On Entropy Production of Repeated Quantum Measurements II. Examples

Abstract

We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation–dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory.

Appendix A Continued Fractions

In this appendix we prove Lemma A.2, which was used in the last section. To this end, we start with a brief summary of some properties of continued fractions (see for example [26, 55] for more details).

Let $(a_n)_{n\in {{\mathbb {N}}}}\subset {{\mathbb {Z}}}$ be such that $a_n\in {{\mathbb {N}}}^*$ for all $n\in {{\mathbb {N}}}^*$.³⁸ Define

$$\begin{aligned} {[}a_0; a_1]\mathrel {\mathop :}=a_0 + \frac{1}{a_1} , \quad [a_0; a_1,a_2] = a_0 + \frac{1}{a_1 + \frac{1}{a_2}}, \end{aligned}$$

and, more generally, for $i\in {{\mathbb {N}}}^*$,

$$\begin{aligned} {[}a_0; a_1, \dots , a_i]\mathrel {\mathop :}=a_0 + \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{\begin{array}{c} \displaystyle a_3+\\ \big . \end{array} ~ \ddots ~ \dfrac{1}{a_{i-1} + \dfrac{1}{a_i}}}}}. \end{aligned}$$

It is well known that the limit

$$\begin{aligned} {[}a_0;a_1, a_2, \dots ]\mathrel {\mathop :}=\lim _{i\rightarrow \infty } [a_0; a_1, \dots , a_i] \end{aligned}$$

exists and is irrational. There is a bijection between the sequences $(a_n)_{n\in {{\mathbb {N}}}}$ such that $a_n\in {{\mathbb {N}}}^*$ for all $n\in {{\mathbb {N}}}^*$ and the irrational numbers. Moreover, for each $\zeta \in {{\mathbb {R}}}\setminus {{\mathbb {Q}}}$, we have

$$\begin{aligned} \zeta =[a_0;a_1, a_2, \dots ], \end{aligned}$$

(A.1)

where

$$\begin{aligned} \zeta _0&\mathrel {\mathop :}=\zeta ,&\qquad a_0&\mathrel {\mathop :}=\lfloor \zeta _0 \rfloor ,\\ \zeta _{i+1}&\mathrel {\mathop :}=\frac{1}{\zeta _i-a_i},&\qquad a_{i+1}&\mathrel {\mathop :}=\lfloor \zeta _{i+1} \rfloor , \qquad i\in {{\mathbb {N}}}. \end{aligned}$$

The right-hand side of (A.1) is called the continued fraction expansion of $\zeta$, and this expansion is unique.

For $i\in {{\mathbb {N}}}^*$, let $p_i \in {{\mathbb {Z}}}$ and $q_i \in {{\mathbb {N}}}^*$ be such that the fraction

$$\begin{aligned} \frac{p_i}{q_i} = [a_0; a_1, \dots , a_i] \end{aligned}$$

(A.2)

is irreducible, and let $p_{-1} \mathrel {\mathop :}=1$, $q_{-1} \mathrel {\mathop :}=0$, $p_0 \mathrel {\mathop :}=a_0$ and $q_0 \mathrel {\mathop :}=1$. We then have for $i\in {{\mathbb {N}}}^*$,

$$\begin{aligned} \begin{bmatrix} p_i &{} p_{i-1}\\ q_i &{} q_{i-1} \end{bmatrix}&= \begin{bmatrix} p_{i-1} &{} p_{i-2}\\ q_{i-1} &{} q_{i-2} \end{bmatrix} \begin{bmatrix} a_i &{} 1 \\ 1&{} 0 \end{bmatrix}, \end{aligned}$$

(A.3)

and, in particular, $q_{i+1} > q_i$. It is also well known that if $\zeta$ is given by (A.1) (so that $\zeta = \lim _{i\rightarrow \infty } \frac{p_i}{q_i}$), then

$$\begin{aligned} \frac{p_{0}}{q_{0}}< \frac{p_{2}}{q_{2}}< \frac{p_{4}}{q_{4}}< \dots< \zeta< \dots< \frac{p_{5}}{q_{5}}<\frac{p_{3}}{q_{3}}< \frac{p_{1}}{q_{1}}, \end{aligned}$$

(A.4)

and

$$\begin{aligned} \frac{1}{2 q_{i+1}}\le \left| {q_i}\zeta - {p_i}\right| \le \frac{1}{q_{i+1}}, \qquad i\in {{\mathbb {N}}}. \end{aligned}$$

(A.5)

We recall here the best approximation property (see for example [26, Theorem 5.9]).

Lemma A.1

Fix $i\in {{\mathbb {N}}}$, and let $(p,q)\in ({{\mathbb {Z}}}\times [\![1, q_{i+1}]\!]) \setminus \{(p_{i}, q_{i}), (p_{i+1}, q_{i+1})\}$. Then

$$\begin{aligned} |\zeta q - p| > |\zeta q_{i}-p_{i}| . \end{aligned}$$

(A.6)

Proof

Let $x,y \in {{\mathbb {Z}}}$ be such that

$$\begin{aligned} \begin{bmatrix} p_{i+1} &{} p_{i}\\ q_{i+1}&{} q_{i} \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} p\\ q \end{bmatrix}. \end{aligned}$$

Such $x,y\in {{\mathbb {Z}}}$ exist, as the determinant of the matrix here is ${\pm } 1$ by (A.3). We consider two cases.

