The classical limit of quantum mechanics through coarse-grained measurements

Our everyday experience provides clear evidence for the validity of the classical physical description at both the kinematic and dynamic levels. Does this description emerge from a more fundamental theory? And, if so, how? These questions are still not fully resolved in physics, despite considerable efforts and a vast literature on the classical limit [Ehr27, Per95, Lan05, LO24]. It is sometimes believed that the problem can be solved by formal derivations in which the limit of Planck’s constant is taken to be zero, i.e. $\hbar\rightarrow 0$. But this cannot be an answer, because $\hbar$ is a dimensional physical constant different from zero, and yet classical physics is an accurate description in the realm of everyday life.

There are two main approaches to understanding the origin of classical physics. In the first approach, it is generally assumed that the classical limit cannot be derived without modifying quantum theory. An example is the introduction of additional collapse mechanisms of quantum states [BDU23], assumed to be due either to a large number of degrees of freedom in a macroscopic superposition [GRW86], or due to the supposed gravitational self-interaction [Dio87, Pen96]. In the second approach, one derives classical physics solely as a limit of quantum theory. Classical physics then arises either through decoherence [Zeh70, Zur81, Zur82, JZK⁺85, Zur03], i.e. interaction between the system with its environment, or through coarse-grained observation of the system [Omn89, Per95, KB07]. The decoherence approach provides a dynamical perspective on the quantum-to-classical transition, emphasizing how interactions with the environment lead to the loss of quantum coherence in a system, thereby inducing classical behavior. In contrast, the transition to classicality via coarse-grained measurements highlights an epistemic perspective, focusing on what can and cannot be observed from the system under limited measurement precision, even if the system’s quantum state is fully coherent. Despite their differing emphases, these two approaches are interconnected. A purification of a coarse-grained measurement can be interpreted as an interaction between the system and the degrees of freedom of the measuring device, which effectively act as an environment. Conversely, every operational statement in quantum mechanics must inherently involve a measurement and its associated precision, including those pertaining to the states of the environmental degrees of freedom. In this paper, we explore coarse-grained measurements as a framework for understanding the quantum-to-classical transition, emphasizing the interplay between the measurement device’s phase-space precision and the system’s inherent quantum uncertainty within phase space.

One of the most pivotal tools for evaluating the compatibility of quantum mechanical systems with classical physics is the set of Bell [Bel64] and Leggett-Garg (LG) [LG85, Leg02] inequalities. The Bell inequalities impose constraints on spatial correlations at a given time, ensuring consistency with the principle of local causality [BCP⁺14], which serves as a formalization of the kinematics inherent in classical systems. Analogously, the LG inequalities are constraints on temporal correlations that are derived from a conjunction of two assumptions, macrorealism per se and non-invasive measurability. Macrorealism per se is the assumption that a macroscopic physical quantity, for example the position of a macroscopic object, has a definite value at any given time prior to and independently of the measurement. Non-invasive measurability is the assumption that the disturbance caused by the measurement of the macroscopic quantity can be assumed to be arbitrarily small. A prime example of a theory that respects both assumptions is classical mechanics and classical statistical mechanics, which evolve according to Liouville dynamics in phase space or through classical stochastic processes. Quantum mechanical correlations are known to violate the LG inequalities [VB23, CK15]; however, experimental demonstrations of these violations have thus far been limited to microscopic system [KSG⁺12, ZWZ⁺23, DBHJ11, GAB⁺11, PLMN⁺10, VST⁺10].

Although significant progress has been made in understanding the origins of classical physics through coarse-grained measurements [Omn89, GMH93, Per95, KB07], many questions remain unanswered. In coarse-graining approaches, measurement results are grouped into slots, each corresponding to a single macroscopic outcome. While it has been shown that coarse-graining outcomes into slots can account for classicality at the kinematical level in specific systems (i.e., a joint probability distribution exists at a given time for all coarse-grained measurements), it is recognized that this explanation depends critically on the specific type of coarse-graining and the measurement precision used – particularly, the way in which outcomes are grouped into slots. To be precise, compatibility with classical kinematics can only be achieved when measurement outcomes that are adjacent to each other are grouped together in slots [Omn89, KB07, RSS11, JLK14, SGS14, Kab21]. In contrast, when non-adjacent outcomes are grouped together, such as in parity measurements in quantum optics, where every odd and every even outcome in the photon-number (Fock) states are grouped separately, they can be exploited to violate the LG [KB07] or Bell’s inequalities even if Wigner’s quasiprobability in phase space is positive [BW98].

Regarding measurement precision, it has been shown that for macroscopic systems composed of many particles, a certain level of precision allows for a classical description of measurement outcomes [GD21]. This phenomenon is closely related to macroscopic locality [NW10]: at a specific coarse-graining threshold, the measurement statistics of certain macroscopic states—specifically, those composed of many independent and identically distributed (IID) correlated pairs—exhibit local correlations, meaning they do not violate Bell inequalities. However, further research [GD21, Gal22, GD24] has demonstrated that nonlocal correlations can emerge when the IID assumption is lifted.

Additionally, classicality has been shown to emerge in the mathematical limit of an infinite number of quantum systems, which can be described using a non-separable Hilbert space, i.e. one with uncountably infinite dimensions. In this framework, the Hilbert space decomposes into superselection sectors that cannot interfere with each other, enforcing a classical-like behavior [BG23a, BG23c, BG23b].

Moving now to dynamics, we note that even under coarse-graining with slots containing adjacent outcomes, a violation of the LG inequalities and a departure from classical dynamics become possible under “non-classical” time evolutions. These evolutions are typically driven by Hamiltonians that generate superpositions between different slots [KB08, JPR09]. The vast majority of Hamiltonians eventually will generate superpositions across multiple slots after a sufficiently long time. Specifically, if we consider coarse-grained measurements in the phase space, all Hamiltonians – except that of a harmonic oscillator – will produce superpositions across multiple slots. This raises the question: how can general Liouville dynamics of the classical physics emerge from the underlying quantum mechanical evolution under the coarse-grained measurements?

A comprehensive approach to understanding the classical limit must address and provide explanations for the following problems:

1. 1.

   Every coarse-grained observation is characterized by its precision (the size of the measurement slot) and the rate of the measurements. Is there a range of these parameters under which the effective evolution of the quantum-mechanical system can be described by a classical Hamiltonian? If so, what form would this Hamiltonian take in the classical limit? Is it possible to generate all possible classical Hamiltonian systems in this way?

