# Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta
functions

Ernesto P. Borges<EMAIL_ADDRESS>Instituto de Física, Universidade Federal da
Bahia, Rua Barão de Jeremoabo s/n, Salvador-BA 40170-115, Brazil  National
Institute of Science and Technology of Complex Systems,
Rua Xavier Sigaud 150, 22290-180 - Rio de Janeiro, RJ, Brazil  Takeshi Kodama
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio de
Janeiro, Rio de Janeiro-RJ 21941-972, Brazil  Instituto de Física,
Universidade Federal Fluminense and National Institute of Science and
Technology for Nuclear Physics and Applications, Campus da Praia Vermelha,
Niterói-RJ 24210-346, Brazil  Constantino Tsallis<EMAIL_ADDRESS>Centro
Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 - Rio de
Janeiro, RJ, Brazil  National Institute of Science and Technology of Complex
Systems,
Rua Xavier Sigaud 150, 22290-180 - Rio de Janeiro, RJ, Brazil  Santa Fe
Institute – 1399 Hyde Park Road, Santa Fe, 87501 NM, United States  Complexity
Science Hub Vienna - Josefstädter Strasse 39, 1080 Vienna, Austria

###### Abstract

The rich history of prime numbers includes great names such as Euclid, who
first analytically studied the prime numbers and proved that there is an
infinite number of them, Euler, who introduced the function
$\zeta(s)\equiv\sum_{n=1}^{\infty}n^{-s}=\prod_{p\,prime}\frac{1}{1-p^{-s}}$,
Gauss, who estimated the rate at which prime numbers increase, and Riemann,
who extended $\zeta(s)$ to the complex plane $z$ and conjectured that all
nontrivial zeros are in the $\mathbb{R}(z)=1/2$ axis. The nonadditive entropy
$S_{q}=k\sum_{i}p_{i}\ln_{q}(1/p_{i})\;(q\in\mathbb{R};\,S_{1}=S_{BG}\equiv-k\sum_{i}p_{i}\ln
p_{i}$, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical
mechanics is based, involves the function
$\ln_{q}z\equiv\frac{z^{1-q}-1}{1-q}\;(\ln_{1}z=\ln z)$. It is already known
that this function paves the way for the emergence of a $q$-generalized
algebra, using $q$-numbers defined as $\langle x\rangle_{q}\equiv
e^{\ln_{q}x}$, which recover the number $x$ for $q=1$. The $q$-prime numbers
are then defined as the $q$-natural numbers $\langle n\rangle_{q}\equiv
e^{\ln_{q}n}\;(n=1,2,3,\dots)$, where $n$ is a prime number $p=2,3,5,7,\dots$
We show that, for any value of $q$, infinitely many $q$-prime numbers exist;
for $q\leq 1$ they diverge for increasing prime number, whereas they converge
for $q>1$; the standard prime numbers are recovered for $q=1$. For $q\leq 1$,
we generalize the $\zeta(s)$ function as follows:
$\zeta_{q}(s)\equiv\langle\zeta(s)\rangle_{q}$ ($s\in\mathbb{R}$). We show
that this function appears to diverge at $s=1+0$, $\forall q$. Also, we
alternatively define, for $q\leq 1$,
$\zeta_{q}^{\Sigma}(s)\equiv\sum_{n=1}^{\infty}\frac{1}{\langle
n\rangle_{q}^{s}}=1+\frac{1}{\langle 2\rangle_{q}^{s}}+\dots$ and
$\zeta_{q}^{\Pi}(s)\equiv\prod_{p\,prime}\frac{1}{1-\langle
p\rangle_{q}^{-s}}=\frac{1}{1-\langle 2\rangle_{q}^{-s}}\frac{1}{1-\langle
3\rangle_{q}^{-s}}\frac{1}{1-\langle 5\rangle_{q}^{-s}}\cdots$, which, for
$q<1$, generically satisfy $\zeta_{q}^{\Sigma}(s)<\zeta_{q}^{\Pi}(s)$, in
variance with the $q=1$ case, where of course
$\zeta_{1}^{\Sigma}(s)=\zeta_{1}^{\Pi}(s)$.

Keywords:  Nonadditive entropies; $q$-prime numbers; $q$-algebras; $q$-zeta
functions

##  1 Introduction

Extending the realm of the Boltzmann-Gibbs-von Neumann-Shannon entropic
functional, many measures of uncertainty have been proposed to handle complex
systems and, ultimately, complexity. All of them are nonadditive, excepting
the Renyi functional. Among them, a paradigmatic one is the entropy $S_{q}$
defined, with the scope of generalizing Boltzmann-Gibbs (BG) statistical
mechanics, as follows [1]:

$S_{q}=k\frac{1-\sum_{i}p_{i}^{q}}{q-1}=k\sum_{i}p_{i}\ln_{q}\frac{1}{p_{i}}\;\;(k>0)$
(1.1)

with $S_{1}=S_{BG}\equiv k\sum_{i}p_{i}\ln\frac{1}{p_{i}}$. The entropy
$S_{q}$ is the most general one which simultaneously is composable and trace-
form [2], and it has been shown to be connected to the Euler-Riemann function
$\zeta(s)$ [3]. The $q$-logarithm function is defined [4] as follows

$\ln_{q}z\equiv\frac{z^{1-q}-1}{1-q}\;\;(\ln_{1}z=\ln z)\,,$ (1.2)

its inverse function being

$e_{q}^{z}=[1+(1-q)z]^{1/(1-q)}\;(e_{1}^{z}=e^{z})\,$ (1.3)

when $[1+(1-q)z]\geq 0$; otherwise it vanishes. The definitions of the
$q$-logarithm and $q$-exponential functions allow consistent generalizations
of algebras [5, 6, 7, 8, 9], calculus [6, 10, 11, 8] (see also [12]) and
generalized numbers [13, 8].

There are different ways of defining generalized $q$-numbers connected with
the pair of inverse ($q$-logarithm, $q$-exponential) functions, namely

$\displaystyle\langle x\rangle_{q}$ $\displaystyle=$ $\displaystyle
e^{\ln_{q}x}\;\;\,(\langle x\rangle_{1}=x>0)\,,$ (1.4)
$\displaystyle{}_{q}\langle x\rangle$ $\displaystyle=$ $\displaystyle
e_{q}^{\ln x}\;\;\;\;({}_{1}\langle x\rangle=x>0)\,,$ (1.5)
$\displaystyle[x]_{q}$ $\displaystyle=$ $\displaystyle\ln
e_{q}^{x}\;\;\;([x]_{1}=x\in\mathbb{R})\,,$ (1.6) $\displaystyle{}_{q}[x]$
$\displaystyle=$ $\displaystyle\ln_{q}e^{x}\;\,(_{1}[x]=x\in\mathbb{R})\,.$
(1.7)

Observe that $\langle\,{}_{q}\langle
x\rangle\,\,\rangle_{q}={}_{q}\langle\>\langle
x\rangle_{q}\,\rangle=[\,{}_{q}[x]\,\,]_{q}={}_{q}[\>[x]_{q}\,]=x$. These four
possibilities are explored in Ref. [8]111Eqs. (1.4), (1.5), (1.6), (1.7) are
equivalent to Eqs. (11a), (11b), (10a), (10b) of Ref. [8] respectively. The
notations introduced in Eqs. (1.4) and (1.5) differ from those used in [8]..
Other generalizations exist in the literature, also referred to as $q$-numbers
[14, 15]. Generalized arithmetic operations follow from each of the
$q$-numbers and, consistently, there are various possibilities. The present
paper will only explore one possibility for $q$-numbers, namely our Eq. (1.4),
equivalent to the iel-number Eq. (11a) of [8]. For this choice, two algebras
will be focused on here, namely,

$\langle x\rangle_{q}\;\Circle^{q}\;\langle
y\rangle_{q}\equiv\langle\>x\>\circ\>y\,\rangle_{q}$ (1.8)

and

$\langle\,x\>\Circle_{q}\>y\,\rangle_{q}\equiv\,\langle
x\rangle_{q}\circ\langle y\rangle_{q}\,,$ (1.9)

the symbol $\circ$ representing any of the ordinary arithmetic operators
$\circ\in\\{+,-,\times,/\\}$, and $\Circle^{q}$ or $\Circle_{q}$ representing
the corresponding generalized operators; naturally,
$\Circle_{1}=\Circle^{1}=\circ$. Possibility (1.8) preserves prime number
factorizability, while possibility (1.9) does not222 The algebra corresponding
to Eq. (1.8) is developed in Section 4, and it was not addressed in Ref. [8].
We use a superscript for its algebraic operators, $\Circle^{q}$, a notation
that was not adopted in Ref. [8]. The other algebra, corresponding to Eq.
(1.9), is presented in Section 5, and it corresponds to the oel-arithmetics
addressed in Section III.D of Ref. [8], where ${}_{\\{q\\}}\Circle$ is here
noted $\Circle_{q}$. .

##  2 Preliminaries

Let as remind, at this point, the so-called Basel problem, which focuses on
the value of the series

$I_{2}=\sum_{n=1}^{\infty}\frac{1}{n^{2}}\,,$ (2.1)

proposed in 1644 by Pietro Mengoli of Bologna, in contrast to the divergent
harmonic series,

$I_{1}=\sum_{n=1}^{\infty}\frac{1}{n}\,.$ (2.2)

This problem was intensively studied by the Bernoulli brothers from Basel
almost half a century later, and became known as the Basel Problem. They
proved that $I_{1}$ is divergent, but $I_{2}$ is finite and smaller than 2,
but failed to obtain the exact value.

The Basel problem was solved by Euler in 1735. He also exhibited the
connection with prime numbers, namely

$I_{2}=\prod_{i=1}^{\infty}\frac{1}{1-p_{i}^{-2}}=\frac{1}{6}\pi^{2},$

where $p_{i}$ is the $i$-th prime numbers $(2,3,5,..)$, thus introducing for
the first time the so-called Euler’s product.

$I_{n}=\prod_{i=1}^{\infty}\frac{1}{1-p_{i}^{-n}}.$ (2.3)

In 1859, Riemann extended the domain of the exponent $n$ to complex numbers,
introducing the notation

$\displaystyle\zeta\left(s\right)$
$\displaystyle\equiv\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ (2.4)
$\displaystyle=\prod_{i=1}^{\infty}\frac{1}{1-p_{i}^{-s}},$ (2.5)

so that $\zeta\left(s\right)$ is often called Riemann’s zeta function (or
Euler-Riemann zeta function). By the way, Gauss made many important
contributions to the field, especially the so-called prime number theorem,
$\pi(N)\sim N/\ln N$ ($N\to\infty$), where $\pi(N)$ is the number of primes up
to the integer $N$.

For the oncoming discussion, it is useful to remind some properties within
integers which are necessary to prove

$\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{i=1}^{\infty}\frac{1}{1-p_{i}^{-s}}.$
(2.6)

For this, let us consider a function $f(z)$, for $z=xy$, which can be written
as

$f\left(z\right)=f\left(x\right)f\left(y\right).$ (2.7)

The function $f\left(z\right)=\frac{1}{z^{s}}$ admits such a property. Indeed,
the power law satisfies

$\displaystyle z^{a+b}$ $\displaystyle=z^{a}z^{b},$ (2.8) $\displaystyle
z^{-a}$ $\displaystyle=\frac{1}{z^{a}},$ (2.9) $\displaystyle z^{ab}$
$\displaystyle=\left(z^{a}\right)^{b}=\left(z^{b}\right)^{a}.$ (2.10)

Now, we know that any integer $n$ can be decomposed into an unique product of
prime numbers333Let us focus on an interesting aspect of the primes. Taking
logarithm of Eq. (2.11), for any positive integer $n\in\mathbb{N},$ we get
$\ln n=\sum_{i}m\left(n\right)_{i}\ \ln p_{i}=\sum_{i}m\left(n\right)_{i}\
\hat{e}_{i}$ where $\hat{e}_{i}\equiv\ln p_{i}$. We can therefore consider the
set $\left\\{\ln n,\ n\in\mathbb{N}\right\\}$ as a kind of infinite
dimensional vector space, whose basis vectors are
$\left\\{\hat{e}_{i},i=1,..,\right\\}.$ However, rigorously, this is not a
vector space, since the coefficients of the linear combination, i.e., the set
of multiplicities, are only integers, and not the set of real numbers.,

$n=p_{{}_{1}}^{m_{1}}p_{2}^{m_{2}}...p_{i}^{m_{i}}\cdots$ (2.11)

where $m_{i}$ is the multiplicity of the prime $p_{i}$ in the product
($p_{0}=1,\,p_{1}=2,\,p_{2}=3,\,\dots$) and they are uniquely determined for a
given $n\in\mathbb{N}.$ That is, the set

$\left\\{m_{1},m_{2},\cdots,m_{i},\cdots\right\\}$ (2.12)

is determined for a given value of $n$ so we can use the notation

$\left\\{m_{1}\left(n\right),m_{2}\left(n\right),\cdots,m_{i}\left(n\right),\cdots\right\\}\,.$
(2.13)

Note that to guarantee the uniqueness of the decomposition, the commutativity
$m\,n=n\,m$ and the associativity $l\,(m\,n)=(l\,m)\,n$ of the product
operation are essential.

