\documentclass[reprint,aps,pra,longbibliography]{revtex4-2}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{amsmath,amssymb}
\usepackage{siunitx}
\usepackage{hyperref}
\usepackage{orcidlink}
\usepackage{xcolor}
\colorlet{green}{green!60!black}
\usepackage{tikz}
\usetikzlibrary{calc,arrows.meta,decorations.pathmorphing,positioning}

\hypersetup{
  colorlinks=true,
  linkcolor=blue,
  citecolor=blue,
  urlcolor=blue
}


%------------------------------------------------------------------
% PACKAGES
%------------------------------------------------------------------
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{braket}      % For \ket, \braket
\usepackage{microtype}
\usepackage{bm}

% Graphics & Plots
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{tikz}

% Math Definitions
\DeclareMathOperator{\sign}{sign}

%------------------------------------------------------------------
% REVISION MARKUP (all changes highlighted)
%------------------------------------------------------------------
\newcommand{\rev}[1]{\textcolor{black}{#1}} % Referee 2 (blue)

%------------------------------------------------------------------
% ADDITIONAL REVISION MARKUP (Referee 1: dark green)
%------------------------------------------------------------------
\definecolor{darkgreen}{rgb}{0.0,0.35,0.0}
\newcommand{\revG}[1]{\textcolor{black}{#1}} % Referee 1 (dark green)

%------------------------------------------------------------------
% AUTHOR REVISION MARKUP (post-referee conceptual refinement: purple)
%------------------------------------------------------------------
\newcommand{\revA}[1]{\textcolor{black}{#1}} % Author's further revision

\begin{document}

%------------------------------------------------------------------
% FRONT MATTER
%------------------------------------------------------------------
\title{Certified Private Relational Time from Entanglement}

\author{Karl Svozil}
\email{karl.svozil@tuwien.ac.at}
% ORCID: 0000-0001-6554-2802
\affiliation{%
    Institute for Theoretical Physics,
    TU Wien,
    Wiedner Hauptstra{\ss}e 8-10/136,
    1040 Vienna, Austria
}

\begin{abstract}
We introduce an ``entangled clock'' in which time is defined operationally by discrete measurement registrations on a singlet state. Locally, each party's tick rate is fixed by the unbiased marginals. The nontrivial resource is the relational (coincidence-tick) stream: because the singlet's information budget is entirely exhausted by joint properties, the only definite temporal structure resides in the correlations between the two parties. Operationally, after exchanging time tags and outcomes, Alice and Bob identify synchronized events (that is, the $++$ channel) and thereby obtain a joint tick record. Comparing the $++$ coincidence rate $R(\theta)=P_{++}(\vec a,\vec b)$ to Peres' isotropic bomb-fragment local-hidden-variable model (yielding $R_{\mathrm{cl}}(\theta)=\theta/(2\pi)$), we find that for obtuse analyzer separations the quantum prediction exceeds this natural classical benchmark, with a maximal relative excess of about $13.6\%$ near $\theta\approx 140.5^\circ$. We emphasize that this ``faster ticking'' refers to the \emph{rate of identified coincidence ticks} under a specific operational convention, not to an improved local clock rate, precision, or stability. Finally, by using multiple settings and a Bell test, we outline ``Certified Private Time'': a device-independent certification of unpredictability/privacy of the relational time-stamp record against adversaries lacking foreknowledge of the settings, analogous to certified randomness generation.
\end{abstract}

\keywords{\rev{quantum clock; entanglement; synchronization; \revA{relational time;} coincidence rate; Bell inequality; device-independent certification; private time}}

\maketitle

\section{Introduction}

The question of the fundamental nature of time and its measurement has recently been revisited through two distinct but complementary lenses: operational metrology in relativity and information-theoretic constraints in quantum foundations.

On the metrological side, the ``trialogue'' regarding the number of fundamental constants~\cite{DuffOkunVeneziano} established a consensus that dimensionless quantities are the ultimate observables. More recently, Matsas, Pleitez, Saa, and Vanzella~\cite{matsas-2024} argued that a relativistic spacetime requires only a single fundamental dimensional constant. By employing ``bona fide clocks,'' one can define the spacetime metric entirely through proper time measurements, rendering spatial rulers redundant.
This ``single-unit universe''---a cosmos whose metric geometry is fixed by
a single dimensional constant, namely proper time---relies on the mathematical rigidity of the Alexandrov-Zeeman theorems~\cite{alex3,zeeman}. These theorems state that the causal structure (light cones) determines the geometry of Minkowski spacetime up to a global conformal factor. The physical clock breaks this dilation symmetry, fixing the scale of the universe~\cite{svozil-2025-lt}.

If a single clock fixes the scale of the universe, a natural question arises: how do two such clocks, spatially separated but sharing a quantum history, measure the flow of time relative to one another? In classical relativity, synchronization is a matter of signaling and convention (Einstein synchronization~\cite{ein-05}). In quantum mechanics, by contrast, synchronization can be mediated by entanglement, which introduces nonclassical correlations between distant systems. These correlations are constrained by no-go theorems such as Bell's, and thus cannot in general be accounted for by local realistic statistics.

This brings us to the second lens: the study of quantum correlations. Zeilinger~\cite{zeil-99} proposed a foundational principle: that the irreducible randomness of quantum events arises because an elementary system carries only one bit of information. In an entangled state, this information is exhausted by the joint properties, leaving the individual constituents completely undefined. Simultaneously, Peres~\cite{peres222} demonstrated that classical correlations are strictly bounded because one can validly imagine the results of ``unperformed experiments.'' In contrast, quantum correlations defy these bounds precisely because such counterfactuals are illegitimate.

\revA{The time defined by the entangled clock is therefore fundamentally \emph{relational}: since the singlet state encodes no information about individual subsystem properties along any axis, the only definite temporal structure is contained in the \emph{joint} correlations between Alice's and Bob's measurement records. We call this ``relational time'' in the general sense that the relevant properties are bipartite/joint rather than marginal---without commitment to any specific interpretive program such as relational quantum mechanics~\cite{Rovelli1996}. Operationally, this relational structure manifests as a coincidence-tick stream in the $++$ channel after classical record exchange (Sec.~\ref{sec:coincidence-ticks}).}

\rev{A brief remark on context: there is a substantial literature on ``quantum clocks'' and time in quantum theory, ranging from early operational analyses of physical clocks and limits to time measurement to modern work on quantum time observables and quantum-enhanced metrology (see, e.g., Refs.~\cite{SaleckerWigner1958,Busch2008Time,Giovannetti2004,Moreva2014}). The present contribution is different in emphasis: rather than optimizing precision of time/phase estimation, we focus on an \emph{operational, Bell-theoretic} notion of a distributed clock in which the key figure of merit is an \emph{informational synchronization rate} (coincidence-tick density) and, in the multi-setting regime, device-independent certification of the unpredictability/privacy of the time-stamp record.}

\revG{A further clarification concerns the role of \emph{measurement} in the present construction. The protocol is formulated at the standard operational level of quantum theory: each run produces (i) a setting choice, (ii) a binary detector outcome, and (iii) a classical time tag. We deliberately do \emph{not} attempt to resolve the quantum measurement problem here (e.g., collapse vs.\ many-worlds vs.\ objective collapse). Instead, we take the experimentally standard viewpoint that a ``click'' constitutes a stable macroscopic record, well modeled by a quantum instrument acting on the measured degrees of freedom and an effectively irreversible amplification chain (cf.\ decoherence-based accounts, e.g.,~\cite{Schlosshauer2007}). The conceptual novelty of this paper is therefore not a new measurement theory, but a reframing of Bell-nonclassicality and device-independent certification in the language of time-stamping and synchronization.}

\revG{Relatedly, the term ``clock'' is used here in an \emph{event-based} operational sense: a clock is any reproducible physical procedure that outputs a time-ordered record of ticks. The entangled clock is thus closer in spirit to a stochastic ``tick generator'' (a point process with correlations) than to a self-contained harmonic oscillator. In particular, no claim is made that it competes with state-of-the-art atomic clocks in precision or stability; rather, it provides an informational synchronization primitive and (in the multi-setting regime) a device-independent privacy and unpredictability certificate for the tick record.}

\revG{For readers of \emph{Entropy}, it is useful to note the connection to Madjid and Myers' paper ``Clocks without `Time' in Entangled-State Experiments''~\cite{MadjidMyers2020}. That work emphasizes how clocks are actually used in entanglement experiments via time tags, record exchange, and post-processing, and it discusses synchronization concepts (including relativity-facing issues) without requiring a global time parameter. The present paper is complementary: it focuses on a specific operational ``tick'' convention and studies the resulting coincidence-tick rate as an informational quantity; moreover, it connects the multi-setting regime explicitly to Bell [e.g., Clauser-Horne-Shimony-Holt (CHSH)] certification and proposes ``Certified Private Time'' as a device-independent statement about the unpredictability/privacy of the time-stamp record.}

In this paper, we unify these perspectives to propose the concept of an \emph{Entangled Clock}. We define the ``tick'' of a clock operationally as the registration of a measurement outcome. \rev{Crucially, we separate (i) the \emph{local} tick streams, whose rates are fixed by the unbiased marginals, from (ii) a \revA{\emph{relational}} (coincidence-tick) stream, obtained only after Alice and Bob compare records and retain synchronized events in a designated coincidence channel.} We show that the synchronization rate of two such clocks is not merely a function of relativistic geometry, but of the information-theoretic constraints on the system. By comparing the quantum prediction against Peres' classical linear model, we identify a regime where the quantum clock \rev{exhibits a higher \emph{identified coincidence-tick rate} than this natural classical benchmark} and, when correlations at several angles are taken into account, \rev{is incompatible with any single local-realistic model reproducing all contexts simultaneously (according to Bell-type tests).}

\section{The Entangled Clock Protocol}

We construct a distributed time standard using a sequence of bipartite quantum systems. Let a source emit pairs of spin-1/2 particles (or polarization-entangled photons) in the singlet Bell state $\ket{\psi^-}$:
\begin{equation}
\ket{\psi^-} = \frac{1}{\sqrt{2}} \left( \ket{+}_A \ket{-}_B - \ket{-}_A \ket{+}_B \right),
\end{equation}
where $A$ and $B$ denote two spatially separated observers (Alice and Bob). Each observer possesses a measurement apparatus oriented along directions defined by unit vectors $\vec{a}$ and $\vec{b}$, respectively.

