Information Loss in Quantum Dynamics Information Loss in Quantum Dynamics

The way data is lost from the wavefunction in quantum dynamics is analyzed. The main results are (A) Quantum dynamics is a dispersive process in which any data initially encoded in the wavefunction is gradually lost. The ratio between the distortion ’ s vari- ance and the mean probability density increases in a simple form. (B) For any given amount of information encoded in the wavefunction, there is a time period, beyond which it is impossible to decode the data. (C) The temporal decline of the maximum information density in the wavefunction has an exact analytical expression. (D) For any given time period there is a specific detector resolution, with which the maximum information can be decoded. (E) For this optimal detector size the amount of informa- tion is inversely proportional to the square root of the time elapsed.

In most quantum computing, the wavefunction is a superposition of multiple binary states (qubits), which can be in spin states, polarization state, binary energy levels, etc. However, since the wavefucntion is a continuous function, it can carry, in principle, an infinite amount of information. Only the detector dimensions and noises limits the information capacity.
The quantum wavefunction, like any complex signal, carries a large amount of information, which can be decoded in the detection process. Its local amplitude can be detected by measuring the probability density in a direct measurement, while its phase can be retrieved in an interferometric detection, just as in optical coherent detection [9].
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 optical communications [10,11].
Nevertheless, unlike dispersion compensating modules in optical communications, there is no way to compensate or "undo" the dispersive process in quantum mechanics. Therefore, the amount of information that can be decoded decreases monotonically with time.
The object of this chapter is to investigate the way information is lost during the quantum dynamics.

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 n = ℜψ n + iℑψ n for n = À ∞, … À 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., where sinc ξ ð Þ sin πξ ð Þ πξ is the well-known "sinc" function.
After a time period t, in which the wavefunctions obeys the free Schrödinger equation. iℏ the wavefunction can be written as a convolution with the Schrödinger Kernel [14].
Due to the linear nature of the problem, Eq. (3) can be solved directly where "dsinc" is the dynamic-sync function Equation (6) is the "sinc" equivalent of the "srect" function, that describes the dynamics of rectangular pulses (see Ref. [15]).
Hereinafter, we adopt the dimensionless variables τ ℏ=m ð Þt=Δx 2 and ξ x=Δx:   Thus, Eq. (2) can be rewritten and Eq. (5) simply reads Therefore, the wavefunction at the detection point of the mth symbol (center of the symbol at ξ = m) is a simple convolution where h n ð Þ dsinc n; τ ð Þand δh n ð Þ dsinc n; τ ð ÞÀδ n ð Þ: Since then Eq. (9) can be written as a linear set of differential equations with the dimensionless 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.

Quantum distortion noise
After a short period of time, the error (distortion) in the wavefunction (i.e., the wavefunction deformation) can be approximated by Then we can define the Quantum Noise as the variance of the error where the triangular brackets stand for spatial averaging, i.e., f x Using the Schrödinger equation, Eq. (17) can be rewritten as follows: Similarly, we can define the average density as Now, from the Parseval theorem [12], the spatial integral (average) can be replaced by a spatial frequency integral over the Fourier transform, i.e., and 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 and with physical dimensions We, therefore, find a universal relation: the relative noise (the ratio between the noise and the density) depends only on a single dimensionless parameter τ (ℏ/m)t/Δx 2 .
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 [À1/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.

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 p different values (this form is equivalent to the Quadrature Amplitude Modulation, QAM, in electrical and optical modulation scheme [18]), i.e., any complex ψ n = ψ(n) = ℜψ n + iℑψ n = Ñv p,q can have one of the values where Ñ is the normalization constant.
Since b = log 2 M is the number of bits encapsulated in each one of the complex symbol, then the difference between adjacent symbol decreases exponentially with the number of bits, i.e.,.
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., in which case the differential Eq. (13) can be written (for short periods) The solution of Eq. (29) is Therefore, each cluster is bounded by a circle whose center is and its radius is Since this result applies only for short periods, then the entire cluster is bounded by the radius which is clearly larger than the cluster's standard deviation σ ¼ π 2 τ= ffiffiffiffiffi 20 p < R.
A simulation based on Eq. (1) with 2 11 À 1 symbols, which were randomly selected from the pool (25) for M = 16 was taken. That is, the probability that ψ n is equal to v p,q is 1/M for all ns, or mathematically P ψ n ¼ v p, q À Á ¼ 1=M, for n ¼ 1, 2, 3, …, 2 11 À 1, and p, q ¼ 1, 2, … ffiffiffiffiffi M p : 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 n are plotted on the complex plain, an ideal constellation image is shown (see the upper left subfigure of Figure 5).
Since the initial distance between centers of adjacent clusters is 2 ffiffiffi ffi , then decoding is impossible for 1 ffiffiffi ffi M p À1 ¼ π 2 τmax 3 , i.e., we finally have an expression for the maximum time τ max , beyond which it is impossible to encode the data (i.e., to differentiate between symbols). This maximum time is It should be noted that this result coincides with the On-Off-Keying (OOK) dispersion limit, for which case ffiffiffiffiffi M p ¼ 2, and then τ max = 1/π ffi 3/π 2 (see Ref. [19]).
Similarly, Eq. (35) can be rewritten to find the maximum M for a given distance, i.e., However, it is clear that this formulae for ffiffiffiffiffiffiffiffiffiffiffi ffi M max p is meaningful only under the constraint that ffiffiffiffiffiffiffiffiffiffiffi ffi M max p is an integer.
Since the number of bits per symbol is log 2 M, then the maximum data density (bit/distance) is , we finally have where F τ ð Þ 2 ffiffi ffi τ p log 2 1 þ 3=π 2 τ À Á is a universal dimensionless function, which is plotted in Figure 6 and receives its maximum value F(x max ) ffi 1.28 for x max ffi 0.0775. However, under the restriction that ffiffiffiffiffiffiffiffiffiffiffi ffi M max p must be an integer, then as can be shown in Figure 6, the maximum bitrate is reached for for which case 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 ffiffiffiffiffi 20 p . Bottom right: The constellation with the circles that represents the bounding circles R = π 2 τ/3.
For this value F τ max ð Þ¼ ffiffi 3 p π log 2 5 ð Þ ffi 1:28, and therefore, the maximum information density that can last after a time period t is 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.

Summary and conclusion
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 where τ (ℏ/m)t/Δx 2 and Δx is the data resolution (the detector size).