Now, the sum over all positive integer values of a function $f(n)$ satisfying
Eq. (2.7) can be written, if it converges absolutely, as

${\sum_{n=1}^{\infty}}f(n)=\sum_{n=1}^{\infty}\prod_{i}^{\infty}f(p_{i}^{m_{i}(n)})\,,$
(2.14)

This comes from the fact that the direct product of the sets
$\left\\{p_{i},p_{i}^{2},p_{i}^{3},\cdots,p_{i}^{m},\cdots\right\\}$ for all
primes is equal to the set of positive natural numbers
$\mathbb{N}^{+}\equiv\\{1,2,3,\cdots\\}$,

$\prod_{i=0}^{\infty}\scalebox{1.5}{$\otimes$}\left\\{p_{i},p_{i}^{2},p_{i}^{3},\cdots,p_{i}^{m},\cdots\right\\}=\mathbb{N}^{+},$
(2.15)

where  $\otimes$ denotes the direct product. The meaning of Eq. (2.14) can be
seen more intuitively here below as

$\displaystyle\begin{array}[c]{ccccccc}\;\;\\{1&2&2^{2}&2^{3}&\cdots&2^{m}\cdots&\\}\\\
\text{\scalebox{1.5}{$\otimes$}}\\{1&3&3^{2}&3^{3}&\cdots&3^{m}\cdots&\\}\\\
\text{\scalebox{1.5}{$\otimes$}}\\{1&\vdots&\vdots&\vdots&\cdots&\vdots&\\}\\\
\text{\scalebox{1.5}{$\otimes$}}\\{1&p_{i}&p_{i}^{2}&p_{i}^{3}&\cdots&p_{i}^{m}&\\}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\end{array}$ (2.21)
$\displaystyle=\left\\{1,1\cdot 2,\cdots,p_{{}_{1}}^{m_{1}}\cdot
p_{2}^{m_{2}}\cdot...\cdot p_{i}^{m_{i}}\cdot,\dots\right\\}=\mathbb{N}\,.$
(2.22)

Note that the multiplicity $m_{i}(n)$ of the $i$-th prime $p_{i}$ takes all
integer values if $n$ runs over all natural numbers. Consistently, for a given
$p_{i},$ the value of $m_{i}(n)$ runs over all nonnegative integers. That is,
$m_{i}(n)$ and $i$ can run over all nonnegative integers independently.
Therefore, by using $l\,(m+n)=l\,m+l\,n$, we can exchange the order of the sum
and the product in Eq. (2.14):

$\sum_{n=1}^{\infty}f\left(n\right)=\prod_{i=1}^{\infty}\sum_{m=0}^{\infty}f\left(p_{i}^{m}\right)$
(2.23)

if $f(x)$ satisfies the property indicated in Eq. (2.7).

Now, let us take the case of the Euler product,

$f\left(n\right)=\frac{1}{n^{s}}.$

Then the summation in $m$ of Eq.(2.23) for $p_{i}$ term is

$S\equiv\sum_{m=0}^{\infty}\left(\left(\frac{1}{p_{i}}\right)^{s}\right)^{m}=1+\frac{1}{p_{i}^{s}}+\left(\frac{1}{p_{i}^{s}}\right)^{2}+\cdots+\left(\frac{1}{p_{i}^{s}}\right)^{k}+\cdots$
(2.24)

which is a geometric series. By writing $r\equiv 1/p_{i}^{s}$, we obtain

$S\equiv
1+r+r^{2}+\cdots=\frac{1}{1-r}=\frac{1}{1-\left(1/p_{i}\right)^{s}}\,.$ (2.25)

Finally, we have the Euler product form as

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{i=1}^{\infty}\sum_{m=0}^{\infty}\left(\frac{1}{p_{i}^{s}}\right)^{m}=\prod_{i=1}^{\infty}\left(\frac{1}{1-\left(1/p_{i}\right)^{s}}\right)\,.$
(2.26)

This connection is known as Euler’s product form. Riemann extended the domain
of $s$ of this function into the complex plane $z$, being since then
frequently referred to as the Riemann zeta function $\zeta(z)$.

##  3 $q$-integers

We focus on the $q$-generalized numbers $\langle n\rangle_{q}$,
$n\in\mathbb{N}$, given by Eq. (1.4), to see how far we can conserve the
essential concept of prime numbers in the set of $q$-numbers. This particular
$q$-number $\langle n\rangle_{q}$ satisfies

$\ln\,\langle n\rangle_{q}=\ln_{q}n.$ (3.1)

Let us consider the set of nonnegative integers $\mathbb{N}$. Eq. (1.4)
defines a mapping from $\mathbb{N}$ to $\mathbb{N}_{q}$, where
$\mathbb{N}_{q}$ denotes the set of all positive $q$-integers. Note that
$\lim_{x\to 1}\langle x\rangle_{q}\to 1$ but, for other general natural
numbers $n$, its $q$-partner

$\langle n\rangle_{q}=\exp\left\\{\frac{n^{1-q}-1}{1-q}\right\\}$ (3.2)

is not an integer, in general. This point is crucial for the question of
factorization. The inverse mapping is then (see Eq. (1.5))

$n=\exp_{q}\left\\{\ln\left(\langle n\rangle_{q}\right)\right\\}\\\
={}_{q}\langle\,\langle n\rangle_{q}\,\rangle.$ (3.3)

The $q$-mapping and its inverse $\mathbb{N}\leftrightarrows\mathbb{N}_{q}$ are
one-to-one [$\exp_{q}(\ln_{q}x)=\ln_{q}(\exp_{q}(x))=x,\,\forall
x\in\mathbb{R}_{+}$]. However, it is clear that, for $q\neq 1$,
$\mathbb{N}\neq\mathbb{N}_{q}$.

Figure 1:  $q$-natural numbers for typical values of $q$. The $q$-prime
numbers are indicated in red. The dotted lines correspond to real values for
the abscissa.

For $q\in(1,\infty)$, the $q$-natural numbers satisfy
$\lim_{n\to\infty}\langle n\rangle_{q}=e^{1/(q-1)}>1\,$. For
$q\in(-\infty,1]$, the $q$-natural numbers satisfy $\lim_{n\to\infty}\langle
n\rangle_{q}=\infty\,$. There are infinitely many $q$-prime numbers for
$q\in(-\infty,\infty)$. See Fig. 1.

##  4  Algebra preserving factorizability of $q$-integer numbers in $q$-prime
numbers

In order to introduce the concept of $q$-primes, we should keep the
factorization concept in $\mathbb{N}_{q}^{+}$. Thus we must first define the
product operation $\mathbb{N}_{q}$, and should be kept invariant under the
$q$-mapping from $\mathbb{N}^{+}$. Following Eq. (1.8), let $\langle
m\rangle_{q}\otimes^{q}\langle n\rangle_{q}$ be such a product444Note that
here $\otimes^{q}$ is not the direct product of sets, but this specific
$q$-generalized product of $q$-numbers. between any pair of $q$-integers
$\langle m\rangle_{q},\,\langle n\rangle_{q}\in\mathbb{N}_{q}$, whose result
give the $q$-transform of $m\,n\ \in\mathbb{N}$, i.e., obeys the following
factorizability:

$\langle m\rangle_{q}\otimes^{q}\langle n\rangle_{q}=\langle mn\rangle_{q}.$
(4.1)

We also define the following generalized summation operator $\oplus^{q}$ in
$\mathbb{N}_{q}$ as

$\langle m\rangle_{q}\oplus^{q}\langle n\rangle_{q}=\langle m+n\rangle_{q}.$
(4.2)

The above definitions correspond to those introduced in Ref. [13] [its Eqs.
(5) and (6)]555The operations defined in [13] follows the same structure of
Eq. (4.1), or, more generally, Eq. (1.8), but with a different deformed
number, namely the Heine number. .

In order that the above operations are meaningful in $\mathbb{N}_{q}^{+}$, it
is necessary that they are closed operations in $\mathbb{N}_{q}$.

The following definitions satisfy properties (4.1) and (4.2):

$\langle m\rangle_{q}\otimes^{q}\langle n\rangle_{q}=\langle
m\rangle_{q}\;\langle n\rangle_{q}\;e^{(1-q)\left(\ln\langle
m\rangle_{q}\right)\left(\ln\langle n\rangle_{q}\right)},$ (4.3) $\langle
m\rangle_{q}\oplus^{q}\langle
n\rangle_{q}=\exp\left\\{\frac{(m+n)^{1-q}-1}{1-q}\right\\}.$ (4.4)

Eq. (4.3) can be rearranged as

$\langle m\rangle_{q}\otimes^{q}\langle n\rangle_{q}=\langle
m\rangle_{q}\;\langle n\rangle_{q}\;e^{\ln\langle
m\rangle_{q}\,\oplus^{q}\,\ln\langle n\rangle_{q}\,-\,\ln\langle
m\rangle_{q}\,-\,\ln\langle n\rangle_{q}}.$ (4.5)

The result of these definitions is a $q$-integer by construction.

By construction, it is clear that this definition of the $q$-product
$\otimes^{q}$ as an operator in $\mathbb{N}_{q}$, conserves the factorization
property under the $q$-transform $n=k\,m\in\mathbb{N}\Longleftrightarrow\
\langle n\rangle_{q}=\langle k\rangle_{q}\otimes^{q}\langle
m\rangle_{q}\in\mathbb{N}_{q}.$ In addition, also by construction, the
operators, $\otimes^{q}$ and $\oplus^{q}$ satisfy the following basic
properties of algebras, valid in $\mathbb{N}_{q}$:

  * •

Closedness of the operation:

$\forall\langle m\rangle_{q},\langle
n\rangle_{q}\in\mathbb{N}_{q}^{+}\Longrightarrow\ \langle
m\rangle_{q}\oplus^{q}\langle n\rangle_{q}\in\mathbb{N}_{q}^{+},\,\langle
m\rangle_{q}\otimes^{q}\langle n\rangle_{q}\in\mathbb{N}_{q}^{+},$ (4.6)

  * •

Commutativity

$\displaystyle\langle m\rangle_{q}\oplus^{q}\langle n\rangle_{q}$
$\displaystyle=\langle n\rangle_{q}\oplus^{q}\langle m\rangle_{q},$ (4.7)
$\displaystyle\langle m\rangle_{q}\otimes^{q}\langle n\rangle_{q}$
$\displaystyle=\langle n\rangle_{q}\otimes^{q}\langle m\rangle_{q},$ (4.8)

  * •

Associativity

$\displaystyle\langle k\rangle_{q}\oplus^{q}\bigl{(}\langle
m\rangle_{q}\oplus^{q}\langle n\rangle_{q}\bigr{)}$
$\displaystyle=\bigl{(}\langle k\rangle_{q}\oplus^{q}\langle
m\rangle_{q}\bigr{)}\oplus^{q}\langle n\rangle_{q},$ (4.9)
$\displaystyle\langle k\rangle_{q}\otimes^{q}\bigl{(}\langle
m\rangle_{q}\otimes^{q}\langle n\rangle_{q}\bigr{)}$
$\displaystyle=\bigl{(}\langle k\rangle_{q}\otimes^{q}\langle
m\rangle_{q}\bigr{)}\otimes^{q}\langle n\rangle_{q},$ (4.10)

  * •

Distributivity of the product $\otimes^{q}$ with regard to the sum
$\oplus^{q}$

$\langle k\rangle_{q}\;\otimes^{q}\,\bigl{(}\langle
m\rangle_{q}\,\oplus^{q}\,\langle n\rangle_{q}\bigr{)}=\bigl{(}\langle
k\rangle_{q}\;\otimes^{q}\,\langle
m\rangle_{q}\bigr{)}\;\oplus^{q}\;\bigl{(}\langle
k\rangle_{q}\;\otimes^{q}\,\langle n\rangle_{q}\bigr{)}.$ (4.11)

  * •

Neutral element of the $q$-addition

$\langle m\rangle_{q}\oplus^{q}\langle 0\rangle_{q}=\langle
m\rangle_{q}\oplus^{q}0=\langle m\rangle_{q}\;\;\;(q\geq 1).$ (4.12)

  * •

Neutral element of the $q$-product

$\langle m\rangle_{q}\otimes^{q}\langle 1\rangle_{q}=\langle
m\rangle_{q}\otimes^{q}1=\langle m\rangle_{q}\,.$ (4.13)

We show that these properties are essential to keep the nature of prime
numbers for $q$-transformed integers, corresponding to Eq. (2.11) in
$\mathbb{N}$.

We also define the following operations:

$\langle x\rangle_{q}\oslash^{q}\langle y\rangle_{q}=\langle x/y\rangle_{q}$
(4.14)

and

$\langle x\rangle_{q}\ominus^{q}\langle y\rangle_{q}=\langle x-y\rangle_{q}$
(4.15)

Besides, for any $a\in\mathbb{R}$, we can define a $a$-power of a $q$-number
by

$\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}a\equiv\Bigl{\langle}x^{a}\Bigr{\rangle}_{q}$
(4.16)

from what follows

$\displaystyle\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}(a+b)$
$\displaystyle=\Bigl{\langle}x^{a+b}\Bigr{\rangle}_{q}$ $\displaystyle=\langle
x^{a}\rangle_{q}\,\otimes^{q}\,\langle x^{b}\rangle_{q}$
$\displaystyle=\Bigl{(}\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}a\Bigr{)}\,\otimes^{q}\,\Bigl{(}\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}b\Bigr{)},$
(4.17) $\displaystyle\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}(ab)$
$\displaystyle=\Bigl{\langle}x^{ab}\Bigr{\rangle}_{q}$
$\displaystyle=\Bigl{\langle}x^{a}\Bigr{\rangle}_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}b$
$\displaystyle=\Bigl{(}\;\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}a\;\Bigr{)}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}b$
$\displaystyle=\Bigl{(}\;\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}b\;\Bigr{)}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}a,$
(4.18)

and

$\displaystyle\Bigl{(}\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}s\Bigr{)}\,\otimes^{q}\,\Bigl{(}\langle
x\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}t\Bigr{)}$
$\displaystyle=\Bigl{(}\langle x\rangle_{q}\Bigr{)}^{\langle
s\rangle_{q}\,\oplus^{q}\,\langle t\rangle_{q}}.$ (4.19)

As we see, for fixed $q$, this algebra is isomorphic to the standard algebra.