\subsection{Operational Time and Zeilinger's Principle}

We define operational time $T$ on each side not as a continuous background parameter, but as the accumulation of discrete events. Specifically, whenever a detector records a specific outcome (say, ``up'' or $+1$), the local clock increments by one unit:
\begin{equation}
T_i \to T_i + 1 \quad \text{if } \sigma_i = +1, \quad i \in \{A, B\}.
\end{equation}
If the outcome is ``down'' ($-1$), the clock implies a ``pause'' or a non-tick for that interval.

\rev{This choice (counting only $+1$ as a local tick) is a convention adopted for definiteness and algebraic simplicity. Other conventions are possible---for instance, counting both outcomes as ticks would make each local clock tick on every trial and would remove any ``speed'' language at the single-stream level. The nonclassical content we discuss resides in \emph{joint} (coincidence) structure, not in an absolute increase of local event rates.}

According to Zeilinger's Foundational Principle~\cite{zeil-99}, an elementary system represents the truth value of a single proposition (1 bit). For a composite system of two spin-1/2 particles, the total information capacity is 2 bits. In the singlet state $\ket{\psi^-}$, these 2 bits are entirely exhausted by the joint propositions (that is, ``spins are opposite along $z$'' and ``spins are opposite along $x$''). Consequently, the system contains \emph{zero} information about individual properties along any specific direction $\vec{n}$.
\begin{equation}
P(+)_{\vec{n}} = P(-)_{\vec{n}} = \frac{1}{2}.
\end{equation}
Thus, the \emph{local} proper time measured by Alice or Bob flows stochastically. Over a large ensemble of $N$ pairs, the local clock will record approximately $N/2$ ticks. \rev{In particular, the \emph{local} tick rate is fixed at $1/2$ per trial for both the quantum singlet and for unbiased classical models; the comparison in this paper concerns \emph{coincidence ticks} (below), not local rates.} This randomness is not merely epistemic; it is ontologically necessary because the individual particle carries no answer to the question ``are you up or down?'' until the measurement is performed. \revA{It is precisely this absence of individual definiteness that makes the clock's temporal content irreducibly \emph{relational}: meaningful time structure exists only in the joint record.}

\subsection{\rev{Coincidence Ticks and Record Exchange (Protocol Step)}}
\label{sec:coincidence-ticks}
\rev{The ``entangled clock'' figure of merit studied here is a \emph{jointly defined} tick stream. Operationally, Alice and Bob each produce a local record
\[
\mathcal{L}_A=\{(t_k^A, x_k^A, a_k)\}_{k=1}^N,\qquad \mathcal{L}_B=\{(t_k^B, x_k^B, b_k)\}_{k=1}^N,
\]
where $x_k^{A,B}\in\{+1,-1\}$ are outcomes, $a_k,b_k$ are settings, and $t_k^{A,B}$ are local time tags. They then \emph{classically exchange} (possibly after the run, offline) sufficient information to (i) match pairs within a coincidence window and (ii) identify which local $+1$ events were part of a designated coincidence channel (that is, $++$) rather than ``wrong plusses'' belonging to $+-$ or $-+$. Only after this comparison does the \emph{coincidence-tick} stream become defined. This is not ``nonlocal'' in the superluminal sense; it is a standard classical postprocessing step required to operationalize coincidences.}

\revA{In the conceptual language introduced above, the coincidence-tick stream is the operational realization of the \emph{relational time} encoded in the entangled state: it is the temporal structure that emerges only when the two individual records are brought together, mirroring the fact that the singlet's information content is entirely relational.}

\subsection{\revG{Measurement operations, destructiveness, and ``clockwork'' repeatability}}

\revG{Each trial of the protocol may be viewed as the following operational pipeline:}
\begin{enumerate}
\item \revG{A source emits one entangled pair (ideally heralded, or time-tagged at the source).}
\item \revG{Alice and Bob choose settings $a_k,b_k$ (possibly at random).}
\item \revG{Each side performs a two-outcome measurement and records a classical triple $(t_k^i,x_k^i,s_k^i)$ consisting of a local time tag $t_k^i$, an outcome $x_k^i\in\{\pm1\}$, and the setting label $s_k^i\in\{a,a'\}$ or $\{b,b'\}$.}
\item \revG{After the run, they exchange sufficient classical information to identify coincidences and, if desired, to compute Bell parameters such as CHSH.}
\end{enumerate}

\revG{In photonic implementations with avalanche photodiodes, a ``click'' is indeed a destructive detection event for the photon. This does not prevent repeated ticking, because the entangled clock is not a single closed two-particle system reused indefinitely; rather, it is a \emph{repeatable preparation--measurement procedure} fed by a stream of newly generated pairs (e.g., Spontaneous Parametric Down-Conversion). In this sense it resembles other event-based clocks such as Geiger-counter clocks or pulsed/Poissonian tick generators: the repeatability resides in the ability to reproduce the preparation and measurement across many trials, not in storing energy in a single oscillator.}

\revG{One may also envision nondestructive or quantum-nondemolition variants in matter-based platforms (e.g., repeated spin measurements or cavity-QED readout), but such refinements are not required for the core conceptual point pursued here: the operational time record is the classical list of registered events, and the nonclassical feature is contained in the multi-setting correlation structure of those events.}

\subsection{Synchronized (Coincidence) Time Rate}

While local time is random, the relative time evolution between Alice and Bob is structured by entanglement. We define the \emph{synchronized coincidence-tick rate} $R(\theta)$ as the probability that both clocks tick simultaneously in a given run \rev{in the designated coincidence channel}:
\begin{equation}
R(\theta) = P_{++}(\vec{a}, \vec{b}),
\end{equation}
where $\theta$ is the angle between the measurement vectors $\vec{a}$ and $\vec{b}$.

For the singlet state, quantum mechanics predicts the joint expectation value $E_{QM}(\theta) = \braket{\sigma_{\vec{a}} \otimes \sigma_{\vec{b}}} = -\vec{a}\cdot\vec{b} = -\cos\theta$. The probability of a coincident tick ($+1, +1$) is derived as follows:
\begin{align}
R_{QM}(\theta) &= \frac{1}{4} \left( 1 + \braket{\sigma_A} + \braket{\sigma_B} + \braket{\sigma_A \sigma_B} \right) \nonumber \\
&= \frac{1}{4} \left( 1 + 0 + 0 - \cos\theta \right) \nonumber \\
&= \frac{1}{2}\sin^2\frac{\theta}{2}.
\label{eq:qm_rate}
\end{align}

\textit{Implementation Note:} While we describe the protocol using spin-1/2 particles, an equivalent implementation uses polarization-entangled photon pairs. In that case, $\ket{+}/\ket{-}$ correspond to horizontal/vertical polarizations, and the detectors are linear polarizers. A ``tick'' corresponds to a photon passing through the polarizer and triggering an avalanche photodiode. \rev{Coincidence ticks are identified by standard time-tag matching and outcome comparison.}

\section{Classical Simulation: Peres' Bomb}

To determine if the behavior of the entangled clock is unique, we must compare it to a classical standard. A robust classical analog for the singlet state was provided by Peres~\cite{peres222} and is often referred to as the ``bomb fragment'' model.