First, if $x=0$ then $(p,q) = (yp_i, yq_i)$. Clearly $y>0$, since $q, q_i > 0$, and in fact the condition $(p,q) \ne (p_i, q_i)$ implies that $y \ge 2$. The result is then obvious, since then $|\zeta q - p| \ge 2|\zeta q_{i}-p_{i}|$.

We now assume that $x\ne 0$. The condition

$$\begin{aligned} q=xq_{i+1}+yq_{i}\in [\![1, q_{i+1}]\!]\end{aligned}$$

(A.7)

implies that $xy<0$. Indeed, clearly (A.7) implies that $xy\le 0$, and if we had $y=0$, then by (A.7) we would find $x=1$, which contradicts the condition $(p,q)\ne (p_{i+1}, q_{i+1})$. This shows that $xy<0$.

Since also $(\zeta q_{i+1}-p_{i+1})(\zeta q_{i}-p_{i}) < 0$ by (A.4), we conclude that $x(\zeta q_{i+1}-p_{i+1})$ and $y(\zeta q_{i}-p_{i})$ have the same sign. But then,

$$\begin{aligned} |\zeta q - p|&= |x(\zeta q_{i+1}-p_{i+1})+y(\zeta q_{i}-p_{i})| \\&= |x||\zeta q_{i+1}-p_{i+1}|+|y||\zeta q_{i}-p_{i}| > |\zeta q_{i}-p_{i}|, \end{aligned}$$

which completes the proof. $\square$

Let $\ell (x)\mathrel {\mathop :}=\min _{p\in {{\mathbb {Z}}}} |x-p|$ as in Sect. 5. Let $i\in {{\mathbb {N}}}^*$. Since $q_{i+1}>q_i$, Lemma A.1 applies to the pairs $(p, q_i)$ for all $p \ne p_i$, from which we conclude that³⁹

$$\begin{aligned} \ell (\zeta q_i) =|\zeta q_i - {p_{i}}|. \end{aligned}$$

(A.8)

In addition, for all $i\in {{\mathbb {N}}}$, applying Lemma A.1 to all pairs $(p,q)\in ({{\mathbb {Z}}}\times [\![1, q_{i+1}-1]\!])\setminus \{(p_i, q_i)\}$ shows that

$$\begin{aligned} \ell (\zeta q)\ge \left| \zeta q_i - {p_{i}}\right| , \qquad q\in [\![1, q_{i+1}-1]\!]. \end{aligned}$$

(A.9)

Lemma A.2

Let $\psi :{{\mathbb {N}}}^*\rightarrow {]}0,1]$ be a decreasing function such that $\sup _{q\in {{\mathbb {N}}}^*}q\psi (q)<\infty$. Then, there exists a dense subset $U\subset {{\mathbb {R}}}$ such that for all $\zeta \in U$,

$$\begin{aligned} 0<\liminf _{q\rightarrow \infty }\frac{\ell (q\zeta )}{\psi (q)}<\infty . \end{aligned}$$

(A.10)

Proof

Fix any open interval $I \subset {{\mathbb {R}}}$ and fix $x \in I\setminus {{\mathbb {Q}}}$. Let $({\widehat{a}}_i)_{i\in {{\mathbb {N}}}}$ be such that $x=[{\widehat{a}}_0; {\widehat{a}}_1, {\widehat{a}}_2, \dots ]$, and let ${\widehat{p}}_i, {\widehat{q}}_i$ be as in (A.2) with $(a_i)$ replaced by $({\widehat{a}}_i)$. Then, since $x = \lim _{i\rightarrow \infty }\frac{\widehat{p}_i}{{\widehat{q}}_i}$, one can choose N large enough so that $\frac{{\widehat{p}}_{N-1}}{{\widehat{q}}_{N-1}} \in I$ and $\frac{\widehat{p}_N}{{\widehat{q}}_N} \in I$. We then consider $\zeta = [a_0; a_1, a_2, \dots ]$, where $a_i = {\widehat{a}}_i$ for $i\le N$, and $a_{i+1}=\min \{n\in {{\mathbb {N}}}\mid nq_i\psi (q_i)\ge 1\}$ for $i\ge N$. Then, since $p_i ={\widehat{p}}_i$ and $q_i ={\widehat{q}}_i$ for $i\le N$, and by (A.4), we obtain that $\zeta \in I$. It remains to show that $\zeta$ satisfies (A.10). For all $i\ge N$, we have by (A.8), (A.5) and (A.3) that

$$\begin{aligned} \ell (\zeta q_i) = \left| \zeta q_i - {p_{i}}\right| \le \frac{1}{q_{i+1}} \le \frac{1}{a_{i+1}q_i} \le \psi (q_i), \end{aligned}$$

so that the second inequality in (A.10) holds. Moreover, by (A.9), (A.5) and (A.3), we have for all $i\ge N$ and all $q\in [\![q_i, q_{i+1}-1]\!]$ that

$$\begin{aligned} \ell (\zeta q)&\ge \left| \zeta q_i - {p_{i}}\right| \ge \frac{1}{2 q_{i+1}} \ge \frac{1}{2(a_{i+1}+1)q_i} \ge \frac{1}{2(\frac{1}{q_i\psi (q_i)}+2)q_i}\\&= \frac{\psi (q_i)}{2(1+2 q_i \psi (q_i))} \ge C^{-1} \psi (q_i) \ge C^{-1} \psi (q), \end{aligned}$$

where $C = 2(1+2 \sup _{q\in {{\mathbb {N}}}}q\psi (q))$. This establishes the first inequality in (A.10), hence the proof is complete. $\square$