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# Open Physics

### formerly Central European Journal of Physics

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Volume 14, Issue 1

# Logical entropy of quantum dynamical systems

• Corresponding author
• Department of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan, Iran (Islamic Republic of)
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Published Online: 2016-02-20 | DOI: https://doi.org/10.1515/phys-2015-0058

## Abstract

This paper introduces the concepts of logical entropy and conditional logical entropy of hnite partitions on a quantum logic. Some of their ergodic properties are presented. Also logical entropy of a quantum dynamical system is dehned and ergodic properties of dynamical systems on a quantum logic are investigated. Finally, the version of Kolmogorov-Sinai theorem is proved.

PACS: 02.10.Ab; 02.30.Cj; 89.70.Cf

## 1 Introduction

Birkhoff and Von Neumann in [1] have introduced the quantum logic approach. Entropy is a tool to measure the amount of uncertainty in random event. Entropy has been applied in a variety of problem areas including physics, computer science, general systems theory, information theory, statistics, biology, chemistry, sociology and many other helds. Hejun Yuan, Mona Khare and Shraddha Roy, using the notion of state of quantum logic, introduced Shanon entropy of hnite partitions on a quantum logic [2-4]. The dehnition of entropy of a dynamical system might be in three stages [2, 5 , 6]. Logical entropy is a measure on set of ordered pairs [7]. In 1982, Rao, Good, Patil and Taillie dehned and studied the notion logical entropy [8-10]. Rao introduced precisely this concept as quadratic entropy [10] and in the years 2009 and 2013, this concept was discussed by Ellerman in [7, 11, 12].

The notion of Shanon entropy of quantum dynamical systems with Bayessian state was studied by Mona Khare and Shraddha Roy in [2]. In this paper, the notion of logical entropy with hnite partitions is dehned and then, logical entropy of quantum dynamical systems with Bayessian state is presented and studied. In Section 2, some basic dehnitions are presented. In Section 3, logical entropy and coditional logical entropy of partitions of a quantum logic with respect to state s are dehned and a few results about them will be presented. In the subsequent section the relations s-refinement and ${\doteq }_{s}$ are defined and logical entropy and coditional logical entropy under the relations are studied. In Section 4, logical entropy of a quantum dynamical system (L,s,φ) is dehned where L is a quantum logic and s is a Bayessian state. At the end, the version of Kolmogorov-Sinai theorem is proved.

## 2 Finite Partitions

At first, some basic dehnitions are presented that will be useful in further considerations.

Definition 1. [4] A quantum logic QL is a σ-orthomodular lattice, i.e., a lattice L (L, ≤, V, Λ, 0,1) with the smallest element 0 and the greatest element 1, an operation ′ : L → L such that the following properties hold for all a, b, c ∈ L:

i)a″ = a, a b ⇒ b≤ a , a V a =1, a Λ a =0;

ii) Given any finite sequence (ai)iεI,a; < a, i /= j, the join Vi∈Nai exists in L;

iii) L is orthomodular: a ≤bb = a V (b Λ a ).

Two elements a, b ∈ QL are called orthogonal if a ≤ b and denoted by a ⊥ b. A sequence (ai)i∈i is said orthogonalifai⊥aj, ∀ii/= j.

Definition 2. [4] Let L be a quantum logic. A map s : L →[0,1] is a state iff s(1) = 1 and for any orthogonal sequence (ai)iI, s(ViℇIai) = ∑iℇIs(ai).

Definition 3. [4] Let P = {a1,..., an} be a finite system of elements of a quantum logic. P is called to be a V- orthogonal system iff ${\vee }_{i=1}^{k}{a}_{i}\perp {a}_{k+1},\forall k$

Definition 4.[4] A system P = {a1,..., an} ⊂ L is said to be a partition of L corresponding to a state s if:

i)P is a V-orthogonal system;

ii)$s\left({\vee }_{i=1}^{n}{a}_{i}\right)=1$

Note that from definition 2, we obtain ${\sum }_{i=1}^{n}s\left({a}_{i}\right)=1$ s(ai)=1.

Definition 5. [4] Let the system (b1,..., bm) be any partition corresponding to a state s and a ε L. The state s is said has Bayes’ Property if s($\left({\vee }_{j=1}^{m}\left(a\wedge {b}_{j}\right)\right)=s\left(a\right)$(a Λ bj)) = s(a).