\rev{We stress at the outset: Peres' model is \emph{not} claimed to be the unique classical comparator. It is used here as a natural, isotropic local-hidden-variable (LHV) construction with unbiased marginals that obeys standard Boole--Bell constraints. For any \emph{single} fixed angle $\theta^{*}$, one may engineer contextual classical mechanisms that reproduce $P_{++}(\theta^{*})$ (cf.\ Sec.~\ref{sec:certify-quantumness}). The discriminating content arises when one demands a \emph{single} LHV model reproducing \emph{all} settings simultaneously, i.e., when one moves from a one-angle fit to the multi-setting correlation polytope.}

Consider a bomb initially at rest (total angular momentum $\vec{J}=0$) that explodes into two fragments with opposite angular momenta $\vec{J}_1$ and $\vec{J}_2 = -\vec{J}_1$. The direction of $\vec{J}_1$ is random and distributed uniformly over the sphere. The observers measure the sign of the projection of the angular momentum onto their axes $\vec{a}$ and $\vec{b}$:
\begin{equation}
r_\alpha = \sign(\vec{a} \cdot \vec{J}_1), \quad r_\beta = \sign(\vec{b} \cdot \vec{J}_2).
\end{equation}
Peres derives the correlation function for this macroscopic local hidden variable (LHV) model geometrically. The correlation depends on the overlap of the hemispheres defined by the vectors $\vec{a}$ and $\vec{b}$ on the unit sphere of possible angular momenta. The classical expectation value is strictly linear in $\theta$:
\begin{equation}
E_{cl}(\theta) = -1 + \frac{2\theta}{\pi} \quad \text{for } \theta \in [0, \pi].
\label{eq:peres_corr}
\end{equation}
The corresponding classical synchronized tick rate is:
\begin{equation}
R_{cl}(\theta) = \frac{1}{4}(1 + E_{cl}) = \frac{1}{4}\left(1 - 1 + \frac{2\theta}{\pi}\right) = \frac{\theta}{2\pi}.
\label{eq:cl_rate}
\end{equation}

\section{Comparison: When Quantum Coincidence Ticks Exceed the Classical Benchmark}
\label{sec:comparison}

\rev{In what follows, any ``faster ticking'' language refers \emph{only} to the \emph{identified coincidence-tick rate} $R(\theta)=P_{++}(\vec a,\vec b)$ under the convention that a tick corresponds to the $+1$ outcome and that the coincidence tick is the $++$ channel after record exchange. The local tick rates at Alice and Bob remain fixed at $1/2$ (unbiased marginals) for both the quantum singlet and for unbiased classical models. No claim is made here about improved metrological precision, stability, or Allan deviation of a physical oscillator.}

We now compare the two synchronization rates. We can distinguish three ``cardinal'' regimes where the physics appears identical, and one ``anomalous'' regime where the quantum clock diverges.

\subsection{The Cardinal Regimes}
\begin{enumerate}
    \item \textbf{Perfect Anti-Synchrony ($\theta=0$):} Detectors are aligned ($\vec{a} = \vec{b}$).
    \[
    R_{QM}(0) = 0, \quad R_{cl}(0) = 0.
    \]
    Because $\ket{\psi^-}$ is perfectly anti-correlated (singlet), if Alice's clock ticks, Bob's does not. The clocks are perfectly out of phase.

    \item \textbf{Non-Correlation ($\theta=\pi/2$):} Detectors are orthogonal ($\vec{a} \perp \vec{b}$).
    \[
    R_{QM}(\pi/2) = 1/4, \quad R_{cl}(\pi/2) = 1/4.
    \]
    Here, $E_{QM} = 0$ and $E_{cl} = 0$. The clocks are statistically independent.

    \item \textbf{Perfect Synchrony ($\theta=\pi$):} Detectors are anti-aligned ($\vec{a} = -\vec{b}$).
    \[
    R_{QM}(\pi) = 1/2, \quad R_{cl}(\pi) = 1/2.
    \]
    Since $\ket{\psi^-}$ implies opposite spins, measuring opposite directions yields identical results ($++$ or $--$). The clocks tick together $50\%$ of the time.
\end{enumerate}
In these regimes, the informational constraint of the quantum bit yields the same statistics as the classical angular momentum conservation.

\subsection{The Anomalous Regime}

The divergence between the single-unit universe governed by quantum information and one governed by classical statistical mechanics appears at intermediate angles. We define the \emph{synchronization excess} $\Delta(\theta)$ as:
\begin{equation}
\Delta(\theta) \equiv R_{QM}(\theta) - R_{cl}(\theta) = \frac{1}{2}\sin^2\frac{\theta}{2} - \frac{\theta}{2\pi}.
\end{equation}
To find the maximum deviation, we find the extrema by setting $d\Delta/d\theta = 0$:
\begin{equation}
\frac{1}{4}\sin\theta - \frac{1}{2\pi} = 0 \implies \sin\theta = \frac{2}{\pi}.
\end{equation}
This yields two solutions in $[0, \pi]$:
\begin{align}
\theta_1 &= \arcsin(2/\pi) \approx 0.69~\text{rad}~(39.5^\circ), \quad (\Delta < 0), \\
\theta_2 &= \pi - \theta_1 \approx 2.45~\text{rad}~(140.5^\circ), \quad (\Delta > 0).
\end{align}

At $\theta_1 \approx 39.5^\circ$, the classical correlation is stronger (more negative), so the quantum clock ticks \emph{less} frequently than the classical prediction. This is the regime where the classical linear correlation drops faster than the quantum cosine.

However, at $\theta_2 \approx 140.5^\circ$, the quantum clock exhibits a significant excess of coincidence ticks (see Fig.~\ref{fig:rates}).
\begin{itemize}
    \item Classical Rate: $R_{cl}(140.5^\circ) \approx 0.390$.
    \item Quantum Rate: $R_{QM}(140.5^\circ) \approx 0.443$.
\end{itemize}
\rev{The relative excess at this angle is}
\[
\rev{\frac{R_{QM}-R_{cl}}{R_{cl}}\bigg|_{\theta\approx 140.5^\circ}\approx \frac{0.443-0.390}{0.390}\approx 0.136.}
\]

\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{axis}[
    width=\columnwidth,
    height=0.7\columnwidth,
    xlabel={$\theta$ (radians)},
    ylabel={Synchronization Rate $R(\theta)$},
    xmin=0, xmax=3.14159,
    ymin=0, ymax=0.55,
    xtick={0, 0.785, 1.571, 2.356, 3.14159},
    xticklabels={$0$, $\frac{\pi}{4}$, $\frac{\pi}{2}$, $\frac{3\pi}{4}$, $\pi$},
    legend pos=north west,
    grid=major,
    grid style={dashed, gray!30},
]
% Quantum rate
\addplot[blue, thick, domain=0:pi, samples=100]
    {0.5*sin(deg(x/2))^2};
\addlegendentry{$R_{QM}(\theta)$}

% Classical rate
\addplot[red, thick, dashed, domain=0:pi]
    {x/(2*pi)};
\addlegendentry{$R_{cl}(\theta)$}

% Mark the critical points
\addplot[only marks, mark=*, mark size=2.5pt, black]
    coordinates {(2.4515, 0.443) (2.4515, 0.390)};

% Labels for markers
\node[anchor=west, font=\scriptsize] at (axis cs:2.5, 0.443) {QM};
\node[anchor=west, font=\scriptsize] at (axis cs:2.5, 0.390) {cl};

% Vertical line at theta_2
\draw[dotted, thick] (axis cs:2.4515,0) -- (axis cs:2.4515,0.5);
\node at (axis cs:2.4515,0.52) [anchor=south] {$\theta_2$};

\end{axis}
\end{tikzpicture}
\caption{\rev{Comparison of the quantum ($R_{QM}$, solid blue) and Peres-benchmark classical ($R_{cl}$, dashed red) $++$ coincidence-tick rates as functions of the relative detector angle $\theta$. The curves coincide at $\theta = 0, \pi/2, \pi$ but diverge maximally at $\theta_2 \approx 140.5^\circ$, where the \emph{identified coincidence-tick rate} predicted by quantum theory exceeds this natural isotropic LHV benchmark.}}
\label{fig:rates}
\end{figure}

The cosine modulation of the quantum correlation function, which is consistent with Zeilinger's information-invariance principle, permits a higher density of coincident events than the linear constraint imposed by the isotropic bomb-fragment model. In particular, at $\theta_2$ the quantum prediction exceeds the linear classical benchmark by about $13.6\%$.
Figure~\ref{fig:delta} explicitly shows this synchronization excess $\Delta(\theta)$. \rev{We emphasize again: this is a statement about the $++$ coincidence channel under an agreed operational convention, not about an absolute increase of local ticking.} The positive lobe represents a regime where, if one demands a single local realistic model that applies to \emph{all} angles simultaneously, the quantum predictions force the clocks to agree more often than such a model can allow.

\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{axis}[
    width=\columnwidth,
    height=0.5\columnwidth,
    xlabel={$\theta$ (radians)},
    ylabel={$\Delta(\theta)$},
    xmin=0, xmax=3.14159,
    ymin=-0.06, ymax=0.06,
    xtick={0, 0.785, 1.571, 2.356, 3.14159},
    xticklabels={$0$, $\frac{\pi}{4}$, $\frac{\pi}{2}$, $\frac{3\pi}{4}$, $\pi$},
    grid=major,
    grid style={dashed, gray!30},
]
\addplot[blue, thick, domain=0:pi, samples=100]
    {0.5*sin(deg(x/2))^2 - x/(2*pi)};
\draw[dashed, black] (axis cs:0,0) -- (axis cs:pi,0);
\addplot[only marks, mark=*, mark size=2pt, red]
    coordinates {(0.69, -0.053) (2.4515, 0.053)};
\node at (axis cs:0.8, -0.04) [anchor=west, font=\scriptsize] {$\theta_1$};
\node at (axis cs:2.35, 0.04) [anchor=east, font=\scriptsize] {$\theta_2$};
\end{axis}
\end{tikzpicture}
\caption{The synchronization excess $\Delta(\theta) = R_{QM} - R_{cl}$. The quantum clock lags at acute angles ($\theta_1 \approx 39.5^\circ$) but leads at obtuse angles ($\theta_2 \approx 140.5^\circ$).}
\label{fig:delta}
\end{figure}

\section{Contextuality, Polytopes, and Unperformed Experiments}

Why does the quantum clock \rev{exceed the Peres benchmark in the $++$ coincidence channel for obtuse angles}? The answer lies in the nature of the information carried by the system and the role of contextuality.