2. 2.

   Quantum Hamiltonians are obtained from classical Hamiltonians by Dirac’s quantisation rule. Is the effective classical Hamiltonian, which is obtained in the classical limit, the same as the one from which the quantum Hamiltonian was generated by the quantisation procedure? In other words, is the circle of quantisation-classical limit-quantisation from Figure 1 consistent?

3. 3.

   One distinguishes between two types of classical systems: (a) macroscopic systems, such as chairs, and cats, which appear to inherently exhibit classical behavior under standard physical conditions; and (b) microscopic systems, such as atoms and electrons, which display classical behavior only under specific conditions. A typical example of the latter are particles leaving traces in the form of classical trajectories in a cloud chamber. Can the classical limit achieved through coarse-grained measurements account for both types of classicality?

Providing comprehensive answers to questions 1–3, would be highly significant for both foundational insights and practical applications in physics. From an applied perspective, understanding dynamics that do not yield a classical limit, even with limited measurement accuracy, is critical to designing experiments that probe macroscopic quantum phenomena [LG85, ANVA⁺99, FPC⁺00, OHA⁺10, GET⁺11, DRD⁺20]. Additionally, non-classical dynamics serve as a valuable resource in quantum information science, as they can optimize memory usage in quantum communication protocols [BTCV04, KGP⁺11]. From a foundational perspective, this research can provide deeper insight into how classical Newtonian mechanics emerges from quantum mechanics or, alternatively, help identify the essential components of one theory required to formulate the other, thereby ensuring the consistency of the cycle depicted in Figure 1. In that respect, it is relevant to mention that while Bohr [Boh49] and the Copenhagen school argued for the necessity of classical physics in explaining quantum mechanics – asserting that classical concepts provide the necessary language for interpreting measurements – Bell [Bel87] and others contended that quantum theory should be self-sufficient, with any classical behavior emerging solely from an underlying quantum framework.

Previous works partially addressed these questions. In particular, the work by Omnès [Omn89] and subsequent follow-up studies [Ana02], as well as the exploration of the classical limit within the consistent history approach [GMH93], investigate the connection between quantum and classical physics in phase space. These works associate projectors in the Hilbert space with large regions of the phase space. One begins with a phase space cell $C_{0}$, chosen sufficiently large so that the localization of a quantum state within it corresponds to a projection onto a subspace of the Hilbert space, defined by the quantum projector $\hat{F}_{0}$. The classical Liouville dynamics evolves this cell into $C_{t}$ over time, with $C_{t}$ assumed to be sufficiently large to correspond to a subspace associated to a projector $\hat{F}_{t}$. The dynamics of the quantum system under unitary evolution $\hat{U}(t)=\exp({-\frac{i}{\hbar}\hat{H}t})$ is said to exhibit a classical limit if $\hat{U}(t)\hat{F}_{0}\hat{U}^{\dagger}(t)=\hat{F}_{t}$. However, this equality generally holds only approximately, with the degree of approximation depending on the form of the potential in $\hat{H}=\frac{\hat{p}^{2}}{2m}+V(\hat{x})$ and the properties of the cell $C_{t}$. More precisely, the framework can associate quantum projector to $C_{t}$ only under specific conditions: the cell must be “macroscopic” and sufficiently regular. This restricts its applicability to systems without highly irregular (e.g., chaotic) or highly localized phase-space regions. Furthermore, the error in approximating quantum dynamics with classical dynamics becomes non-negligible for potentials that are polynomials of degree higher than two. This raises questions about whether the method can account for the emergence of these higher-order potentials in classical physics from quantum mechanics. Finally, to provide an operational explanation of how classical mechanics arises through coarse-grained measurements, it is natural to assume that the POVM elements represented by cells are independent of the quantum Hamiltonian. However, the current approach requires adapting the shape of the cells at each measurement time step, thereby limiting its practical applicability. Despite these limitations, these works represent a groundbreaking effort to bridge classicality and quantum mechanics through an innovative yet constrained methodology.

In this work, we provide a comprehensive and rigorous proof of the classical limit of quantum mechanics through coarse-grained measurements, deriving both effective classical kinematics and dynamics, as well as addressing the three questions 1.-3. outlined in the text above. We demonstrate that when the product of the measurement precisions of position and momentum in phase space (defining a single “slot”) is much larger than the Planck constant, all quantum states can be effectively described by classical probability distributions in phase space. Furthermore, if the quantum Hamiltonian can be approximated linearly in terms of “quantum fluctuations” around its classical values within the slots, the effective dynamics of the probability distribution is governed by classical Liouville dynamics. Thereby, the effective classical Hamiltonian has the same functional form in terms of phase variables as the underlying quantum Hamiltonian has in terms of the corresponding phase-space operators. Hence, the methodological loop from Figure 1 is consistent if the classical Hamiltonian is defined as an effective description of the evolution governed by the quantum Hamiltonian under coarse-grained measurements and the linear approximation.

We estimate the Ehrenfest time as the minimum duration after which errors from the linear approximation become significant. The classicality of the entire dynamics over an extended time can only be ensured if consecutive coarse-grained measurements are performed before the Ehrenfest time is reached between the measurements. This analysis is applied to two typical scenarios where classical behavior emerges:

1. (a)

   Macroscopic systems under standard conditions, where the Ehrenfest time is found to be extremely long.

2. (b)

   Microscopic systems under repeated measurements, such as particles in a cloud chamber, where the Ehrenfest time is very short but still longer than the interval between successive collisions with molecules in the chamber. These collisions can be interpreted as successive measurements.

We outline the structure of the paper. Section 1 introduces the problem of the classical limit of quantum mechanics and situates our approach within the broader context of existing literature. Section 2 defines the classical limit in terms of both kinematics and dynamics, establishing the criteria for transitioning from quantum to classical descriptions. In Section 3, we introduce coarse-grained measurements in phase space and demonstrate their compatibility with classical kinematics. Section 4 presents the derivation of Liouville dynamics under coarse-grained observations, highlighting the conditions under which classical dynamics emerge. In Section 5, we analyze the Ehrenfest time, estimating its role in determining the time scales for the validity of classical approximations. Section 6 discusses the emergence of classical trajectories and their dependence on the precision of the measurement and the rate of consecutive measurements. Section 7 extends the derivation of Liouville dynamics to a classical stochastic process in phase space. Section 8 applies these findings to various physical systems, comparing the behavior of macroscopic and microscopic systems under coarse-grained measurements. Finally, Section 9 concludes the paper by summarizing the key findings and proposing directions for future research. The appendices provide supporting proofs of the results presented in the main text.