Lemma 6. [4] Let Q = (b1,..., bm) be a partition on L, and a ε L, and the state s has Bayes’Property. Then $\sum {}_{j=1}^{m}s\left(a\wedge {b}_{j}\right)=s\left(a\right)$ s(a Λ bj ) = s(a).

## 3 Logical entropy of finite partitions

Let P = {a1,..., an} and Q = {b1,..., bm} be two finite partitions of a quantum logic corresponding to a state s. The common refinement of these partitions is:

P ∨ Q = {ai ∧ bj : ai ε P, bj ε R}.

Definition 7. Let P = {a1,..., an} be a partition of a quantum logic corresponding to a state s. The logical entropy of P with respect to state s is defined by: $hsl(p)=∑i=1ns(ai)(1−s(ai)).$ Since ${\sum }_{i=1}^{n}s\left({a}_{i}\right)=1,$ we have ${\sum }_{i=1}^{n}{\left(s\left({a}_{i}\right)\right)}^{2}$

Definition 8. Let P = {a1,..., an} and Q ={b1,..., bm} be two partitions of a quantum logic corresponding to a state s. The conditional logical entropy of P given Q with respect to state s is defined as:  In the next theorem an upper bound for logical entropy on a quantum logic is presented.

Theorem 9. Let P be a finite partition of a quantum logic corresponding to a state s. Then $0≤0hsl(P)≤1−1n.$

Proof. Let P = (p1,...,pn) ε Rn be a probability distribution, then from [7], maximum value of the logical entropy is $0\le {h}_{s}^{l}\left(P\right)\le 1-\frac{1}{n}$ $1−1n.∑i=1ns(ai)=1,P=(s(a1),.....,s(an))$ is a probability distribution and hence the proof is complete.

In the following theorem the conditional logical entropy under the common refinement of partitions is studied.

Theorem 10. Let P, Q and R be finite partitions of a quantum logic corresponding to a state s having Bayes’ Property. Then hls(P ∨ Q|R) = h1s(P|R) + h1s(Q|P ∨ R).

Proof. Let P = {a1,..., an}, Q = {b1,..., bm} and R = {c1,..., cr}. Since s has Bayes’ Property, by Lemma 6 we have $hsl(P∨Q|R)=∑i=1n∑j=1m∑k=1rs(ai∧bj∧ck)(s(ck)−s(ai∧bj∧ck))=∑i=1n∑j=1m∑k=1rs(ai∧bj∧ck)(s(ck)−∑i=1n∑j=1m∑k=1r(s(ai∧bj∧ck)(s(ck))2=∑i=1n∑j=1m∑k=1rs(ai∧bj∧ck)(s(ck)−∑i=1n∑j=1m∑k=1r(s(ai∧bj∧ck)(s(ck))2+∑i=1n∑k=1r(s(ai∧ck))2−∑i=1n∑j=1m∑k=1r(s(ai∧bj∧ck)(s(ck))2.$ On the other hand we can write $hsl(P|R)=∑i=1n∑k=1rs(ai∧bj∧ck)(s(ck)−∑i=1n∑k=1r(s(ai∧ck))2$ and $hsl(Q|P∨R)=∑i=1n∑j=1m∑k=1rs(ai∧bj∧ck)(s(ck)(s(ai∧ck)−∑i=1n∑j=1m∑k=1r(s(ai∧bj∧ck)(s(ck))2.=∑i=1n∑k=1r(s(ai∧ck))2−∑i=1n∑j=1m∑k=1r(s(ai∧bj∧ck)(s(ck))2$ Thus the proof is complete. □

Now the assertion of the following theorem will be proved that will be useful in further theorems.

Theorem 11. Let P and Q be finite partitions of a quantum logic corresponding to a state s having Bayes’ Property. Then .

Proof. Let P = {a1,..., an} and Q = {b1,..., bm}. From Lemma 6 we can write

$hsl(P∨Q)=∑i=1n∑j=1ms(ai∧bj)(1−s(ai∧bj))=∑i=1n∑j=1ms(ai∧bj)−∑i=1n∑j=1m(s(ai∧bj))2=∑j=1ms(bj)−∑i=1n∑j=1m(s(ai∧bj))2$

Hence the proof is complete.

