{E}^{\min}=\min_{t\in[0,T]}b_{E}.$

Let us discuss situations in which the bound becomes tight. First, assume that one of the measurement basis vectors is an energy eigenvector, i.e., $\ket{i}=\ket{E}$ for some $i$ and $E$. Then the bound gives $p_{E}=p_{i}$ for this specific $E$, as intuitively expected. Second, consider a situation in which we always obtain a single outcome when measuring at a specific time, i.e., $p_{i}=1$ for some $i$ at some time $t$. Doing this is akin to identifying the state of the system as being equal to $\ket{i}$. As a result, the bound gives an identity $p_{E}=\absolutevalue{\innerproduct{i}{E}}$ for all energies $E$ so the entire energy distribution is determined exactly.

The two extreme cases just discussed suggest two possible scenarios in which the bounds perform well. The bounds are relatively tight when either the measurement basis resembles the eigenbasis of the estimated observable (in this case, the Hamiltonian), or when the state of the system comes close to one of the measurement basis vectors during its time evolution.

In addition to optimization over time, the inequalities can be further tightened by performing measurements in different bases. Defining a set of performed measurements, $\mathcal{M}=\{{\mathcal{C}}_{m}\}$, each measurement bounds the $p_{E}$ independently, so we can take

$$a_{E}^{\max}=\max_{{\mathcal{C}}_{m}\in\mathcal{M},t\in[0,T]}a_{E}^{m},\quad b_{E}^{\min}=\min_{{\mathcal{C}}_{m}\in\mathcal{M},t\in[0,T]}b_{E}^{m},$$

(4)

for the bound (3). $a_{E}^{m}(t)$ and $b_{E}^{m}(t)$ are defined by Eqs. (2) for each measurement ${\mathcal{C}}_{m}$. This may be helpful when there are limits on the types of measurements we can perform. For example, we can be experimentally limited to using only one- and two-qubit gates due to many-qubit gates having a low fidelity.

For the Hamiltonian evolution, extremizing over times of measurement yields tighter bounds

$${\sqrt{\boldsymbol{p}}}^{T}\boldsymbol{A}_{i}{\sqrt{\boldsymbol{p}}}\ \leq\ p_{i}^{\min}\ \leq p_{i}^{\max}\ \leq\ {\sqrt{\boldsymbol{p}}}^{T}\boldsymbol{B}_{i}{\sqrt{\boldsymbol{p}}},$$

(7)

where $p_{i}^{\min}=\min_{t\in[0,T]}p_{i}$ and $p_{i}^{\max}=\max_{t\in[0,T]}p_{i}$.

These collective inequalities are generally non-linear in $p_{E}$ and neither convex nor concave. There are as many as the number of measurement outcomes. While they do not bound each energy probability separately, they provide relationships between their respective sizes. For example, one can derive quantitative statements of type: if $p_{E_{1}}$ is high, then $p_{E_{2}}$ must be low. They might require numerical methods to work with due to their non-linearity. However, they can provide a robust improvement in estimating energy in some cases. See Appendix B for such an example of coarse-grained energy measurements.

Similar to Eq. (4), one can employ measurements in different bases. These generate more conditions for probabilities $p_{E}$ of type (7), thus making quantitative relations between them stricter.

The inner bound is computed by optimizing the mean value of energy, $\langle E\rangle=\sum_{E}E\,p_{E}$, as

$$E_{\min}=\min_{\{p_{E}\}\in S}\langle E\rangle,\quad E_{\max}=\max_{\{p_{E}\}\in S}\langle E\rangle,$$

(9)

over the set of probability distributions consistent with our observations, i.e., over the set that satisfies all the required inequalities

$$S=\left\{\{p_{E}\}\bigg{|}\sum_{E}p_{E}=1,\ \mathrm{Eq.}~{}\eqref{eq:linear_inequalities},\text{ and }\mathrm{Eq.}~{}\eqref{eq:sum_inequalities_extremized}\right\}.$$

(10)