In Peres' model~\cite{peres222}, the angular momentum vector $\vec{J}$ exists independently of the measurement. Peres argues that Bell's inequality can be derived by constructing a table of outcomes for experiments actually performed (settings $\vec{a}, \vec{b}$) and hypothetical experiments \emph{not} performed (settings $\vec{a}', \vec{b}'$). For a classical system, this table must be internally consistent because the value $\vec{J}$ pre-exists and determines the outcome for any possible question. In the specific bomb-fragment model, the overlap of the corresponding hemispheres on the unit sphere yields a strictly linear correlation function, Eq.~\eqref{eq:peres_corr}, which in turn strictly limits the synchronized tick rate.

In the quantum case, following Zeilinger~\cite{zeil-99}, the bipartite system carries only 2 bits of information, which are already exhausted by the specific joint state. There is no ``spare'' information to encode definite answers to all possible, mutually incompatible measurement questions along directions $\vec{a}',\vec{b}'$. From this information-theoretic perspective, the system is not constrained to maintain consistency across a full table of actual and counterfactual outcomes, and thus is not subject to the Boole--Bell-type constraints~\cite{froissart-81,pitowsky-86} that enforce the existence of a single joint probability distribution for all settings and, in particular, lead to linear behavior in models like Peres' bomb. This ``freedom from counterfactual definiteness'' is one way to understand why the quantum correlation $E_{QM}(\theta)$ follows the cosine law and exceeds the linear bomb-model bound at $\theta \approx 140^\circ$.

\rev{Equivalently (and more geometrically), the set of correlation families compatible with a single LHV model forms a convex polytope in the space of multi-setting correlations~\cite{froissart-81,pitowsky-86}. A one-angle agreement is easy to engineer; what is constrained is the \emph{simultaneous} reproduction of several angles by one underlying joint distribution. The ``cosine-vs-linear'' mismatch visible already at the level of Fig.~\ref{fig:rates} is one slice of this broader polytope structure, which is operationalized by Bell-type inequalities.}

\subsection{Certifying Quantumness}
\label{sec:certify-quantumness}

While the rate $R_{QM} \approx 0.44$ is notably higher than the standard classical rate $R_{cl} \approx 0.39$, one must be cautious. A single measurement setting cannot distinguish quantum from classical clocks. As noted in recent work on contextuality~\cite{svozil-2022-epr}, for any \emph{single} fixed angle $\theta^*$, one can construct a local classical model (e.g., Aerts' ``broken elastic band'' model~\cite{aerts-69}) that reproduces the quantum probability $R_{QM}(\theta^*)$ exactly. In such models, the hidden variable space is deformed or the probability density is non-uniform to mimic the cosine statistics locally.

The signature of quantum synchronization is \emph{contextual}: the same underlying state produces the \emph{entire} cosine curve across all angles simultaneously. To certify quantumness, Alice and Bob must therefore vary their angles and compare several correlations at once. Any local realistic model must admit a single joint probability distribution that reproduces all these correlations and thus obeys Bell-type inequalities, whereas the quantum clock yields a family of cosine values that violates these bounds. The enhanced synchronized tick rate in the obtuse regime is then seen not as an isolated anomaly, but as one facet of a globally nonclassical correlation structure.

The excess synchronization rate $\Delta(\theta) > 0$ in the obtuse regime is directly related to violations of Bell-type inequalities such as the Clauser--Horne--Shimony--Holt (CHSH) inequality~\cite{chsh}.
The CHSH parameter $S$ is composed of four correlation terms:
\begin{equation}
S = |E(\vec{a},\vec{b}) - E(\vec{a},\vec{b}') + E(\vec{a}',\vec{b}) + E(\vec{a}',\vec{b}')|.
\end{equation}
The classical bound $S \le 2$, obtained from the convex hull computation of the corresponding correlation polytope~\cite{froissart-81,pitowsky-86}, arises from the requirement that all four correlations be marginals of a single joint probability distribution, that is, from the assumption that the table of actual and counterfactual outcomes is internally consistent.
The linear correlation function~\eqref{eq:peres_corr} is one explicit realization of such a local realistic model.
The quantum violation (approaching the Tsirelson bound $2\sqrt{2}$) and the enhanced synchronization rate in the obtuse regime are two manifestations of the same underlying phenomenon.

\rev{\subsection{Explicit CHSH violation from the obtuse-angle excess}}
\rev{To make the connection concrete, choose coplanar analyzer directions
specified by angles (in radians the violation is maximized as long as the following conditions are met: (i)  Alice's two settings are orthogonal ($\pi/2$ apart); (ii)  Bob's two settings are orthogonal ($\pi/2$ apart); and (iii) Bob's basis is rotated by $\pi/4$ relative to Alice's basis)
\[
a=0,\qquad a'=\frac{\pi}{2},\qquad b=\frac{\pi}{4},\qquad b'=\frac{3\pi}{4}.
\]
For the singlet, $E_{QM}(x,y)=-\cos(x-y)$, hence
\begin{align}
E(a,b)&=-\cos\frac{\pi}{4}=-\frac{\sqrt2}{2},
\nonumber \\
E(a,b')&=-\cos\!\Big(\!-\frac{3\pi}{4}\Big)=+\frac{\sqrt2}{2},
\nonumber \\
E(a',b)&=-\cos\frac{\pi}{4}=-\frac{\sqrt2}{2},
\nonumber \\
E(a',b')&=-\cos\!\Big(\!-\frac{\pi}{4}\Big)=-\frac{\sqrt2}{2}. \nonumber
\end{align}
Therefore
\[
S_{QM}=\left|-\frac{\sqrt2}{2}-\frac{\sqrt2}{2}-\frac{\sqrt2}{2}-\frac{\sqrt2}{2}\right|=2\sqrt2>2.
\]
This standard choice includes an obtuse relative angle of $|a-b'|=3\pi/4=135^\circ$, which lies in the positive lobe of Fig.~\ref{fig:delta} (indeed $\Delta(3\pi/4)>0$). Thus, the same cosine-vs-linear mismatch that yields an excess $++$ coincidence-tick rate at obtuse angles also underlies the familiar CHSH violation.}

\section{Experimental Considerations}

Implementing this protocol requires addressing practical constraints. In an ideal scenario, every particle pair results in a potential tick. In reality, photon detectors have finite efficiency $\eta < 1$.

If $\eta < 1$, the clocks will frequently ``miss'' ticks even when the quantum outcome would have been $+1$. However, the synchronization rate $R(\theta)$ is typically measured as a coincidence rate normalized by the total number of emitted pairs (or inferred from singles counts). If we define the measured rate $R_{exp}(\theta)$, we have:
\begin{equation}
R_{exp}(\theta) \approx \eta_A \eta_B R_{QM}(\theta).
\end{equation}
This reduced efficiency lowers the absolute rate of ticking for both classical and quantum models. The crucial comparison is the \emph{relative} rate or the visibility of the interference fringe. A high-visibility experiment (using, e.g., spontaneous parametric down-conversion and superconducting nanowire single-photon detectors with $\eta > 0.9$) would clearly resolve the difference between the linear classical prediction and the sinusoidal quantum prediction.

Furthermore, the definition of ``simultaneity'' requires a coincidence window $\tau_c$. If the path lengths to Alice and Bob differ, or if there is electronic jitter, ticks might not register as simultaneous. Standard quantum optical techniques allow $\tau_c$ to be on the order of nanoseconds. \rev{As emphasized in Sec.~\ref{sec:coincidence-ticks}, the coincidence-tick stream is obtained only after classical record exchange (time tags, outcomes, and in the multi-setting case also the settings) and coincidence matching.} The ``Entangled Clock'' is thus a statistical construct: it does not provide a continuous readout for a single run but establishes a time scale over an ensemble of events, where the density of synchronized events serves as the metrological standard.

\rev{We note again that this notion of ``metrological standard'' is meant here in the operational and informational sense of a reproducible \emph{rate of identified coincidence events}, not as a claim of superior oscillator performance.}

\revG{From the perspective of physical implementation, the relevant ``system'' is therefore intrinsically open: it includes the source (pair generation), transmission channels, detectors, and the classical time-tagging electronics. The protocol does not assume coherent unitary evolution of a single isolated clockwork between ticks; instead, it assumes repeatable preparation and measurement producing a stable classical event record. This places the construction closer to the operational practice of time-tagged entanglement experiments (where post-processing defines coincidence structure) than to autonomous oscillator clocks.}

\section{Certified Private Time}
\label{sec:certified-private-time}

The discussion above highlights a crucial distinction: while a classical model can be engineered to reproduce quantum statistics for a single context, it fails to reproduce the full correlation structure across multiple contexts. This brings us to a stronger application of the entangled clock: the certification of ``private time.''