In the next theorem it is proved subadditivity of logical entropy of partitions on a quantum logic.

Theorem 12. Let P and Q be finite partitions of a quantum logic corresponding to a state s having Bayes’Property. Then

i)

ii) max{h1s(P), h1s(Q)} ≤ h1s(P ∨ Q) ≤ h1s(P) + h1s(Q).

Proof. i) Let P = {a1,..., an} and Q = {b1,..., bm}. For each ai εP, we can write

$∑j=1ms(ai∧bj)(sbj)−s(ai∧bj))≤∑j=1ms(ai∧bj)(∑j=1ms(bj)−s(ai∧bj))=s(ai)(∑j=1ms(bj)−s(ai∧bj))=s(ai)(1−∑j=1ms(ai∧bj)=(s(ai)(1−s(ai)).$

ii) From Theorem 11 and part i),



By Theorem 11,

Let s be a state. Two finite partitions P and Q of a quantum logic are called s-independent if s(a ∧ b) = s(a)s(b) for all a ε P, and b ε Q.

In the next theorem you observe that, for two s- independent finite partitions P and Q of a QL, h1s(P ∨ Q).= h1s(P) + h1s(Q) necessarily. Also, in this case h1s(P|Q)= h1s(P) necessarily.

Theorem 13. Let s be a state and let P and Q be s- independent finite partitions of a quantum logic. Then

i) ;

ii) 

Proof. i) Since

$hsl(P)=1−∑i=1n(s(ai))2≤1−∑i=1n(s(bj))2=hsl(Q)$

and P, Q are s-independent, we can write

$hsl(P∨Q)=1−∑i=1n∑j=1m(s(ai∧bj))21−∑i=1n∑j=1m(s(ai)2(s(bj))21−(∑i=1n(s(ai))2)(∑j=1m(s(bj))2)1−(∑i=1n(s(ai))2)(∑j=1m(s(bj))2)−∑i=1n(s(ai))2+∑i=1n(s(ai))2=hsl(P)+∑i=1n(s(ai))2(1−∑j=1m(s(bj))2)=hsl(P)+(1−hsl(P)hsl(Q)=hsl(P)+hsl(Q)−hsl(P)hsl(Q)$

ii) Follows from i) and Theorem 11. □

Definition 14. Let P = {a1,..., an} and Q = {b1,..., bm} be two partitions of a quantum logic corresponding to a state s. We say Q is a s-rehnement of P, denoted by P ≾s Q, if there exists a partition I(1),..., I(n) of the set {1,..., m} such that ai = VjεI(i)bj for every i = 1,..., n.

Now the relation between the s-rehnement and the logical entropy of hnite partitions will be studied.

Theorem 15. Let P = {a1,..., an}, Q = {b1,..., bm} and R = {c1,..., cr} be partitions of a quantum logic corresponding to a state s. Then

i) P ≾s Q implies that 

ii) If P ≾s Q and the quantum logic be distributive then 

Proof. i)Since P ≾s Q, there exists a partition I(1),..., I(n) of the set {1,...,m} such that ai = VjεI(i)sbj for every i = ..., n. So from dehnition 2, s(ai) = ∑jεI(i)s(bj), therefore $s\left({a}_{i}\right)\left({\sum }_{j=1}^{m}s\left({b}_{j}\right)-s\left({a}_{i}\wedge {b}_{j}\right)\right)$ so

$hsl(P)=1−∑i=1n(s(ai))2≤1−∑i=1n(s(bj))2=hsl(Q)$

ii) P ≾s Q implies that P ∨ R ≾s Q ∨ R, because let ai ∧ c be an arbitrary element of P ∨ R, then there exists a partition I(1),..., I(n) of the set {1,...,m} such that ai = VjεI(i)bj for every i = 1,..., n. Therefore ai ∧ c = (VjεI(i)bj) ∨ c = VjεI(i)(bj ∨ c), hence P ∨ R ≾s Q ∨ R. Now by Theorems 11 and 15, it will be obtained 

Definition 16. Let P = {a1,..., an} and Q = {b1,..., bm} be two partitions of a quantum logic corresponding to a state s. Q if for each bj ε Q there exists ai ε P with s(ai ∧ bj) = s(bj). P ≗s Q if P εOs Q and Q εOs P.

