2. Quantum dynamics of a random sequence

The general idea is to encode the data on the initial wavefunction. In accordance to signals in coherent optical communications, in every point in space the data can be encoded in both the real and imaginary parts of the wavefunction.

The amount of distortion determines the possibility to differentiate between similar values, and therefore, it determines the maximum amount of information that the wavefunction carries.

The detector width Δx determines the highest volume of data that can be stored in a given space, i.e., it determines the data density. All spatial frequencies beyond 1/Δx cannot be detected and cannot carry information. Moreover, due to this constrain, there is no point in encoding the data with spatial frequency higher than 1/Δx.

A wavefunction, which consists of the infinite random complex sequence ψ<sup>n</sup> = ℜψ<sup>n</sup> + iℑψ<sup>n</sup> for n = � ∞, … � 1, 0, 1, 2, … ∞, which occupies the spatial spectral bandwidth 1/Δx (higher frequencies cannot be detected by the given detector) can be written initially as an infinite sequence of overlapping Nyquist-sinc functions [12, 13] (see Figure 1), i.e.,

$$\psi(\mathbf{x}, t=0) = \sum\_{n=-\text{ss}}^{\text{ss}} \psi\_n \text{sinc}(\mathbf{x}/\Delta \mathbf{x} - n),\tag{1}$$

ψð Þ¼ x; t

K x � x<sup>0</sup> ð Þ¼ , t

Due to the linear nature of the problem, Eq. (3) can be solved directly

n¼�∞

exp �i

ξ2 2τ

� � erf � <sup>ξ</sup> � πτ

ffiffiffiffiffiffi <sup>i</sup>2<sup>τ</sup> <sup>p</sup> � �

<sup>ψ</sup>ð Þ¼ x, t <sup>&</sup>gt; <sup>0</sup> <sup>X</sup><sup>∞</sup>

ffiffiffiffiffiffiffiffi i 2πτ r

where "dsinc" is the dynamic-sync function

1 2

dsincð Þ� ξ; τ

with the Schrödinger Kernel [14].

normalization constant of the wavefucntion.



(*x*)

(*x*)

ð ∞

K x � x<sup>0</sup> ð Þ , t ψ x<sup>0</sup> ð Þ , 0 dx<sup>0</sup> (3)

<sup>ψ</sup>ndsinc <sup>x</sup>=Δ<sup>x</sup> � n,ð Þ <sup>ℏ</sup>=<sup>m</sup> <sup>t</sup>=Δx<sup>2</sup> � � (5)

� erf � <sup>ξ</sup> <sup>þ</sup> πτ

� � � � : (6)

ffiffiffiffiffiffi <sup>i</sup>2<sup>τ</sup> <sup>p</sup>

: (4)

Information Loss in Quantum Dynamics http://dx.doi.org/10.5772/intechopen.70395 101

<sup>x</sup> � <sup>x</sup><sup>0</sup> ð Þ<sup>2</sup> t " #

�∞

r

ffiffiffiffiffiffiffiffiffiffiffi m 2πiℏt

exp im 2ℏ

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

*x*/ *x*

Figure 1. Illustration of the way the data is encoded in the wavefucntion. In every Δx, there is a single complex number ψ<sup>n</sup> = ℜψ<sup>n</sup> + iℑψ<sup>n</sup> (the circles), while the continuous wavefunction is a superposition of these numbers multiplied by sinc's functions (three of which are presented by the dashed curves). The values in the y-axis should be multiplied by the

where sincð Þ� <sup>ξ</sup> sin ð Þ πξ πξ is the well-known "sinc" function.

After a time period t, in which the wavefunctions obeys the free Schrödinger equation.

$$i\hbar\frac{\partial\psi(\mathbf{x},t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi(\mathbf{x},t)}{\partial\mathbf{x}^2},\tag{2}$$

the wavefunction can be written as a convolution

Figure 1. Illustration of the way the data is encoded in the wavefucntion. In every Δx, there is a single complex number ψ<sup>n</sup> = ℜψ<sup>n</sup> + iℑψ<sup>n</sup> (the circles), while the continuous wavefunction is a superposition of these numbers multiplied by sinc's functions (three of which are presented by the dashed curves). The values in the y-axis should be multiplied by the normalization constant of the wavefucntion.

$$
\psi(\mathbf{x},t) = \int\_{-\infty}^{\infty} K(\mathbf{x} - \mathbf{x}',t)\psi(\mathbf{x}',0)d\mathbf{x}' \tag{3}
$$

with the Schrödinger Kernel [14].

The amount of information depends on the detector's capabilities, i.e., it depends on the detector's spatial resolution and its inner noise level. Therefore, the maximum amount of information that can be decoded from the wavefunction is determined by the detector's characteristics. However, unlike the classical wave equation, the quantum Schrödinger dynamics is a dispersive process. During the quantum dynamics, the wavefunction experiences distortions. These distortions increase in time just like the dispersion effects on signals in

Nevertheless, unlike dispersion compensating modules in optical communications, there is no way to compensate or "undo" the dispersive process in quantum mechanics. Therefore, the

The object of this chapter is to investigate the way information is lost during the quantum

The general idea is to encode the data on the initial wavefunction. In accordance to signals in coherent optical communications, in every point in space the data can be encoded in both the

The amount of distortion determines the possibility to differentiate between similar values, and therefore, it determines the maximum amount of information that the wavefunction

The detector width Δx determines the highest volume of data that can be stored in a given space, i.e., it determines the data density. All spatial frequencies beyond 1/Δx cannot be detected and cannot carry information. Moreover, due to this constrain, there is no point in

A wavefunction, which consists of the infinite random complex sequence ψ<sup>n</sup> = ℜψ<sup>n</sup> + iℑψ<sup>n</sup> for n = � ∞, … � 1, 0, 1, 2, … ∞, which occupies the spatial spectral bandwidth 1/Δx (higher frequencies cannot be detected by the given detector) can be written initially as an infinite

n¼�∞

After a time period t, in which the wavefunctions obeys the free Schrödinger equation.