The mean energy itself is a linear function. While $\sum_{E}p_{E}=1$ and Eq. (3) are linear constraints, Eq. (7) is in general non-linear. Computing $E_{\min}$ and $E_{\max}$ is, therefore, a non-linear constrained optimization problem. These problems are considered to be computationally demanding, although they are difficult to characterize within computational complexity theory Hochbaum (2007). They can be solved only approximately by various methods ¹¹1These are, for example, Nelder-Mead algorithm Luersen and Le Riche (2004), random search Price (1983), differential evolution Storn (1996), machine learning methods Sivanandam and Deepa (2008), simulated Bertsimas and Tsitsiklis (1993) or quantum Das and Chakrabarti (2005) annealing, and modified linear programming methods Powell (1998). The time to find an exact solution typically scales exponentially with the number of variables, in our case, the dimension of the system.

However, we can solve an easier problem by including only linear constraints in the optimization, i.e., by removing the requirement for satisfying Eqs. (7). This makes the bound looser but allows for solving this optimization problem analytically. The reasoning behind the following derivation is explained in Fig. 1 and performed in detail in Appendix A. We assume that energy eigenvalues are ordered in increasing order as $E_{j}\leq E_{j+1}$ with $E_{1}$ representing the ground state energy. We have

$$E_{\min}^{\mathrm{lin}}=\sum_{j=1}^{N}E_{j}u_{j},$$

(11)

where probabilities $u_{j}\equiv u_{E_{j}}$ are computed recursively starting from the ground state as

	$\displaystyle u_{1}$	$\displaystyle=\min\left\{b_{1}^{\min},\ 1-\sum_{l=2}^{N}a_{l}^{\max}\right\},$		(12)
	$\displaystyle u_{j}$	$\displaystyle=\min\left\{b_{j}^{\min},\ 1-\sum_{l=1}^{j-1}u_{l}-\sum_{l=j+1}^{N}a_{l}^{\max}\right\},\quad 2\leq j\leq N.$

(We simplified lower indices as $l\equiv E_{l}$, and the dimension of the system is $N$.) Similarly, we obtain

$$E_{\max}^{\mathrm{lin}}=\sum_{j=1}^{N}E_{j}w_{j},$$

(13)

where starting from the highest energy state, we have

	$\displaystyle w_{N}$	$\displaystyle=\min\left\{b_{N}^{\min},\ 1-\sum_{l=1}^{N-1}a_{l}^{\max}\right\},$		(14)
	$\displaystyle w_{j}$	$\displaystyle=\min\left\{b_{j}^{\min},\ 1-\sum_{l=j+1}^{N}w_{l}-\sum_{l=1}^{j-1}a_{l}^{\max}\right\},\quad 1\leq j\leq N-1.$

Bounds for the higher moments are computed by replacing eigenvalues $E$ with $E^{k}$ in Eq. (9).

The first case is when the state of the system comes close to one of the measurement basis vectors during its time evolution. If the state becomes exactly one of the basis vectors, i.e., $p_{i}(t)=1$ at some time $t$, we can identify the state exactly as $\ket{i}$ and so all of its properties, energy included. In this case, bounds (3) become tight and we obtain $\langle E\rangle=E_{\max}^{\mathrm{lin}}=E_{\min}^{\mathrm{lin}}$. The most informative times of measurement are those of a low value of observational entropy von Neumann (2010); Šafránek et al. (2019a, b); Strasberg and Winter (2021); Šafránek et al. (2021); Buscemi et al. (2022), due to the state wandering into a small subspace of the Hilbert space Šafránek et al. (2021); Šafránek and Thingna (2020) recognizable by the measurement. This can be advantageous for energy estimation in systems exhibiting recurrences and Loschmidt echo Usaj et al. (1998); Sánchez et al. (2016); Pastawski et al. (1995); Levstein et al. (2004); Rauer et al. (2018), which return close to their original state after some time.

The second is when the measurement is close to the energy measurement itself. This happens, for example, in the localized phase of many-body localized systems, in which the energy eigenvectors tend to localize in small portions of the Fock space Abanin et al. (2019). Thus, measuring local particle numbers is almost as good as measuring the energy itself. This is mathematically justified below Eq. (3).