Definition 1  A $q$-integer $\langle n\rangle_{q}$ (i.e., an element of
$\mathbb{N}_{q}^{+}$) is called “$q$-prime” and written as $\langle
p\rangle_{q}$ if it can not be written as a $q$-factorized form in terms of
two smaller $q$-integers as

$\langle n\rangle_{q}=\langle m\rangle_{q}\otimes^{q}\langle k\rangle_{q},$
(4.20)

with $\langle m\rangle_{q},\langle k\rangle_{q}\in\mathbb{N}_{q}^{+}$, except
for the trivial factorization case, i.e., either one of $\langle m\rangle_{q}$
or $\langle k\rangle_{q}$ is the unity, $\langle 1\rangle_{q}$.

It is evident from the definition of $q$-integers that all the set of
$q$-primes $\left\\{\langle p_{i}\rangle_{q},i=1,\dots\right\\}$ are
$q$-partners of the prime numbers in $\mathbb{N}^{+}$,
$\left\\{p_{i},i=1,...\right\\}$. For any integer $n\in\mathbb{N}^{+}$ which
is not a prime, there exists the non-trivial factors, $k,m\in\mathbb{N}^{+}$
such that $n=k\times m$. But from the definition of the $q$-product, Eq.
(4.1), $\langle n\rangle_{q}=\langle km\rangle_{q}=\langle
k\rangle_{q}\,\otimes^{q}\,\langle m\rangle_{q}$, showing that $\langle
n\rangle_{q}$ has a non-trivial $q$-factorization and it is not a $q$-prime.
As we mentioned, the $q$-correspondence between the two sets, $\mathbb{N}^{+}$
and $\mathbb{N}_{q}^{+}$ is one-to-one, $q$-primes in $\mathbb{N}_{q}^{+}$ are
$q$-transformations of the normal primes in $\mathbb{N}^{+}$.

The basic property of primes is that any natural number $n\geq 2$ can be
written uniquely as the products of primes $\left\\{p_{i}\right\\}$ as

$n=\prod_{i}p_{i}^{m_{i}(n)},$ (4.21)

where

$\left\\{p_{1},p_{2},p_{3},p_{4},p_{5}\ \cdots\ \ \right\\}=\left\\{2,3,5,7,\
\cdots\right\\}$

is the set of primes and for a given $n$,

$\left\\{m_{1}(n),m_{2}(n),\cdots\right\\}$

is the set of multiplicities $m_{i}(n)$ of the prime $p_{i}$ is uniquely
determined for given $n$.

Now, for $q$-integers, $\langle n\rangle_{q}\in\mathbb{N}_{q}^{+}$, the
corresponding decomposition property is valid in $\mathbb{N}_{q}^{+}$ in terms
$q$-integers with $q$-products, satisfying the properties of commutativity and
distributivity. By construction, we can write for any $\langle n\rangle_{q}$
the $q$-prime decomposition as

$\langle n\rangle_{q}={\displaystyle\prod_{i}}^{q}\;\;\Bigl{(}\langle
p_{i}\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}m_{i}(n)\Bigr{)}$
(4.22)

for the sake of isomorphism of the product operations in $\mathbb{N}$ and
$\mathbb{N}_{q}$. The symbol ${\prod}^{q}$ represents the generalized product
(4.1) of a number of terms,
$\overset{\\!\\!n}{{\underset{\\!\\!i}{\sideset{}{{}^{q}}{\prod}}}}x_{i}\equiv
x_{1}\otimes^{q}x_{2}\otimes^{q}\cdots\otimes^{q}x_{n}$.

Since we have the isomorphisms of the operations of sum and product in
$\mathbb{N}^{+}$ and in $\mathbb{N}_{q}^{+}$, we can write down the
$q$-version of the Euler product:

${\sum_{\langle
n\rangle_{q}\in\mathbb{N}_{q}^{+}}}^{\\!\\!\\!\\!\\!q}\;\left(\frac{1}{\langle
n\rangle_{q}}\right)^{m\,\oplus^{q}\,s}={\sum_{\langle
n\rangle_{q}\in\mathbb{N}_{q}^{+}}}^{\\!\\!\\!\\!\\!q}\;\prod_{i}\;\Bigl{(}\langle
p_{i}\rangle_{q}\Bigr{)}^{m_{i}(n)},$ (4.23)

with the symbol ${\sum}^{q}$ representing the generalized summation according
to Eq. (4.2).

Now, the multiplicity $m_{i}(n)$ of the $i$-th prime $p_{i}$ should take all
integer values if $n$ runs over all the integers. Inversely, for a given
$p_{i}$, the value of $m_{i}(n)$ runs over all integers. That is, $m_{i}(n)$
and $i$ can run over all integers independently. This comes from the fact that
the direct product of the sets,
$\left\\{p_{i},p_{i}^{2},p_{i}^{3},\cdots,p_{i}^{m},\cdots\right\\}$, for
every prime is equal to the set of natural numbers $\mathbb{N}^{+}$ (see Eq.
(2.15)). We can therefore see that exchange the order of the sum and the
product666 Due to the distributivity of the ordinary multiplication with
regard to the ordinary addition, $l\,(m+n)=l\,m+l\,n$.  in Eq. (2.14) as

$\sum_{n\in\mathbb{N}^{+}}f(n)=\sum_{n=1}^{\infty}\prod_{i}f\left(p_{i}^{m_{i}(n)}\right)=\prod_{i=1}\sum_{m=0}^{\infty}f\left(p_{i}^{m}\right)$
(4.24)

if $f\left(x\right)$ satisfies the property Eq. (2.7).

The definitions of the $q$-algebra introduced here is sufficient to write down
the $q$-version of the Euler sum and the Euler product in $q$-representation
formally as

$\displaystyle\zeta_{q}(s)$ $\displaystyle=\underset{\forall\langle
n\rangle_{q}\in\mathbb{N}_{q}}{\sum}^{\\!\\!\\!\\!\\!\\!q}\;\;\Bigl{\langle}\;\langle
1\rangle_{q}\,\oslash^{q}\,\left(\langle
n\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}s\right)\;\Bigr{\rangle}_{q}$
(4.25)
$\displaystyle={\underset{i\in\mathbb{N}}{\sideset{}{{}^{q}}{\prod}}}\;\Bigl{\langle}\;\langle
1\rangle_{q}\,\oslash^{q}\,\langle\,1-1/p_{i}^{s}\,\rangle_{q}\Bigr{\rangle}_{q},$
(4.26)

and

$\zeta_{q}(s)\equiv\langle\zeta(s)\rangle_{q}$ (4.27)

(with $\zeta_{1}(s)\equiv\zeta(s)$). In other words, the Euler product form is
preserved for all values of $q$. Fig. 2 depicts $\zeta_{q}(s)$ for different
values of $q$.

Figure 2:  $\langle\zeta(s)\rangle_{q}=\zeta_{q}(s)$ for typical values of
$q$. These values have been calculated from Eq. (4.27) with the first $10^{5}$
prime numbers. The $q$-number $\langle n\rangle_{q}$, Eq. (3.2), with $q>1$
presents an upper limit, $\lim_{n\to\infty}\langle
n\rangle_{q>1}=e^{1/(q-1)}$, and this prevents the divergence of
$\zeta_{q}(s)$ for $s<1$.

We numerically identify the location of the divergence by fixing an arbitrary
value of $\zeta_{q}(s)$ noted as $\zeta_{q}^{\text{div}}$ and identify the
corresponding the value of $s$ (noted as $s^{\text{div}}$) with increasing
number of primes. The procedure is repeated with increasing values of
$\zeta_{q}^{\text{div}}$. The three top panels of Fig. 3 illustrate the
procedure for $q=1$, and the three bottom panels for $q=-1$. Each curve in
Fig. 3(top left) displays the value of $s^{\text{div}}$ for which
$\zeta_{q}(s^{\text{div}})\equiv\zeta_{q}^{\text{div}}=\langle\zeta_{1}(s^{\text{div}})\rangle_{q}=10^{3},10^{4},\dots,10^{8}$
with increasing numbers of primes ($10^{3},10^{4},\dots,10^{6}$ primes, shown
with solid circles). The representation with $1/\log_{10}(\text{number of
primes})$ in the abscissa is not a straight line, and we empirically found
that introducing a power $\sigma$ ($\sigma$ depends on
$\zeta_{q}^{\text{div}}$) as shown in Fig. 3(top middle), straight lines
emerge, which can be extrapolated (dashed lines) to infinite number of primes
— the open circles at the ordinate axis. These extrapolations correspond to
infinite number of primes, but, nevertheless, the values of
$\zeta_{q}^{\text{div}}$ are still finite. Finally, the limit
$\zeta_{q}^{\text{div}}\to\infty$ is achieved as illustrated in Fig. 3(top
right). The open circles at the ordinate axis of Fig. 3(top middle) are
represented in Fig. 3(top right) for each value of $\zeta_{q}^{\text{div}}$,
identified with their respective colors, with the change of variables
$1/[\log_{10}(\zeta_{q}^{\text{div}})]^{\mu}$ in the abscissa. $\mu$ is an
empirical power that transforms the curves into straight lines ($\mu$ depends
on $q$; $\mu(q=1)=1$). A final extrapolation is then allowed, identifying the
location of the divergence (open square). The difference
$\lim_{\zeta_{1}^{\text{div}}\to\infty}\lim_{\text{number of
primes}\to\infty}s^{\text{div}}-1$ characterizes the numerical error. We use
the same procedure for $q<1$, and the three bottom panels illustrate the case
$q=-1$.

Fig. 4 shows the final step of the procedure (see Fig. 3 top right and bottom
right panels) for different values of $q\leq 1$, and the maximal estimated
numerical error is less than 3% for the values of $q$ that we have checked.
This result definitely differs from what a brief glance at Fig. 2 might induce
one to think.

Figure 3:  Top left panel: Dependence of
$s^{\text{div}}=s(\zeta_{1}^{\text{div}})$ with $1/\log_{10}(\text{number of
primes})$ in Eq. (2.5); $\zeta_{1}^{\text{div}}$ is a proxy for the divergence
of $\zeta_{1}(s)$. Top middle panel: Power-law rescaling of
$s^{\text{div}}=s(\zeta_{1}^{\text{div}})$ with
$1/\left[\log_{10}(\text{number of primes})\right]^{\sigma}$. The curves point
towards $s^{\text{div}}$ with infinite number of primes (open circles); Top
right panel: Extrapolated values of Fig. 3(top middle) linearly rescaled with
$1/\left[\log_{10}(\zeta_{q}^{\text{div}})\right]^{\mu}$ point towards the
analytically exact value $s^{\text{div}(\infty)}=1$ (open square)
($\lim_{\zeta_{1}^{\text{div}}\to\infty}\lim_{\text{number of
primes}\to\infty}s^{\text{div}}=1$) within a numerical error less than 2% for
$q=1$. The colors of the open circles refer to the values of
$\zeta_{1}^{\text{div}}$ identified in Fig. 3(top middle). Bottom panels: the
same as top panels for $q=-1$, Eq. (4.27).  Figure 4:  The same as Fig. 3(top
right and bottom right) for typical values of $q\leq 1$. The colors of the
open circles identify the value of $\zeta_{q}^{\text{div}}$ as used in Fig.
3(top middle and bottom middle). The colors of the solid lines identify the
value of $q$ according to the legend in the present figure. The divergence of
$\zeta_{q}(s)$ occurs at $s^{\text{div}(\infty)}=1$ (open squares) within a
numerical error less than 3% for the values of $q$ that we have checked.

The numerical procedure can be taken in the inverse order, taking
$\zeta_{q}^{\text{div}}\to\infty$ as the first step, and then taking
increasing number of primes: see Figs. 5 and 6. Each curve in Fig. 5(top left
and bottom left) displays the value of $s^{\text{div}}$ calculated with the
same number of primes in Eq. (4.27) ($10^{3},10^{4},10^{5},10^{6}$ primes) as
a function of $1/\log_{10}\zeta_{q}^{\text{div}}$. Here, similarly to Fig.
3(top left and bottom left), the curves are not straight lines, so they can
hardly be extrapolated. The empirical power-law rescaling shown in Fig. 5(top
middle and bottom middle) indicates that $s^{\text{div}}$ linearly scales with
$1/[\log_{10}\zeta_{q}^{\text{div}}]^{\rho}$ ($\rho$ depends on the number of
primes), and the generated straight lines point to the corresponding values of
$s^{\text{div}}$ with infinite number of primes in the $\zeta_{q}(s)$ function
(open circles of the top middle and bottom middle panels). These extrapolated
values are rescaled according to a power-law shown in Fig. 5(top right and
bottom right), with the empirical power $\nu$ depending on $q$.