<sup>∂</sup><sup>t</sup> ¼ � <sup>ℏ</sup><sup>2</sup>

2m

<sup>∂</sup><sup>2</sup>ψð Þ <sup>x</sup>; <sup>t</sup>

ψnsincð Þ x=Δx � n ; (1)

<sup>∂</sup>x<sup>2</sup> ; (2)

amount of information that can be decoded decreases monotonically with time.

2. Quantum dynamics of a random sequence

encoding the data with spatial frequency higher than 1/Δx.

the wavefunction can be written as a convolution

sequence of overlapping Nyquist-sinc functions [12, 13] (see Figure 1), i.e.,

<sup>ψ</sup>ð Þ¼ x, t <sup>¼</sup> <sup>0</sup> <sup>X</sup><sup>∞</sup>

πξ is the well-known "sinc" function.

<sup>i</sup><sup>ℏ</sup> <sup>∂</sup>ψð Þ <sup>x</sup>; <sup>t</sup>

real and imaginary parts of the wavefunction.

optical communications [10, 11].

100 Advanced Technologies of Quantum Key Distribution

dynamics.

carries.

where sincð Þ� <sup>ξ</sup> sin ð Þ πξ

$$K(\mathbf{x} - \mathbf{x}', t) = \sqrt{\frac{m}{2\pi i \hbar t}} \exp\left[\frac{im}{2\hbar} \frac{(\mathbf{x} - \mathbf{x}')^2}{t}\right]. \tag{4}$$

Due to the linear nature of the problem, Eq. (3) can be solved directly

$$\psi(\mathbf{x}, t > 0) = \sum\_{n = -\infty}^{\infty} \psi\_n \text{dscnc} \{ \mathbf{x} / \Delta \mathbf{x} - n\_\prime (\hbar / m) \mathbf{t} / \Delta \mathbf{x}^2 \} \tag{5}$$

where "dsinc" is the dynamic-sync function

$$\text{dscinc}(\xi, \tau) \equiv \frac{1}{2} \sqrt{\frac{i}{2\pi\tau}} \exp\left(-i\frac{\xi^2}{2\tau}\right) \left[ \text{erf}\left(-\frac{\xi - \pi\tau}{\sqrt{i2\pi}}\right) - \text{erf}\left(-\frac{\xi + \pi\tau}{\sqrt{i2\pi}}\right) \right]. \tag{6}$$

Equation (6) is the "sinc" equivalent of the "srect" function, that describes the dynamics of rectangular pulses (see Ref. [15]).

Note that lim<sup>τ</sup>!<sup>0</sup> ½ �¼ dsincð Þ ξ; τ sincð Þ ξ :

Some of the properties of the dsinc function are illustrated in Figures 2 and 3. As can be seen, the distortions form dsinc(n, 0) = δ(n) gradually increase with time.

Hereinafter, we adopt the dimensionless variables

$$
\pi \equiv (\hbar/m)t/\Delta \mathbf{x}^2 \\
\text{and } \xi \equiv \mathbf{x}/\Delta \mathbf{x}.\tag{7}
$$

Thus, Eq. (2) can be rewritten

and Eq. (5) simply reads

ξ = m) is a simple convolution

with the dimensionless

w mð Þ� <sup>⋯</sup> <sup>1</sup>

where

Since

i ∂ψ ξð Þ ; τ <sup>∂</sup><sup>τ</sup> ¼ � <sup>1</sup>

ψ ξð Þ¼ , <sup>τ</sup> <sup>&</sup>gt; <sup>0</sup> <sup>X</sup><sup>∞</sup>

<sup>ψ</sup>ð Þ¼ <sup>m</sup>; <sup>τ</sup> <sup>X</sup>

<sup>∂</sup>2sincð Þ <sup>ξ</sup> ∂ξ<sup>2</sup>

� � � � τ¼n6¼0

dψð Þ m; τ <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>i</sup>

<sup>32</sup> � <sup>1</sup>

is local and consequently w(m) is time-independent.

3. Quantum distortion noise

deformation)

n

¼ 2

X n

� �

<sup>6</sup> <sup>1</sup> � <sup>1</sup>

22

It should be noted that the fact that Eq. (14) is a universal sequence, i.e. it is independent of time, is not a trivial one. It is a consequence of the properties of the sinc function. Unlike rectangular pulses, which due to their singularity has short time dynamics is mostly nonlocal (and therefore, time-dependent) [15, 16], sinc pulses are smooth and therefore, their dynamics

After a short period of time, the error (distortion) in the wavefunction (i.e., the wavefunction

1 <sup>32</sup> <sup>⋯</sup>

then Eq. (9) can be written as a linear set of differential equations

<sup>22</sup> <sup>1</sup> � <sup>π</sup><sup>2</sup>

2

n¼�∞

Therefore, the wavefunction at the detection point of the mth symbol (center of the symbol at

<sup>ψ</sup>nh mð Þ¼ � <sup>n</sup> <sup>ψ</sup><sup>m</sup> <sup>þ</sup><sup>X</sup>

<sup>n</sup><sup>2</sup> ð Þ �<sup>1</sup> <sup>n</sup>þ<sup>1</sup> and <sup>∂</sup>2sincð Þ <sup>ξ</sup>

<sup>∂</sup><sup>2</sup>ψ ξð Þ ; <sup>τ</sup>

n

h nð Þ� dsincð Þ n; τ and δh nð Þ� dsincð Þ� n; τ δð Þ n : (11)

∂ξ<sup>2</sup>

� � � � τ¼0

w mð Þ � n ψð Þ� n; τ iw mð Þ ∗ψð Þ m; τ (13)

<sup>¼</sup> ð Þ �<sup>1</sup> <sup>m</sup>þ<sup>1</sup>

(

¼ � <sup>π</sup><sup>2</sup>

<sup>∂</sup>ξ<sup>2</sup> (8)

Information Loss in Quantum Dynamics http://dx.doi.org/10.5772/intechopen.70395 103

ψndsincð Þ ξ � n, τ : (9)

ψnδh mð Þ � n (10)

<sup>3</sup> ; (12)

<sup>=</sup>m<sup>2</sup> <sup>m</sup> 6¼ <sup>0</sup> �π<sup>2</sup>=<sup>6</sup> <sup>m</sup> <sup>¼</sup> <sup>0</sup>

:

(14)

Figure 2. Several plots of the real and imaginary parts of the dsinc function for different discrete values of ξ = 0, 1, 2, … 5.