In this section, we formalize the concept of the classical limit in terms of both kinematics and dynamics. Kinematically, the classical limit is characterized by the condition that, in the limit, the probability distributions describing measurements at a fixed time can be understood as marginalizations of an underlying classical phase-space probability distribution. Dynamically, the classical limit requires that these phase-space probabilities evolve in accordance with classical laws, specifically through Liouville’s equation. For simplicity, we restrict our discussion to a one-dimensional system. However, the conclusions drawn here can readily be extended to systems with higher dimensions.

Classical kinematics is the limit of quantum mechanics where the following conditions hold:

1. 1.

   For any quantum state and time $t$ there exists a non-negative, normalized phase-space probability distribution $P(x,p,t)$, satisfying:

   $$P(x,p,t)\geq 0,\quad\iint_{-\infty}^{\infty}dx\,dp\,P(x,p,t)=1.$$

   (1)

2. 2.

   There exists a subset of quantum observables $\mathcal{O}=\{\hat{A}\}$, such that the phase-space representation $A_{\text{cl}}(x,p,t)$ of any observable $\hat{A}\in\mathcal{O}$ satisfies:

   $$P(a\in\mathcal{R},t)=\iint_{\Gamma_{\mathcal{R}}}P(x,p,t)\,dx\,dp,$$

   (2)

   where $P(a\in\mathcal{R},t)$ is the probability of obtaining any outcome $a$ from set $\mathcal{R}$ at time $t$, and $\Gamma_{\mathcal{R}}=\{(x,p)\mid A_{\text{cl}}(x,p,t)\in\mathcal{R}\}$ is the corresponding subset of phase space.

The two conditions above correspond directly to the Kolmogorov axioms of classical probability theory, ensuring that the sample space formed by the joint eigenvalues of the observables in the classical limit adheres to classical probability. Specifically, Condition (1) guarantees non-negativity and normalization of the probability distribution, while Condition (2) ensures additivity over disjoint sets of outcomes: For any countable sequence of such sets of outcomes $\mathcal{R}_{1},\mathcal{R}_{2},\dots$ (i.e., $\mathcal{R}_{i}\cap\mathcal{R}_{j}=\emptyset$ for $i\neq j$), the probability of finding an outcome in their union equals the sum of the probabilities for finding it in individual sets, i.e. $P\left(\bigcup_{i=1}^{\infty}\mathcal{R}_{i}\right)=\iint_{\cup_{i=1}^{\infty}\Gamma_{\mathcal{R}_{i}}}dxdp\;P(x,p,t)=\sum_{i}\iint_{\Gamma_{\mathcal{R}_{i}}}dxdp\;P(x,p,t)=\sum_{i=1}^{\infty}P(\mathcal{R}_{i}),$ where $\Gamma_{\mathcal{R}_{1}},\Gamma_{\mathcal{R}_{2}},\dots$ ($\Gamma_{\mathcal{R}_{i}}\cap\Gamma_{\mathcal{R}_{j}}=\emptyset$ for $i\neq j$) are disjoint regions in the phase space, corresponding to the mutually exclusive sets of outcomes.

Although these conditions highlight a key distinction between the kinematics of quantum and classical mechanics, it is known that quasi-probability distributions are often used in quantum theory to calculate measurement probabilities. However, none of these distributions satisfy both classical conditions simultaneously. For example, the Wigner function satisfies Condition (2) but violates Condition (1) since it can take negative values, while the Husimi $Q$-function satisfies Condition (1) but fails to fulfill Condition (2). This failure reflects the incompatibility of quantum mechanics with a classical, non-contextual hidden variable model. Nevertheless, as we will demonstrate, our framework shows that these classical conditions can be satisfied when the system is restricted to sufficiently coarse-grained observables. Our next step is the definition of the dynamical classical limit.

Classical (deterministic) dynamics for a closed system is defined as the limit of quantum mechanics where the probability distribution $P(x,p,t)$, satisfying Eqs. (1) and (2), evolves according to Liouville’s equation in phase space:

$$\frac{\partial P}{\partial t}=-\{P,H\},$$

(3)

where $\{...\,,...\}$ denotes the Poisson bracket, and $H(x,p,t)$ is the classical Hamiltonian governing the system.

We will next define the coarse-grained measurement in the phase space.

In this section, we define a class of coarse-grained phase-space measurements in the form of positive operator-valued measures (POVMs). We then demonstrate that, under appropriate conditions, these POVMs are approximately projective. Building upon this, we construct a discrete phase-space probability distribution and define a subset of classical observables that satisfy the kinematic conditions introduced earlier. Finally, we discuss under which conditions the discrete phase-space probability distributions can be approximated by continuous functions.

Here, $\ket{x\,p}$ is a coherent state centered at the phase-space point $(x,p)$, where $\ket{x\,p}$ is defined as: $\ket{x\,p}=\hat{D}(x,p)\ket{0},$ with $\hat{D}(x,p)=e^{\frac{i}{\hbar}(x\hat{p}-p\hat{x})}$ being the displacement operator, and $\ket{0}$ representing the ground state of the harmonic oscillator. These coherent states are minimal wave packets, which means that their uncertainties in position $\sigma_{x}$ and momentum $\sigma_{p}$ satisfy the minimum uncertainty relation: $\sigma_{x}\sigma_{p}=\frac{\hbar}{2}.$ Further, the first (dimensionless) index, $i$, represents the discrete position index, while the second, $j$, denotes the momentum index. The corresponding (dimensional) position and momentum centers of the slots will be denoted by $x_{i}\equiv\left(i+\frac{1}{2}\right)\Delta x$ and $p_{j}\equiv\left(j+\frac{1}{2}\right)\Delta p$, and they will be more convenient to use in certain contexts. Figure 2 illustrates the regions of the POVM elements.

The coarse-grained POVMs in phase-space have been previously analyzed in Ref. [Omn89], where it was shown that for sufficiently large coarse-graining, the POVM elements become nearly orthogonal, satisfying the approximate projectivity condition.

