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.

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Notes

  1. 1.

    Many additional examples are considered in [13].

  2. 2.

    A density matrix \(\rho \) is a non-negative operator whose trace satisfies \({{\text {tr}}}(\rho )=1\).

  3. 3.

    Such an instantaneous direct measurement is, of course, an idealization, most real measurements are indirect and require some finite time to complete (see [41] for a concrete study).

  4. 4.

    See [70, Section V.3]. In modern parlance, this follows from the operator-convexity of the function \(x\mapsto x\log x\), see, e.g., [29].

  5. 5.

    For an integer T, \({{\mathcal {A}}}^T\) denotes the T-fold cartesian product of the set \({{\mathcal {A}}}\) with itself.

  6. 6.

    \({{\mathbb {N}}}\) denotes the set of natural integers including 0, and \({{\mathbb {N}}}^*\mathrel {\mathop :}={{\mathbb {N}}}\setminus \{0\}\). \({{\mathcal {A}}}^{{{\mathbb {N}}}^*}\) is the set of sequences \(\omega =(\omega _1,\omega _2,\ldots )\).

  7. 7.

    For any integers \( i\le j\) we set \([\![i,j]\!]=[i,j]\cap {{\mathbb {Z}}}\).

  8. 8.

    By convention, the cylinder with empty base is \(\Omega \).

  9. 9.

    see [2] for a general introduction to this topic.

  10. 10.

    \(\sigma _T\) is often called the log-likelihood ratio in the framework of hypothesis testing, and relative information random variable in information theory.

  11. 11.

    A sufficient condition for ergodicity is that the completely positive map \(\Phi =\sum _{a\in {{\mathcal {A}}}}\Phi _a\) is irreducible.

  12. 12.

    \(\partial ^{\mp }\) denotes the left/right derivative.

  13. 13.

    \(\text {int}(S)/\text {cl}(S)\) denotes the interior/closure of the set S.

  14. 14.

    We recall that I is a rate function if it is non-negative, lower semicontinuous, and not everywhere infinite. We call I a good rate function if, in addition, it has compact level sets.

  15. 15.

    Note that the formula given in [21] is different, but coincides with this one since \(e(\alpha ) = e(1-\alpha )\). The expression given here is convenient in view of the generalizations we discuss at the end of this section.

  16. 16.

    There are pairs of instruments such that \((\Omega ,{\mathbb {P}},\phi )\) is ergodic and \(\text {ep}({\mathbb {P}},{\widehat{{\mathbb {P}}}})=0\), yet \({\mathbb {P}}\ne {\widehat{{\mathbb {P}}}}\) and \(\text {ep}({\widehat{{\mathbb {P}}}},{\mathbb {P}})>0\), even when \({\text {supp}}\,{\mathbb {P}}_T = {\text {supp}}\,{\widehat{{\mathbb {P}}}}_T\) for all \(T\in {{\mathbb {N}}}^*\). This happens for example if \({\mathbb {P}}\), \({\mathbb {Q}}\) are two distinct, fully supported ergodic measures, and \({\widehat{{\mathbb {P}}}}= \frac{1}{2} {\mathbb {P}}+ \frac{1}{2} {\mathbb {Q}}\).

  17. 17.

    \({{\text {tr}}}_{{{\mathcal {H}}}_p}\) denotes the partial trace over \({{\mathcal {H}}}_p\).

  18. 18.

    \((P_l)_{l\in {{{\mathcal {L}}}}}\) is the family of eigenprojections of an observable commuting with \(\rho _p\), not necessarily the ones of \(\rho _p\) itself. In particular, we do not assume the projections \(P_l\) to be rank one.

  19. 19.

    All the entropic notions we use are reviewed in [21, Section 2.1]. We recall that \(S(\mu _1|\mu _2)\ge 0\) and that the equality holds iff \(\mu _1=\mu _2\).

  20. 20.

    Note that if \({\mathbb {Q}}\) is the unraveling of some quantum instrument, then this definition is consistent with the notation introduced in Sect. 1.2.

  21. 21.