Figure 3. The dependence of the absolute value of the dsinc function on τ for different discrete values of ξ = 0, 1, 2, … 5.

Thus, Eq. (2) can be rewritten

$$i\frac{\partial\psi(\xi,\tau)}{\partial\tau} = -\frac{1}{2}\frac{\partial^2\psi(\xi,\tau)}{\partial\xi^2} \tag{8}$$

and Eq. (5) simply reads

$$\psi(\xi,\tau>0) = \sum\_{n=-\infty}^{\infty} \psi\_n \text{dsimc}(\xi - n, \tau). \tag{9}$$

Therefore, the wavefunction at the detection point of the mth symbol (center of the symbol at ξ = m) is a simple convolution

$$
\psi(m,\tau) = \sum\_{n} \psi\_n h(m-n) = \psi\_m + \sum\_{n} \psi\_n \delta h(m-n) \tag{10}
$$

where

Equation (6) is the "sinc" equivalent of the "srect" function, that describes the dynamics of

Some of the properties of the dsinc function are illustrated in Figures 2 and 3. As can be seen,


10-1 <sup>100</sup> <sup>101</sup> <sup>102</sup> 10-3

Figure 3. The dependence of the absolute value of the dsinc function on τ for different discrete values of ξ = 0, 1, 2, … 5.

[dsinc( , )]

=3

Figure 2. Several plots of the real and imaginary parts of the dsinc function for different discrete values of ξ = 0, 1, 2, … 5.

=4

=5

<sup>τ</sup> � ð Þ <sup>ℏ</sup>=<sup>m</sup> <sup>t</sup>=Δx2and <sup>ξ</sup> � <sup>x</sup>=Δx: (7)

=1

=2

=0

the distortions form dsinc(n, 0) = δ(n) gradually increase with time.

rectangular pulses (see Ref. [15]).

Note that lim<sup>τ</sup>!<sup>0</sup> ½ �¼ dsincð Þ ξ; τ sincð Þ ξ :

102 Advanced Technologies of Quantum Key Distribution

Hereinafter, we adopt the dimensionless variables


10-2


100

=0

=1

=2

=3 =4 =5

[dsinc( , )]

$$h(n) \equiv \text{d}\text{sinc}(n, \tau) \text{and } \delta h(n) \equiv \text{d}\text{sinc}(n, \tau) - \delta(n). \tag{11}$$

Since

$$\left. \frac{\partial^2 \text{sinc}(\xi)}{\partial \xi^2} \right|\_{\mathfrak{r} = n \neq 0} = \frac{2}{n^2} (-1)^{n+1} \quad \text{and} \quad \left. \frac{\partial^2 \text{sinc}(\xi)}{\partial \xi^2} \right|\_{\mathfrak{r} = 0} = -\frac{\pi^2}{3}, \tag{12}$$

then Eq. (9) can be written as a linear set of differential equations

$$\frac{d\psi(m,\tau)}{d\tau} = \mathrm{i}\sum\_{n} w(m-n)\psi(n,\tau) \equiv \mathrm{i}w(m) \* \psi(m,\tau) \tag{13}$$

with the dimensionless

$$w(m) \equiv \left[ \begin{array}{cccc} \dots & \frac{1}{3^2} & -\frac{1}{2^2} & 1 & -\frac{\pi^2}{6} & 1 & -\frac{1}{2^2} & \frac{1}{3^2} & \cdots \end{array} \right] = \begin{cases} (-1)^{m+1}/m^2 & m \neq 0\\ -\pi^2/6 & m = 0 \end{cases} . \tag{14}$$

It should be noted that the fact that Eq. (14) is a universal sequence, i.e. it is independent of time, is not a trivial one. It is a consequence of the properties of the sinc function. Unlike rectangular pulses, which due to their singularity has short time dynamics is mostly nonlocal (and therefore, time-dependent) [15, 16], sinc pulses are smooth and therefore, their dynamics is local and consequently w(m) is time-independent.

#### 3. Quantum distortion noise

After a short period of time, the error (distortion) in the wavefunction (i.e., the wavefunction deformation)

$$
\Delta\psi(\xi,\tau) \equiv \psi(\xi,\tau) - \psi(\xi,0) \tag{15}
$$

and with physical dimensions

information must decrease gradually.

4. The rate of information loss

where Ñ is the normalization constant.

difference between adjacent symbol

between the symbols.

amplitude, i.e.,

vp, <sup>q</sup> <sup>¼</sup> <sup>2</sup><sup>p</sup> � ffiffiffiffiffi

decreases exponentially with the number of bits, i.e.,.

<sup>Δ</sup><sup>v</sup> <sup>¼</sup> <sup>2</sup> ffiffiffiffiffi M <sup>p</sup> � <sup>1</sup>

[�1/Δx, 1/Δx].

values

N <sup>ρ</sup> <sup>¼</sup> <sup>t</sup> 2 Δx<sup>4</sup>

density) depends only on a single dimensionless parameter <sup>τ</sup> � (ℏ/m)t/Δx<sup>2</sup>

values. In this case, both the real and imaginary parts can have ffiffiffiffiffi

M <sup>p</sup> � <sup>1</sup> ffiffiffiffiffi M <sup>p</sup> � <sup>1</sup>

þ i

<sup>¼</sup> <sup>2</sup> <sup>2</sup><sup>b</sup>=<sup>2</sup> � <sup>1</sup>

ℏ m � �<sup>2</sup> π4

We, therefore, find a universal relation: the relative noise (the ratio between the noise and the

It should be stressed that this is a universal property, which emerges from the quantum dynamics. This relation is valid regardless of the specific data encoded in the wavefunction provided the data's spectral density is approximately homogenous in the spectral bandwidth