Following the approach from Ref. [Omn89], we quantify the error of approximate projectivity as

$$\varepsilon=\frac{||\hat{P}_{ij}^{2}-\hat{P}_{ij}||_{\text{1}}}{||\hat{P}_{ij}||_{\text{1}}},$$

(5)

where $||A||_{\text{1}}=\operatorname{Tr}\left[\sqrt{A^{\dagger}A}\right]$ is the trace norm. In Appendix A we show that in the limit of large side lengths of the slots, $\Delta x\gg\sigma_{x}$ and $\Delta p\gg\sigma_{p}$ (and hence $\frac{\Delta x\Delta p}{\hbar}\gg 1$), it holds

$$\varepsilon\sim O\left(\frac{\sigma_{x}}{\Delta x}+\frac{\sigma_{p}}{\Delta p}\right).$$

(6)

This shows that the error in our approximation decreases as the cell size increases, confirming that $\hat{P}_{ij}$ can be effectively approximated as projectors in the limit of sufficiently coarse-grained measurements. The behavior can be intuitively understood by noting that the error is approximately proportional to the ratio of the “boundary area” to the area $\Delta x\Delta p$ of the entire slot. The former is defined as the product of the slot’s perimeter and the approximate area ($\sim\hbar$) covered by a coherent state, i.e. $2\hbar\left(\frac{\Delta x}{\sigma_{x}}+\frac{\Delta p}{\sigma_{p}}\right)$. The ratio between these two areas reflects the probability of randomly selecting a coherent state that lies precisely on the slot’s boundary, where the approximation fails. Indeed, the ratio scales as $\sim 2\left(\frac{\sigma_{x}}{\Delta x}+\frac{\sigma_{p}}{\Delta p}\right)$. As the slot size increases, the ratio decreases, rendering the mean error negligible.

We now demonstrate that the probability distribution associated with this POVM approximately satisfies the classical kinematical and dynamical conditions outlined earlier. To this end, the probability of observing a single POVM element $\hat{P}_{ij}$ at time $t$ for a quantum state $\hat{\rho}(t)$ is given by:

$$p_{ij}(t)\equiv p(x_{i},p_{j},t)=\text{Tr}\left[\hat{P}_{ij}\hat{\rho}(t)\right]=\iint_{\Omega_{ij}}dx\,dp\,Q(x,p,t),$$

(7)

where the Husimi function $Q(x,p,t)$ is defined as:

$$Q(x,p,t)=\frac{1}{2\pi\hbar}\braket{x,p|\hat{\rho}(t)|x,p}.$$

(8)

Thus, the discrete probability distribution $p_{ij}(t)$ satisfies the non-negativity and normalization condition in Eq. (1). This has already been derived for spin systems in Ref. [KB07].

The proof of Statement 2 is straightforward. We have thus established the classical kinematics for the discrete probability distribution associated with coarse-grained measurements. The next step is to determine the conditions under which the discrete probability distribution $p(x_{i},p_{j})$, as defined in Eq. (7), can be approximated by a continuous phase-space probability density $P(x,p)$, ensuring consistency with the classicality criteria outlined in Section 2. In the following discussion we omit the explicit time dependence whenever it is not relevant.

To analyze the transition from a discrete to a continuous description, we introduce an extrapolating function $P(x,p)$ that smoothly extends the discrete distribution across the phase space. Specifically, we define $P(x,p)$ as an integrable and continuous function over a region $R$ of phase space with the volume $|R|\equiv(\bar{x}-\underaccent{\bar}{x})(\bar{p}-\underaccent{\bar}{p})$, with bounded first partial derivatives, ensuring that it does not exhibit large fluctuations within $R$.

We introduce the following notation for the phase-space boundaries:

$$\underline{x}=\underline{n}\Delta x,\quad\overline{x}=\overline{n}\Delta x,\quad\underline{p}=\underline{m}\Delta p,\quad\overline{p}=\overline{m}\Delta p.$$

Additionally, we define the normalized discrete probability distribution:

$$\tilde{p}(x_{i},p_{j})\equiv\frac{1}{\Delta x\Delta p}\,p(x_{i},p_{j}).$$

By interpolating $\tilde{p}(x_{i},p_{j})$ across the phase space using $P(x,p)$, the discrete sum can be approximated by an integral over the region $R$:

$$\int_{\underline{x}}^{\overline{x}}\int_{\underline{p}}^{\overline{p}}P(x,p)\,dx\,dp\approx\sum_{i=\underline{n}}^{\overline{n}}\sum_{j=\underline{m}}^{\overline{m}}\tilde{p}(x_{i},p_{j})\Delta x\Delta p=\sum_{i=\underline{n}}^{\overline{n}}\sum_{j=\underline{m}}^{\overline{m}}p(x_{i},p_{j}).$$

Using known results (see, e.g., Ref. [Ste12]), the error in this approximation as quantified by

$$E=\left|\int_{\underline{x}}^{\overline{x}}\int_{\underline{p}}^{\overline{p}}P(x,p)\,dx\,dp-\sum_{i=\underline{n}}^{\overline{n}}\sum_{j=\underline{m}}^{\overline{m}}p(x_{i},p_{j})\right|,$$

can be bounded as:

$$E\leq\max_{x,p\in R}\left|\frac{\partial P}{\partial x}\right|\cdot\frac{(\overline{x}-\underline{x})^{2}(\overline{p}-\underline{p})}{n}+\max_{x,p\in R}\left|\frac{\partial P}{\partial p}\right|\cdot\frac{(\overline{p}-\underline{p})^{2}(\overline{x}-\underline{x})}{m},$$

(9)

where $n=\bar{n}-\underaccent{\bar}{n}$ and $m=\bar{m}-\underaccent{\bar}{m}$ represent the number of discrete slots in the position and momentum directions, respectively. From the above, we conclude that the error decreases as the number of slots within the region $R$ increases. Therefore, if the discrete probability distribution does not exhibit significant fluctuations (i.e., the first derivatives of the extrapolated function $P(x,p)$ are not too large) and extends over a sufficiently large region relative to the slot size, the discrete probability distribution can be well approximated by a continuous probability density. We summarize this finding in the following statement, and represent it in Figure 3.

In the previous section, we demonstrated that probabilities arising from sufficiently coarse-grained measurements can be interpreted classically at each fixed time. In this section, we shift our focus to the dynamics of these probabilities, showing that under certain conditions they approximately follow the classical Liouville evolution.