The next theorem shows that, the logical entropy and logical conditional entropy of hnite partitions of a quantum logic corresponding to a state s having Bayes’ Property, are invariant under the relation ≗s .

Theorem 17. Let P and Q be finite partitions of a quantum logic corresponding to a state s having Bayes’ Property. Then

i) P εOs Q if and only if h1s(P|Q) = 0;

ii) if P ≗s Q then h1s(P) = h1s(Q);

iii) if P ≗s Q then h1s(P|R) =h1s(Q|R);

iv) if Q ≗s R then h1s(P|Q) = h1s(P|R).

Proof. i) Let P = {a1,..., an} and Q = {b1,..., bm} and P εOs Q, then for each bj ε Q there exists ai0 ε P such that s(bj) = s(ai0 Λ bj). Since s(bj) = ΣÌu s(ai Λ bj), we obtain s(bj) = s(ai0 Λ bj) and for each i/ = j0, s(ai ∧ bj) = 0 and so h1s(P|Q) = 0. Conversely, if h1s(P|Q) = 0 then for each i, j, s(bj) = s(ai ∧ bj) or s(ai ∧ bj) = 0. For an arbitrary element bj Q, since 0= s(bj) = ΣÌu s(a; ∧ bj) we deduce that there exists an q, 1 ≤ i1 ≤ n, such that s(bj) = s(ai1 ∧ bj). ii) Since P εOs Q, by i) we have h1s(P|Q) = 0. So by theorem 11, h1s(P|Q) = h1s (P ∨ Q)- h1s (Q) = 0 and therefore h1s (P ∨ Q) = h1s (Q). Similarly if OsP, then h1s (QIP) = h1s (P ∨ Q) - h’s(P) = 0 and so h’s(P ∨ Q) = h’s(P). Hence we imply that h1s (P) = h1s (Q).

iii) We first show that OsQ implies that P ∨ RεOsQ V R. Let bj0 ∧ cko be an arbitrary element of Q ∧ R. Since OsQ, there exists aio e P such that s(aio ∧ bj0) = s(bj0). Now s((aio ∧ ck0) ∧ (bj0 ∧ c^)) = s(ai0 ∧ bj0 ∧ (ck0 ∧ ck0)) = s(aio ∧ bj0 ∧ ck0), it is sufficient to show that s(aio ∧ bj0 ∧ ck0) = s(bj0 ∧ck0). Since s has Bayes’ Property, s(aio ∧bj0) = s(bjrj) = En=i s(ai ∧ bj0). So for each i/ = io, s(ai ∧ bj0) = 0, therefore for each i/ = i0, s(ai ∧ bj0 ∧ ck0) = 0 and this implies that

$s\left({a}_{{i}_{o}}\vee {b}_{{j}_{o}}\vee {c}_{{k}_{o}}\right)=\sum _{i=1}^{n}s\left({a}_{{i}_{o}}\vee {b}_{{j}_{o}}\vee {c}_{{k}_{o}}\right)=s\left({b}_{{j}_{o}}\vee {c}_{{k}_{o}}\right).$

Thus P ∨ RεOsQ ∧ R. By changing the role of P and Q, P —s Q implies that P ∨ R ≗sQ ∧ R. Hence from ii), h1s (PR) = h1s (PVR)- h1s (R) = h1s (Q VR)- h1s (R) = h1s (QR). iv)Weneed toshowthat OsR implies that PVQεOsPVR. Let aio ∧cko be an arbitrary element of P ∨ R. Since OsR, there exists bjo e Q such that s(bjo ∧ ck0) = s(ck0). Now s((ah ∧ bj0) ∧ (aio ∧ cko)) = s((aio ∧ aio) ∧ bjo ∧ cko) = s(aio ∧ bjo ∧ cko), it is sufficient to show that s(aio ∧bj0 ∧cko) = s(aio ∧cko). Since s has Bayes’ Property, s(bjrj ∧ cko) = s(cko) = Eh s(bj ∧ cko). So for each j = j0, s(bj ∧ cko) = 0, therefore for each j = j0, s(aio ∧ bj ∧ cko) = 0 and this implies that

$s(aio∧bjo∧cko)=∑j=1ms(aio∧bjo∧cko)=s(bjo∧cko).$

Thus P ∨ Q c° P ∨ R. By changing the role of Q and R, Q —s R implies that P V Q —s P ∨ R. Now from ii), h1s (P|Q) = h1s (P ∨ Q) - h1s (Q) = h1s (P ∨ R) - h1s (R) = h1s (PR).