Clearly, since the noise increases gradually, it will becomes more difficult to decode the data from the wavefucntion. In fact, as is well known from Shannon celebrated equation [17], the amount of noise determines the data capacity that can be decoded. Therefore, the amount of

We assume that at every Δx interval the wavefunction can have one of M different complex

form is equivalent to the Quadrature Amplitude Modulation, QAM, in electrical and optical modulation scheme [18]), i.e., any complex ψ<sup>n</sup> = ψ(n) = ℜψ<sup>n</sup> + iℑψ<sup>n</sup> = Ñvp,<sup>q</sup> can have one of the

> <sup>2</sup><sup>q</sup> � ffiffiffiffiffi M <sup>p</sup> � <sup>1</sup> ffiffiffiffiffi M <sup>p</sup> � <sup>1</sup>

Since b = log2 M is the number of bits encapsulated in each one of the complex symbol, then the

Therefore, as the number of bits per symbol increases, it becomes more difficult to distinguish

Clearly, maximum distortion occurs, when all the other symbols oscillates with maximum

<sup>20</sup> : (24)

Information Loss in Quantum Dynamics http://dx.doi.org/10.5772/intechopen.70395 105

.

M

ffi 21�b=<sup>2</sup> <sup>¼</sup> 2 exp½ � �bð Þ ln 2=<sup>2</sup> (27)

M

for p, q <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, … ffiffiffiffiffi

Δv ¼ ℜvp, <sup>q</sup> � ℜvp�1, <sup>q</sup> ¼ ℑvp, <sup>q</sup> � ℑvp, <sup>q</sup>�<sup>1</sup> (26)

<sup>p</sup> different values (this

<sup>p</sup> (25)

can be approximated by

$$
\Delta\psi(\xi,\tau) \equiv \psi(\xi,\tau) - \psi(\xi,0) \cong \tau \frac{\partial\psi(\xi,\tau)}{\partial\tau}\Big|\_{\tau=0}.\tag{16}
$$

Then we can define the Quantum Noise as the variance of the error

$$N = \left\langle \left| \Delta \psi(\xi, \tau) \right|^2 \right\rangle \cong \tau^2 \left\langle \left| \frac{\partial \psi(\xi, \tau)}{\partial \tau} \right|\_{\tau=0} \right|^2 \right\rangle \tag{17}$$

where the triangular brackets stand for spatial averaging, i.e., h i f xð Þ � <sup>1</sup> X ð<sup>X</sup>=<sup>2</sup> �X=2 f x<sup>0</sup> ð Þdx<sup>0</sup> .

Using the Schrödinger equation, Eq. (17) can be rewritten as follows:

$$N = \left\langle \left| \Delta \psi(\xi, \tau) \right|^2 \right\rangle \cong \frac{\tau^2}{4} \left\langle \left| \frac{\partial^2 \psi(\xi, 0)}{\partial^2 \xi} \right|^2 \right\rangle. \tag{18}$$

Similarly, we can define the average density as

$$\rho = \left\langle \left| \psi(\xi, \tau) \right|^2 \right\rangle. \tag{19}$$

Now, from the Parseval theorem [12], the spatial integral (average) can be replaced by a spatial frequency integral over the Fourier transform, i.e.,

$$N = \frac{1}{2\pi} \left\langle |\Delta \psi(\kappa, \tau)|^2 \right\rangle \tag{20}$$

and

$$\rho = \frac{1}{2\pi} \left\langle |\psi(\kappa, \tau)|^2 \right\rangle \tag{21}$$

where

$$\psi(\kappa,\tau) \equiv \left(2\pi\right)^{-1} \int d\xi \exp(-i\kappa\xi)\psi(\xi,\tau) \text{ and } \Delta\psi(\kappa,\tau) \equiv \left(2\pi\right)^{-1} \int d\xi \exp(-i\kappa\xi)\Delta\psi(\xi,\tau). \tag{22}$$

Therefore, the ratio between the noise and the density (i.e., the reciprocal of the Signal-to-Noise Ratio, SNR) satisfies the surprisingly simple expression

$$\frac{N}{\rho} = \frac{\left< |\Delta \psi(\xi, \tau)|^2 \right>}{\left< |\psi(\xi, 0)|^2 \right>} \cong \tau^2 \frac{\frac{1}{2\pi} \int d\kappa \frac{\kappa^4}{4} |\psi(\kappa, 0)|^2}{\frac{1}{2\pi} \int d\kappa |\psi(\kappa, 0)|^2} = \tau^2 \frac{\tau \tau^4}{20} \tag{23}$$

and with physical dimensions

Δψ ξð Þ� ; τ ψ ξð Þ� ; τ ψ ξð Þ ; 0 (15)

� � � � τ¼0

� � � �

> X ð<sup>X</sup>=<sup>2</sup> �X=2

: (19)

: (16)

f x<sup>0</sup> ð Þdx<sup>0</sup> .

: (18)

dξ expð Þ �iκξ Δψ ξð Þ ; τ : (22)

<sup>20</sup> (23)

(17)

(20)

(21)

∂ψ ξð Þ ; τ ∂τ

> � � � � τ¼0

<sup>2</sup> \* +

<sup>∂</sup><sup>2</sup>ψ ξð Þ ; <sup>0</sup> ∂<sup>2</sup>ξ

<sup>2</sup> \* +

� � � �

ð

<sup>¼</sup> <sup>τ</sup><sup>2</sup> <sup>π</sup><sup>4</sup>

<sup>4</sup> j j ψ κð Þ ; <sup>0</sup> <sup>2</sup>

<sup>d</sup>κψκ j j ð Þ ; <sup>0</sup> <sup>2</sup>

� � � �

ffi <sup>τ</sup><sup>2</sup> <sup>∂</sup>ψ ξð Þ ; <sup>τ</sup> ∂τ

� � � �

ffi τ2 4

ρ ¼ j j ψ ξð Þ ; τ <sup>2</sup> D E

<sup>N</sup> <sup>¼</sup> <sup>1</sup>

<sup>ρ</sup> <sup>¼</sup> <sup>1</sup>

<sup>d</sup><sup>ξ</sup> expð Þ �iκξ ψ ξð Þ ; <sup>τ</sup> and <sup>Δ</sup>ψ κð Þ� ; <sup>τ</sup> ð Þ <sup>2</sup><sup>π</sup> �<sup>1</sup>