To achieve this, we will make a certain approximation of the Hamiltonian $\hat{H}$ that governs time evolution of the quantum system under the restriction of coarse-grained measurements. For this purpose, we define the “fluctuation operators” $\delta\hat{x}$ and $\delta\hat{p}$, representing the differences between the position and momentum operators and their coarse-grained counterparts. The latter are highly degenerate operators in the limit of large coarse-graining, with eigenvalues corresponding to the positions $x_{i}$ and momenta $p_{j}$ of the centers of the slots $(i,j)$. We define:

	$\displaystyle\delta\hat{x}$	$\displaystyle\equiv\hat{x}-\sum_{i}x_{i}\hat{P}_{x_{i}},$		(10)
	$\displaystyle\delta\hat{p}$	$\displaystyle\equiv{}\hat{p}-\sum_{j}p_{j}\hat{P}_{p_{j}}.$		(11)

Here $\hat{P}_{x_{i}}=\sum_{j}\hat{P}_{ij}$ is the projector associated to the stripe centered at position $x_{i}$, and similarly $\hat{P}_{p_{j}}=\sum_{i}\hat{P}_{ij}$ is the projector associated to the stripe centered at momentum $p_{j}$.

We will consider physical situations where the power functions of the position and momentum operators can be approximated using a linear expansion in the fluctuation operators around their classical values in the slots. Under this approximation, the powers of the position and momentum operators become

	$\displaystyle\hat{x}^{n}$	$\displaystyle\approx\sum_{i}\left(x_{i}^{n}+n\,x_{i}^{n-1}\delta\hat{x}\right)\hat{P}_{x_{i}},$		(12)
	$\displaystyle\hat{p}^{m}$	$\displaystyle\approx\sum_{j}\left(p_{j}^{m}+m\,p_{j}^{m-1}\,\delta\hat{p}\right)\hat{P}_{p_{j}}.$		(13)

These operator relations are to be understood in terms of the vanishing trace norms of the differences between the left- and right-hand sides of Eqs. (12) and (13) under certain physical conditions. The analysis of these conditions will be presented in detail in the next section.

From now on, we will consider the standard quantum-mechanical Hamiltonian: $\hat{H}=\frac{\hat{p}^{2}}{2m}+\sum_{n}c_{n}\hat{x}^{n}.$ The effective Hamiltonian obtained under the linear approximation in quantum fluctuations is expressed as

$$\hat{H}_{\text{eff}}=\sum_{j}\left(\frac{p_{j}^{2}}{2m}+\frac{p_{j}}{m}\delta\hat{p}\right)\hat{P}_{p_{j}}+\sum_{i,n}c_{n}\left(x_{i}^{n}+n\,x_{i}^{n-1}\delta\hat{x}\right)\hat{P}_{x_{i}}.$$

(14)

For the purposes of further derivation we need to compute the following commutators:

$$[\delta\hat{x},\hat{P}_{ij}]\approx-i\hbar\frac{\hat{P}_{i,j+1}-\hat{P}_{ij}}{\Delta p},\quad[\delta\hat{p},\hat{P}_{ij}]\approx i\hbar\frac{\hat{P}_{i+1,j}-\hat{P}_{ij}}{\Delta x}.$$

(15)

They demonstrate that $\delta\hat{x}$ and $\delta\hat{p}$ act on $\hat{P}_{ij}$ approximately as the generators of discrete momentum and position translations, respectively. We present the computation for the second commutator in Eq. (15); the first can be treated analogously. First, note that $[\delta\hat{p},\hat{P}_{ij}]=[\hat{p},\hat{P}_{ij}]$. We can calculate the commutator $[\hat{p},\hat{P}_{ij}]$ by studying the action of the translation operator $\hat{T}(a)=e^{-\frac{i}{\hbar}a\hat{p}}$ on the projectors $\hat{P}_{ij}$. Specifically, direct calculation yields:

$$\hat{T}(a)\hat{P}_{ij}\hat{T}^{\dagger}(a)=\hat{P}_{x_{i}+a,p_{j}}.$$

(16)

Note that for the projector $\hat{P}_{x_{i}+a,p_{j}}$ to be a valid POVM element of our coarse-grained measurement, $a$ must be an integer multiple of $\Delta x$. Expanding the left-hand side of Eq. (16) in a Taylor series in $a$, and the right-hand side using a discrete Taylor series in $a$, and keeping only the linear term, we find: $\hat{P}_{ij}-a\frac{i}{\hbar}[\hat{p},\hat{P}_{ij}]\approx\hat{P}_{ij}+a\frac{\hat{P}_{i+1,j}-\hat{P}_{ij}}{\Delta x}.$ We estimate the error arising from using the discrete Taylor expansion instead of the continuous one in Appendix B. From this, we finally deduce $[\hat{p},\hat{P}_{ij}]\approx i\hbar\frac{\hat{P}_{i+1,j}-\hat{P}_{ij}}{\Delta x}.$

Next, we derive how the projector $\hat{P}_{ij}$ evolves over time under the effective Hamiltonian in the Heisenberg picture:

	$\displaystyle\partial_{t}\hat{P}_{ij}(t)$	$\displaystyle=\frac{i}{\hbar}[\hat{H},\hat{P}_{ij}]\approx\frac{i}{\hbar}[\hat{H}_{\text{eff}},\hat{P}_{ij}]=\frac{i}{\hbar}\left(\frac{p_{j}}{m}[\delta\hat{p},\hat{P}_{ij}]+\sum_{n}c_{n}n\,x_{i}^{n-1}[\delta\hat{x},\hat{P}_{ij}]\right)$
		$\displaystyle\approx\frac{h_{i+1,j}-h_{ij}}{\Delta x}\frac{\hat{P}_{i,j+1}-\hat{P}_{ij}}{\Delta p}-\frac{h_{i,j+1}-h_{i,j}}{\Delta p}\frac{\hat{P}_{i+1,j}-\hat{P}_{i,j}}{\Delta x},$		(17)

where we introduce classical Hamiltonian taking value $h_{ij}=\frac{p_{j}^{2}}{2m}+\sum_{n}c_{n}x_{i}^{n}$ in the slot $(i,j)$.

We now use the evolution law of the projectors (17) to determine how the probabilities $p_{ij}$ to find the system in slot $(i,j)$ evolve over time. We arrive at the following:

We see that the discrete probability distribution satisfies the discrete Liouville equation, generated by the classical Hamiltonian that takes the values $h_{ij}$ in the slots $(i,j)$. Note that the functional dependence of the classical Hamiltonian $H(x_{i},p_{j})$ on phase-space variables is the same as that of the original quantum Hamiltonian $\hat{H}(\hat{x},\hat{p})$ on the phase-space operators. We next analyze whether, and if so, under what conditions, the discrete Liouville equation can be approximated by the continuous one, as given in Eq. (3).