Fig. 6 is equivalent to top right and bottom right panels of Fig. 5 for
different values of $q\leq 1$. All these cases indicate $\lim_{\text{number of
primes}\to\infty}\lim_{\zeta_{q}^{\text{div}}\to\infty}s^{\text{div}}=1$
within a numerical error less than 4%

The empirical powers ($\sigma$, $\rho$, $\mu$, $\nu$) have been estimated by
fitting a parabola $y=a+bx+cx^{2}$ to the corresponding curve, $x$ is the
variable of the abscissa of the corresponding middle and right panels,
$y=s^{\text{div}}$, and the fitting value of the power is that for which
$c\approx 0$, estimated with four digits for the power parameter. The
coefficient $a$ of the fitting of the parabola is the extrapolated value of
the corresponding curve (open circles of the top middle and bottom middle
panels of Figs. 3 and 5, open squares of the top right and bottom right panels
of Figs. 3 and 5 and open squares of Figs. 4 and 6).

Similar behavior is expected for $\zeta_{q}(s)$ evaluated with the version
with summations, Eq. (4.25).

Figure 5:  Top left panel: Dependence of
$s^{\text{div}}=s(\zeta_{1}^{\text{div}})$ with
$1/\log_{10}(\zeta_{1}^{\text{div}})$ in Eq. (2.5); Top middle panel: Power-
law rescaling of $s^{\text{div}}=s(\zeta_{1}^{\text{div}})$ with
$1/\left[\log_{10}(\zeta_{1}^{\text{div}})\right]^{\rho}$. The curves point
towards $s^{\text{div}}$ with $\zeta_{q}^{\text{div}}\to\infty$ (open
circles); Top right panel: Extrapolated values of Fig. 5(top middle) linearly
rescaled with $1/\left[\log_{10}(\text{number of primes})\right]^{\nu}$ point
towards $s^{\text{div}(\infty)}=1$ (open square) ($\lim_{\text{number of
primes}\to\infty}\lim_{\zeta_{1}^{\text{div}}\to\infty}s^{\text{div}}=1$)
within a numerical error less than 3%. The colors of the open circles refer to
the number of primes identified in Fig. 5(top middle). Bottom panels: the same
as top panels with $q=-1$, Eq. (4.27).  Figure 6:  The same as Fig. 5(top
right and bottom right) for typical values of $q\leq 1$. The colors of the
open circles identify the number of primes as used in Fig. 5(top middle and
bottom middle). The colors of the solid lines identify the value of $q$
according to the legend in the present figure. The divergence of
$\zeta_{q}(s)$ occurs at $s^{\text{div}(\infty)}=1$ (open squares) within a
numerical error less than 4% for the values of $q$ that we have checked.

##  5  Algebra violating factorizability of $q$-integer numbers in $q$-prime
numbers

The $q$-product corresponding to Eq. (1.9),

$\langle x\otimes_{q}y\rangle_{q}=\langle x\rangle_{q}\,\langle y\rangle_{q},$
(5.1)

is defined as777Eq. (7) of [6], Eq. (48) of [8].

$x\,\otimes_{q}\,y\equiv\Bigl{[}x^{1-q}+y^{1-q}-1\Bigr{]}^{\frac{1}{1-q}}\;\;\;\;(x\geq
0,\,\,y\geq 0;\,x\otimes_{1}y=xy)$ (5.2)

or, equivalently,

$x\otimes_{q}y\equiv e_{q}^{\ln_{q}x+\ln_{q}y}\,,$ (5.3)

and the $q$-sum is defined as888Eq. (A19) of [8].

$\displaystyle x\oplus_{q}y$ $\displaystyle\equiv$ $\displaystyle
e_{q}^{\ln\,[e^{\ln_{q}x}+e^{\ln_{q}y}]}$ (5.4) $\displaystyle=$
$\displaystyle\Bigl{\\{}1+(1-q)\ln\Bigl{[}e^{\frac{x^{1-q}-1}{1-q}}+e^{\frac{y^{1-q}-1}{1-q}}\Bigr{]}\Bigr{\\}}^{1/(1-q)}.$

If $q\neq 1$, $\langle x\rangle_{q}\langle y\rangle_{q}\neq\langle
x\,y\rangle_{q}$. Eq. (5.4) with the symbol $\oplus_{q}$ is equivalent to Eq.
(44) and (A19) of [8] with the symbol ${}_{\\{q\\}}\oplus$, called oel-
addition999 The notation $\oplus_{q}$ adopted in Eq. (5.4) was used as Eq. (4)
of [6] with a different meaning than here, namely $x\oplus_{q}y\equiv
x+y+(1-q)xy$, which is usually referred to as $q$-sum. $x+y+(1-q)xy$ is
denoted in Ref. [8] (Eq. (25)) with the symbol $x\,{}_{[q]}\\!\oplus y$ and is
called ole-addition. .

The properties (4.6)–(4.13) also hold for the $\Circle_{q}$ operations (see
[8]).

$\zeta^{\Sigma^{\prime}}_{q}(s)\equiv\overset{\\!\\!\infty}{{\underset{\\!\\!n=1}{\sideset{}{{}_{q}}{\sum}}}}\;\frac{1}{n^{s}}=1\oplus_{q}\frac{1}{2^{s}}\oplus_{q}\frac{1}{3^{s}}\oplus_{q}\dots$
(5.5)

and

$\displaystyle\zeta^{\Pi^{\prime}}_{q}(s)$ $\displaystyle\equiv$
$\displaystyle\overset{}{{\underset{p\,prime}{\sideset{}{{}_{q}}{\prod}}}}\frac{1}{1-p^{-s}}$
(5.6) $\displaystyle=$
$\displaystyle\frac{1}{1-2^{-s}}\otimes_{q}\frac{1}{1-3^{-s}}\otimes_{q}\frac{1}{1-5^{-s}}\otimes_{q}\cdots$

In the next Section we present details on a specific generalization.

##  6 $q$-Generalizations of the $\zeta(s)$ function directly stemming from
$q$-numbers

We define the following $q$-generalizations of the Riemann $\zeta$ function:

$\zeta^{\Sigma}_{q}(s)\equiv\sum_{n=1}^{\infty}\frac{1}{\langle
n\rangle_{q}^{s}}=1+\frac{1}{\langle 2\rangle_{q}^{s}}+\frac{1}{\langle
3\rangle_{q}^{s}}+\dots\;(s\in\mathbb{R})$ (6.1)

and

$\displaystyle\zeta^{\Pi}_{q}(s)$ $\displaystyle\equiv$
$\displaystyle\prod_{p\,prime}\frac{1}{1-\langle p\rangle_{q}^{-s}}$ (6.2)
$\displaystyle=$ $\displaystyle\frac{1}{1-\langle
2\rangle_{q}^{-s}}\frac{1}{1-\langle 3\rangle_{q}^{-s}}\frac{1}{1-\langle
5\rangle_{q}^{-s}}\cdots\,,$

where $\langle n\rangle_{q}$ is the $q$-number defined in Eq. (1.4); see Figs.
7 and 8.

Figure 7:  Influence on $\zeta^{\Sigma}_{1}(x)=\zeta^{\Pi}_{1}(x)=\zeta(x)$ of
the number of terms in the sum and in the product, where we have used
respectively Eq. (6.1) and (6.2).

Figure 8:  $\zeta^{\Sigma}_{q}(s)$ and $\zeta^{\Pi}_{q}(s)$ for typical values
of $q\leq 1$.

Other possibilities may naturally be considered for generalizing $\zeta(s)$,
for instance

$\zeta^{\Sigma^{\prime\prime}}_{q}(s)\equiv\sum_{n=1}^{\infty}\Bigl{\langle}\frac{1}{n^{s}}\Bigr{\rangle}_{q}=1+\Bigl{\langle}\frac{1}{2^{s}}\Bigr{\rangle}_{q}+\Bigl{\langle}\frac{1}{3^{s}}\Bigr{\rangle}_{q}+\dots$
(6.3)

and

$\displaystyle\zeta^{\Pi^{\prime\prime}}_{q}(s)$ $\displaystyle\equiv$
$\displaystyle\prod_{p\,prime}\Bigl{\langle}\frac{1}{1-p^{-s}}\Bigr{\rangle}_{q}$
(6.4) $\displaystyle=$
$\displaystyle\Bigl{\langle}\frac{1}{1-2^{-s}}\Bigr{\rangle}_{q}\Bigl{\langle}\frac{1}{1-3^{-s}}\Bigr{\rangle}_{q}\Bigl{\langle}\frac{1}{1-5^{-s}}\Bigr{\rangle}_{q}\cdots$

Let us however clarify that it is not the scope of the present paper to
systematically study all such possibilities.

##  7 Final remarks

Let us now illustrate the two algebraic approaches focused on in the present
paper (see Fig. 9):

$\langle 18\rangle_{q}=\langle 2\rangle_{q}\otimes^{q}\langle
3\rangle_{q}\otimes^{q}\langle 3\rangle_{q}=\langle
2\rangle_{q}\otimes^{q}\bigl{(}\langle
3\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}^{q}2\bigr{)}\;\;\;(\forall
q)$ (7.1)

whereas

$\langle 18\rangle_{q}\neq\langle 2\rangle_{q}\otimes_{q}\langle
3\rangle_{q}\otimes_{q}\langle 3\rangle_{q}\neq\langle
2\rangle_{q}\otimes_{q}\bigl{(}\langle
3\rangle_{q}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\scriptstyle{\wedge}}{\medcirc}}_{q}2\bigl{)}\;\;\;(\forall
q\neq 1).$ (7.2)

Analogously we have

$\langle 5\rangle_{q}=\langle 2\rangle_{q}\oplus^{q}\langle
3\rangle_{q}\;\;\;(\forall q)$ (7.3)

whereas

$\langle 5\rangle_{q}\neq\langle 2\rangle_{q}\oplus_{q}\langle
3\rangle_{q}\;\;\;(\forall q\neq 1).$ (7.4)

Figure 9:  Left panel: Factorization is preserved through the generalized
product defined by (4.1), illustrated with $\langle
2\rangle_{q}\otimes^{q}\langle 3\rangle_{q}\otimes^{q}\langle 3\rangle_{q}$
for different values of $q$ (black curve); the generalized product defined by
(5.3), $\langle 2\rangle_{q}\otimes_{q}\langle 3\rangle_{q}\otimes_{q}\langle
3\rangle_{q}$, does not preserve factorization (green curve). The red curve is
the corresponding $q$-number, Eq. (1.4). Right panel: instance of the $q$-sum
of $q$-numbers, with the $q$-sums given by (4.2), $\langle
2\rangle_{q}\oplus^{q}\langle 3\rangle_{q}$, (black curve) and (5.4), $\langle
2\rangle_{q}\oplus_{q}\langle 3\rangle_{q}$, (green curve). The red curve is
the corresponding $q$-number, Eq. (1.4). Notice that there exists a nontrivial
value of $q\neq 1$ for which $\langle 2\rangle_{q}\otimes_{q}\langle
3\rangle_{q}\otimes_{q}\langle 3\rangle_{q}=18$ and, similarly, $\langle
2\rangle_{q}\oplus_{q}\langle 3\rangle_{q}=5$.

The algebra preserving the factorizability of $q$-integer numbers into
$q$-prime numbers (see equality (7.1)) achieves this remarkable property
essentially because it is isomorphic to the usual prime numbers. On the other
hand, precisely because of that, it is unable to properly $q$-generalize the
concept of a vectorial space in terms of nonlinearity. In contrast, the
algebra which violates the factorizability of $q$-integer numbers into
$q$-prime numbers (see inequality (7.2)), or some similar algebra, emerges as
a possible path for achieving the concept of nonlinear vector spaces, which
has the potential of uncountable applications in theoretical chemistry and
elsewhere.

Since the inequality relation between $q$-primes remains the same as that for
$q=1$, it is plausible that the nontrivial zeros in the analytic extension
behaves similarly. More precisely, it might well be that, by extending
$\zeta_{q}(s)$, $\zeta^{\Sigma}_{q}(s)$ and $\zeta^{\Pi}_{q}(s)$ to the
complex plane $z$, all nontrivial zeros belong to specific single continuous
curves, $\mathbb{R}(z)=f_{q}(\mathbb{I}(z))$, thus $q$-generalizing the $q=1$
Riemann’s 1859 celebrated conjecture $\mathbb{R}(z)=1/2$.

We have here explored generalizations of the $\zeta(s)$ function based on a
specific type of $q$-number, Eq. (1.4), and two associated generalized
algebras (Sections 4 and 5). Three additional forms of $q$-generalized
numbers, Eqs. (1.5)–(1.7), are identified in Ref. [8]. To each of these
$q$-numbers, we can associate two consistently generalized algebras, one of
them violating the factorizability in prime numbers (see Ref. [8]), the other
one following along the lines of Section 4. Similarly, various other
generalizations of the $\zeta(s)$ function may of course be developed.
Naturally, the extension of the present $q$-generalized $\zeta(s)$ functions
to complex $z$ surely is interesting, but does not belong to the aim of the
present effort. In any case, the intriguing fact that various infinities
appear to linearly scale with negative powers of logarithms might indicate
some general tendencies.