Now, from the Parseval theorem [12], the spatial integral (average) can be replaced by a spatial

<sup>2</sup><sup>π</sup> j j <sup>Δ</sup>ψ κð Þ ; <sup>τ</sup> <sup>2</sup> D E

<sup>2</sup><sup>π</sup> j j ψ κð Þ ; <sup>τ</sup> <sup>2</sup> D E

Therefore, the ratio between the noise and the density (i.e., the reciprocal of the Signal-to-Noise

1 2π ð dκ κ4

> 1 2π ð

Δψ ξð Þ� ; τ ψ ξð Þ� ; τ ψ ξð Þffi ; 0 τ

Then we can define the Quantum Noise as the variance of the error

N ¼ j j Δψ ξð Þ ; τ <sup>2</sup> D E

where the triangular brackets stand for spatial averaging, i.e., h i f xð Þ � <sup>1</sup>

Using the Schrödinger equation, Eq. (17) can be rewritten as follows:

N ¼ j j Δψ ξð Þ ; τ <sup>2</sup> D E

Similarly, we can define the average density as

frequency integral over the Fourier transform, i.e.,

and

where

ψ κð Þ� ; <sup>τ</sup> ð Þ <sup>2</sup><sup>π</sup> �<sup>1</sup>

ð

Ratio, SNR) satisfies the surprisingly simple expression

j j Δψ ξð Þ ; τ <sup>2</sup> D E

j j ψ ξð Þ ; <sup>0</sup> <sup>2</sup> D E ffi <sup>τ</sup><sup>2</sup>

N ρ ¼

can be approximated by

104 Advanced Technologies of Quantum Key Distribution

$$\frac{N}{\rho} = \frac{t^2}{\Delta \mathbf{x}^4} \left(\frac{\hbar}{m}\right)^2 \frac{\pi^4}{20}. \tag{24}$$

We, therefore, find a universal relation: the relative noise (the ratio between the noise and the density) depends only on a single dimensionless parameter <sup>τ</sup> � (ℏ/m)t/Δx<sup>2</sup> .

It should be stressed that this is a universal property, which emerges from the quantum dynamics. This relation is valid regardless of the specific data encoded in the wavefunction provided the data's spectral density is approximately homogenous in the spectral bandwidth [�1/Δx, 1/Δx].

Clearly, since the noise increases gradually, it will becomes more difficult to decode the data from the wavefucntion. In fact, as is well known from Shannon celebrated equation [17], the amount of noise determines the data capacity that can be decoded. Therefore, the amount of information must decrease gradually.

#### 4. The rate of information loss

We assume that at every Δx interval the wavefunction can have one of M different complex values. In this case, both the real and imaginary parts can have ffiffiffiffiffi M <sup>p</sup> different values (this form is equivalent to the Quadrature Amplitude Modulation, QAM, in electrical and optical modulation scheme [18]), i.e., any complex ψ<sup>n</sup> = ψ(n) = ℜψ<sup>n</sup> + iℑψ<sup>n</sup> = Ñvp,<sup>q</sup> can have one of the values

$$
\sigma\_{p,q} = \frac{2p - \sqrt{M} - 1}{\sqrt{M} - 1} + i \frac{2q - \sqrt{M} - 1}{\sqrt{M} - 1} \text{ for } p, q = 1, 2, \dots \\
\sqrt{M} \tag{25}
$$

where Ñ is the normalization constant.

Since b = log2 M is the number of bits encapsulated in each one of the complex symbol, then the difference between adjacent symbol

$$
\Delta \mathbf{v} = \Re v\_{p,q} - \Re v\_{p-1,q} = \mathfrak{S}v\_{p,q} - \mathfrak{S}v\_{p,q-1} \tag{26}
$$

decreases exponentially with the number of bits, i.e.,.

$$
\Delta v = \frac{2}{\sqrt{M} - 1} = \frac{2}{2^{b/2} - 1} \cong 2^{1 - b/2} = 2 \exp[-b(\ln 2/2)] \tag{27}
$$

Therefore, as the number of bits per symbol increases, it becomes more difficult to distinguish between the symbols.

Clearly, maximum distortion occurs, when all the other symbols oscillates with maximum amplitude, i.e.,

$$\psi\_n = \psi(n, 0) = \begin{cases} \psi(m, 0) & n = m \\ (-1)^{n-m} & n \neq m \end{cases},\tag{28}$$

in which case the differential Eq. (13) can be written (for short periods)

$$\frac{d\psi\_{\text{max/min}}(m,\tau)}{d\tau} = -i w(0)\psi\_{\text{max/min}}(m,\tau) \mp i \sum\_{n \neq 0} w(m-n)(-1)^n = i \frac{\pi^2}{6} \psi\_{\text{max/min}}(m,\tau) \mp i \tau^2/3. \tag{29}$$

The solution of Eq. (29) is

$$
\psi\_{\text{max}/\text{min}}(m,\tau) = \psi(m,0)\exp\left(i\pi^2\tau/6\right) \pm 2\left(1 - \exp\left(i\pi^2\tau/6\right)\right).
\tag{30}
$$

Therefore, each cluster is bounded by a circle whose center is

$$\psi(m,0)\exp\left(i\pi^2\tau/6\right)\tag{31}$$

and its radius is

$$R = 2\left|1 - \exp\left(i\pi^2 \tau/6\right)\right| = 4\left|\sin\left(\pi^2 \tau/12\right)\right|.\tag{32}$$

Since this result applies only for short periods, then the entire cluster is bounded by the radius

$$R = \pi^2 \tau / 3,\tag{33}$$

which it is impossible to encode the data (i.e., to differentiate between symbols). This maxi-

Figure 4. Plot of the SNR as a function of τ. The solid curve represents the simulation result, and the dashed line

10-3 10-2 10-1 <sup>100</sup> 10-1

It should be noted that this result coincides with the On-Off-Keying (OOK) dispersion limit, for