## 4 Logical entropy of quantum dynamical systems

Definition 18. [2] Let L be a quantum logic and φ : L → L be a map with the following properties:

i) φ(a ∨ b) = φ(ά) ∨ φα, ∀a, b ε L;

ii) φ(a ∧ b) = φ(a) ∧ φ(b), ∀a, b ε L;

iii) φ(a) = φ(a)), ∀a ε L.

φ : L → L with respect to a state s is called state preserving if s(φ(a)) = s(a) for every a ε L. Then the triple (L, s, φ) is said a quantum dynamical system where the state s having Bayes’ Property. In the following theorem the existence of the limit in Definition 20 is shown.

Theorem 19. Let (L, s, φ) bea quantum dynamical system and P be a partition of (L, s), then limn→∞) exists.

Proof. Let an = h1s1s . It will be shown that for p ε N, an+p ≤ an + ap and then by Theorem 4.9 in [13], $limn→∞1nan$ exists and equals infn $ann$ . By Theorem 12, ii) we have

$an+p=hsl(Vi=1n+p−1φiP)≤hsl(Vi=1n−1φiP)+hsl(Vi=nn+p−1φiP)=an+hsl(Vi=1p−1φn+iP)=an+hsl(Vi=1p−1φiP)=an+ap.$

The second stage and the final stage of the definition of the logical entropy of a quantum dynamical system (L, s, φ) is given in the next definition.

Definition 20. Let (L, s, φ) be a quantum dynamical system and P be a partition of (L, s). The logical entropy of T respect to P is defined by:

$hsl(φ,P)=limn→∞1nhsl(Vi=1nφiP)$

The logical entropy of φ is defined as:

$hsl(φ)=suphsl(φ,P)$

where the supremum is taken over all finite partitions of (L, s).

In the following proposition some ergodic properties of h1s (φ, P) and h1s (φ) will be studied.

Proposition 21. If (L, s, φ) is a quantum dynamical system and P is a partition of (L, s), then

i)h’s (φ, P) = h’s (φ, Vk=i φ’Ρ);

ii) For k ∈ N, h’s^k) = kh’s(φ).

Proof.

$hsl(φ,P)=limn→∞1nhsl(Vi=1nφiP)$

ii) Let P be an arbitrarary finite partition of (L, s). we can write

$hsl(φk,∨i=1nφiP)=limn→∞1nhsl(∨j=1n(φk)j(∨i=1nφiP))=limn→∞1nhsl(∨j=1n∨i=1kφkj+iP)=limn→∞1nhsl(∨i=1nk−1φiP)=limn→∞nkn1nkhsl(∨i=1nk−1φiP)=khsl(φ,p)$

Definition 22. Let (L, s, φ) be a quantum dynamical system. A finite partition R of (L, s), is said to be an s- generator ofφ, if there exists r∈ N such thatP ≾s Vri-1 φiR for each finite partition P of (L, s).

The main aim of this theorem is to prove an analogue of the Kolmogorov-Sinaj theorem on logical entropy and generators.

Theorem 23. Let (L, s, φ) bea quantum dynamical system and let R be an s-generator of φ, then h1s (φ) = h1s (φ, R).

Proof. Let P be an arbitrary finite partition of (L, s). Since R is an s-generator, P ≾s Vri-1R. By Theorem 15, you get $hsl(φ,P)≤hsl(φ,Vi=1rR)=hsl(φ,R)Hencehsl(φ)=suphsl(φ,P)≤hsl(φ,R).$ Hence  .On the other hand .

## 6 Conclusion

This paper has introduced logical entropy and conditional logical entropy of hnite partitions on a quantum logic and has presented some of their ergodic properties. Also, logical entropy of a quantum dynamical system with hnite partitions studied and some of its properties proved.

## 1 References

Accepted: 2015-12-08

Published Online: 2016-02-20

Published in Print: 2016-01-01

Citation Information: Open Physics, Volume 14, Issue 1, Pages 1–5, ISSN (Online) 2391-5471,

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