It is well known that both random matrices and quantum chaos (classically
corresponding to strong chaos, i.e., positive maximal Lyapunov exponent) [16,
17, 18, 19, 20] are related to the Riemann $\zeta$-function and prime numbers.
On the other hand, both random matrices and strong chaos have been
conveniently $q$-generalized, in [21] and [22, 23, 24] respectively. These
facts open the door for possible applications of the present
$q$-generalizations of prime numbers and of the $\zeta$-function to $q$-random
matrices and to weak chaos (classically corresponding to vanishing maximal
Lyapunov exponent, which recovers strong chaos in the $q\to 1$ limit).
Moreover, connections of the present developments within the realm of the
theory of numbers, or, more specifically, the theory of prime numbers, remain,
at this stage, out of our scope. Further work along these lines would
naturally be very welcome.

## Acknowledgments

This work has received partial financial support by CNPq, CAPES and FAPERJ
(Brazilian agencies).

## References

  * [1] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479-487 (1988).
  * [2] A. Enciso and P. Tempesta, Uniqueness and characterization theorems for generalized entropies, J. Stat. Mech. 123101 (2017).
  * [3] P. Tempesta, Group entropies, correlation laws, and zeta functions, Phys. Rev. E 84, 021121 (2011).
  * [4] C. Tsallis, What are the numbers that experiments provide?, Quimica Nova 17, 468 (1994).
  * [5] L. Nivanen, A. Le Méhauté and Q. A. Wang, Generalized algebra within a nonextensive statistics, Rep. Math. Phys. 52, 437 (2003).
  * [6] E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A 340, 95–101 (2004).
  * [7] L. Nivanen et al, Hierarchical structure of operations defined in nonextensive algebra, Rep. Math. Phys. 63, 279 (2009).
  * [8] E. P. Borges and B. G. da Costa, Deformed mathematical objects stemming from the $q$-logarithm function, (2021), arxiv 2105.01549 [math.GM].
  * [9] I. S. Gomez and E. P. Borges, Algebraic structures and position-dependent mass Schrödinger equation from group entropy theory, Lett. Math. Phys. 11, 43 (2021).
  * [10] N. Kalogeropoulos, Algebra and calculus for Tsallis thermo-statistics, Physica A 356, 408 (2005).
  * [11] F. D. Nobre, M. A. Rego-Monteiro and C. Tsallis, Nonlinear Relativistic and Quantum Equations with a Common Type of Solution, Phys. Rev. Lett. 106, 140601 (2011).
  * [12] M. Czachor, Unifying Aspects of Generalized Calculus, Entropy 22, 1180 (2020).
  * [13] T.C.P. Lobao, P.G.S. Cardoso, S.T.R. Pinho, E.P. Borges, Some properties of deformed $q$-numbers, Braz. J. Phys 39, 402–407 (2009).
  * [14] M. J. S. Haran, The Mysteries of the Real Prime, (Oxford University Press, 2001).
  * [15] V. Kac and P. Cheung, Quantum Calculus, (Springer, 2002).
  * [16] M. V. Berry, Riemann’s zeta function: A model for quantum chaos?, in Quantum Chaos and Statistical Nuclear Physics, T. H. Seligman and H. Nishioka, eds., Lecture Notes in Phys. 263 (Springer-Verlag, New York, 1986).
  * [17] M. V. Berry and J.P. Keating, The Riemann Zeros and Eigenvalue Asymptotics, SIAM Review 41, 236-266 (1999).
  * [18] J.P. Keating and N.C. Snaith, Random matrix theory and $\zeta(1/2+it)$, Commun. Math. Phys. 214, 57-89 (2000).
  * [19] F. W. K. Firk and S. J. Miller, Nuclei, primes and the random matrix connection, Symmetry 1, 64-105 (2009).
  * [20] T. Kriecherbauer, J. Marklof and A. Soshnikov, Random matrices and quantum chaos, PNAS 98 (19), 10531-10532 (2001).
  * [21] F. Toscano, R.O. Vallejos and C. Tsallis, Random matrix ensembles from nonextensive entropy, Phys. Rev. E 69, 066131 (2004).
  * [22] Y.S. Weinstein, S. Lloyd and C. Tsallis, Border between between regular and chaotic quantum dynamics, Phys. Rev. Lett. 89, 214101 (2002).
  * [23] Y. S. Weinstein, C. Tsallis and S. Lloyd, On the emergence of nonextensivity at the edge of quantum chaos, in Decoherence and Entropy in Complex Systems, ed. H.-T. Elze, Lecture Notes in Physics 633 (Springer-Verlag, Berlin, 2004), page 385.
  * [24] S. M. D. Queiros and C. Tsallis, Edge of chaos of the classical kicked top map: Sensitivity to initial conditions, in Complexity, Metastability and Nonextensivity, Proc. 31st Workshop of the International School of Solid State Physics (20-26 July 2004, Erice-Italy), eds. C. Beck, G. Benedek, A. Rapisarda and C. Tsallis (World Scientific, Singapore, 2005), page 135.


# Affinity-based geometric discord and quantum speed limits of its creation
and decay

Rajendran Muthuganesan111Corresponding author Department of Physics, Faculty
of Nuclear Sciences and Physical Engineering, Czech Technical University in
Prague, Br̆ehová 7, 115 19 Praha 1-Staré Mĕsto, Czech Republic, Email:
<EMAIL_ADDRESS>S. Balakrishnan Department of Physics, School of
Advanced Sciences, Vellore Institute of Technology, Vellore– 632014, India.

###### Abstract

In this article, we define a faithful quantifiers of bipartite quantum
correlation, namely geometric version of quantum discord using affinity based
metric. It is shown that the newly-minted measure resolves the local ancilla
problem of Hilbert-Schmidt measures. Exploiting the notion of affinity-based
discord, we derive Margolus-Levitin (ML) and Mandelstamm-Tamm (MT) bounds for
the quantum speed limit time for the creation and decay of quantum
correlation. The dynamical study suggests that the affinity measure is a
better resource compared to entanglement. Finally, we study the role of
quantum correlation on quantum speed limit.

Keywords : Quantum speed limit; Quantum correlationa; Affinity; Uncertainty
relation; Ornstein-Uhlenbeck noise.

##  1 Introduction

Entanglement and uncertainty, foundational aspects of the physical systems in
the quantum regime, are the building blocks for the development of modern
quantum technology. Entanglement is a peculiar phenomena emerges from the
superposition of quantum states and found to be an important for advantages of
various quantum tasks such as teleportation, super dense coding, quantum
communication etc [1]. Owing to the rigorous framework of quantum resource
theory, different entanglement measures have been proposed in both bipartite
and multipartite scenarios [2]. Over the past decades, the research have been
carried out on the nonlocal features and the analysis centered around the
violation of Bell inequality [3, 4, 5]. The characterization of quantum
systems from the perspective of nonlocality is a formidable task and reveals
that the entanglement is not a whole indicators of the nolocal attributes of
the quantum system [6]. Further development in the quantum resource theory
identifies the different measures such as quantum discord [7], measurement-
induced nonlocality [8], quantum coherence [9, 10] etc.

On the other hand, the uncertainty relation (UR), originally introduced by
Heisenberg, places the accuracy while measuring the observable [11]. In other
words, it is shown that the outcomes for two incompatible measurements cannot
be accurately measured at the same time. In recent times, the entropy based
uncertainty relation is powerful tool in the characterization of quantum
system dynamics in the different environment. Later, energy-time UR also
interpreted as quantum speed limits (QSLs) of unitary evolution of physical
systems and QSL is defined as the bounds on the shortest time required to
evolve between two quantum states [12].

QSL provides a key understanding on the evolution and controlling of quantum
system, and also unifies the framework of the both computational and evolution
speed. Due to the wide range of applications of QSL in quantum information
theory, the researchers got enormous interest on characterization of QSL for
nonunitary dynamics. QSL has been studied from the different perspectives such
as decay rate of quantum states, parameter estimation, rate of information
transfer and processing, entropy production [28], precision in quantum
metrology, state preparation and time scales of quantum optimal control, many-
body open system [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].
Recently, QSL for the states with a bounded energy spectrum is also studied
[28] and realizable using different candidates such as on single photons [29],
trapped ions [30], trapped atoms [31], and artificial atoms in superconducting
quantum circuits [32, 33]. In recent times, to quantify the maximal quantum
speed of systems interacting with their environments different approaches have
been proposed in terms of Fisher information [34], relative purity [35] and
geometric view point [36]. It is shown that the entanglement “speed up” the
time evolution of composite systems [37, 38, 39]. Identifying the relation
between QSL and resource theory is a promising field with in the framework of
QIT. Ornstein-Uhlenbeck (OU) noise is an important non-Markovian decoherence
in the framework of quantum information theory which provides a mathematical
description of Brownian motion [40]. The Ornstein–Uhlenbeck noise has been
widely studied in different perspectives [41, 42, 43, 44]. The OU noise could
be helpful to control the loss of quantum resources in quantum system’s
dynamical maps. Hence, the understanding of the loss of information and
control of resources under OU noise is very essential in the context of
quantum information theory.

In view of this, in this present work, we derive Margolus-Levitin (ML) and
Mandelstamm-Tamm (MT) like bound of QSL time for the decay and creation of
quantum correlation using newly minted affinity based geometric discord. The
affinity based quantum discord measure resolves the local ancilla problem of
Hilbert-Schmidt based quantifiers [8, 45]. It is shown that the robustness of
affinity-based discord under Ornstein-Uhlenbeck (OU) dephasing noise suggest
it would be a better resource than the entanglement. We investigate the QSL
time for the decay and creation of quantum correlations for the nonunitary
evolution under OU dephasing noise.

The paper is structured as follows. In Sec. 2, first we review the notion of
quantum discord, then we define affinity-based geometric version of quantum
discord. In Sec. 3, we derive ML and MT like bound of QSL time through the
newly-minted discord. The dynamics of quantum correlation under OU dephasing
noise and QSL time are calculated in Sec. 4. Finally, the conclusions are
presented in Sec. 5.

##  2 Affinity-based discord

It is widely accepted fact that the entanglement could not capture all the
nonlocal aspects of a quantum system. To provide complete the spectrum of the
nonlocal aspects of quantum system, Olivier and Zurek identified a new
quantifiers from the perspective measurement is called quantum discord (QD).
First, we recall some basic notations and definitions related the discord
measure. Let $\rho_{ab}$ be a two-qubit state shared between the parties $a$
and $b$ in the separable Hilbert space
$\mathcal{H}^{a}\otimes\mathcal{H}^{b}$. Then the mutual information of
$\rho_{ab}$ is defined as

$\displaystyle\mathcal{I}(\rho_{ab})=S(\rho^{a})+S(\rho^{b})-S(\rho_{ab})$ (1)

where $\rho^{i}=\text{Tr}_{j}(\rho_{ab})\leavevmode\nobreak\
(\\{i,j\\}\in\\{a,b\\},i\neq j)$ are the marginal states of $\rho_{ab}$. Here,
$S(\rho)=-\text{Tr}\rho\text{log}\rho$ is von Neumann entropy of quantum state
and logarithm is base 2. The quantum conditional entropy of the bipartite
state is given by

$\displaystyle S_{a|b}=\leavevmode\nobreak\
^{\text{min}}_{\Pi^{b}_{k}\in\mathcal{M}}\sum_{k}p_{k}S(\rho_{a|k})$

where the minimization is taken over all quantum measurements
$\\{\Pi^{b}_{k}\\}$ on $b$ and $\mathcal{M}$ is set of all such measurements.
The post-measurement state is
$\rho_{a|k}=\text{Tr}_{b}[(\mathds{1}\otimes\Pi_{k}^{b})\rho_{ab}(\mathds{1}\otimes\Pi_{k}^{b})]/p_{k}$
with
$p_{k}=\text{Tr}(\mathds{1}\otimes\Pi_{k}^{b})\rho_{ab}(\mathds{1}\otimes\Pi_{k}^{b})$.
The equivalent expression of mutual information eq. (1), defined as

$\displaystyle\mathcal{J}_{a|b}=S(\rho^{a})-S_{a|b}.$ (2)

The mutual information eq. (1) is always greater than eq. (2) and both are
equal in the classical regime. The QD is defined as [7]

$\displaystyle\mathcal{D}=\mathcal{I}(\rho_{ab})-\mathcal{J}_{a|b}$ (3)

the difference between two inequivalent expressions of mutual information in
the quantum regime. The above quantity also demonstrate the presence of
correlation in separable state. Even though, the complete manifestation of
nonlocality, the computation of QD is a non-deterministic polynomial even for
a simple bipartite state.

Motivated by the notion of quantum discord, the geometric version of discord
is defined as [45, 46]

$\displaystyle\mathcal{D}_{G}(\rho_{ab})=\leavevmode\nobreak\
\leavevmode\nobreak\
^{\text{min}}_{\\{\Pi_{k}^{a}\\}}\lVert\rho_{ab}-\Pi^{a}(\rho_{ab})\rVert^{2}$
(4)

where the optimization is taken over von Neumann projective measurements and
$\Pi^{a}(\rho)=\sum_{k}(\Pi_{k}^{a}\otimes\mathds{1})\rho(\Pi_{k}^{a}\otimes\mathds{1})$
with $\\{\Pi_{k}^{a}\\}=\\{|k\rangle\langle k|\\}$ are the projectors acting
on state space $a$ and $\mathds{1}$ is the identity operator in state space
$b$. Here
$\lVert\mathcal{O}\rVert^{2}=\text{Tr}\mathcal{O}\mathcal{O}^{\dagger}$ is the
Hilbert-Schmidt norm of operator $\mathcal{O}$. The quantum discord based on
the Hilbert-Schmidt is not a faithful measure of quantum correlation due to
the non-cotractvity and local ancilla problem. To combat this local ancilla
issue, the different versions of geometric discord is studied using Schatten
1-norm [48], Bures distance [49] and Hellinger distance [50].