Since the number of bits per symbol is log2M, then the maximum data density (bit/distance) is

ffiffiffiffiffiffiffiffiffiffiffi <sup>M</sup>max <sup>p</sup> <sup>¼</sup> <sup>2</sup>

3 π<sup>2</sup>τ

<sup>Δ</sup><sup>x</sup> log2 <sup>1</sup> <sup>þ</sup> <sup>3</sup>=π<sup>2</sup>

<sup>p</sup> � <sup>1</sup> � �π<sup>2</sup> (35)

<sup>M</sup>max <sup>p</sup> is meaningful only under the constraint that

: (36)

Information Loss in Quantum Dynamics http://dx.doi.org/10.5772/intechopen.70395 107

τ � �: (37)

<sup>τ</sup>max <sup>¼</sup> <sup>3</sup> ffiffiffiffiffi M

Similarly, Eq. (35) can be rewritten to find the maximum M for a given distance, i.e.,

ffiffiffiffiffiffiffiffiffiffiffi <sup>M</sup>max <sup>p</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

<sup>p</sup> <sup>¼</sup> 2, and then <sup>τ</sup>max = 1/<sup>π</sup> ffi 3/π<sup>2</sup> (see Ref. [19]).

mum time is

which case ffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffi

Using Δx ¼

<sup>M</sup>max <sup>p</sup> is an integer.

ffiffiffiffiffiffiffiffiffiffi ð Þ ℏ=m t τ q

M

100

101

102

SNR=

/*N*

103

104

105

106

However, it is clear that this formulae for ffiffiffiffiffiffiffiffiffiffiffi

represents the approximation for short τ (the reciprocal of Eq. (23)).

<sup>S</sup>max <sup>¼</sup> <sup>2</sup>

, we finally have

<sup>Δ</sup><sup>x</sup> log2

which is clearly larger than the cluster's standard deviation <sup>σ</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup>τ<sup>=</sup> ffiffiffiffiffi <sup>20</sup> <sup>p</sup> <sup>&</sup>lt; <sup>R</sup>.

A simulation based on Eq. (1) with 2<sup>11</sup> � 1 symbols, which were randomly selected from the pool (25) for M = 16 was taken. That is, the probability that ψ<sup>n</sup> is equal to vp,<sup>q</sup> is 1/M for all ns, or mathematically

$$P(\psi\_n = v\_{p,q}) = 1/M,\text{ for } n = 1, 2, 3, \dots, 2^{11} - 1,\text{ and } p, q = 1, 2, \dots \\ \sqrt{M}.\tag{34}$$

The temporal dependence of the calculated SNR is presented in Figure 4. As can be seen, Eq. (23) is indeed an excellent approximation for short τ.

Since the symbols were selected randomly (with uniform distribution), then when all the symbols ψ(n, 0) = ψ<sup>n</sup> are plotted on the complex plain, an ideal constellation image is shown (see the upper left subfigure of Figure 5).

In Figure 5, a numerical simulation for a QAM 16 scenario is presented initially and after a time period, τ = 0.1. Moreover, the dashed circles represents the standard deviation, i.e., the noise level (radius π<sup>2</sup>τ= ffiffiffiffiffi <sup>20</sup> <sup>p</sup> ), and the bounding circles (radius <sup>R</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup>τ=<sup>3</sup> <sup>&</sup>gt; <sup>π</sup><sup>2</sup>τ<sup>=</sup> ffiffiffiffiffi <sup>20</sup> <sup>p</sup> ).

Since the initial distance between centers of adjacent clusters is <sup>2</sup>ffiffiffi M <sup>p</sup> �<sup>1</sup> , then decoding is impossible for <sup>1</sup>ffiffiffi M <sup>p</sup> �<sup>1</sup> <sup>¼</sup> <sup>π</sup>2τmax <sup>3</sup> , i.e., we finally have an expression for the maximum time τmax, beyond

Figure 4. Plot of the SNR as a function of τ. The solid curve represents the simulation result, and the dashed line represents the approximation for short τ (the reciprocal of Eq. (23)).

which it is impossible to encode the data (i.e., to differentiate between symbols). This maximum time is

$$
\tau\_{\text{max}} = \frac{3}{\left(\sqrt{M} - 1\right)\pi^2} \tag{35}
$$

It should be noted that this result coincides with the On-Off-Keying (OOK) dispersion limit, for which case ffiffiffiffiffi M <sup>p</sup> <sup>¼</sup> 2, and then <sup>τ</sup>max = 1/<sup>π</sup> ffi 3/π<sup>2</sup> (see Ref. [19]).

Similarly, Eq. (35) can be rewritten to find the maximum M for a given distance, i.e.,

$$
\sqrt{M\_{\text{max}}} = 1 + \frac{3}{\pi^2 \pi}.\tag{36}
$$

However, it is clear that this formulae for ffiffiffiffiffiffiffiffiffiffiffi <sup>M</sup>max <sup>p</sup> is meaningful only under the constraint that ffiffiffiffiffiffiffiffiffiffiffi <sup>M</sup>max <sup>p</sup> is an integer.

Since the number of bits per symbol is log2M, then the maximum data density (bit/distance) is

$$S\_{\text{max}} = \frac{2}{\Delta \mathbf{x}} \log\_2 \sqrt{M\_{\text{max}}} = \frac{2}{\Delta \mathbf{x}} \log\_2 \left(1 + 3/\pi^2 \tau\right). \tag{37}$$

Using Δx ¼ ffiffiffiffiffiffiffiffiffiffi ð Þ ℏ=m t τ q , we finally have

<sup>ψ</sup><sup>n</sup> <sup>¼</sup> <sup>ψ</sup>ð Þ¼ <sup>n</sup>; <sup>0</sup> <sup>ψ</sup>ð Þ <sup>m</sup>; <sup>0</sup> <sup>n</sup> <sup>¼</sup> <sup>m</sup>

�

X n6¼0

<sup>ψ</sup>ð Þ <sup>m</sup>; <sup>0</sup> exp <sup>i</sup>π<sup>2</sup>

Since this result applies only for short periods, then the entire cluster is bounded by the radius