As reviewed by Ac\'in and Masanes~\cite{Acin2016CertifiedRandomness}, the violation of Bell inequalities serves as a device-independent certificate for randomness. In the context of random number generation (QRNG), this certification ensures that the output bits were not pre-determined by any hidden variable $\lambda$ available to an adversary (Eve). \rev{Here we adapt the same logic to a \emph{time-stamp record}: what is certified is the unpredictability (relative to side information) of the ordered sequence of identified coincidence ticks, under the standard device-independent assumptions (in particular, no adversarial foreknowledge of future settings).}

\revG{In particular, ``certification'' here has the same logical status as in device-independent randomness: it is a statement about the incompatibility of the observed input--output statistics with pre-programmed local-hidden-variable strategies, given standard assumptions (that is, measurement independence and no information leakage of future settings). It is not a claim to have derived a fundamental solution of the measurement problem, nor a claim that the devices implement a closed dynamical time observable.}

\subsection{The Memory-Stick Clock}
Consider the temporal equivalent of the ``memory-stick attack'' described in Ref.~\cite{Acin2016CertifiedRandomness}. An adversary, Eve, manufactures two clocks and gives them to Alice and Bob. These clocks are classical devices containing a pre-recorded sequence of ticks derived from a hidden variable $\lambda$. To Alice and Bob, the ticks appear stochastic (e.g., mimicking radioactive decay), and they may even exhibit correlations. However, because the sequence is pre-determined, Eve possesses a ``lookup table'' for time. She knows exactly when Alice's clock will tick before it happens. In such a scenario, Alice's time is not her own; it is fully correlated with the adversary's timeline.

In a classical universe (or one governed by local hidden variables), one cannot prove that a clock is not a playback device. The flow of time could be merely the reading of a script written at the Big Bang (super-determinism aside).

\subsection{Protocol for Time Certification}
The entangled clock allows Alice and Bob to rule out this ``playback'' scenario. By switching between measurement settings (angles $\vec{a}, \vec{a}'$ for Alice and $\vec{b}, \vec{b}'$ for Bob) and accumulating the statistics of their coincident ticks, they can estimate the CHSH parameter $S$.

If they observe $S > 2$, they certify that the joint state does not admit a local realistic description. Physically, this implies that the specific outcomes (the ticks) did not exist prior to the measurement events.
Consequently, the time signal generated by the clock is:
\begin{enumerate}
    \item \textbf{Intrinsic:} The ticks are generated \emph{in situ} by the act of measurement, not read from a memory.
    \item \textbf{Private:} \rev{the coincidence-tick record is unpredictable to any adversary whose side information is not correlated with the (future) setting choices, in the same device-independent sense as certified randomness. In particular, the certification concerns the \emph{ordering/time-stamping record} of identified coincidence ticks, not the performance of a physical oscillator.}
\end{enumerate}

\rev{\subsection{Cryptographic vs.\ metrological meaning of ``private time''}}
\rev{The intended meaning of ``private time'' is cryptographic and informational: Alice and Bob can maintain a time-stamped record (a sequence of coincidence ticks with associated time tags) whose fine-grained content cannot be reconstructed by an external observer without access to the settings and without violating the device-independent constraints implied by $S>2$. This should not be confused with a metrological claim such as improved phase resolution, reduced timing jitter, or improved Allan deviation.}

\subsection{Escaping the Block Universe}
This leads to a radical reinterpretation of the ``flow'' of time. In a standard operational view, time is a parameter inferred from correlations between a clock system and the rest of the universe. However, if the clock's ticks are certified random via Bell violation, the clock is dynamically decoupled from the rest of the universe until the moment of the tick.

A ``Certified Private Clock'' thus ticks at a rate that is not only statistically distinct \rev{at the level of the $++$ coincidence-tick benchmark comparison} (the ``faster'' rate at obtuse angles discussed in Sec.~\ref{sec:comparison}) but ontologically distinct. It generates a local history that no external observer could have predicted. Just as device-independent protocols use Bell violation to expand a short private random seed into a distinct, private key, the entangled clock expands a short interval of private operational agency (the choice of settings) into a sustained, private flow of time.

This certification is robust against the ``middleman attack'' often cited in foundational debates. Even if Eve builds the source and controls the channels, she cannot force the devices to violate the CHSH inequality with pre-programmed ticks unless she knows the measurement settings (the context) in advance. To prevent this, Alice and Bob must ensure ``measurement (parameter)independence''~\cite{shimony_1993}. \rev{This is the same assumption used in standard device-independent QRNG and QKD; no stronger assumption is invoked here.} This can be achieved either through genuine free choice or, alternatively, by consuming a short, predefined private random key to govern the settings. In the latter case, the system functions as a temporal implementation of Ac\'in's expansion protocol~\cite{Acin2016CertifiedRandomness}, growing the initial finite seed into a sustained, sovereign timeline that is mathematically guaranteed to be uncorrelated with any master clock Eve might hold.

\section{Conclusion}

We have proposed an operational protocol for \emph{Certified Private Time}, in which spatially separated quantum clocks are synchronized via singlet-state correlations. By adapting device-independent certification methods~\cite{Acin2016CertifiedRandomness}, we showed that if observers consume a private seed to ensure measurement independence, the resulting timeline is certified to be ``sovereign''---generated \emph{in situ} and mathematically uncorrelated with any pre-existing external script.

Crucially, \rev{the nontrivial ``rate'' comparison in this work concerns a \emph{jointly defined coincidence-tick stream} (that is, $++$) obtained after classical record exchange and coincidence identification. Locally, the marginal tick rates remain fixed by unbiased statistics.} For instance, at obtuse relative angles ($\theta \approx 140^\circ$), the synchronized $++$ coincidence-tick rate exhibits a ``temporal excess'' of ${\sim}13.6\%$ relative to Peres' isotropic bomb-fragment model. \rev{This does not constitute an absolute claim that quantum clocks ``run faster'' in the usual metrological sense; rather, it is an operational statement about the density of identified synchronized events under a specified convention.}

This deviation is a direct signature of contextuality. Unlike the classical model, the quantum system is not constrained to maintain consistency with unperformed measurements. The violation of Bell-type inequalities thus serves a dual purpose: it certifies the privacy of the time stream against a middleman adversary, and it confirms that the \rev{multi-setting} correlation structure cannot be embedded into any single local-realistic model. \rev{In this sense, the obtuse-angle ``excess'' lobe and the CHSH violation are two views of the same nonclassical resource.}

\revA{We close by returning to the conceptual core. The time defined by the entangled clock is \emph{relational} in a precise sense: because the singlet state encodes no definite properties of either subsystem individually, the only meaningful temporal structure is contained in the joint record---the coincidence-tick stream that emerges when Alice's and Bob's local records are brought together. This relational character has no single-party analogue and no classical counterpart that could simultaneously reproduce the full angular dependence of the coincidence rate. In the multi-setting (Bell-violating) regime, this relational time furthermore acquires a device-independent privacy guarantee: the joint tick record is certified to be unpredictable to any adversary who lacks foreknowledge of the measurement settings. The certified private time is thus irreducibly relational---it exists only in the ``between,'' not in either party alone.}

\revG{Finally, we contrast this construction with practical atomic clocks. Atomic clocks are engineered oscillators stabilized by coherent interrogation and feedback, and their performance is judged by frequency stability/accuracy metrics. The present entangled-clock protocol is different in kind: it produces an event-based time-stamp record whose \emph{joint} coincidence structure can exceed a natural isotropic LHV benchmark at certain angles, and whose \emph{privacy and unpredictability} can be device-independently certified in the multi-setting (Bell-violating) regime.}


\begin{acknowledgments}
This text was partially created and revised with assistance from large language models. All content, ideas, and prompts were provided by the author.

The entangled clock metaphor was suggested by Adrienne Toman during a hike to the Michelberg.

This research was funded in whole or in part by the Austrian Science Fund (FWF), grant number DOI: 10.55776/PIN5424624. The author acknowledges TU Wien Bibliothek for financial support through its Open Access Funding Programme.

\textbf{Data Availability:} No new data were created or analyzed in this study. Data sharing is not applicable to this article.

\textbf{Conflicts of Interest:} The author declares no conflict of interest.

\end{acknowledgments}

\bibliography{svozil}

\end{document}
\begin{thebibliography}{999}

\bibitem[Duff et~al.(2002)Duff, Okun, and Veneziano]{DuffOkunVeneziano}
Duff, M.J.; Okun, L.B.; Veneziano, G.
\newblock Trialogue on the number of fundamental constants.
\newblock {\em Journal of High Energy Physics} {\bf 2002}, {\em 2002},~023,
  \href{http://arxiv.org/abs/arXiv:physics/0110060}{{\normalfont
  [arXiv:physics/0110060]}}.
\newblock {\url{https://doi.org/10.1088/1126-6708/2002/03/023}}.

\bibitem[Matsas et~al.(2024)Matsas, Pleitez, Saa, and Vanzella]{matsas-2024}
Matsas, G.E.A.; Pleitez, V.; Saa, A.; Vanzella, D.A.T.
\newblock The number of fundamental constants from a spacetime-based
  perspective.
\newblock {\em Scientific Reports} {\bf 2024}, {\em 14},~22594,
  \href{http://arxiv.org/abs/arXiv:2311.09249}{{\normalfont
  [arXiv:2311.09249]}}.
\newblock {\url{https://doi.org/10.1038/s41598-024-71907-0}}.

\bibitem[Alexandrov(1975)]{alex3}
Alexandrov, A.D.
\newblock Mappings of Spaces with Families of Cones and Space-Time
  Transformations.
\newblock {\em Annali di Matematica Pura ed Applicata} {\bf 1975}, {\em
  103},~229--257.
\newblock {\url{https://doi.org/10.1007/BF02414157}}.

\bibitem[Zeeman(1964)]{zeeman}
Zeeman, E.C.
\newblock Causality implies the {L}orentz group.
\newblock {\em Journal of Mathematical Physics} {\bf 1964}, {\em 5},~490--493.
\newblock {\url{https://doi.org/10.1063/1.1704140}}.