In this context, we introduce a new version of geometric QD based on affinity
as

$\displaystyle\mathcal{D}(\rho_{ab})=\leavevmode\nobreak\ \leavevmode\nobreak\
^{\text{min}}_{\\{\Pi_{k}^{a}\\}}d_{\mathcal{A}}(\rho_{ab},\Pi^{a}(\rho_{ab}))$
(5)

where $\mathcal{A}(\rho,\sigma)=\text{Tr}\sqrt{\rho}\sqrt{\sigma}$ is the
affinity between the states $\rho$ and $\sigma$ [47, 51] and
$d_{\mathcal{A}}(\rho,\sigma)=1-\sqrt{\mathcal{A}(\rho,\sigma)}$ is a
affinity-induced metric. In recent times, the usefulness of affinity in
characterization of quantum correlation of bipartite state [52], bilocal state
[53] and quantum coherence [54, 55] is also demonstrated. A similar quantum
correlation measure is characterized using Hellinger distance [50]. The
properties of the affinity-based measure are

  1. (i)

$\mathcal{D}(\rho_{ab})$ is non-negative and positive for entangled state.

  2. (ii)

$\mathcal{D}(\rho_{ab})=0$ for any product state and the classical-quantum
state in the form $\rho_{ab}=\sum_{k}p_{k}|k\rangle\langle k|\otimes\rho_{k}$
with nondegenerate marginal state $\rho^{a}=\sum_{k}p_{k}|k\rangle\langle k|$.

  3. (iii)

$\mathcal{D}(\rho_{ab})$ is locally unitary invariant i.e.,
$\mathcal{D}\left((U\otimes V)\rho_{ab}(U\otimes
V)^{\dagger}\right)=\mathcal{D}(\rho_{ab})$ for any local unitary operators
$U$ and $V$.

  4. (iv)

$\mathcal{D}(\rho_{ab})$ is invariant under the addition of any local ancilla
to the unmeasured party $b$.

  5. (v)

For any arbitrary pure state
$|\Psi\rangle=\sum_{i}s_{i}|\alpha_{i}\rangle\otimes|\beta_{i}\rangle$ with
$m\leq n$, the
$\mathcal{D}(|\Psi\rangle\langle\Psi|)=1-\sqrt{\sum_{i}s_{i}^{2}}$.

  6. (vi)

Any $2\times n$ dimensional mixed state, the measure $\mathcal{D}(\rho_{ab})$
has a closed formula:

$\displaystyle\mathcal{D}(\rho_{ab})=1-\leavevmode\nobreak\
\leavevmode\nobreak\
^{\text{max}}_{r_{ki}}\sqrt{r_{ki}T_{ij}r_{ki}}=1-\sqrt{T_{1}}$ (6)

where $T_{1}$ is the latgest eigenvalue of the matrix $T$ with the elements
$T_{ij}=\text{Tr}[\sqrt{\rho_{ab}}(\sigma_{i}\otimes\mathds{1})\sqrt{\rho_{ab}}(\sigma_{j}\otimes\mathds{1})]$.

The properties (i) - (iv) guarantee that the affinity-based discord is
legitimate measure of quantum correlation.

Now, we proceed to sketch the proof of the above properties. The property (i)
is straight forward to understand. To prove the property (ii), we consider the
eigenprojectors of the marginal state
$\rho^{a}=\text{Tr}_{b}(\rho_{ab})=\sum_{k}p_{k}|k\rangle\langle k|$ as
$\\{\Pi_{k}^{a}\\}=\\{|k\rangle\langle k|\\}$. Then the state and measured
state for the product and classical-quantum state are equal i.e.,
$\Pi^{a}(\rho_{ab})=\rho_{ab}$ and the desired results follows. Property (iii)
follows from the unitary invarience of affinity.

In order to prove the property (iv), it is sufficient to prove the affinity do
not affected by the addition of local ancilla. First, we define a map
$\Gamma^{\sigma}:X\rightarrow X\otimes\sigma$, i.e., the map adding a noisy
ancillary state to the party $b$. Under such an operation [56, 57]

$\displaystyle\|X\|\rightarrow\|\Gamma^{\sigma}X\|=\|X\|\sqrt{\text{Tr}\sigma^{2}}.$
(7)

Due to the addition of local ancilla $\rho^{c}$, the resultant state is
$\rho_{a:bc}=\rho_{ab}\otimes\rho_{c}$. Here : represents the partition
between the system. Hence the geometric discord eq.(4 ) of the resultant state
is

$\displaystyle\mathcal{D}_{G}(\rho^{a:bc})=\mathcal{D}_{G}(\rho^{ab})\cdot\text{Tr}(\rho^{c})^{2}$

implying that Hilbert-Schmidt based discord is not an ideal candidate in
measuring quantum correlation. To show the affinity based measure resolves
this issue, we recall the multiplicative property of the affinity:

$\displaystyle\mathcal{A}(\rho_{1}\otimes\rho_{2},\sigma_{1}\otimes\sigma_{2})=\mathcal{A}(\rho_{1}\otimes\sigma_{1})\cdot\mathcal{A}(\rho_{2}\otimes\sigma_{2}).$
(8)

The affinity between the pre– and post–measurement states is written as

$\displaystyle\mathcal{A}\left(\rho_{a:bc},\Pi^{a}(\rho_{a:bc})\right)=\mathcal{A}\left(\rho_{ab}\otimes\rho^{c},\Pi^{a}(\rho_{ab})\otimes\rho^{c}\right).$

Using properties of affinity such as multiplicativity and the affinity of same
state is unity, we show that

$\displaystyle\mathcal{A}\left(\rho_{a:bc},\Pi^{a}(\rho_{a:bc})\right)=\mathcal{A}\left(\rho_{ab},\Pi^{a}(\rho_{ab})\right),$

the affinity do not altered by the addition of local ancilla. Here, the
contractivity of the distance measure $d_{\mathcal{A}}(\rho,\sigma)$ is shown
in the context of local ancilla problem. In general, we can show that
contractivity of the affinity distance from the monotonicity property of the
affinity [55]. Hence the contractive nature evident that the affinity based
measure is a faithful measure of quantum correlation.

To show the property (v), we follow the Schmidt decomposition of the pure
state as
$|\Psi\rangle=\sum_{i}\sqrt{s_{i}}|\alpha_{i}\rangle\otimes|\beta_{i}\rangle$
and the corresponding density matrix is
$\rho=|\Psi\rangle\langle\Psi|=\sum_{ij}\sqrt{s_{i}s_{j}}|\alpha_{i}\rangle\langle\alpha_{j}|\otimes|\beta_{i}\rangle\langle\beta_{j}|$.
The von Neumann projective measurements on the subsystem $a$ can be
represented as
$\\{\Pi_{k}^{a}\\}=\\{U|\alpha_{k}\rangle\langle\alpha_{k}|U^{\dagger}\\}$ and
the subsystem $\rho^{a}$ can be written as

$\displaystyle\rho^{a}=\sum_{k}\langle\alpha_{k}|U^{\dagger}\rho^{a}U|\alpha_{k}\rangle
U|\alpha_{k}\rangle\langle\alpha_{k}|U^{\dagger}$ (9)

the spectral decomposition in the measurement basis
$\\{U|\alpha_{k}\rangle\\}$ with the eigenvalues
$\langle\alpha_{k}|U^{\dagger}\rho^{a}U|\alpha_{k}\rangle$.

Exploiting the identity $\Pi^{a}f(\rho)\Pi^{a}=f(\Pi^{a}\rho\Pi^{a})$ [58],
the square root of post-measurement state is written as
$\sqrt{\Pi^{a}(\rho_{ab})}=\Pi^{a}(\sqrt{\rho_{ab}})$. After the straight
forward algebra, the affinity between the state under our consideration and
the measured state is computed as

$\displaystyle\mathcal{A}(\rho_{ab},\Pi^{a}(\rho_{ab}))=\sum_{k}s_{k}^{2}.$
(10)

Then the affinity based geometric discord for pure state is computed as

$\displaystyle\mathcal{D}(\rho_{ab})=1-\sqrt{\sum_{k}s_{k}^{2}}.$ (11)

To derive the closed formula of affinity based measure (property vi), we
define the mixed quantum state in the separable Hilbert space
$\mathcal{H}^{a}\otimes\mathcal{H}^{b}$. Assume
$\\{X_{i}:i=0,1,2,\cdots,m^{2}-1\\}\in\mathcal{L}(\mathcal{H}^{a})$ be a set
of orthonormal operators for the state space $\mathcal{H}^{a}$ and
$\\{Y_{j}:j=0,1,2,\cdots,n^{2}-1\\}\in\mathcal{L}(\mathcal{H}^{b})$ for the
state space $\mathcal{H}^{b}$. The operators $X_{i}$ and $Y_{j}$ are
satisfying the conditions
$\text{Tr}(X_{k}^{\dagger}X_{l})=\text{Tr}(Y_{k}^{\dagger}Y_{l})=\delta_{kl}$.
Any arbitrary $m\times n$ dimensional state of a bipartite system in the
composite space $\mathcal{L}(\mathcal{H}^{a}\otimes\mathcal{H}^{b})$ can be
written as

$\sqrt{\rho_{ab}}=\sum_{i,j}\gamma_{ij}X_{i}\otimes Y_{j},$ (12)

where $\Gamma$ is the real matrix with the elements
$\gamma_{ij}=\text{Tr}(\sqrt{\rho}\leavevmode\nobreak\ X_{i}\otimes Y_{j})$.
After a straight forward calculation, we compute the affinity between pre- and
post- measurement state as

$\displaystyle\mathcal{A}(\rho_{ab},\Pi^{a}(\rho_{ab}))=\text{Tr}(R\Gamma\Gamma^{t}R^{t}).$
(13)

where $R$ is the measurement operators with $r_{ki}=\text{Tr}(|k\rangle\langle
k|X_{i})$. Then the affinity based measure is

$\mathcal{D}(\rho_{ab})=1-\leavevmode\nobreak\ \leavevmode\nobreak\
^{\text{max}}_{R}\sqrt{\text{Tr}(R\Gamma\Gamma^{t}R^{t})}.$ (14)

In order to obtain the closed formula for any $2\times n$ dimensional mixed
state, we follow the optimization technique given in Ref.[57]. The Bloch
representation of the eigenprojectors is given by

$\displaystyle\Pi_{k}^{a}=\frac{1}{2}[\mathds{1}_{2}+\vec{r}\cdot\vec{\sigma}]$

with $\vec{r}\cdot\vec{\sigma}=\sum_{i=1}^{3}r_{ki}\sigma_{i}$ and
$\sum_{i}r_{ki}=1$. Then the measurement operator $R$ is

$\displaystyle R=\begin{pmatrix}1&r_{01}&r_{02}&r_{03}\\\
1&-r_{01}&-r_{02}&-r_{03}\end{pmatrix}$

and substituting in eq. (14), we obtain the closed formula of affinity based
discord as

$\displaystyle\mathcal{D}(\rho_{ab})=1-\leavevmode\nobreak\
^{\text{max}}_{r_{ki}}\sqrt{r_{ki}T_{ij}r_{ki}}=1-\sqrt{T_{1}}$ (15)

where $T_{1}$ is the largest eigenvalue of the matrix $T$ with the elements
$T_{ij}=\text{Tr}[\sqrt{\rho_{ab}}(\sigma_{i}\otimes\mathds{1})\sqrt{\rho_{ab}}(\sigma_{j}\otimes\mathds{1})]$.
Due to the analytical formula, the affinity based correlation measure is easy
to compute while compared with the quantum discord.