A simulation based on Eq. (1) with 2<sup>11</sup> � 1 symbols, which were randomly selected from the pool (25) for M = 16 was taken. That is, the probability that ψ<sup>n</sup> is equal to vp,<sup>q</sup> is 1/M for all ns, or

� � <sup>¼</sup> <sup>1</sup>=M, for <sup>n</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, …, 211 � <sup>1</sup>, and p, q <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, … ffiffiffiffiffi

The temporal dependence of the calculated SNR is presented in Figure 4. As can be seen,

Since the symbols were selected randomly (with uniform distribution), then when all the symbols ψ(n, 0) = ψ<sup>n</sup> are plotted on the complex plain, an ideal constellation image is shown

In Figure 5, a numerical simulation for a QAM 16 scenario is presented initially and after a time period, τ = 0.1. Moreover, the dashed circles represents the standard deviation, i.e., the

<sup>p</sup> ), and the bounding circles (radius <sup>R</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup>τ=<sup>3</sup> <sup>&</sup>gt; <sup>π</sup><sup>2</sup>τ<sup>=</sup> ffiffiffiffiffi

<sup>3</sup> , i.e., we finally have an expression for the maximum time τmax, beyond

M <sup>p</sup> �<sup>1</sup>

<sup>R</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup>

in which case the differential Eq. (13) can be written (for short periods)

<sup>ψ</sup>max=minð Þ¼ <sup>m</sup>; <sup>τ</sup> <sup>ψ</sup>ð Þ <sup>m</sup>; <sup>0</sup> exp <sup>i</sup>π<sup>2</sup>

<sup>R</sup> <sup>¼</sup> 2 1 � exp <sup>i</sup>π<sup>2</sup>

which is clearly larger than the cluster's standard deviation <sup>σ</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup>τ<sup>=</sup> ffiffiffiffiffi

<sup>τ</sup>=<sup>6</sup> � � � � �

Therefore, each cluster is bounded by a circle whose center is

<sup>d</sup><sup>τ</sup> ¼ �iwð Þ<sup>0</sup> <sup>ψ</sup>max=minð Þ <sup>m</sup>; <sup>τ</sup> <sup>∓</sup><sup>i</sup>

106 Advanced Technologies of Quantum Key Distribution

dψmax=minð Þ m; τ

and its radius is

mathematically

P ψ<sup>n</sup> ¼ vp, <sup>q</sup>

(see the upper left subfigure of Figure 5).

noise level (radius π<sup>2</sup>τ= ffiffiffiffiffi

M <sup>p</sup> �<sup>1</sup> <sup>¼</sup> <sup>π</sup>2τmax

sible for <sup>1</sup>ffiffiffi

Eq. (23) is indeed an excellent approximation for short τ.

20

Since the initial distance between centers of adjacent clusters is <sup>2</sup>ffiffiffi

The solution of Eq. (29) is

ð Þ �<sup>1</sup> <sup>n</sup>�<sup>m</sup> <sup>n</sup> 6¼ <sup>m</sup> ;

w mð Þ� � <sup>n</sup> ð Þ<sup>1</sup> <sup>n</sup> <sup>¼</sup> <sup>i</sup>

<sup>τ</sup>=<sup>6</sup> � � � 2 1 � exp <sup>i</sup>π<sup>2</sup>

� <sup>¼</sup> 4 sin <sup>π</sup><sup>2</sup>

<sup>τ</sup>=<sup>12</sup> � � � � �

π2

<sup>6</sup> <sup>ψ</sup>max=minð Þ <sup>m</sup>; <sup>τ</sup> <sup>∓</sup>iπ<sup>2</sup>

τ=6 � � � � : (30)

�: (32)

M

<sup>p</sup> : (34)

20 <sup>p</sup> ).

, then decoding is impos-

τ=6 � � (31)

τ=3; (33)

20 <sup>p</sup> <sup>&</sup>lt; <sup>R</sup>. (28)

=3:

(29)

Figure 5. Upper left: the initial constellation of the data in the wavefunction. Upper right: the data constellation after <sup>τ</sup> = 0.1. Bottom left: the constellation with the circles that stands for the standard deviation <sup>σ</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup>τ<sup>=</sup> ffiffiffiffiffi <sup>20</sup> <sup>p</sup> . Bottom right: The constellation with the circles that represents the bounding circles R = π<sup>2</sup> τ/3.

$$S\_{\text{max}} = \frac{1}{\sqrt{(\hbar/m)t}} F(\pi) \tag{38}$$

<sup>Δ</sup>xmax <sup>¼</sup> <sup>2</sup><sup>π</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

10-2 10-1 <sup>100</sup> 0.7

log2ð Þ5 π

It should be stressed that this expression is universal and the only parameter, which it depends on, is the particle's mass. The higher the mass is, the longer is the distance the information can last.

We investigate the decay of information from the wavefunction in the quantum dynamics.

SNR <sup>¼</sup> <sup>ρ</sup>

where <sup>τ</sup> � (ℏ/m)t/Δx<sup>2</sup> and <sup>Δ</sup><sup>x</sup> is the data resolution (the detector size).

A. The signal-to-noise ratio, i.e., the ratio between the mean probability and the variance of

<sup>N</sup> <sup>¼</sup> <sup>20</sup> τ<sup>2</sup>π<sup>4</sup>

ffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð Þ ℏ=m t

s

). The closest circle to the maximum point is M = 5<sup>2</sup>

For this value Fð Þ¼ τmax

(M = 22 , 32 , 4<sup>2</sup> , … 10<sup>2</sup>

that can last after a time period t is

Figure 6. Plot of the function F xð Þ�<sup>2</sup> ffiffiffi

5. Summary and conclusion

The main conclusions are the following:

ffiffi 3 p

0.8

0.9

1

1.1

*F*( ) 1.2

1.3

1.4

Smax ¼

This equation reveals the loss of information from the wave function.

the distortion, has a simple analytical expression for short times

ℏt=3m p : (41)

M p

109

ð Þ <sup>ℏ</sup>=<sup>m</sup> <sup>t</sup> <sup>p</sup> : (42)

<sup>π</sup> log2ð Þffi 5 1:28, and therefore, the maximum information density

<sup>x</sup> <sup>p</sup> log2 <sup>1</sup> <sup>þ</sup> <sup>3</sup>=π<sup>2</sup><sup>x</sup> � �. The circles stands for different values of integer ffiffiffiffiffi

.