\bibitem[Svozil(2025)]{svozil-2025-lt}
Svozil, K.
\newblock Causal Rigidity and the Single-Unit Universe: {I}ntegrating the
  {A}lexandrov-{Z}eeman and {U}nruh Clock Scales,  2025,
  \href{http://arxiv.org/abs/arXiv:2511.20728}{{\normalfont
  [arXiv:2511.20728]}}.
\newblock {\url{https://doi.org/10.48550/arXiv.2511.20728}}.

\bibitem[Einstein(1905)]{ein-05}
Einstein, A.
\newblock {Z}ur {E}lektrodynamik bewegter {K}{\"{o}}rper.
\newblock {\em Annalen der Physik} {\bf 1905}, {\em 322},~891--921.
\newblock {\url{https://doi.org/10.1002/andp.19053221004}}.

\bibitem[Zeilinger(1999)]{zeil-99}
Zeilinger, A.
\newblock A Foundational Principle for Quantum Mechanics.
\newblock {\em Foundations of Physics} {\bf 1999}, {\em 29},~631--643.
\newblock {\url{https://doi.org/10.1023/A:1018820410908}}.

\bibitem[Peres(1978)]{peres222}
Peres, A.
\newblock Unperformed experiments have no results.
\newblock {\em American Journal of Physics} {\bf 1978}, {\em 46},~745--747.
\newblock {\url{https://doi.org/10.1119/1.11393}}.

\bibitem[Rovelli(1996)]{Rovelli1996}
Rovelli, C.
\newblock Relational Quantum Mechanics.
\newblock {\em International Journal of Theoretical Physics} {\bf 1996}, {\em
  35},~1637--1678,  \href{http://arxiv.org/abs/quant-ph/9609002}{{\normalfont
  [quant-ph/9609002]}}.
\newblock {\url{https://doi.org/10.1007/BF02302261}}.

\bibitem[Salecker and Wigner(1958)]{SaleckerWigner1958}
Salecker, H.; Wigner, E.P.
\newblock Quantum Limitations of the Measurement of Space-Time Distances.
\newblock {\em Physical Review} {\bf 1958}, {\em 109},~571--577.
\newblock {\url{https://doi.org/10.1103/PhysRev.109.571}}.

\bibitem[Busch(2008)]{Busch2008Time}
Busch, P.
\newblock The Time--Energy Uncertainty Relation. In {\em Time in Quantum
  Mechanics}; Muga, J.G.; Ruschhaupt, A.; del Campo, A., Eds.; Springer:
  Berlin, Heidelberg,  2008; Vol. 734, {\em Lecture Notes in Physics}, pp.
  73--105.
\newblock {\url{https://doi.org/10.1007/978-3-540-73473-4_3}}.

\bibitem[Giovannetti et~al.(2004)Giovannetti, Lloyd, and
  Maccone]{Giovannetti2004}
Giovannetti, V.; Lloyd, S.; Maccone, L.
\newblock Quantum-Enhanced Measurements: Beating the Standard Quantum Limit.
\newblock {\em Science} {\bf 2004}, {\em 306},~1330--1336.
\newblock {\url{https://doi.org/10.1126/science.1104149}}.

\bibitem[Moreva et~al.(2014)Moreva, Brida, Gramegna, Giovannetti, Maccone, and
  Genovese]{Moreva2014}
Moreva, E.; Brida, G.; Gramegna, M.; Giovannetti, V.; Maccone, L.; Genovese, M.
\newblock Time from quantum entanglement: an experimental illustration.
\newblock {\em Physical Review A} {\bf 2014}, {\em 89},~052122.
\newblock {\url{https://doi.org/10.1103/PhysRevA.89.052122}}.

\bibitem[Schlosshauer(2007)]{Schlosshauer2007}
Schlosshauer, M.
\newblock {\em Decoherence and the Quantum-to-Classical Transition}; Springer:
  Berlin,  2007.
\newblock {\url{https://doi.org/10.1007/978-3-540-35775-9}}.

\bibitem[Madjid and Myers(2020)]{MadjidMyers2020}
Madjid, F.H.; Myers, J.M.
\newblock Clocks without ``Time'' in Entangled-State Experiments.
\newblock {\em Entropy} {\bf 2020}, {\em 22},~434.
\newblock {\url{https://doi.org/10.3390/e22040434}}.

\bibitem[Froissart(1981)]{froissart-81}
Froissart, M.
\newblock Constructive generalization of {B}ell's inequalities.
\newblock {\em Il Nuovo Cimento B (11, 1971-1996)} {\bf 1981}, {\em
  64},~241--251.
\newblock {\url{https://doi.org/10.1007/BF02903286}}.

\bibitem[Pitowsky(1986)]{pitowsky-86}
Pitowsky, I.
\newblock The range of quantum probability.
\newblock {\em Journal of Mathematical Physics} {\bf 1986}, {\em
  27},~1556--1565.
\newblock {\url{https://doi.org/10.1063/1.527066}}.

\bibitem[Svozil(2024)]{svozil-2022-epr}
Svozil, K.
\newblock On the Complete Description Of Entangled Systems Part {I} ---
  {E}xploring Hidden Variables and Context Communication Cost in Simulating
  Quantum Correlations.
\newblock {\em International Journal of Theoretical Physics} {\bf 2024}, {\em
  63},~1--18,  \href{http://arxiv.org/abs/arXiv:2209.09590}{{\normalfont
  [arXiv:2209.09590]}}.
\newblock {\url{https://doi.org/10.1007/s10773-023-05544-0}}.

\bibitem[Aerts(1986)]{aerts-69}
Aerts, D.
\newblock A possible explanation for the probabilities of quantum mechanics.
\newblock {\em Journal of Mathematical Physics} {\bf 1986}, {\em 27},~202--210.
\newblock {\url{https://doi.org/10.1063/1.527362}}.

\bibitem[Clauser et~al.(1969)Clauser, Horne, Shimony, and Holt]{chsh}
Clauser, J.F.; Horne, M.A.; Shimony, A.; Holt, R.A.
\newblock Proposed Experiment to Test Local Hidden-Variable Theories.
\newblock {\em Physical Review Letters} {\bf 1969}, {\em 23},~880--884.
\newblock {\url{https://doi.org/10.1103/PhysRevLett.23.880}}.

\bibitem[Ac{\'i}n and Masanes(2016)]{Acin2016CertifiedRandomness}
Ac{\'i}n, A.; Masanes, L.
\newblock Certified randomness in quantum physics.
\newblock {\em Nature} {\bf 2016}, {\em 540},~213--219.
\newblock {\url{https://doi.org/10.1038/nature20119}}.

\bibitem[Shimony(1993)]{shimony_1993}
Shimony, A.
\newblock Controllable and uncontrollable non-locality. In {\em The Search for
  a Naturalistic World View. {V}olume {II}}; Shimony, A., Ed.; Cambridge
  University Press: Cambridge, England, UK,  1993; Vol.~2, pp. 130--139.
\newblock {\url{https://doi.org/10.1017/CBO9781139172196.010}}.

\end{thebibliography}


expose

tentativer title: quantum clocks tick faster

* intro section on recent developments (input manuscript I)

* prime section: compare two readings of an ``entangled clock'' on different spatial locations.  Singlet state of two photos or electrons.

Whenever the outcome records an ``up'' click, the clock adds ``1'' to the time counter.

** readdoff: number of ``up'' readings determines time on each side. ``synchronicity'' is depending on the relative

** consider perfect opposite direction reading observables (detectors \pi apart): clocks will be in perfect synchrony.
** consider perfect same direction reading observables (detectors 0 apart): clocks will be in perfect anti-synchrony.
** consider perfect unbiuased direction reading observables (detectors \pi/2 apart): clocks will be in perfect non-correllation.
** in all those cases single events and outcomes on either one side will occur totally random, as entanglement is defined vie relational encoding (cf. Zeilinger's foundational principle, paper enclosed)

in the there ``special'' cases this is also possible classically; cf. Peres' bomb explosion example.

*** now consider relative measurement angles not equal to the ``special cases'' 0, Pi/2 and \pi, but, say, somewhere in-between \pi/2 and \pi; ideally at an angle that maximises  the absolute difference between quantum and classical (linear) correllation) for a Bell basis singlet state---please compute.

** in such a case the clocks on both sides will record ``more'' clicks than ``usual classical clocks clocks'', because of the difference between classical and quantum correllations.
Same could be done with a reagin between relative angle 0 and \pi/2, only the time measures is less that the classical time recorden.

New section: not so fast! However for SINGLE unique relative measurement angle measurement the cosine correlation function can be reproduced classically---cf Aerts' rubber band model. But not for 4 direwctions and 4 measurement contexts, as is the case for the CHSH ineaquality.

So only if more contexts are taken into account, Bell-type inequalities could assure ``quantumness'' of  the clock readings.

###########################################################################################################################################

RR 1:

This is a very original and very engaging article, well written and clear. Indeed, I suggest that it be considered for particular attention upon publication. I suggest some additional clarifications, hence minor revisions, that would increase the impact of this paper.

Although the paper suggests operational definitions for the exhibition of time flow by the quantum "clocks" it describes, it would be helpful to provide a more complete characterization of these operations. Measurements are essential to the functioning of this method, but exactly how a measurement takes place is not dealt with in much detail---the quantum measurement problem looms just beyond what is discussed.

This seems to me significant, because the best clocks we currently use in practice, the atomic clocks, involve measurements of frequencies within quantum systems but the operation of the clock involves time indications via measurements on large electromagnetic cavities and doesn't obviously involve an irreversible measurement. Thus, the quantum measurement problem is avoided.  More fully explicating the relation between the quantum state and the measurement example described in the paper---avalanche photodiode clicks (a destructive measurement, which makes one wonder about the closed-ness of the system and its ability to operate for more than one click as described so far)---would be very helpful to ensure the system being a true time-exhibiting clock above an beyond the metaphorical: There is mention in the acknowledgments  of an "entangled clock metaphor." So, it would be good to clarify to what extent the paper deals with a true fundamental definition of quantum clock, which the abstract and introduction suggest is the case, or a toy model or a metaphorical clock.

This paper is, to me, very nice extension of what Bell-inequality violating correlations can be considered to mean. So it would be very helpful to clarify the above points before publication.

It would be good also if the author could comment briefly on the relation of the method considered in this paper to the concerns and methods of following paper on the subject of quantum clocks and entanglement published previously in Entropy the delves also more explicitly into relativity; this might be made in the introduction as an aid to readers of Entropy: https://www.mdpi.com/1099-4300/22/4/434

RR2:

The manuscript proposes an “entangled clock” where local time is defined operationally by discrete detection events (ticks) generated from measurements on a singlet state. You compare the synchronized tick (++) coincidence rate (R_{\mathrm{QM}}(\theta) = P_{++}(\vec a,\vec b)) with a classical benchmark derived from Peres’ bomb-fragment model, (R_{\mathrm{cl}}(\theta)=\theta/(2\pi)). You show that for obtuse relative analyzer angles, (R_{\mathrm{QM}}(\theta)) exceeds the linear benchmark, with a maximum excess around (\theta\approx 140.5^\circ) (about 13–14% relative increase). You then interpret this as a “quantum clock ticking faster than any classical clock,” and build on Bell/CHSH ideas to propose the notion of “Certified Private Time,” in analogy with device-independent randomness generation.

The work is original and nicely written, with a clear operational construction and closed-form rates that are easy to understand and reproduce. The narrative—recasting Bell nonclassicality as a story about clock ticks and time—is appealing and suitable for Entropy. However, the strongest claims (“faster than any classical clock” and “certified private time”) currently overreach the formal content of the analysis and need to be scoped more carefully. With a clearer definition of the classical benchmark, a sharper logical separation between single-angle “speedup” and multi-setting Bell certification, and a more precise formulation of what is meant by “private time,” I believe the paper can become a solid and publishable contribution.



i) The headline claim, that the quantum clock “ticks faster than any classical clock”—is catchy but, as written, a bit too loose. Throughout the text, “ticking faster” is defined in terms of the synchronized +1,+1 coincidence probability R(?)=P++?(a,b) for a chosen convention where only +1 is counted as a tick. It would help if you state very explicitly, early on, that:

The local tick rate is fixed at ~N/2N/2N/2 for both quantum and classical models (because of the unbiased marginals).

The comparison concerns only the coincidence rate in the +1,+1 channel under your operational definition of a “tick.”

You may want to add a short discussion on the conventionality of this choice: with a different convention (that is, counting both +1 and -1 on each side as ticks, or defining a tick when either side fires rather than both), the “speedup” could be reframed or disappear. The conceptual message is still valid, but this will prevent readers from over-interpreting the phrase “faster clock” as something absolute or metrologically superior in the usual sense of precision or stability.

Right now the text gestures at this (that is, Section 4) but I think a few sentences explicitly disentangling “clock rate,” “coincidence channel,” and “Bell contextuality” would significantly strengthen the paper and avoid misunderstandings.



ii) I do not think you need a long literature survey, but one short paragraph acknowledging that there is a substantial body of work on “quantum clocks” and explaining how your contribution is different in focus (operational, Bell-based, and centered on synchronization rate as an informational quantity rather than on optimal phase estimation or time-energy uncertainty) would situate the paper more clearly.



iii)  Using Peres’ bomb fragment model as a classical counterpart is elegant and pedagogical, but the way it is presented risks suggesting that the linear correlation function is somehow the unique or canonical classical benchmark. As You already note (Sec. 5.1) that for any fixed angle one can engineer a contextual classical model (that is, Aerts’ broken elastic band) to reproduce the quantum statistics. It might be worth explicitly stating that:

a) Peres’ model is used as a natural, isotropic, local hidden-variable model that obeys standard Boole–Bell constraints.

b) The central point is not that this particular classical model fails, but that any single local realistic model reproducing all angles simultaneously must live inside the classical correlation polytope and hence exhibit linear-type constraints that your cosine violates.

A couple of sentences making this “polytope vs. single-angle fit” distinction more explicit would further connect the synchronization-rate story to the usual convex-geometry intuition behind Bell inequalities.



iv)Section 5.1 and 5.2 already qualitatively connect the “synchronization excess” ?(?)\Delta(\theta)?(?) to Bell-type inequalities and the CHSH parameter SSS. However, the current text remains somewhat verbal. You might consider: Giving one explicit choice of four angles (a,a',b,b') that both yield a sizeable positive lobe of ?(?)\Delta(\theta)?(?) in the obtuse range, and Produce a CHSH violation S>2. Or, at least, add a sentence that explicitly states: “The same cosine-vs-linear mismatch that produces a positive lobe of ?(?)\Delta(\theta)?(?) in Fig. 2 is what underlies the familiar CHSH violation: for the standard choice of angles …, the quantum correlation function yields S = 2\sqrt{2}S while any local realistic model compatible with Peres’ linear dependence satisfies S=2.”

This would help readers see that your faster ticking is not an independent phenomenon but a re-expression of contextuality/Bell violation in the time-synchronization language, which I think is one of the conceptual strengths of the paper.



v) The Certified Private Time section should clarify that what is really being certified is the unpredictability of the tick sequence relative to any side information not correlated with future settings, just as in device-independent QRNG. You should distinguish clearly between a cryptographic interpretation (a time-stamped record whose ordering cannot be reconstructed by an adversary without the settings) and a metrological one (no claim of improved Allan deviation or phase precision). Emphasize that the novelty is informational/device-independent certification of the time stamps, rather than a superior physical clock.
Finally, state explicitly that the needed assumption of measurement (parameter) independence is exactly the same as in standard DI-QRNG and QKD, so you are not requiring anything stronger.



vi) You could also clarify a few secondary points to improve readability.
First, briefly explain why a “tick” is defined only for outcome (+1) (that is it keeps marginals unbiased and makes the benchmark simple), and note that local tick rates are similar in quantum and classical cases—the difference is in ++ coincidences.
Second, when quoting the “13.6%” excess at (\theta \approx 140.5^\circ), say explicitly that it is ((R_{\text{QM}}-R_{\text{cl}})/R_{\text{cl}} \approx 0.136) (or specify your chosen convention).
Finally, add a short paragraph placing your work within existing research on quantum clocks/time observables and device-independent randomness, making clear that your contribution is a new operational reframing of Bell nonclassicality in terms of time-stamping and synchronization, not a competing precision clock.



in its current form, I would recommend acceptance after a substantial but focused revision. The core construction and calculations are sound and interesting, but the strongest claims need to be tightened and aligned with what is actually proven. With the mnetioned changes, I believe the manuscript will offer a well-grounded and engaging contribution to the foundations and information-theory readership of Entropy.


####################################################



Response to Referee Report 1

The author thanks the referee for the generous assessment of the manuscript and for the thoughtful suggestions, which have materially improved the paper. Below, each point is addressed in detail, indicating where the corresponding revisions appear. All changes responding to this report are marked in dark green (`\revG`) in the revised manuscript.

1. The quantum measurement problem

Concern regarding measurement details: The referee noted that while measurements are essential to the method, the text did not treat the quantum measurement problem in detail.

Response and revision: A new paragraph has been added to the Introduction (Page 2, dark green) that explicitly addresses this point. The protocol is formulated at the standard operational level of quantum theory (setting choice $\to$ binary outcome $\to$ classical time tag). The text clarifies that there is no attempt to resolve the measurement problem (collapse, many-worlds, objective collapse, etc.). Instead, the experimentally standard viewpoint is adopted: a "click" constitutes a stable macroscopic record, well modeled by a quantum instrument with an effectively irreversible amplification chain. Schlosshauer's decoherence monograph [Ref. 14] is cited as an anchor. It is clarified that the conceptual novelty of the paper is a reframing of Bell-nonclassicality in the language of time-stamping, not a new measurement theory.

2. Destructive measurements, closed-ness, and repeated ticking

Concern regarding system closed-ness: The referee raised a question regarding avalanche photodiode clicks being destructive measurements, and how this affects the closed-ness of the system and its ability to operate for more than one click.

Response and revision: A new subsection (Sec. 2.3, "Measurement operations, destructiveness, and 'clockwork' repeatability," entirely in dark green) has been added. The key clarification is that the entangled clock is not a single closed two-particle system reused indefinitely; it is a repeatable preparation–measurement procedure fed by a stream of newly generated pairs (e.g., Spontaneous Parametric Down-Conversion). Repeatability resides in the ability to reproduce the preparation and measurement across many trials, not in storing energy in a single oscillator. Analogies are drawn to other event-based clocks (Geiger-counter clocks, pulsed tick generators). The text also notes that nondestructive or QND variants are conceivable but not required for the core conceptual point.

Additionally, a paragraph in the Experimental Considerations section (Sec. 6, dark green) makes explicit that the relevant "system" is intrinsically open (source + channels + detectors + classical electronics), placing the construction closer to the operational practice of time-tagged entanglement experiments than to autonomous oscillator clocks.

3. True clock vs. metaphor

Concern regarding the term "clock": The referee requested clarification on whether the paper deals with a true fundamental definition of a quantum clock or a metaphorical one, noting the acknowledgement's reference to a "metaphor."

Response and revision: A paragraph has been added to the Introduction (Page 2, dark green) addressing this. It states that "clock" is used in an event-based operational sense: a clock is any reproducible physical procedure that outputs a time-ordered record of ticks. The entangled clock is closer to a stochastic "tick generator" (a point process with correlations) than to a self-contained harmonic oscillator. No claim is made that it competes with state-of-the-art atomic clocks in precision or stability; rather, it provides an informational synchronization primitive and (in the multi-setting regime) a device-independent privacy and unpredictability certificate. The word "metaphor" in the Acknowledgments refers to the initial conversational spark; the paper itself develops the concept into a concrete operational protocol.

Furthermore, the concluding paragraph (Page 13, dark green) explicitly contrasts this construction with practical atomic clocks, which are engineered oscillators judged by frequency stability/accuracy metrics, making clear that the entangled-clock protocol is different in kind.

4. Relation to Madjid and Myers (2020), Entropy 22, 434

Request for comparison: The referee suggested commenting on the relation of the method to the work of Madjid & Myers [Entropy 22, 434 (2020)].

Response and revision: A dedicated paragraph has been added to the Introduction (Page 2, bottom, dark green) discussing this paper. It is noted that Madjid and Myers emphasize how clocks are used in entanglement experiments—via time tags, record exchange, and post-processing—and discuss synchronization concepts without requiring a global time parameter. The present paper is described as complementary: it focuses on a specific operational "tick" convention and studies the resulting coincidence-tick rate as an informational quantity; moreover, it connects the multi-setting regime explicitly to Bell [e.g., Clauser-Horne-Shimony-Holt (CHSH)] certification. The paper is now cited as [Ref. 15].

5. Additional author-initiated refinements

In the course of revision, the following changes I also made:

* I introduced the term "relational time" (marked in purple, `\revA`) at key conceptual junctures (title, abstract, introduction, conclusion) to emphasize that the entangled clock's temporal content is irreducibly joint/bipartite. A brief disclaimer distinguishes this usage from Rovelli's specific "relational quantum mechanics" program [Ref. 9].
* I corrected the CHSH calculation in Sec. 5.2: the original choice $b'=-\pi/4$ produced $S=0$, not $2\sqrt{2}$. It was replaced with $b'=3\pi/4$, which yields $S=2\sqrt{2}$ and preserves the obtuse angle $135^\circ$ in the positive lobe of $\Delta(\theta)$.




Response to Referee Report 2

The author thanks the referee for the detailed and constructive report. The suggestions have substantially sharpened the paper's claims, improved its internal logic, and strengthened the connection between the single-angle "speedup" narrative and the multi-setting Bell certification. Below, each numbered point is addressed. All changes responding to this report are marked in blue (`\rev`) in the revised manuscript.

Point (i): Local rate fixed; coincidence channel only; conventionality

Concern: The referee suggested stating explicitly that the local tick rate is fixed at $N/2$ for both quantum and classical models, and that the comparison concerns only the coincidence rate in the $+1,+1$ channel.

Revisions made (all in blue):

1.  Abstract: Now states explicitly that "locally, each party's tick rate is fixed by the unbiased marginals" and that "this 'faster ticking' refers to the rate of identified coincidence ticks under a specific operational convention, not to an improved local clock rate, precision, or stability."
2.  Sec. 2.1 (Operational Time): A new paragraph (Page 3) explains that counting only $+1$ as a tick is a convention for definiteness and algebraic simplicity; counting both outcomes would remove any "speed" language at the single-stream level; the nonclassical content resides in the joint (coincidence) structure.
3.  Sec. 2.1, after Eq. (3): A sentence states: "In particular, the local tick rate is fixed at $1/2$ per trial for both the quantum singlet and for unbiased classical models; the comparison in this paper concerns coincidence ticks (below), not local rates."
4.  Sec. 4 opening: A paragraph reiterates that any "faster ticking" language refers only to the identified coincidence-tick rate $R(\theta)=P_{++}(\vec a,\vec b)$.
5.  Sec. 4.2, after Fig. 1: The disclaimer is repeated: "We emphasize again: this is a statement about the $++$ coincidence channel under an agreed operational convention, not about an absolute increase of local ticking."

Point (ii): Literature positioning

Concern: The referee requested an acknowledgement of the body of work on "quantum clocks" to situate the paper's focus.

Revision (blue): A new paragraph in the Introduction (Page 2, paragraph 4) cites Salecker & Wigner (1958), Busch (2008), Giovannetti et al. (2004), and Moreva et al. (2014) as representative of the existing literature on quantum clocks, time observables, and quantum-enhanced metrology. It explains that the present contribution differs in emphasis: rather than optimizing precision of time/phase estimation, the focus is on an operational, Bell-theoretic notion of a distributed clock whose figure of merit is an informational synchronization rate. These are Refs. [10]–[13].

Point (iii): Peres model—not unique; polytope vs. single-angle fit

Concern: The referee noted that the Peres model is just one example, and that the central point is that any single local realistic model reproducing all angles must live inside the classical correlation polytope.

Revisions (blue):

1.  Sec. 3, opening paragraph: A new paragraph states explicitly that Peres' model is not claimed to be the unique classical comparator; it is used as a natural, isotropic LHV construction with unbiased marginals. For any single fixed angle, one may engineer contextual classical mechanisms to reproduce $P_{++}$. The discriminating content arises when one demands a single LHV model reproducing all settings simultaneously.
2.  Sec. 5, new paragraph: The polytope language is made explicit: "the set of correlation families compatible with a single LHV model forms a convex polytope in the space of multi-setting correlations... A one-angle agreement is easy to engineer; what is constrained is the simultaneous reproduction of several angles by one underlying joint distribution."

Point (iv): Explicit CHSH angles and connection to the obtuse excess

Concern: The referee requested one explicit choice of four angles yielding a sizeable positive lobe of $\Delta(\theta)$ in the obtuse range while producing a CHSH violation $S>2$.

Revision (blue): New subsection 5.2, "Explicit CHSH violation from the obtuse-angle excess." The choice is:
$$a=0,\quad a'=\pi/2,\quad b=\pi/4,\quad b'=3\pi/4.$$
The text computes all four correlations $E_{QM}(x,y)=-\cos(x-y)$ explicitly, yielding $S_{QM}=2\sqrt{2}>2$. It is noted that this choice includes the obtuse relative angle $|a-b'|=3\pi/4=135^\circ$, which lies in the positive lobe of $\Delta(\theta)$, stating: "Thus, the same cosine-vs-linear mismatch that yields an excess $++$ coincidence-tick rate at obtuse angles also underlies the familiar CHSH violation."

Point (v): Certified Private Time—unpredictability, crypto vs. metro, measurement independence

Concern: The referee requested clarification that what is being certified is unpredictability (cryptographic interpretation) rather than metrological precision, and that the measurement independence assumption is standard.

Revisions (blue):

1.  Sec. 7, opening: A sentence now states: "what is certified is the unpredictability (relative to side information) of the ordered sequence of identified coincidence ticks, under the standard device-independent assumptions (in particular, no adversarial foreknowledge of future settings)."
2.  Sec. 7.2, item 2 ("Private"): Rewritten to specify that the coincidence-tick record is unpredictable to any adversary whose side information is not correlated with future setting choices, "in the same device-independent sense as certified randomness."
3.  New Sec. 7.3, "Cryptographic vs. metrological meaning of 'private time'": Explicitly distinguishes the cryptographic and informational meaning (unpredictability of the fine-grained time-stamp record) from metrological claims (phase resolution, timing jitter, improved Allan deviation).
4.  Sec. 7.4 (Escaping the Block Universe): A sentence states: "This is the same assumption used in standard device-independent QRNG and QKD; no stronger assumption is invoked here."

Point (vi): Secondary clarifications

Responses to specific sub-points:

1.  (vi-a) Why tick = +1: Addressed in Sec. 2.1 (blue paragraph on conventionality).
2.  (vi-b) 13.6% explicit formula: A displayed equation is now given in Sec. 4.2:
    $$\frac{R_{QM}-R_{cl}}{R_{cl}}\bigg|_{\theta\approx 140.5^\circ}\approx \frac{0.443-0.390}{0.390}\approx 0.136.$$
3.  (vi-c) Positioning paragraph: Addressed via the new Introduction paragraph on quantum-clock literature (see Point (ii) above).

Additional author-initiated refinements

In the course of revision, the following changes were also made (marked in purple, `\revA`):

Title: Changed from "Certified Private Coincidence Time from Entanglement" to "Certified Private Relational Time from Entanglement." The term "relational" captures the physical content more precisely. A disclaimer in the Introduction distinguishes this usage from Rovelli's specific program [Ref. 9].

Abstract, Introduction, Sec. 2.1, Sec. 2.2, Conclusion: Strategic placements of the "relational" framing, bridged explicitly to the operational "coincidence-tick" language.