##  3 Quantum Speed Limit

In this section, we aim to define quantum speed time $\tau$ for the decay or
creation of quantum correlation in a system for a non-unitary evolution. To
derive the Margolus-Levitin (ML) and Mandelstamm-Tamm (MT) bounds for the
quantum speed limit time for the creation and decay of quantum correlation, we
follow the procedure given in Ref. [59]. It is defined as the minimum time
required to change by an amount

$\displaystyle\Delta\mathcal{Q}=\lvert\mathcal{D}(\rho_{0})-\mathcal{D}(\rho_{t})\rvert$
(16)

is the difference between the initial state correlation
$\mathcal{D}(\rho_{0})$ and final state correlation $\mathcal{D}(\rho_{t})$
after the nonunitary evolution. Here, we consider the nonunitary evolution of
the form

$\displaystyle\dot{\rho}_{t}=\mathcal{L}_{t}\rho_{t}$

where $\mathcal{L}_{t}$ is the generator of the evolution. In order to
quantify the quantum speed limit (QSL) time for $\Delta\mathcal{Q}$, we
exploit the notion of affinity-based geometric discord introduced in Eq. (5).
The difference between discord of the initial and final state after the
nonunitary evolution can be written as

$\Delta\mathcal{Q}=\begin{cases}\sqrt{\mathcal{A}(\rho_{t})}-\sqrt{\mathcal{A}(\rho_{0})}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{D}(\rho_{0})\geq\mathcal{D}(\rho_{t}),\\\
\sqrt{\mathcal{A}(\rho_{0})}-\sqrt{\mathcal{A}(\rho_{t})}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{otherwise}.\end{cases}$ (17)

Here $\mathcal{A}(\cdot)$ represents the optimal affinity between the state
under our consideration and corresponding measured state. From the above, the
temporal rate of change of $\Delta\mathcal{Q}$ is computed as

$\displaystyle\frac{d}{dt}\Delta\mathcal{Q}=\frac{\dot{\mathcal{A}}(\rho_{t})}{2\sqrt{\mathcal{A}(\rho_{t})}}.$

It is obvious that
$\frac{d}{dt}\Delta\mathcal{Q}\leq\lvert\frac{d}{dt}\Delta\mathcal{Q}\rvert$,
then

$\displaystyle
2(1-\mathcal{D}(\rho_{t}))\frac{d}{dt}\Delta\mathcal{Q}\leq\lvert\dot{\mathcal{A}}(\rho_{t})\rvert.$
(18)

Let $\rho_{\psi}(t)=|\psi(t)\rangle\langle\psi(t)|$ and
$\sigma_{\phi}(t)=|\phi(t)\rangle\langle\phi(t)|$ are the purification of the
states $\rho$ and $\Pi^{a}(\rho)$ respectively. Then, we have

$\displaystyle
2(1-\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q})\frac{d}{dt}\Delta\mathcal{Q}\leq\lvert\langle\phi|\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}|\phi\rangle+\langle\psi|\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2}|\psi\rangle\rvert$
(19)

where $\mp$ indicates the decay or creation of quantum correlation under
nonunitary evolution and $\mathcal{L}_{t}$ is generator of the evolution.

###  3.1 Margolus-Levitin bound:

Using the trace property $\text{Tr}|m\rangle\langle n|A=\langle n|A|m\rangle$,
the above equation can be recast as

$\displaystyle
2(1-\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q})\frac{d}{dt}\Delta\mathcal{Q}\leq\lvert\text{Tr}\\{|\phi\rangle\langle\phi|\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}\\}+\text{Tr}\\{|\psi\rangle\langle\psi|\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2}\\}\rvert$
(20)

Let $A$ and $B$ are the two complex matrices. Then the von Neumann trace
inequality is $\text{Tr}|AB|\leq\sum_{i=1}^{n}\lambda_{1,i}\lambda_{2,i}$ with
$\lambda_{1,i}(\lambda_{2,i})$ are the singular values of $A(B)$ arranged in
decreasing order and using the triangle inequality of absolute values, the
above inequality can be written as

$\displaystyle
2(1-\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q})\frac{d}{dt}\Delta\mathcal{Q}\leq\lambda_{1,1}+\lambda_{2,1}$
(21)

where $\lambda_{1,i}(\lambda_{2,i})$ is the singular values of
$\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}(\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2})$.
Then, we have

$\displaystyle
2(1-\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q})\frac{d}{dt}\Delta\mathcal{Q}\leq\|\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}\|_{op}+\|\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2})\|_{op}$
(22)
$\displaystyle\leq\|\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}\|_{tr}+\|\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2})\|_{tr}.$
(23)

Integrating over the time, we obtain the QSL limit

$\displaystyle\tau\geq\tau_{QC}=\text{max}\\{\frac{1}{\Lambda_{op}},\frac{1}{\Lambda_{tr}}\\}2\Delta\mathcal{Q}\left(1-\frac{2\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q}}{2}\right)$
(24)

where
$\Lambda_{op,tr}^{\tau}=1/\tau\int_{0}^{\tau}dt(\|\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}\|_{op,tr}+\|\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2})\|_{op,tr})$.

###  3.2 Mandelstamm-Tamm bound:

To derive MT bound, we rewrite the inequality Eq. (19) as

$\displaystyle
2(1-\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q})\frac{d}{dt}\Delta\mathcal{Q}\leq\sqrt{\text{Tr}\\{\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}\mathcal{L}_{t}\rho_{\psi}(t)^{1/2\dagger}\\}}+\sqrt{\text{Tr}\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2}\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2\dagger}}.$

Here, purity of the states $\rho_{\psi}(t)$ and $\sigma_{\phi}(t)$ are unity.
Using the definition of Hilbert-Schmidt norm of an operator, the MT bound as

$\displaystyle
2(1-\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q})\frac{d}{dt}\Delta\mathcal{Q}\leq\|\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}\|_{HS}+\|\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2}\|_{HS}$

and integrating over time, we obtain the bound as

$\displaystyle\tau\geq\tau_{QC}=\frac{1}{\Lambda_{HS}}2\Delta\mathcal{Q}\left(1-\frac{2\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q}}{2}\right)$
(25)

where
$\Lambda_{HS}=1/\tau\int_{0}^{\tau}dt(\|\mathcal{L}_{t}\rho_{\psi}(t)^{1/2}\|_{HS}+\|\mathcal{L}_{t}\sigma_{\phi}(t)^{1/2}\|_{HS})$.
Combining the above two bounds, we have

$\displaystyle\tau_{QC}=\text{max}\\{\frac{1}{\Lambda_{op}},\frac{1}{\Lambda_{tr}},\frac{1}{\Lambda_{HS}}\\}2\Delta\mathcal{Q}\left(1-\frac{2\mathcal{D}(\rho_{0})\mp\Delta\mathcal{Q}}{2}\right)$
(26)

##  4 Dynamics under OU dephasing noise

In what follows, we study the dynamics of newly-minted discord under OU
dephasing noise and compare with the entanglement (quantified by concurrence)
[61]. Further, we demonstrate the speed limit time bound on the decay and
creation of quantum correlation in terms of affinity discord. In general, the
influence of environmental noise on the system is accounted by completely
positive trace-preserving operation namely called Kraus operator sum
representation and can be written as [1]

$\displaystyle\rho(t)=\sum_{\mu}K_{\mu}(t)\rho(0)K_{\mu}^{\dagger}(t)$ (27)

where $\rho(0)$ is the initial state, which is assumed to be free from the
noise and the Kraus operators $K_{\mu}$ satisfy
$\sum_{\mu}K_{\mu}K_{\mu}^{\dagger}=\mathds{1}$. The Kraus operators of the
Ornstein-Uhlenbeck dephasing noise: $K_{1}=E_{1}^{a}\otimes E_{1}^{b}$,
$K_{2}=E_{1}^{a}\otimes E_{2}^{b}$, $K_{3}=E_{2}^{a}\otimes E_{1}^{b}$ and
$K_{4}=E_{2}^{a}\otimes E_{2}^{b}$ and the operators are

$E_{1}^{a(b)}=\begin{pmatrix}p_{a(b)}&0\\\
0&1\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ E_{2}^{a(b)}=\begin{pmatrix}q_{a(b)}&0\\\
0&0\end{pmatrix}.$

where $q_{a(b)}=\sqrt{1-p^{2}_{a(b)}}$, $p_{a(b)}=\text{exp}(-f(t))$ and
$f(t)=\frac{\Gamma t}{2}[1+(e^{-\gamma t}-1)/\gamma t]$. For our detailed
analysis, we consider the Bell diagonal state

$\displaystyle\rho(0)=\frac{1}{4}\left(\mathds{1}_{a}\otimes\mathds{1}_{b}+\sum_{j=1}^{3}c_{j}\sigma^{j}_{a}\otimes\sigma^{j}_{b}\right),$
(28)

where $-1\leq c_{i}=\langle\sigma_{i}\otimes\sigma_{i}\rangle\leq 1$ are the
elements of $\vec{c}$ specifies the quantum state. It is worth pointing that
the evolution of the under the OU noise preserve the structure of the initial
state. The coefficients of the time evolved state are [60]

$\displaystyle
c_{1}(t)=c_{1}(0)e^{-2f(t)},c_{2}(t)=c_{2}(0)e^{-2f(t)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ c_{3}(t)=c_{3}(0).$

The entanglement of the above mixed state is computed using the concurrence
[61] and is defined as
$C(\rho)=\text{max}\\{0,\eta_{1}-\eta_{2}-\eta_{3}-\eta_{4}\\}$ with
$\eta_{i}$ are the eigenvalues of the matrix
$\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$,
$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$
is spin flipped density matrix and * denotes the complex conjugation in the
computational basis. Now, we investigate the dynamics of affinity-based
discord under OU dephasing noise and compare with the entanglement quantified
by the concurrence. For this purpose, we consider the maximally entangled
state as an initial state for our investigation.

Figure 1: (color online) The dynamical behaviors of entanglement and affinity
based measure under OU dephasing noise for the initial states (a) $(1,1,-1)$,
(b) $(1,-1,1)$, and (c) $(1,0.5,-0.5)$. The fixed parameter $\gamma=1$.

We have plotted the dynamical behaviors of entanglement measured through
concurrence and discord in Fig.1 for different initial conditions. In all the
cases, we observe that the correlation measures decrease monotonically and the
OU dephasing noise cause sudden death in entanglement whereas the affinity
measure exhibit more robustness than the entanglement. The better robustness
suggests that the affinity-based discord may provide advantages in information
processing over the entanglement.

Figure 2: (color online) QSL time for the creation and decay of quantum
correlation under OU dephasing noise.The fixed parameter $\gamma=1$.

Considering the pure state with $(1,1,-1)$, we now study the QSL time limit of
decay and creation of quantum correlation by an amount $\Delta\mathcal{Q}$
through the affinity based measure. Here also we consider the above maximally
entangled state for investigation. For pure state, the affinity is computed as
$\mathcal{A}(\rho,\Pi^{a}(\rho))=\sum_{k}s_{k}^{2}$ with $s_{k}$ are the
Schmidt coefficients and discord of maximally entangled state is
$\mathcal{D}(\rho)=1-1/\sqrt{2}$. In Fig. (2), we have plotted the QSL time
for the decay and creation of affinity based quantum correlations under the OU
dephasing noise. Both decay and creation QSL decreases with the increase of
coupling strength.

##  5 Conclusions

To summarize, we have defined a new variant of geometric version of quantum
correlation measure using affinity and remedied the local ancilla problem of
Hilbert-Schmidt based measures. We have derived the Margolus-Levitin (ML) and
Mandelstamm-Tamm (MT) bounds for the quantum speed limit time for the creation
and decay of quantum correlation through the affinity measure. The better
robustness of affinity based quantum discord suggests that implementing some
quantum information processing tasks with quantum correlation could be better
than with entanglement under Ornstein-Uhlenbeck noisy environment.

Moreover, the influence of quantum correlation on quantum speed limit time is
also studied under non-markovian evolution. It is worth-pointing that the
affinity based metric used here also useful in quantification of coherence.
Hence, the relation between the affinity based quantum coherence and QSL time
is worth investigation.

## Acknowledgements

Authors are indebted to the referees for their critical comments to improve
the contents of the manuscript. RM is grateful for the financial support from
MŠMT RVO 14000.