    We denote the transpose of an arbitrary matrix A, by \(A^{\mathsf {T}}\).

  22. 22.

    M is called irreducible whenever, for any index pair ij, there is a power \(M^n\) with non-vanishing ij entry.

  23. 23.

    \(m_{xy}(a) = p_{xy}\delta _{ya}\) is an equally valid choice.

  24. 24.

    We use the convention \(0/0=0\).

  25. 25.

    We mention here that the link between HM and matrix products was used in [53] for the study of the Kolmogorov–Sinai entropy.

  26. 26.

    In this context, see also Remark 4.7.

  27. 27.

    Here and below, \(\Rightarrow \) denotes convergence in law.

  28. 28.

    For PMP unravelings \(e(\alpha ) <\infty \) for all \(\alpha \in {{\mathbb {R}}}\).

  29. 29.

    Recall that \(n_{ab} = n_{ab}(\xi )\).

  30. 30.

    Here \(\lambda \cdot X\) denotes the Euclidean inner product on \({{\mathbb {R}}}^3\).

  31. 31.

    We note that (3.20) does not follow from (3.12) and (3.19) alone. Some estimate on the speed of convergence is required.

  32. 32.

    \(f^*\) denotes the Legendre transform of f.

  33. 33.

    Note that, since the third component of \(\nabla Q(0)\) vanishes, we cannot get rid of the absolute value of \(W_T\) in the same way.

  34. 34.

    The symmetry (3.32) allows to conveniently estimate \(\overline{J}^{(\varepsilon )}\) in terms of \(\text {ep}^{(\varepsilon )}\), but its role is not fundamental. One can, in principle, also obtain (3.34) by writing \(J_T^{(\varepsilon )}\) in terms of the vector \(X_T\) introduced in Sect. 3.3 and then using the LDP that it obeys (with respect to \({\mathbb {P}}^{(\varepsilon )}\)).

  35. 35.

    Recall our convention \({\text {sign}}(0)=1\).

  36. 36.

    Note that \(|a_{\pm }|^2+2b_{\pm }^2=1\).

  37. 37.

    This case only happens when \(b_-=b_+=0\).

  38. 38.

    We recall the convention chosen in Sect. 1.2 about \({{\mathbb {N}}}\) (which includes 0) and \({{\mathbb {N}}}^*\) (which does not).

  39. 39.

    Note that (A.8) does not hold for \(i=0\) in general, because we may have \(q_1 = q_0=1\) (for example if \(\zeta = \pi /4\)) and so Lemma A.1 may not apply to all pairs \((p, q_i)\), \(p \ne p_i\).

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Acknowledgements

This research was supported by the Agence Nationale de la Recherche through the grant NONSTOPS (ANR-17-CE40-0006-01, ANR-17-CE40-0006-02, ANR-17-CE40-0006-03), and the CNRS collaboration grant Fluctuation theorems in stochastic systems. Additionally, this work received funding by the CY Initiative of Excellence (grant “Investissements d’Avenir” ANR-16-IDEX-0008) and was developed during VJ’s stay at the CY Advanced Studies, whose support is gratefully acknowledged. The research of TB was supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. NC and TB were also supported by the ANR grant QTRAJ (ANR-20-CE40-0024-01). VJ acknowledges the support of NSERC. The work of CAP has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). We wish to thank D. Roy and D. Jakobson for useful discussions.

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Appendix A Continued Fractions

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 \(n\in {{\mathbb {N}}}^*\).Footnote 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 thatFootnote 39

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

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Benoist, T., Cuneo, N., Jakšić, V. et al. On Entropy Production of Repeated Quantum Measurements II. Examples. J Stat Phys 182, 44 (2021). https://doi.org/10.1007/s10955-021-02725-1

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Keywords

  • Repeated quantum measurements
  • Quantum instrument
  • Arrow of time
  • Matrix product measure
  • Hidden Markov model
  • Equilibrium state
  • Non-Gibbsian measure
  • Entropy production
  • Fluctuation relations
  • Large deviations
  • Hermodynamic formalism
  • Shift space