We follow considerations similar to those in Statement 3. We assume that both the discrete probabilities $p_{ij}$ and the values $h_{ij}$ of the effective Hamiltonian in the slots are not highly fluctuating. Additionally, we assume that the probability distribution is non-negligible over a region with a volume significantly larger than the slot size $\Delta x\Delta p$. Under these conditions, we can approximate the discrete values with continuous functions, extrapolating values $p_{ij}\approx P(x_{i},p_{j})\,dx\,dp$ and $h_{ij}\approx H(x_{i},p_{j})\,dx\,dp$, in the sense of Eq. (9), where functions $P(x,p)$ and $H(x,p)$ have bounded first partial derivatives. It is then easy to see that in this limit one has: $\frac{\Delta_{x}\,p_{ij}}{\Delta x}\approx\frac{\partial P}{\partial x}dxdp$, $\frac{\Delta_{x}\,h_{ij}}{\Delta x}\approx\frac{\partial H}{\partial x}dxdp$ and similarly for partial derivatives after momentum.

All these assumptions ensure that the transition from the discrete to the continuous description preserves to the dynamics of the probability distribution over time. Hence, we arrive at the continuous Liouville dynamical equation (3) in the classical limit.

We note that the derived Liouville equation for the probability of finding a particle in a slot does not assume that the system is continuously measured (although we will consider that case in Section 6). Instead, $p_{ij}(t_{k})$ is the probability for detecting the particle at a specific time $t_{k}$ in a phase-space slot $(i,j)$, provided the coarse-grained measurement is performed at $t_{k}$. To test the predicted evolution at several time instances $t_{k}$ $(k=1,2,...)$, one would need to collect statistics by repeatedly measuring the particle at $t_{k}$ without measuring it at earlier times $t_{l}$, $l<k$. Only through extrapolation between the phase-space slots, where the statistics were collected at different times, can we deduce the time evolution of the system as if it were continuously following the predicted classical dynamics, as illustrated in Figure 4. However, we will consider a sequence of coarse-grained measurements on a single system, the results of which will be discussed in Section 6.

In our derivation of the Liouville dynamics, we assumed that the powers of the position and momentum operators, and hence the quantum Hamiltonian, are linear in the quantum fluctuations around their classical values in the slots. In this section, we estimate the minimal time after which this approximation is no longer valid, leading to a deviation between quantum and classical evolutions. We refer to this time as the Ehrenfest time, consistent with its interpretation in other approaches [Ehr27, LL77, Per84, ZP94, JSB03, Lan05] as the time within which the classical approximation of quantum evolution remains valid.

During time evolution, the probabilities of finding a particle in a slot, when evolved according to the quantum Hamiltonian $\hat{H}$ or the effective Hamiltonian $\hat{H}_{\text{eff}}$, will differ if the former contains terms higher than linear in quantum fluctuations. We will denote by $\delta\hat{H}=\hat{H}-\hat{H}_{\text{eff}}$ the difference between the two Hamiltonians up to the second order in quantum fluctuations $\delta\hat{x}$ and $\delta\hat{p}$. The difference $\delta p_{ij}$ in the corresponding probabilities depends on time $\delta t$ and can be estimated as:

$$\delta p_{ij}\approx\frac{1}{\hbar}\operatorname{Tr}\left[\hat{\rho}(0)[\delta\hat{H},\hat{P}_{ij}]\right]\delta t$$

for sufficiently short $\delta t$, where $\hat{\rho}(0)$ is the initial quantum state.

We define the Ehrenfest time $t_{E}$ as the time when the probability difference becomes significant, i.e. $|\delta p_{ij}|\sim 1$, for the Ehrenfest time, $\delta t\equiv t_{E}$. Hence,

$$\frac{1}{t_{E}}\approx\frac{1}{\hbar}\left|\operatorname{Tr}\left[\hat{\rho}(0)[\delta\hat{H},\hat{P}_{ij}]\right]\right|.$$

(19)

To estimate the lower bound on $t_{E}$ we make use of Hölder’s inequality:

$$\left|\langle\hat{A},\hat{B}\rangle_{HS}\right|=\left|\text{Tr}[\hat{A}^{\dagger}\hat{B}]\right|\leq\|\hat{A}\|_{p}\|\hat{B}\|_{q}\quad\text{ with }\frac{1}{p}+\frac{1}{q}=1.$$

(20)

Here $\langle\hat{A},\hat{B}\rangle_{HS}$ is the Hilbert-Schmidt inner product between operators $\hat{A}$ and $\hat{B}$, $\|A\|_{p}=\left(\sum_{i=1}^{\infty}\sigma_{i}^{p}\right)^{\frac{1}{p}}$, $p\in[1,\infty)$, is the Schatten $p$-norm and $\sigma_{i}$ are the singular values of $\hat{A}$, i.e., the square roots of the eigenvalues of $\hat{A}^{\dagger}\hat{A}$ (or $\hat{A}\hat{A}^{\dagger}$ in the case of non-square matrices). Specifically, the Schatten 1-norm, also known as the trace norm, is defined as the sum of the singular values of the operator: $\|\hat{A}\|_{1}=\sum_{i=1}^{\infty}\sigma_{i}$, and the Schatten $\infty$-norm, also called the spectral norm, is defined as the largest singular value of the operator: $\|\hat{A}\|_{\infty}=\max_{i}\sigma_{i}.$

Choosing $\hat{A}=\hat{\rho}(0)$, $\hat{B}=[\delta\hat{H},\hat{P}_{ij}]$, $p=1$ and $q=\infty$ in Eq. (20), we obtain

$$\left|\operatorname{Tr}\left[\hat{\rho}(0)[\delta\hat{H},\hat{P}_{ij}]\right]\right|\leq\|\hat{\rho}(0)\|_{1}\|[\delta\hat{H},\hat{P}_{ij}]\|_{\infty}=\|[\delta\hat{H},\hat{P}_{ij}]\|_{\infty}$$