We can address this heuristically given the points introduced in the previous section. Generally, the ideal number and times of measurement depend on the initial state: if the state does not evolve much, or at all, which is the case for any energy eigenstate (such as the ground state), then only a single measurement is required. Additional measurements will not yield any improvement.

On the other hand, if a nontrivial evolution occurs, then more times of measurement might be advantageous. The rule of thumb is to measure for as long as the observational entropy related to the measurement grows until it reaches its equilibrium value. This is because, bigger dips in observational entropy give more information, while small dips do not provide as much. The same criterium could be applied to identify the times of measurement within this interval. There should be as many as to reproduce the medium-sized dips in the observational entropy evolution.

The Hamiltonian is given by

$${\hat{H}}=\sum_{i}\big{(}\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+1}^{x}+\hat{\sigma}_{i}^{y}\hat{\sigma}_{i+1}^{y}+\hat{\sigma}_{i}^{z}\hat{\sigma}_{i+1}^{z}\big{)}+\sum_{i}h_{i}\hat{\sigma}_{i}^{z},$$

(15)

where $\hat{\sigma}_{i}^{a}$, $a=x,y,z$, are the Pauli operators acting at the site $i$. The constants $h_{i}$ are randomly extracted within the interval $\left[-W,W\right]$ with $W$ being the disorder strength. We show the case $W=0.5$ for the chaotic (delocalized) regime and $W=10$ for the localized regime Pal and Huse (2010); Luitz et al. (2015). See Appendix E for the Bethe integrable regime $W=0$ Bethe (1931).

We choose a complete measurement in the local number basis,

$${\mathcal{C}}=\{\ket{i_{1}}\otimes\cdots\otimes\ket{i_{L}}\}_{i_{1},\dots,i_{L}},$$

(16)

for small systems simulations. There, $i_{j}=0,1$ for all $j$ and $L$ is the length of the chain (such that the dimension of the Hilbert space is $N=2^{L}$). For example, for the chain of $L=3$ sites, the measurement basis is

$${\mathcal{C}}=\{\ket{000},\ket{001},\ket{010},\ket{011},\ket{100},\ket{101},\ket{110},\ket{111}\}.$$

(17)

This is an example of a one-local measurement, meaning that the measurement basis does not consist of states entangled between two or more sites. For large system simulations, we also add optimized $k$-local measurements. $k$-local measurements are those that project onto states that are allowed to be entangled between $k$ neighboring sites. An example of a two-local measurement on the chain of $L=4$ sites is the measurement in the local Bell basis,

$$\begin{split}{\mathcal{C}}&=\{\ket{\Phi^{+}}\!\ket{\Phi^{+}},\ket{\Phi^{+}}\!\ket{\Phi^{-}},\ket{\Phi^{+}}\!\ket{\Psi^{+}},\ket{\Phi^{+}}\!\ket{\Psi^{-}},\dots\},\end{split}$$

(18)

where $\ket{\Phi^{\pm}}=(\ket{00}\pm\ket{11})/\sqrt{2}$ and $\ket{\Psi^{\pm}}=(\ket{01}\pm\ket{10})/\sqrt{2}$.

We consider three types of initial states: First the ground state (G). The second is a “pure thermal” state (C),

$$\ket{\psi_{\beta}}=\frac{1}{\mathcal{N}}\sum_{E}e^{-\beta E/2}\ket{E},$$

(19)

where $\mathcal{N}$ is the normalization factor, and the inverse temperature is chosen as $\beta=6/(E_{N}-E_{1})$. We take this choice of $\beta$ to imitate a cold state. Third, we randomly choose a pure state from the Hilbert space with the Haar measure (H), imitating an infinite temperature state.

To compute the bounds, we evolve them with the Hamiltonian for the total time of $T=160$.

In Figure 2, we show estimates of energy in small systems, taking three particles on the chain of $L=6$ sites. The Heisenberg Hamiltonian is particle conserving, so the initial state explores only a subspace of the full Hilbert space. We analytically solve for the looser bound using Eqs. (11) and (13). This solution serves as a starting point for the COBYLA optimization algorithm Powell (1994, 1998) to compute the tighter bound, Eq. (9). In Figure 3, we plot estimates of energy in a large system, 5 particles on $L=10$ sites, in which computing the numeric bound is prohibitively difficult. Instead, we add more measurements and calculate the corresponding analytical bounds in the increasing degree of non-locality.

POVM elements admit a spectral decomposition $\hat{\Pi}_{i}=\sum_{k}\gamma_{i}^{k}|i^{k}\rangle\langle i^{k}|$. There, $0<\gamma_{i}^{k}\leq 1$ and $\ket{i^{k}}$ are orthogonal to each other for different $k$’s. We define its “volume” as $V_{i}=\tr[{\hat{\Pi}}_{i}]$. We further define $x_{i}^{E}=\min_{k}\gamma_{i}^{k}\absolutevalue{\innerproduct{i^{k}}{E}}^{2}$, $y_{i}^{E}=\max_{k}\absolutevalue{\innerproduct{i^{k}}{E}}^{2}$, and $\gamma_{i}=\min_{k}\gamma_{i}^{k}$. Note that these extrema are taken only over $k$ for which $\gamma_{i}^{k}$ is positive, i.e., non-zero.

The results of this paper generalize to mixed states and general measurements by taking

	$\displaystyle a_{E}$	$\displaystyle=\max\left\{\sum_{i}p_{i}\big{(}x_{i}^{E}+\gamma_{i}y_{i}^{E}\big{)}-\left(\sum_{i}\sqrt{p_{i}y_{i}^{E}V_{i}}\right)^{2}\!\!\!,0\right\},$
	$\displaystyle b_{E}$	$\displaystyle=\bigg{(}\sum_{i}\sqrt{p_{i}\bra{E}{\hat{\Pi}}_{i}\ket{E}}\bigg{)}^{2},$		(20)

in Eq. (1), and by taking

$$\begin{split}(\boldsymbol{A}_{i})_{EE^{\prime}}&=(-1)^{1+\delta_{EE^{\prime}}}\absolutevalue{\bra{E}{\hat{\Pi}}_{i}\ket{E^{\prime}}},\\ (\boldsymbol{B}_{i})_{EE^{\prime}}&=\absolutevalue{\bra{E}{\hat{\Pi}}_{i}\ket{E^{\prime}}},\end{split}$$

(21)

in Eq. (6). See Appendix A for the proofs. Results that come after do not depend on the specific form of the bounds. Thus these proceed identically. For a complete projective measurement, we have ${\hat{\Pi}}_{i}=\outerproduct{i}{i}$, which implies $x_{i}^{E}=y_{i}^{E}=\absolutevalue{\innerproduct{i}{E}}^{2}$ and $\gamma_{i}=V_{i}=1$, from which the initial bounds easily follow.

Quantum measurements provide more information than one would initially think. We developed a method that allows us to measure one observable and predict bounds on the distribution of outcomes and expectation values of every other observable. In this method, it is assumed that we have enough copies of the initial state so we can determine the entire probability distribution of outcomes of the measured observable. The method works well either when the measured and the estimated observables resemble each other or when the system state is close to one of the measurement basis states. In those cases, the bounds will be very tight. On the other hand, if the measurement cannot distinguish between two eigenstates with very different eigenvalues, and the system state has considerable overlap with one of them, the method naturally cannot give a good estimate. However, this can be overcome by combining measurements in different bases. Additionally, when estimating conserved quantities, better estimates are obtained by measuring at different times.

It is interesting to compare the presented method with the recent work of Huang et al. (2020). There, an algorithm is provided in which it is possible to approximate the mean value of an observable with high probability by applying random measurements, called classical shadows. This idea has been extended in subsequent literature both theoretically Zhao et al. (2021); Hadfield et al. (2022); Bu et al. (2022); Sack et al. (2022); Ippoliti (2023); Hu and You (2022); Hu et al. (2023); Gresch and Kliesch (2023); Akhtar et al. (2023); Seif et al. (2023), and experimentally Zhang et al. (2021); Struchalin et al. (2021).

In classical shadows literature, it is assumed that performing any type of measurement is possible. These measurements are sampled randomly from a tomographically complete set, meaning that with this set of measurements, quantum tomography is possible. The goal is to estimate the expectation value of an observable and achieve an error lower than $\epsilon$, with as few measurements as possible. In other words, it is assumed that only a limited number of copies of the state are available to be measured.

In contrast, in the method presented here, it is assumed that only a single type of measurement can be performed. This measurement has been chosen from a limited set of measurements that are experimentally available. At the same time, it is assumed that infinitely many copies of the initial state are available. Thus, we can perform as many repetitions of the same measurement as necessary to fully specify its probability distribution.

In our method, since we do not sample from a tomographically complete set, we always have a finite error in estimating the expectation value. This error comes from the misalignment of the chosen measurement basis and the estimated observable and the eigenbasis of the density matrix.

Thus, while attempting to address a very similar goal, the two approaches are different in their assumptions and outcomes. Our method shines in exactly those situations in which not every measurement can be performed. This is motivated by the experimental capabilities of current state-of-the-art quantum simulators , which allow for the application of only one and possibly two-qubit gates. For this reason, we focused on local measurements. Of course, the method will work better with the improvement of the experimental capabilities. Its main strength, though, is to be able to give a prediction of the mean value of observables together with its error even in cases when the experimental capabilities are very low and other methods cannot be used.

We argued for using this method for estimating moments of energy, which have a wide range of applications while being difficult to measure directly in many-body systems. For instance, using this method, one can bound the characteristic timescale through the Mandelstam-Tamm and Margolus-Levitin bounds Mandelstam and Tamm (1991); Margolus and Levitin (1998); Deffner and Campbell (2017), to estimate the amount of extractable work from an unknown source of states Šafránek et al. (2022), or to estimate temperature. Moreover, the latest can be used to benchmark the cooling function of quantum annealers Benedetti et al. (2016); Pino and García-Ripoll (2020); Hauke et al. (2020); Imoto et al. (2021) and adiabatic quantum computers Mohseni et al. (2019). This could be particularly suitable for systems with area-law entanglement scaling Eisert et al. (2010); Abanin et al. (2019), in which local measurements should be more powerful given the absence of long-range correlations in the eigenstates of such systems. We confirmed this numerically in two gapped models, which are proven to have area-law ground states Hastings (2007). In these, estimating the ground state energy using only local measurements works especially well. The method can be used equally well and proceeds identically for estimating observables other than energy, with the only exception that if such an observable does not commute with the Hamiltonian, then also its mean value is time-dependent, therefore, one has to pick a specific time for the analysis. (In contrast, when measuring energy, we could improve the bounds by measuring at different times, using that its mean value is conserved.) The method can also be used to prove entanglement through an entanglement witness (operator $\hat{A}$), without measuring the witness itself: to prove entanglement, it is enough to show that the expectation value of this operator is negative Terhal (2000); Barbieri et al. (2003); Lewenstein et al. (2000).

On a theoretical ground, this research instigates new possible paths of exploration. How to choose a measurement given some restriction, for example, on its locality or the number of elementary gates it consists of, that leads to the tightest possible bound? Is it possible to apply machine learning models to find this optimal strategy? Will the bound give an exact value when the set of measurements is tomographically complete? Can this method be modified to identify the properties of channels instead of states? Given that this method bounds the entire probability distribution of outcomes, is it possible to modify it to calculate estimates of functions of a state other than expectation values, such as entanglement entropy?

This Appendix provides methods and proofs of the bounds, several examples, and additional numerical experiments on experimentally realized many-body systems. It contains App. A: Methods and proofs of the bounds. App. B: Example in which the bound on the collection of energy probabilities provides a much better estimate of the mean energy value. App. C: Introduction of quality factors — two measures of performance. App. D: Simple analytic example of the mean energy estimation of a qubit. App. E: Estimation of the mean energy in experimentally relevant models: Heisenberg, Ising, XY, and PXP models.