*M*=4

Information Loss in Quantum Dynamics http://dx.doi.org/10.5772/intechopen.70395

*M*=9

*<sup>M</sup>*=16 *<sup>M</sup>*=25

1:28 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where <sup>F</sup>ð Þ� <sup>τ</sup> <sup>2</sup> ffiffiffi <sup>τ</sup> <sup>p</sup> log2 <sup>1</sup> <sup>þ</sup> <sup>3</sup>=π<sup>2</sup><sup>τ</sup> � � is a universal dimensionless function, which is plotted in Figure 6 and receives its maximum value F(xmax) ffi 1.28 for xmax ffi 0.0775. However, under the restriction that ffiffiffiffiffiffiffiffiffiffiffi <sup>M</sup>max <sup>p</sup> must be an integer, then as can be shown in Figure 6, the maximum bitrate is reached for

$$M\_{\text{max}} = 25,\tag{39}$$

for which case

$$
\tau\_{\text{max}} = \frac{3}{4\pi^2} \cong 0.076,\tag{40}
$$

Which means that for a given time of measurement t, the largest amount of information would survive provided the detector size (i.e., the sampling interval) is equal to

Figure 6. Plot of the function F xð Þ�<sup>2</sup> ffiffiffi <sup>x</sup> <sup>p</sup> log2 <sup>1</sup> <sup>þ</sup> <sup>3</sup>=π<sup>2</sup><sup>x</sup> � �. The circles stands for different values of integer ffiffiffiffiffi M p (M = 22 , 32 , 4<sup>2</sup> , … 10<sup>2</sup> ). The closest circle to the maximum point is M = 5<sup>2</sup> .

$$
\Delta \mathbf{x}\_{\text{max}} = 2\pi \sqrt{\hbar t/3m}. \tag{41}
$$

For this value Fð Þ¼ τmax ffiffi 3 p <sup>π</sup> log2ð Þffi 5 1:28, and therefore, the maximum information density that can last after a time period t is

$$S\_{\text{max}} = \sqrt{\frac{3}{(\hbar/m)t}} \frac{\log\_2(5)}{\pi} \cong \frac{1.28}{\sqrt{(\hbar/m)t}}.\tag{42}$$

This equation reveals the loss of information from the wave function.

It should be stressed that this expression is universal and the only parameter, which it depends on, is the particle's mass. The higher the mass is, the longer is the distance the information can last.

#### 5. Summary and conclusion

<sup>S</sup>max <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>τ</sup> = 0.1. Bottom left: the constellation with the circles that stands for the standard deviation <sup>σ</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup>τ<sup>=</sup> ffiffiffiffiffi

Figure 5. Upper left: the initial constellation of the data in the wavefunction. Upper right: the data constellation after

Figure 6 and receives its maximum value F(xmax) ffi 1.28 for xmax ffi 0.0775. However, under the

Which means that for a given time of measurement t, the largest amount of information would

<sup>τ</sup>max <sup>¼</sup> <sup>3</sup>

survive provided the detector size (i.e., the sampling interval) is equal to

<sup>τ</sup> <sup>p</sup> log2 <sup>1</sup> <sup>þ</sup> <sup>3</sup>=π<sup>2</sup><sup>τ</sup> � � is a universal dimensionless function, which is plotted in

τ/3.


*n*


*n*

<sup>p</sup> must be an integer, then as can be shown in Figure 6, the maximum bit-

where <sup>F</sup>ð Þ� <sup>τ</sup> <sup>2</sup> ffiffiffi



*n*

*n*

rate is reached for

for which case

restriction that ffiffiffiffiffiffiffiffiffiffiffi

Mmax


108 Advanced Technologies of Quantum Key Distribution


*n*

The constellation with the circles that represents the bounding circles R = π<sup>2</sup>

*n*

ð Þ <sup>ℏ</sup>=<sup>m</sup> <sup>t</sup> <sup>p</sup> <sup>F</sup>ð Þ<sup>τ</sup> (38)



*n*

20

<sup>p</sup> . Bottom right:

*n*

Mmax ¼ 25; (39)

<sup>4</sup>π<sup>2</sup> ffi <sup>0</sup>:076; (40)

We investigate the decay of information from the wavefunction in the quantum dynamics.

The main conclusions are the following:

A. The signal-to-noise ratio, i.e., the ratio between the mean probability and the variance of the distortion, has a simple analytical expression for short times

$$\text{SNR} = \frac{\rho}{N} = \frac{20}{\pi^2 \pi^4}$$

where <sup>τ</sup> � (ℏ/m)t/Δx<sup>2</sup> and <sup>Δ</sup><sup>x</sup> is the data resolution (the detector size).

B. When there are M possible symbols (as in QAM M), then the maximum time, beyond which the data cannot be decoded is <sup>τ</sup>max <sup>¼</sup> <sup>3</sup> ffiffiffi M p ð Þ �<sup>1</sup> <sup>π</sup><sup>2</sup>

[10] Agrawal GP. Fiber-Optic Communication Systems. New-York: J. Wiley& Sons; 2002

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[11] Granot E, Luz E, Marchewka A. Generic pattern formation of sharp-boundaries pulses propagation in dispersive media. Journal of the Optical Society of America B. 2012;29:763

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[13] Soto MA, Alem M, Shoaie MA, Vedadi A, Brès CS, Thévenaz L, Schneider T. Optical sincshaped Nyquist pulses of exceptional quality. Nature Communications. 2013;4:2898 [14] Feynman RP, Hibbs AR. Quantum Mechanics and Path Integrals. 1st ed. New-York:

[15] Granot E, Luz E, Marchewka A. Generic pattern formation of sharp-boundaries pulses propagation in dispersive media. Journal of the Optical Society of America B. 2012;29:

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