(21)

due to $\|\hat{\rho}(0)\|_{1}=1$. Since the largest contributions to the difference between the two Hamiltonians come from quadratic powers in quantum fluctuations $\delta\hat{x}$ and $\delta\hat{p}$, we can bound the right-hand side of the equation in the following way:

	$\displaystyle\|[\delta\hat{H},\hat{P}_{ij}]\|_{\infty}$	$\displaystyle\approx\left\|\left[\frac{(\delta\hat{p})^{2}}{2m}+\sum_{k}\frac{\Delta_{x}^{2}V(x_{k})}{\Delta x^{2}}(\delta\hat{x})^{2}\hat{P}_{x_{k}},\hat{P}_{ij}\right]\right\|_{\infty}$
		$\displaystyle\leq 2\left\|\frac{(\delta\hat{p})^{2}}{2m}\right\|_{\infty}+2\left|\frac{\Delta_{x}^{2}V(x_{i})}{\Delta x^{2}}\right|\left\|(\delta\hat{x})^{2}\right\|_{\infty},$		(22)

where in the last line we have used the triangle inequalities for the operator norms $\|\hat{A}+\hat{B}\|\leq\|\hat{A}\|+\|\hat{B}\|$, and $\|\hat{P}_{x_{k}}\|_{\infty}=\|\hat{P}_{ij}\|_{\infty}=1$. Finally, combining this with Eq. (19), and noticing that $\|\delta\hat{p}\|_{\infty}=\Delta p$ and $\|\delta\hat{x}\|_{\infty}=\Delta x$, we obtain the following lower bound on the Ehrenfest time:

$$t_{E}\geq\frac{\hbar}{\frac{(\Delta p)^{2}}{m}+2\left|\frac{\Delta_{x}^{2}V(x_{i})}{\Delta x^{2}}\right|(\Delta x)^{2}}.$$

(23)

Note the dependence of the Ehrenfest time on the position coordinate $x_{i}$ of the specific slot $(x_{i},p_{j})$ where the system is initially assumed to reside.

We now discuss the qualitative behavior of the lower bound on Ehrenfest time, and in Section 8 we quantify it for various physical systems.

The expression on the right-hand side resembles the textbook formulas for the standard Ehrenfest time if $\Delta x$ and $\Delta p$ would not quantify our measurement precisions, but instead would represent the uncertainties $\sigma_{x}$ in position and $\sigma_{p}$ in momentum assigned to the quantum state of the system, respectively. Indeed, in the case of a free particle, the textbook expression for the Ehrenfest time is given by:

$$t_{E}\sim\frac{m\hbar}{(\sigma_{p})^{2}}.$$

Furthermore, considering evolutions under the presence of a potential, Ehrenfest’s theorem [Ehr27, Per95] states that the mean values of phase space observables evolve according to classical Hamiltonian equations when the mean value of the derivative of the potential coincides with the derivative of the potential evaluated at the mean value of the position, i.e. $\braket{\frac{dV(\hat{x})}{d\hat{x}}}\sim\frac{dV(\braket{\hat{x}})}{d\braket{\hat{x}}}$. The textbook scenario in which this condition is verified occurs when the position uncertainty of the state is much smaller than the characteristic length scale over which the potential varies significantly. Eq. (23) shows a similar result in the context of coarse-grained measurements: when the potential varies significantly over the side length $\Delta x$ of the slot, the bound of the Ehrenfest time decreases.

From the point of view of the results of this paper, these textbook conditions for the classicality of time evolution become understandable because larger uncertainties in position and momentum worsen the linear approximation of the Hamiltonian around its classical value within the slots.

We have seen that, after the Ehrenfest time, the dynamics of a quantum mechanical system begin to noticeably deviate from the classical Liouville evolution. Depending on the parameters in Eq. (23), this implies that a system, after a sufficiently long time, would exhibit signatures of quantum mechanical evolution. Why do we not observe such behavior under normal physical conditions? The following statement provides an answer.

For simplicity of notation, we present the proof for measurements performed at uniform time intervals $\tau_{k}=\tau$, noting that the general case follows analogously. Assume a quantum system initially in the state $\hat{\rho}(0)$, undergoing a sequence of coarse-grained measurements interspersed with evolution intervals governed by the quantum Hamiltonian $\hat{H}\equiv H(\hat{x},\hat{p})$. After $n$ such measurements, the state at time $t_{n}=n\tau$ is given by

	$\displaystyle\hat{\rho}(t_{n})=\sum_{\begin{subarray}{c}\mu_{0},\dots,\mu_{n}\\ \nu_{0},\dots,\nu_{n}\end{subarray}}\hat{P}_{\mu_{n}\nu_{n}}\,$	$\displaystyle\hat{U}(\tau)\,\cdots\hat{P}_{\mu_{1}\nu_{1}}\,\hat{U}(\tau)\,\hat{P}_{\mu_{0}\nu_{0}}\,\hat{\rho}(0)$
		$\displaystyle\hat{P}_{\mu_{0}\nu_{0}}\,\hat{U}^{\dagger}(\tau)\,\hat{P}_{\mu_{1}\nu_{1}}\cdots\hat{U}^{\dagger}(\tau)\,\hat{P}_{\mu_{n}\nu_{n}},$		(24)

where $\hat{U}(\tau)=e^{-i\hat{H}\tau/\hbar}$.

A classical trajectory emerges if, for any initially measured slot $(\mu_{0},\nu_{0})$, the subsequent measurement results $\{(\mu_{k},\nu_{k})\}_{k=1,\dots,n}$ can be calculated deterministically as solutions of Liouville dynamics, given the initial condition at the phase-space position $(\mu_{0},\nu_{0})$.

As shown in Section 4, the unitary evolution $\hat{U}(\tau)$ is given by the effective one $\hat{U}_{\text{eff}}(\tau)\equiv e^{-i\hat{H}_{\text{eff}}\,\tau/\hbar}$ provided that the time interval between measurements is shorter than the Ehrenfest time, i.e., $\tau<t_{E}$. This induces Liouville classical evolution for the phase-space coordinates of the slot in which the particle would be found if measured:

$$\hat{U}_{\text{eff}}(\tau)\hat{P}_{\mu_{0}\nu_{0}}\hat{U}_{\text{eff}}^{\dagger}(\tau)=\hat{P}_{\mu(\mu_{0},\nu_{0},\tau),\,\nu(\mu_{0},\nu_{0},\tau)},$$

(25)

where $\mu(\mu_{0},\nu_{0},\tau)$ and $\nu(\mu_{0},\nu_{0},\tau)$ are the solutions of the discrete classical Poisson equations

$$\dot{\mu}=\frac{\Delta H(\mu,\nu)}{\Delta\nu},\quad\dot{\nu}=-\frac{\Delta H(\mu,\nu)}{\Delta\mu},$$

(26)

for the initial conditions $\mu(0)=\mu_{0}$ and $\nu(0)=\nu_{0}$.