## References

## References

  * [1] Nielsen M and Chuang I 2010 Quantum Computation and Quantum Information, Cambridge University Press, Cambridge.
  * [2] Plenio M B and Virmani S 2007  An introduction to entanglement measures, Quantum Inf. Comput. 7 1–51.
  * [3] Schrodinger E 1935 Discussion of probability relations between separated systems, Proc. Camb. Philos. Soc. 31 555\.
  * [4] Schrodinger E 1936 Probability relations between separated systems, Proc. Camb. Philos. Soc. 32 446\.
  * [5] Bell J S 1964 On the Einstein Podolsky Rosen paradox, Physics 1 195\.
  * [6] Werner R F 1989 Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model Phys. Rev. A 40 4277\.
  * [7] Ollivier H, and Zurek W H 2001 Quantum Discord: A Measure of the Quantumness of Correlations, Phys. Rev. Lett. 88 017901\.
  * [8] Luo S, and Fu S 2011 Measurement-induced nonlocality, Phys. Rev. Lett. 106 120401\.
  * [9] Baumgratz T, Cramer M and Plenio M B 2014 Quantifying coherence, Phys. Rev. Lett. 113 140401\.
  * [10] Girolami D 2014 Observable measure of quantum coherence in finite dimensional systems, Phys. Rev. Lett. 113 170401\.
  * [11] Auletta G, Fortunato M and Parisi G 2009 Quantum Mechanics , Cambridge University Press, Cambridge.
  * [12] Mandelstam L and Tamm I G 1945 The uncertainty relation between energy and time in non-relativistic quantum mechanics, J. Phys. (Moscow) 9 249\.
  * [13] Bekenstein J D 1981 Energy cost of information transfer, Phys. Rev. Lett. 46, 623.
  * [14] Lloyd S 2000 Ultimate physical limits to computation, Nature (London) 406, 1047.
  * [15] Giovannetti V, Lloyd S, and Maccone L 2011 Advances in quantum metrology, Nat. Photonics 5, 222 (2011).
  * [16] Deffner S and Lutz E 2010 Generalized Clausius inequality for nonequilibrium quantum processes, Phys. Rev. Lett. 105, 170402.
  * [17] Caneva T, Murphy M, Calarco T, Fazio R, Montangero S, Giovannetti V, and Santoro G E 2009 Optimal control at the quantum speed limit, Phys. Rev. Lett. 103, 240501.
  * [18] Fleming G N 1973 A unitarity bound on the evolution of nonstationary states, Nuovo Cimento A 16, 232.
  * [19] Bhattacharyya K 1983 Quantum decay and the Mandelstam-Tamm-energy inequality, J. Phys. A 16, 2993.
  * [20] Anandan J, and Aharonov Y 1990 Geometry of quantum evolution, Phys. Rev. Lett. 65, 1697.
  * [21] Vaidman L 1992 Minimum time for the evolution to an orthogonal quantum state, Am. J. Phys. 60, 182.
  * [22] Uffink J 1993 The rate of evolution of a quantum state, Am. J. Phys. 61, 935.
  * [23] Brody D C 2003 Elementary derivation for passage times, J. Phys. A 36, 5587.
  * [24] Luo S, and Zhang Z,2005 On decaying rate of quantum states, Lett. Math. Phys. 71 1\.
  * [25] Boixo S, Flammia S T, Caves C M, and Geremia J M, 2007 Generalized limits for single-parameter quantum estimation, Phys. Rev. Lett. 98 090401\.
  * [26] Aifer M, and Deffner S, 2022 From quantum speed limits to energy-efficient quantum gates, New J. Phys. 24 055002\.
  * [27] Bukov M, Sels D, and Polkovnikov A, 2019 Geometric speed limit of accessible many-body state preparation, Phys. Rev. X 9 011034\.
  * [28] Ness G, Alberti A, and Sagi Y 2022 Quantum speed Limit for states with a bounded energy spectrum, Phys. Rev. Lett. 129 140403\.
  * [29] Langford N K, Dalton R B, Harvey M D, O’Brien J L, Pryde G J, Gilchrist A, Bartlett S D, and White A G, 2004 Measuring entangled qutrits and their use for quantum bit commitment, Phys. Rev. Lett. 93 053601\.
  * [30] Klimov A B, Guzmán R, Retamal J C, and Saavedra C, 2003 Qutrit quantum computer with trapped ions, Phys. Rev. A 67, 062313 (2003).
  * [31] Belmechri N, Förster L, Alt W, Widera A, Meschede D, and Alberti A, 2013 Microwave control of atomic motional states in a spin-dependent optical lattice, J. Phys. B 46 104006\.
  * [32] Vepsäläinen A, Danilin S, and Paraoanu G S, 2019 Superadiabatic population transfer in a three-level superconducting circuit, Sci. Adv. 5, eaau5999.
  * [33] Cervera-Lierta A, Krenn M, Aspuru-Guzik A, and Galda A, 2022 Experimental high-dimensional Greenberger-Horne-Zeilinger entanglement with superconducting Transmon qutrits, Phys. Rev. Applied 17, 024062.
  * [34] Taddei M M, Escher B M, Davidovich L and de Matos Filho R L 2013 Quantum speed limit for physical processes, Phys. Rev. Lett. 110 050402\.
  * [35] del Campo A, Egusquiza I L, Plenio M B and Huelga S F 2013 Quantum speed limits in open system dynamics, Phys. Rev. Lett. 110 050403
  * [36] Deffner S and Lutz E 2013 Quantum speed limit for non-Markovian dynamics, Phys. Rev. Lett. 111 010402\.
  * [37] Giovannetti V, Lloyd S, and Maccone L 2003 The role of entanglement in dynamical evolution, Europhys. Lett. 62 615\.
  * [38] Giovannetti V, Lloyd S, and Maccone L 2003 Quantum limits to dynamical evolution, Phys. Rev. A 67 052109\.
  * [39] Zander C, Plastino A R, Plastino A and Casas M 2007 Entanglement and the speed of evolution of multi-partite quantum systems, J. Phys A 40 2861\.
  * [40] Uhlenbeck G E, and Ornstein L S 1930 On the Theory of the Brownian Motion, Phys. Rev. 36 823\.
  * [41] Chiara G D, Palma G M, 2003 Berry phase for a spin 1/2 particle in a classical fluctuating field, Phys. Rev. Lett. 91 090404\.
  * [42] Fiasconaro A, Spagnolo B, 2009 Stability measures in metastable states with Gaussian colored noise, Phys. Rev. E 80 041110\.
  * [43] Grotz T, Heaney L, Strunz W T, 2006 Quantum dynamics in fluctuating traps: Master equation, decoherence, and heating, Phys. Rev. A 74 022102\.
  * [44] Altintas F, Eryigit F, 2010 Quantum correlations in non-Markovian environments, Phys. Lett. A 374 4283\.
  * [45] Dakić B, Vedral V, and Brukner C, 2010 Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett. 105 190502\.
  * [46] Luo S, and Fu S 2010 Geometric measure of quantum discord, Phys. Rev. A 82 034302\.
  * [47] Holevo A S 1972 On quasiequivalence of locally normal states, Theor. Math. Phys. 13 1071–1082.
  * [48] Paula F M, de Oliveira T R and Sarandy M S, 2013 Geometric quantum discord through the Schatten 1-norm, Phys. Rev. A 87 064101\.
  * [49] Spehner D, and Orszag M 2013 Geometric quantum discord with Bures distance, New J Phys. 15 103001
  * [50] Roga W, Spehner D, and Illuminati F 2016 Geometric measures of quantum correlations, J Phys. A: Math. Theor. 49 235301\.
  * [51] Luo S and Zhang Q 2004 Informational distance on quantum-state space, Phys. Rev. A 69 032106
  * [52] Muthuganesan R and Chandrasekar V K 2019 Characterizing nonclassical correlation using affinity, Quantum Inf Process 18 223\.
  * [53] Muthuganesan R, Balakrishnan S and Chandrasekar V K 2022 Characterizing nonbilocal correlation: A geometric perspective, Quantum Inf Process 21 216\.
  * [54] Xiong C, Kumar A and Wu J, 2018 Family of coherence measures and duality between quantum coherence and path distinguishability, Phys. Rev. A 98, 032324.
  * [55] Muthuganesan R, Chandrasekar V K and Sankaranarayanan R 2021 Quantum coherence measure based on affinity, Phys. Lett A 394 127205\.
  * [56] Piani M 2012 Problem with geometric discord, Phys. Rev. A 86 034101
  * [57] Chang L and Luo S 2013 Remedying the local ancilla problem with geometric discord, Phys. Rev. A 87 062303\.
  * [58] Girolami D and Adesso G 2012 Observable Measure of Bipartite Quantum Correlations, Phys. Rev. Lett. 108 150403
  * [59] Paulson K G, and Banerjee S 2022 Quantum speed limit for the creation and decay of quantum correlations, arxiv: 2205.11882.
  * [60] Ting Y and Eberly J H 2010 Entanglement evolution in a non-Markovian environment, Opt. Commun. 283 676\.
  * [61] Hill S and Wootters W K 1997 Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78 5022\.


[CAMKT] [CAMKT] [CAMKT] [CAMKT] [CAMK] CAMKT]Nicolaus Copernicus Astronomical
Center, Polish Academy of Sciences, Rabiańska 8, 87–100 Toruń, Poland
CAMK]Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
Bartycka 18, 00–716 Warsaw, Poland

# Evolution and Dynamics of Tight Triple Systems

Ayush Moharana  Krzysztof G. Hełminiak  Frédéric Marcadon  Tilaksingh Pawar
Maciej Konacki  [  [

###### Abstract

Tight Triple Systems have stars in a hierarchical configuration with a third
star orbiting the inner binary with a period of fewer than 1000 days. Such
systems are important for understanding the formation and evolution of stars
in multiple systems. Having a detached eclipsing binary (DEB) as one of its
components allows us to obtain precise stellar and orbital parameters of these
systems. We discuss the process to obtain accurate parameters of these systems
using high-resolution spectroscopy, radial velocity measurements, and precise
space-based photometry. This enables us to have a 3D geometrical picture as
well as the metallicity, age, and evolutionary status of these systems.

##  1 Introduction

Hierarchical triple systems have the third star at a greater distance compared
to the inner binary separation. Most of these systems have long outer periods
and therefore their dynamics can have timescales of decades or centuries.
Meanwhile, a subset of these triples, called Tight Triple Systems (TTS) have
an outer orbit period of fewer than 1000 days (Hajdu et al. 2017) and hence
their dynamics can be observed in timescales of years. Detached eclipsing
binaries (DEBs) are the source of the most accurate (<1%) stellar parameters
which are robust and independent of different models and methods (Maxted et
al. 2020; Korth et al. 2021). Therefore, TTS with DEB would serve as an ideal
system to extract accurate stellar parameters. Our study probes a sample of
20-25 TTS (Fig.1) which are spectroscopically double-lined (SB2) or triple-
lined (SB3) systems. The systems were extracted from literature, the spectra
database of the CRÈME project (P.I.: K.G. Hełminiak) and the photometric
eclipse timing campaign of the Solaris project (P.I.: M. Konacki).

##  2 Methodology

We obtained TESS and/or Kepler observations for our targets from the the
Mikulski Archive for Space Telescopes (MAST). PHOEBE2.3 (Conroy et al. 2020)
eclipsing binary modelling code was used for the light curve modelling. The
new version of code offers advantages of modelling stellar spots (Fig.1:
Left). The errors of the parameters were calculated using MCMC sampling
through the code. The radial velocities were extracted from spectra taken from
the CRÈME project. We used the TODCOR algorithm to extract radial velocities
which were modelled with the V2FIT code (Konacki et al. 2010). The centre-of-
mass velocity of the inner-binary (Fig. 1: Right) was then modelled along with
with the tertiary radial velocity (if any). The total mass of the inner binary
was used to get the mass estimate of the tertiary.

Figure 1: (Left) Phase-folded TESS light curve (grey) with best-fit PHOEBE2.3
model (red) for BD+44 5528 (SB3 system). (Right) Phase-folded radial
velocities of centre-of-mass of the binary (filled circles) and the tertiary
(hollow boxes). Best fit V2FIT model is in blue.

##  3 Evolution and Dynamics

The measurements of mass and radius give us an opportunity to test evolution
scenarios for triple stars. Usually for TTS, we see no interactions between
the stars themselves (unless their orbits are very dynamic). This would mean
that all three stars in a TTS will evolve separately. Therefore we expect
similar metallicity and age for the three stars in this “clean” system (Fig.2:
Left). Assuming a “clean” system, we can estimate a range for the radius of
the tertiary from the mass-radius relation obtained from the isochrone fitting
using the binary. But sometimes this “clean” system assumption fails and all
the three stars do not follow the same isochrone (Fig.2: Right). This calls
for additional probes by independent estimates of metallicity and other
relevant stellar parameters (e.g. from spectral analysis of disentangled
spectra).

Figure 2: (Left) MESA isochrone in comparison to the parameters obtained for
system EPIC 249432662 obtained by Borkovits et al. (2019). All the three stars
are of the same age and metallicity, hence form a “clean" system. (Right)
Parameters of HD 144548 obtained by Alonso et al. (2015) against different
MESA isochrones. The binary stars and tertiary disagree on age given they have
the same metallicity. The shaded space corresponds to the age estimate of the
OB association in Upper Scorpius to which the system belongs.

In TTS, higher-order dynamics are much more significant than wide hierarchical
triples. Following the treatment in Toonen et al. (2020), we use the masses of
the three stars and the orbital parameters to predict the possible dynamical
interactions in TTS. We found that most of the systems lie in the octuple
regime (Fig. 3) , in which the inner-binary can undergo inclination flips.
Three systems are seen to lie in the semi-secular region and are prone to
stellar collisions. But the errors on the positions are too high to be
conclusive and hence more observations are needed.

##  4 Conclusion

Triple systems are now easier to probe with DEBs using high-precision
photometry and high-resolution spectroscopy. The accurate parameters of these
systems enable us to test scenarios of triple-star evolution and are good
checkpoints for future models of multiple star evolution. The accurate mass
distributions and orbital architectures provide a platform to test and improve
various theories of multiple star formation.

Figure 3: The plot is a visualisation of parameter space of different
dynamical regimes in a triple system, based on Toonen et al. (2020), where
$e_{out}$ is outer orbit eccentricity and period ratio of outer and inner
orbits is $P_{out}/P_{in}$ The proposed systems are over-plotted to this
configuration. Alpha Centauri triple system is plotted as a reference.

###### Acknowledgements.

A.M. acknowledges the support provided by the Polish National Science Centre
(NCN) with the grant 2021/41/N/ST9/02746. A.M., F.M., T.P., and M.K. are
supported by NCN through grant no. 2017/27/B/ST9/02727.

## References

  * Alonso et al. (2015) Alonso, R., et al., _A &A_ 584, L8 (2015)
  * Borkovits et al. (2019) Borkovits, T., et al., _MNRAS_ 483, 2, 1934 (2019)
  * Conroy et al. (2020) Conroy, K. E., et al., _ApJS_ 250, 2, 34 (2020)
  * Hajdu et al. (2017) Hajdu, T., et al., _MNRAS_ 471, 1, 1230 (2017)
  * Konacki et al. (2010) Konacki, M., Muterspaugh, M. W., Kulkarni, S. R., Hełminiak, K. G., _ApJ_ 719, 2, 1293 (2010)
  * Korth et al. (2021) Korth, J., et al., _Contributions of the Astronomical Observatory Skalnate Pleso_ 51, 1, 58 (2021)
  * Maxted et al. (2020) Maxted, P. F. L., et al., _MNRAS_ 498, 1, 332 (2020)
  * Toonen et al. (2020) Toonen, S., Portegies Zwart, S., Hamers, A. S., Bandopadhyay, D., _A &A_ 640, A16 (2020)