04-ilang-time / 04-clock-comparison
04clock comparisonverified
Determines the joint readings of two clocks, their rate ratio through readings, their agreement and synchronization, and how a contract with an environment changes a clock's rate.
# Question: 04-clock-comparison
- **Subproject:** 04-ilang-time
- **Package:** 04-clock-comparison
- **Equation tags:** (4.k)
- **Created:** 2026-10-10
## Goal
Determine how the readings of two clocks compare:
- their joint statistics;
- the ratio of their rates, expressed through readings;
- when two clocks agree, and how readings can synchronize them;
- how a contract with an environment changes the rate of a clock, for one clock and for two clocks of different types.
This is criterion 3. Time as a relation between a clock and the rest of a system is criterion 2 and not part of this package.
**Setting.**
- **Objects.** Two clock objects $K_1$ and $K_2$, each a part of its own; the rest of the system; and, in item 3, an environment part $E$. No two objects of the system have the same type, so every partition of joint places is an admissible reading.
- **Clocks.** $K_i$ has $N_i\ge2$ places $m_i\in\mathbb Z_{N_i}$, a tick $\lambda_{0i}>0$, and the contract
$$C_{K_i}=\frac{2\pi}{N_i\lambda_{0i}}\sum_{k=0}^{N_i-1}k\,\lvert f^{(i)}_k\rangle\langle f^{(i)}_k\rvert,\qquad \lvert f^{(i)}_k\rangle=\frac1{\sqrt{N_i}}\sum_{m\in\mathbb Z_{N_i}}e^{2\pi ikm/N_i}\lvert m\rangle .$$
By (2.9) and (2.12) of 02-clock, quoted under "Inputs", each is a uniform ring clock with $\theta_m=0$ and minimal cost spread (the case $\alpha_m=0$, $\varepsilon=0$, $\ell_k=k$); call such a clock a **minimal ring clock**. Its reading is the place reading.
- **Free clocks (items 1–2).** No contract term acts across $K_1$, $K_2$ and the rest, so each clock is self-driven by (2.1). The start is $\lvert0\rangle\otimes\lvert0\rangle\otimes\chi$, with $\chi$ a state of the rest.
- **Joint reading.** The place reading on $K_1\cup K_2$, with one class per joint place $(m_1,m_2)$, applied at a stated value of $\lambda$ (A8).
1. **Joint readings of two free clocks.**
- (a) Determine the joint distribution of $(m_1,m_2)$ at $\lambda$.
- (b) Determine the set of $\lambda>0$ at which both readings are certain, and the readings there. Determine a necessary and sufficient condition on $\lambda_{01}$ and $\lambda_{02}$ under which this set is nonempty.
- (c) At $\lambda=n\lambda_{01}$, $n\in\mathbb Z$, the reading of $K_1$ is certain. Determine the distribution of the reading of $K_2$ there, and its places of maximal probability, as functions of $n$, $\lambda_{01}/\lambda_{02}$ and $N_2$.
2. **Identical clocks.** Let $N_1=N_2=N$ and $\lambda_{01}=\lambda_{02}=\lambda_0$.
- (a) Determine the probability that the two readings coincide ($m_1=m_2$) as a function of $\lambda$, its minimum over $\lambda$, and the set of $\lambda$ at which it equals $1$.
- (b) A joint reading at $\lambda_s$ gives $(m_1,m_2)$. For a second joint reading at $\lambda>\lambda_s$, determine the joint distribution of its outcomes, and the probability that $m_2-m_1$ (mod $N$) is the same as at $\lambda_s$.
3. **A clock in an environment.** Let $C=C_{K_1}\otimes\mathbb 1_E+\mathbb 1_{K_1}\otimes C_E+A\otimes B$, with $A=A^\dagger$ on $\mathcal H_{K_1}$, $B=B^\dagger$ on $\mathcal H_E$ and $[B,C_E]=0$; the contracts of $K_2$ and of the rest are omitted. The start is $\lvert0\rangle\otimes\lvert b\rangle$, where $\lvert b\rangle$ is a common eigenvector of $B$ and $C_E$ with $B\lvert b\rangle=b\lvert b\rangle$.
- (a) Determine the view of $K_1$ for every $\lambda$. Determine whether $K_1$ is self-driven in the sense of 02-clock.
- (b) Let $\mathfrak B\subset\mathbb R$ have at least two elements. Determine all $A$ for which $C_{K_1}+bA$ is, for every $b\in\mathfrak B$, the contract of a minimal ring clock with $N_1$ places and some tick $\lambda_{01}(b)>0$. Determine $\lambda_{01}(b)$.
- (c) Let both clocks be coupled to the same environment: $C=C_{K_1}+C_{K_2}+C_E+A_1\otimes B+A_2\otimes B$, each term extended by the identity, with $A_i$ of the form found in (b) for $K_i$, and the start $\lvert0\rangle\otimes\lvert0\rangle\otimes\lvert b\rangle$. Determine a necessary and sufficient condition under which the ratio $\lambda_{01}(b)/\lambda_{02}(b)$ is the same for all $b\in\mathfrak B$.
## Inputs
From 02-clock@v2. Setting and notation fixed there:
- a part $K$ with companion $R$, and $C=C_K\otimes\mathbb 1_R+\mathbb 1_K\otimes C_R+C_\partial$ with $C_\partial=\sum_nA_n\otimes B_n$ the terms across the cut;
- $K$ is self-driven if $V_K(\lambda)=e^{-iC_K\lambda}V_K(0)e^{iC_K\lambda}$ for every start and every $\lambda$;
- a uniform ring clock is a single object with $N\ge2$ places $m\in\mathbb Z_N$, with $e^{-iC_K\lambda_0}\lvert m\rangle=e^{i\theta_m}\lvert m+1\rangle$ for every $m$, where $\lambda_0>0$ is its tick;
- $S\lvert m\rangle=\lvert m+1\rangle$, $\lvert f_k\rangle=N^{-1/2}\sum_me^{2\pi ikm/N}\lvert m\rangle$, $V_0:=V_K(0)$, and $\Delta C_K$ is the spread of $C_K$ in the view of $K$.
Eq. (2.1), a sufficient condition for $K$ to be self-driven:
$$
C_\partial=\mathbb 1_K\otimes B,\qquad B=B^\dagger\ \text{on}\ \mathcal H_R\quad(\text{in particular } C_\partial=0).
$$
Eq. (2.4), for a self-driven $K$ and a reading on $K$ with projectors $\Pi_r$:
$$
p(r;\lambda)=\operatorname{Tr}\bigl(\Pi_r\,e^{-iC_K\lambda}V_0\,e^{iC_K\lambda}\bigr),
$$
Eq. (2.9), "where the $\alpha_m$ and $\varepsilon$ are arbitrary reals. The phases are then $\theta_m=\alpha_{m+1}-\alpha_m-\varepsilon\lambda_0$ (mod $2\pi$, with $\alpha_N:=\alpha_0$). If $\theta_m=0$ for all $m$, then $e^{-iC_K\lambda_0}=S$. The same argument applied to $S$, whose eigenvalues $e^{-2\pi ik/N}$ on $\lvert f_k\rangle$ are distinct, gives exactly $C_K=\sum_k\frac{2\pi\ell_k}{N\lambda_0}\lvert f_k\rangle\langle f_k\rvert$ with $\ell_k\equiv k$."
$$
C_K=\sum_{k\in\mathbb Z_N}E_k\lvert v_k\rangle\langle v_k\rvert,\quad
\lvert v_k\rangle=\frac{1}{\sqrt N}\sum_{m}e^{2\pi ikm/N}e^{i\alpha_m}\lvert m\rangle,\quad
E_k=\varepsilon+\frac{2\pi\ell_k}{N\lambda_0},\quad \ell_k\in\mathbb Z,\ \ell_k\equiv k\ (\mathrm{mod}\ N),
$$
Eq. (2.12), for the start $\lvert0\rangle$:
$$
\min\Delta C_K=\frac{2\pi}{\lambda_0}\sqrt{\frac{N^2-1}{12N^2}},\quad\text{attained iff } \{\ell_k\}=\{c,c+1,\dots,c+N-1\},\ c\in\mathbb Z .
$$
Eq. (2.13), the place-reading distribution of a self-driven uniform ring clock of a minimal member, started at $\lvert0\rangle$:
$$
p(m;\lambda)=\frac{\sin^2(\pi\lambda/\lambda_0)}{N^2\sin^2\bigl(\pi(\lambda/\lambda_0-m)/N\bigr)}\qquad(\text{value }1\text{ where the denominator vanishes}).
$$
Eq. (2.14), under the condition that the state after a reading is admissible (it holds when no object of $R$ has the type of the clock object): "A place reading with outcome $m$ acts by $\Pi_m\otimes\mathbb 1_R$ (1.3). The partial-trace rules give the new view"
$$
V_K\ \longmapsto\ \frac{\Pi_mV_K\Pi_m}{\operatorname{Tr}(\Pi_mV_K)}=\lvert m\rangle\langle m\rvert .
$$
## Assumptions
- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5).
- Readings are applied at stated values of $\lambda$ (A8). Comparing readings taken at equal $\lambda$ uses the order of $\lambda$; whether that order is detectable is criterion 8 and not part of this package.
## Scope
- In scope:
- two minimal ring clocks, free (items 1–2) or coupled to one environment part through one contract each (item 3);
- environment states that are common eigenvectors of $B$ and $C_E$.
- Out of scope:
- time as a relation between a clock and the rest (criterion 2);
- environment states that are superpositions of eigenvectors of $B$;
- contracts between the two clocks, and clocks of other forms;
- moving clocks, inertia and maximal speed (criteria 4–6);
- any physical meaning of $\lambda$, $A$ or $B$ beyond the definitions (M1).
## Depth
- Item 1: (a) short argument; (b) and (c) derive.
- Item 2: (a) derive; (b) short argument.
- Item 3: (a) short argument; (b) derive; (c) short argument.
## Expected result
- Item 1: a closed form; a set, the readings there, and an equivalence; a closed form and the places of maximal probability.
- Item 2: a closed form, its minimum, and a set; a closed form and a probability.
- Item 3: a closed form and a yes/no answer; a characterization and a closed form; an equivalence.
Give every main result a tag $(4.k)$.
Consistency checks, at most three: for example $N_1=N_2=2$, $\lambda_{01}=\lambda_{02}$, and $b=0$.
## Code
None.