Inserting the identity relation $\hat{U}_{\text{eff}}^{\dagger}(\tau)\,\hat{U}_{\text{eff}}(\tau)=\mathbb{1}$ to the right of $\hat{P}_{\mu_{0},\nu_{0}}$ in its first appearance in Eq. (24) and to the left of $\hat{P}_{\mu_{0},\nu_{0}}$ in its second appearance, and using Eq. (25), we obtain

	$\displaystyle\hat{\rho}(t_{n})=\sum_{\begin{subarray}{c}\mu_{0},\dots,\mu_{n}\\ \nu_{0},\dots,\nu_{n}\end{subarray}}\hat{P}_{\mu_{n}\nu_{n}}\,$	$\displaystyle\hat{U}_{\text{eff}}(\tau)\,\cdots\hat{P}_{\mu_{1}\nu_{1}}\,\hat{P}_{\mu(\mu_{0},\nu_{0},\tau),\,\nu(\mu_{0},\nu_{0},\tau)}\hat{U}_{\text{eff}}(\tau)\,\hat{\rho}(0)$
		$\displaystyle U_{\text{eff}}^{\dagger}(\tau)\hat{P}_{\mu(\mu_{0},\nu_{0},\tau),\,\nu(\mu_{0},\nu_{0},\tau)}\,\hat{P}_{\mu_{1}\nu_{1}}\cdots\hat{U}_{\text{eff}}^{\dagger}(\tau)\,\hat{P}_{\mu_{n}\nu_{n}},$		(27)

The product of projectors can be written as

$$\hat{P}_{\mu_{1},\nu_{1}}\hat{P}_{\mu(\mu_{0},\nu_{0},\tau),\,\nu(\mu_{0},\nu_{0},\tau)}=\delta_{\mu_{1},\,\mu(\mu_{0},\nu_{0},\tau)}\,\delta_{\nu_{1},\,\nu(\mu_{0},\nu_{0},\tau)}\hat{P}_{\mu_{1},\nu_{1}},$$

(28)

showing that the only nonzero contribution to the sum arises when the measurement cell $(\mu_{1},\nu_{1})$ coincides with the cell obtained by the Liouville evolution of the initial cell $(\mu_{0},\nu_{0})$ after a time $\tau$. We repeat the same procedure for the remaining projectors. In the $k$-th repetition we insert the identity relation $\hat{U}_{\text{eff}}^{\dagger}(\tau)\,\hat{U}_{\text{eff}}(\tau)=\mathbb{1}$ to the right of $\hat{P}_{\mu_{k},\nu_{k}}$ in its first appearance in the state and to the left of $\hat{P}_{\mu_{k},\nu_{k}}$ in its second appearance, respectively. After $n$ repetitions we arrive at the following form

	$\displaystyle\hat{\rho}(t_{n})=\sum_{\begin{subarray}{c}\mu_{0},\dots,\mu_{n}\\ \nu_{0},\dots,\nu_{n}\end{subarray}}$	$\displaystyle\hat{P}_{\mu_{n},\nu_{n}}\,\hat{U}_{\text{eff}}(t_{n})\,\rho(0)\,\hat{U}_{\text{eff}}^{\dagger}(t_{n})\,\hat{P}_{\mu_{n},\nu_{n}}$		(29)
		$\displaystyle\prod_{i=1}^{n}\delta_{\mu_{i},\,\mu(\mu_{i-1},\nu_{i-1},\tau)}\,\delta_{\nu_{i},\,\mu(\mu_{i-1},\nu_{i-1},\tau)},$		(30)

where we have made the replacement $\hat{U}_{\text{eff}}(t_{n})=\hat{U}^{n}_{\text{eff}}(\tau)$.

These Kronecker delta conditions enforce a chain of position and momentum updates consistent with the discrete set of observation points of classical Liouville dynamics governed by the classical Hamiltonian $H(x,p)$ as illustrated in Figure 5. Indeed, each measurement result $(\mu_{k},\nu_{k})$ can be found deterministically by solving the discrete Liouville equations in Eq. (26), evolving the initial condition $(\mu_{0},\nu_{0})$ for a time $t_{k}=k\tau$: this shows the emergence of the classical trajectory. Notably, the classical Hamiltonian in Eq. (26) has the same functional dependence on the classical variables of position and momentum as the underlying Hamiltonian $H(\hat{x},\hat{p})$ has on the position and momentum operators. As shown in the previous sections, in the limit of highly frequent measurements and if the trajectory occupies a volume of the phase space that is much larger than the size of a single slot, we may approximate the discrete trajectory with a classical continuous one, recovering classical Liouville evolution (3).

In Section 4 we demonstrated that the evolution is approximately classical up to the Ehrenfest time. However, in this case, there is no such restriction: due to the presence of repeated measurements, classical trajectories emerge even for times longer than the Ehrenfest time, $t_{n}>t_{E}$, as long as the time between measurements is shorter. This concludes our proof of the derivation of classical Liouville dynamics as an effective description of quantum mechanical evolution under coarse-grained measurements.

In the preceding sections, we have demonstrated that Liouville dynamics can emerge as an effective description through the interplay of unitary quantum evolution (with the standard Hamiltonian consisting of the kinetic and potential term), coarse-grained phase-space measurements, and the linear approximation of the underlying Hamiltonian. We extend our analysis to the case where the evolution is not strictly unitary but is instead described by a statistical mixture of possible unitary evolutions $\hat{U}_{n}(t)$, each occurring with probability $p_{n}$. The overall dynamics can no longer be represented by a single deterministic trajectory but instead follows a statistical mixture of these evolutions. This probabilistic evolution can be effectively captured using a completely positive trace-preserving (CPTP) map that describes the evolution of the quantum state:

$$p_{ij}(t)=\operatorname{Tr}\Bigl{[}\mathcal{E}_{t}(\rho(0))\hat{P}_{ij}\Bigr{]},$$

(31)

where the CPTP map $\mathcal{E}_{t}$ has a Kraus decomposition: