The Measurement Problem in Statistical Signal Processing ^†

Abstract

Discussing quantum theory foundations, von Neumann noted that the measurement process should not be regarded in terms of a temporal evolution. A reason for their claim is the insurmountability of the gap between reversible and irreversible processes. The time operator formalism that goes beyond such a gap is an adequate framework to elaborate the measurement problem. It considers signals to be stochastic processes, regardless of whether they correspond to variables or distribution densities. Signal processing that utilizes statistical properties to perform tasks is statistical signal processing. The hierarchy of the measurement process is indicated by crossing between states and devices, which implies an evolution in the temporal domain. The concept has been generalized to an open system by the use of duality in frame theory.

1. Introduction

The uncertainty relation is regarded as a fundamental principle of quantum theory. Although it has a long history, starting from Sommerfeld and Heisenberg [1,2], the problem came into focus due to the discovery of wave mechanics by Schrödinger, which led to its formulation in terms of mathematical physics. The concept of the wave function was utilized by Gabor in order to establish communication theory upon decomposing signals into elementary quanta of information [3]. In that respect, uncertainty comes down to the commutator relation

$$\left[Q,P\right]=i$$

(1)

concerning a pair of canonically conjugate operators.

The current paper is aimed at reformulating the measurement problem in the same manner. It considers signals in relation to stochastic processes, whether corresponding to variables or distribution densities. Signal processing that utilizes statistical properties to perform its tasks is statistical signal processing. The climax of such a trail should be quantum theory of information, in which the measurement is a fundamental conception [4].

The phrase experimental mathematics comes up a lot in the fields of chaos, fractals and non-linear dynamics [5]. It emerged during the last century, notwithstanding that mathematicians had always used some experiments in order to identify properties and patterns. The measurement is therefore a basic concept not only of geometry, but of mathematics overall. A link between the measurement problem and experimental mathematics has already been elaborated [6]. Thw current paper should complement such a discussion and revise some oversights that appeared in the previous one. The multidisciplinary framework it has implied corresponds to the time operator formalism of complex systems physics. The theory originated from the Brussels School of Thermodynamics, proposing a unification of reversible and irreversible processes. A relation to the problem appears in respect to its definition that was postulated by von Neumann [7].

Measurement is argued to be the fundamental conception of science [8]. Elaborating issues it raises is therefore significant for the epistemology and methodology of scientific research. Interrelating some aspects such as states, devices, probabilities, etc., a hierarchy which is designed in that respect should coincide to the principle of psychophysical parallelism. It is indicated by crossing between states and devices, which implies an evolution in the temporal domain [6]. A paradigmatic measurement corresponds to commensuration of magnitudes by the Euclidean algorithm, which is an intensional procedure producing real numbers of the unit interval. Regarding that, one comes to a general definition of the process concerning a time series of binary digits [9].

The paradigm asserts the significance of time for elaboration of the measurement problem, which has explicated a substantial relation between signals and stochastic processes. In that regard, a signal corresponds to the ensemble which is originated by a measurement. The problem is formulated in terms of mathematical physics, notwithstanding any interpretation of physical theories such as QBism or many worlds. A comparison to the uncertainty relation (1) is a picturesque instance, since it appears in statistical signal processing regardless of the interpretation imposed. The measurement problem is therefore a predominantly mathematical issue which is related to the very foundation of geometry, analysis, probability and other topics. It concerns intentionality that is the manner in which mathematics has always applied [10]. This was the reason for it to be termed the reality problem by Philip Pearle [11].

After the Introduction, Section 2 presents the time operator formalism of complex systems. The concept of ensembles is defined, as well as a link between reversible and irreversible processes. The measurement hierarchy is elaborated in Section 3, following a paradigm which corresponds to commensuration of magnitudes by the Euclidean algorithm. It presents a general definition of the problem in statistical signal processing, which has related an ensemble to the distribution density of a time series. Section 4 considers projective measurements in the hierarchical base, constituting a measurable space that is the domain of an observable. A hierarchy that has complemented the von Neumann definition arises from a temporality of the domain.

The main advancement concerns a consistent realization of psychophysical parallelism that is a principle which Bohr and von Neumann have already pointed out [12]. It is realized due to a change in representation which is the operator function of time. General measurements are considered in Section 5, replacing self-duality of the Hilbert space by duality in frame theory [13]. In that manner, crossing between states and devices should generalize to an open system which is partially described by the stochastic process [14].

2. Time and Complexity in Physical Science

2.1. Time in Quantum Theory

Von Neumann has indicated two fundamentally diverse types of interventions in a system, the first of which corresponds to a temporal evolution that is reversible and the second one to an irreversible measurement [12]. He was puzzled by the fact that an entropy increase not representing any temporal evolution follows the measurement process, which is totally opposite to thermodynamics, relating the increase of entropy to an evolution in the temporal domain. The reason for such an odd situation is the fake concept of time in quantum theory which is a classical one, considering that it is represented by linear parameterization just like in Newtonian mechanics [7]. Von Neumann has admitted an essential weakness of quantum theory, which concerns the fact that it is non-relativistic, whereas spatial coordinates are represented by operators and time is a mere parameter, making the Poincaré symmetry impossible. The time operator, which should be a chief link between quantum and relativity theories [15], is substantially related to the measurement problem.

The uncertainty between time and energy has been discussed frequently [16]. In the classical formulation of quantum theory, however, there is no operator that satisfies the commutator relation (1) in respect to a Hamiltonian corresponding to the energy of a system. A reason for this is that the Hamiltonian governs evolution via the Schrödinger equation of the wave function, which is a stationary state, like orbits in Newtonian mechanics [7]. The time operator is definable in the Liouville–von Neumann mechanics, which considers density operators of ensembles. In that regard, T implements the commutator relation

$$\left[T,L\right]=i$$

(2)

wherein L governs the evolution of density operators by the Liouville equation.

An ensemble is defined by the mapping $P\mapsto \pi \left(P\right)$ which assigns a probability $\pi \left(P\right)$ to each projector P, such that

$$\pi \left(0\right)=0$$

$$\pi \left(1\right)=1$$

$$\pi \left(\sum _{P\in \mathcal{P}}P\right)=\sum _{P\in \mathcal{P}}\pi \left(P\right)$$

for orthogonal constituents of the sum [17]. According to the Gleason theorem, there is a density operator $\rho$ that satisfies $\pi \left(P\right)=〈\rho |P〉$, which should be positive semidefinite $\rho \ge 0$, Hermitian ${\rho }^{†}=\rho$ and unity traced $Tr\rho =1$. It follows that $\rho =F{F}^{†}$, whereby F is the root operator which is unity normed, considering that $Tr\rho ={\parallel F\parallel }^2$.

2.2. Physics of Complex Systems

If $\rho =\left|f\right.〉\left.〈f\right|$ for a unity normed signal f, the density is coincident to $\rho ={\left|f\right|}^2$, provided that tracing the operator corresponds to integrating the function, since $Tr\rho =〈\rho |1〉$. The projector $Pf=cf$, which multiplies signals by a characteristic function, has the probability $\pi \left(P\right)=〈\rho |c〉$ that is an expected value of the variable c. The Koopman–von Neumann mechanics, which has been postulated in that manner, considers the evolution of densities and variables due to the action of a one-parameter group $G^t$ onto the measurable space that should preserve a probability measure. Variables upon the probability space evolve by the group of unitary operators $U^t:f\mapsto f\circ G^t$ and densities are governed by adjoints $U^{t†}=U^{-t}$. In that instance, there is an infinitesimal generator L such that $U^{t†}=e^{iLt}$, wherefrom it follows the Liouville equation

$$\frac{\partial \rho }{i\partial t}=L\rho$$

which is governing an evolution of the density [18]. In terms of the evolutionary group, the commutator relation (2) is equivalent to $\left[T,U^t\right]=t{U}^t$, which comes down to

$$\left[T,U\right]=U$$

(3)

supposing the cyclic group generated by U.

If there is an operator T satisfying the commutator relation (3), such a system is termed complex. The time operator formalism of complex systems originated from the Brussels School of Thermodynamics, which was investigating a link between reversible and irreversible processes [7]. It is realized due to a change in representation $\Lambda =\lambda \left(T\right)$ that is the operator function of time, which transfigures the Lie group $U^{t†}$ into a Markov semigroup

$$W^{t†}=\Lambda U^{t†}{\Lambda }^{-1},t\ge 0$$

(4)

The semigroup (4) indicates irreversibility, since operators $W^{t†}$ for $t<0$ are not positivity-preserving and therefore not related to the evolution of density. The change in representation should preserve the positivity $\rho \ge 0\Rightarrow \Lambda \rho \ge 0$, the trace $Tr\rho =Tr\Lambda \rho$, and the uniform density $1=\Lambda 1$ and it should be invertible in a dense subset. Terms of the change imply that $\Lambda$ maps a density into a density without any information loss [19].

The link between reversible and irreversible processes is substantial for elucidation of the measurement problem. The evolutionary group $U^t$ has become the semigroup $W^t$ for $t\ge 0$ due to a change in representation (4). In that respect, irreversible evolution corresponds to an increase of the information entropy, which is a measurement characterized by [7].

3. Wavelets and the Measurement Hierarchy

3.1. Paradigm of the Measurement Process

In book V of Elements, Euclid elaborates the doctrine of proportion concerning commensuration of magnitudes. Due to the Euclidean algorithm, magnitudes $a\le b$ measure each other in terms of the continued fraction

$$\frac{a}{b}=\frac{1}{n_1+\frac{1}{n_2+\frac{1}{\ddots }}}$$

(5)

which should indicate a process that takes place step by step over time [6].

The Euclidean algorithm is a paradigmatic measurement. The process corresponds to a sequence

$${\xi }_j=\frac{1}{n_1+\frac{1}{\ddots +\frac{1}{n_j}}}$$

whose elements ${\xi }_j=\frac{h_j}{k_j}$ are obtained by the recurrence equation

$$h_{j+1}=n_{j+1}{h}_j+h_{j-1},k_{j+1}=n_{j+1}{k}_j+k_{j-1}$$

considering the initial conditions $h_0=0$, $h_1=1$, $k_0=1$, $k_1=n_1$. The difference between successive elements

$$\Delta {\xi }_j={\xi }_{j+1}-{\xi }_j=\frac{h_{j+1}}{k_{j+1}}-\frac{h_j}{k_j}=\frac{{(-1)}^j}{k_j{k}_{j+1}}$$

expands the continued fraction (5) to the alternating series

$$\Delta {\xi }_0+\cdots +\Delta {\xi }_j+\cdots =\frac{1}{k_0{k}_1}-\cdots \frac{{(-1)}^j}{k_j{k}_{j+1}}\cdots$$

(6)

which is a sparse representation [20], composed of terms from the redundant dictionary $\frac{1}{1},\frac{1}{2}\cdots$

The expansion (6) corresponds to a binary code, wherein 0 is assigned to terms of the dictionary that do not participate in the series and an alternating value $\pm 1$ is assigned to those that do participate. Such a representation of the measurement process is highly redundant, since the complete dictionary cannot be involved in a series. One should therefore eliminate excess zeros, which is realized by coding the sequence $n_1,n_2,\cdots$ The code is composed of alternative $\pm 1$ values at positions $n_1,n_1+n_2,\cdots$, which gives rise to the Minkowski function

$$?:\frac{1}{n_1+\frac{1}{n_2+\frac{1}{\ddots }}}\mapsto \frac{1}{2^{n_1-1}}-\frac{1}{2^{n_1+n_2-1}}+\cdots$$

that is an automorphism transfiguring the continued fraction into the binary representation. The process is codified by a real number

$$x=\underset{n_1}{\underbrace{0.0\dots 0}}\underset{n_2}{\underbrace{1\dots 1}}\underset{\dots }{\underbrace{0\dots 0}}1\dots$$

(7)

wherein digital positions concern its temporality [9].

In that regard, the measurement is considered to be a time series of binary digits (7), which is a general definition of the problem. Time is related to a scale j of the binary tree whose nodes $0<\frac{2k-1}{2^{j+1}}<1$ correspond to both states and devices of the measurement process. A step concerns the Rényi map

$$R\left(x\right)=\left\{\begin{array}{cc}2x,& 0\le x<\frac{1}{2}\\ 2x-1,& \frac{1}{2}\le x<1\end{array}\right.$$

(8)

representing a shift in terms of binary digits. It is a self-similarity of the binary tree, which maps both left and right subtrees to the entire one [6].

3.2. Hierarchical Bases of the Signal Space

The binary structure reflects the hierarchy of the signal space representing the measurement process. It concerns both states $\Sigma$ and devices $\Delta$ which should be considered in a dual manner [6]. If the signal space $\Sigma$ has identified states, the topological dual $\tilde{\Sigma }=\Delta$ corresponds to measurement devices. Starting from devices $\Delta$ on the other hand, the topological dual $\tilde{\Delta }=\Sigma$ concerns measurement states. These options may differ in more than a conceptual sense; taking the dual of the dual does not necessarily bring back to the departure. Even if it does, there may be some reasons to favor one of them since an aspect of the process has been obscured [14].

A sensible solution should consider signals to be both states and devices concurrently, which leads to a source–detector interchangeability that is termed crossing in quantum theory [14]. It is a reason for regarding $\Sigma =\Delta$ as self-dual, which applies to the Hilbert space $L_{\mu }^2$, wherein $\mu$ is the Lebesgue measure over the unit interval. A hierarchy is realized by wavelets which correspond to orthonormal bases of the signal space [6]. The Haar base is paradigmatically designed by translation and normalized dilatation of the mother wavelet $\chi \left(x\right)=\left\{\begin{array}{cc}-1,& 0\le x<\frac{1}{2}\\ +1,& \frac{1}{2}<x\le 1\end{array}\right.$ in the manner of

$${\chi }_{j,k}\left(x\right)=\left\{\begin{array}{cc}-2^{j/2},& \frac{k}{2^j}\le x<\frac{k+1/2}{2^j}\\ +2^{j/2},& \frac{k+1/2}{2^j}<x\le \frac{k+1}{2^j}\end{array}\right.$$

(9)

implying that basic elements are zero-valued almost everywhere else.

Wavelets on the unit interval have arisen from those on the real line, which are the orthonormal bases

$${\Psi }_{j,k}\left(x\right)=2^{j/2}\Psi (2^{j}x-k)$$

obtained by translation and normalized dilatation of a mother wavelet $\Psi$. They reappear in the signal space $L_{\mu }^2$ due to ${\psi }_{j,k}\left(x\right)={\sum }_n{\Psi }_{j,k}(x+n)$, which gives rise to the periodization axiom

$${\psi }_{j,k}={\psi }_{j,k+2^j}$$

(10)

and also the annihilation

$$j<0\Rightarrow {\psi }_{j,k}=0$$

(11)

In this manner, one obtains the pyramid ${\psi }_{j,k}$ for $j\ge 0$ and $1\le k\le 2^j$ which is an orthonormal base of $L_{\mu }^2\ominus \mathbb{1}$ representing the orthocomplement of constants $\mathbb{1}$ [21]. Signals are decomposed in a hierarchical base due to the resolution of identity

$$1=\left|1\right.〉\left.〈1\right|+\sum _{j\ge 0}\sum _{k=1}^{2^j}\left|{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right|$$

wherein $\left.〈·\right|$ corresponds to a state and $\left|·\right.〉$ to a device of the measurement process. The translation axiom

$${\psi }_{j,k}(x-\frac{m}{2^j})={\psi }_{j,k+m}\left(x\right)$$

(12)

is satisfied as well, which means that variables are equally distributed within each scale.

The evolution of wavelets concerns the operator $U:f\mapsto f\circ R$ which is induced by the Rényi map (8). The evolutionary axiom holds in terms of its adjoint

$$U^{†}{\psi }_{j,k}=\frac{1}{\sqrt{2}}{\psi }_{j-1,k}$$

(13)

which comes down to

$$U{\psi }_{j,k}=\frac{1}{\sqrt{2}}{\psi }_{j+1,k}+\frac{1}{\sqrt{2}}{\psi }_{j+1,k+2^j}$$

Since R is a measure preserving transformation of the unit interval, the operator U preserves distribution of a variable. The orthogonality implies that variables are decorrelated, considering that $E{\psi }_{j,k}=〈1|{\psi }_{j,k}〉=0=E\overline{{\psi }_{j,k}}$ and

$$(j,k)\ne (l,m)\Rightarrow E\overline{{\psi }_{j,k}}{\psi }_{l,m}=〈{\psi }_{j,k}|{\psi }_{l,m}〉=0=E\overline{{\psi }_{j,k}}E{\psi }_{l,m}$$

The absolute square $|{\psi }_{j,k}{|}^2$ is a density function as well, which makes the base generate both states and devices concurrently.

Time of the measurement hierarchy corresponds to a scale, which implies the operator

$$T=\sum _{j\ge 0}\sum _{k=1}^{2^j}j\left|{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right|$$

(14)

that is defined on a dense subset of $L_{\mu }^2\ominus \mathbb{1}$ [22]. The commutator relation (3) follows immediately from the evolutionary axiom, considering that

$$\left[U^{†},T\right]{\psi }_{j,k}=U^{†}j{\psi }_{j,k}-(j-1)U^{†}{\psi }_{j,k}=U^{†}{\psi }_{j,k}$$

3.3. Space of Ensembles

A problem might occur concerning the generation of an evolutionary group, since the operator U is not invertible. However, it extends naturally to an invertible operator $U_{\chi }:F\mapsto F\circ B$ which is induced by the baker map

$$B(x,y)=\left\{\begin{array}{cc}\left(2x,\frac{y}{2}\right),& 0\le x<\frac{1}{2}\\ \left(2x-1,\frac{y+1}{2}\right),& \frac{1}{2}\le x<1\end{array}\right.$$

(15)

that is a measure preserving transformation of the unit square [23]. This is a reason to embed the signal space $L_{\mu }^2$ into an extended one $L_{{\mu }^2}^2=L_{\mu }^2\otimes L_{\mu }^2$, wherein ${\mu }^2=\mu \otimes \mu$ is the product measure [6].

The space of ensembles $L_{{\mu }^2}^2=\Delta \otimes \Sigma$ is a tensor product of devices and states. The resolution of identity gives rise to the decomposition

$$F=\left|1\right.〉\left.〈A\right|+\sum _{j\ge 0}\sum _{k=1}^{2^j}\left|{\psi }_{j,k}\right.〉\left.〈D_{j,k}\right|$$

wherein $\left.〈A\right|=〈1|F$ is the approximation coefficient and $\left.〈D_{j,k}\right|=〈{\psi }_{j,k}|F$ are detail coefficients at a certain scale of the measurement hierarchy, implying the matrix multiplication

$$F_1{F}_2(x,y)=\int F_1(x,t)F_2(t,y)dt$$

The time operator $T_{\chi }$ of the system evolving by $U_{\chi }$ has been explicitly constructed [17]. Its projection onto the signal space $L_{\mu }^2$ concerns the hierarchy of the Haar base (9). The time operator of any wavelet (14) is obtained through conjugation $T=C{T}_{\chi }{C}^{†}$ by $C:\left|{\chi }_{j,k}\right.〉\left.〈{\chi }_{l,m}\right|\mapsto \left|{\psi }_{j,k}\right.〉\left.〈{\psi }_{l,m}\right|$, which transforms the Haar base to the other one. It corresponds to a system whose evolution is governed by $U=C{U}_{\chi }{C}^{†}$, which is also an extension of the evolutionary operator U, which is a reason for it to be denoted in the same manner.

One defines the density operator of an ensemble to be $\rho =F{F}^{†}$, wherein the root F is unity normed. The density evolves by an adjoint of $\mathfrak{U}\rho =\left(UF\right){\left(UF\right)}^{†}$, which is the superoperator ${\mathfrak{U}}^{†}\rho =U^{†}\rho U$. The time operator T that concerns the evolution of U is relevant to $\mathfrak{U}$ as well, considering that the commutator relation (3) is satisfied

$$\left[T,\mathfrak{U}\right]\rho =\left[T,U\right]\rho U^{†}=\mathfrak{U}\rho$$

It induces a change in representation $\Lambda =\lambda \left(T\right)$, which should transfigure the evolutionary group generated by ${\mathfrak{U}}^{†}$ to a semigroup (4) generated by

$${\mathfrak{W}}^{†}=\Lambda {\mathfrak{U}}^{†}{\Lambda }^{-1}$$

(16)

4. Orthonormal Wavelets and Projective Measurements

4.1. Measurements in the Hierarchical Base

The von Neumann measurement corresponds to a complete set of orthogonal projectors in the Hilbert space. Considering a paradigmatic measurement, one should assume a hierarchy that is realized by the time series of binary digits. This is a reason to represent orthonormal wavelets ${\psi }_{j,k}$ in terms of projectors $P_{j,k}=\left|{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right|$, which concerns an embedment of $L_{\mu }^2\ominus \mathbb{1}$ into ${(L_{\mu }^2\ominus \mathbb{1})}^2$.

Projectors constitute the Boolean algebra, which is isomorphic to an algebra of sets due to the Stone representation theorem. It is the measurable space corresponding to devices, which an observable has been defined on [9,10]. A measurement state on the other hand corresponds to a density $\rho =F{F}^{†}$ which is defined upon the same domain. One concludes that it should commute with each of projectors, which comes down to the requirement $\rho ={\sum }_{j,k}{P}_{j,k}\rho P_{j,k}$. In that respect, the density is reduced to the subspace of commutative operators

$$\mathfrak{M}\rho =\sum _{j,k}{P}_{j,k}\rho P_{j,k}=\sum _{j,k}{\parallel D_{j,k}\parallel }^2{P}_{j,k}$$

and the measurement problem concerns the issue of how such a reduction has taken place.

It is obvious that the problem occurs only if the measurable space does not fit to the state. If one measures the density itself, there is no reduction, since devices are generated by eigenprojectors $P_{j,k}^o$ of the density operator. Such a measurement

$${\mathfrak{M}}^o\rho =\sum _{j,k}{P}_{j,k}^o\rho P_{j,k}^o=\sum _{j,k}{\parallel D_{j,k}^o\parallel }^2{P}_{j,k}^o$$

is termed optimal, considering that the density operator $\rho ={\mathfrak{M}}^o\rho$ is an invariance of the process.

Starting from the decomposition $F={\sum }_{j,k}\left|{\psi }_{j,k}^o\right.〉\left.〈D_{j,k}^o\right|$ of an ensemble from ${(L_{\mu }^2\ominus \mathbb{1})}^2$, one obtains $\rho ={\sum }_{j,k,l,m}\left|{\psi }_{j,k}^o\right.〉〈D_{j,k}^o|D_{l,m}^o〉\left.〈{\psi }_{l,m}^o\right|$ as well as ${\mathfrak{M}}^o\rho ={\sum }_{j,k}{\parallel D_{j,k}^o\parallel }^2\left|{\psi }_{j,k}^o\right.〉\left.〈{\psi }_{j,k}^o\right|$. It follows that $(j,k)\ne (l.m)\Rightarrow 〈D_{j,k}^o|D_{l,m}^o〉=0$, meaning that detail coefficients are decorrelated in the optimal base. In regard to another base ${\psi }_{l,m}$ that is suboptimal, the same ensemble is composed of coefficients

$$\left.〈D_{l,m}\right|=\sum _{j,k}〈{\psi }_{l,m}|{\psi }_{j,k}^o〉\left.〈D_{j,k}^o\right|$$

Since the basic elements ${\psi }_{j,k}^o$ and ${\psi }_{l,m}$ are almost entirely supported by segments $[\frac{k-1}{2^j},\frac{k}{2^j}]$ and $[\frac{l-1}{2^m},\frac{l}{2^m}]$, respectively, values $〈{\psi }_{l,m}|{\psi }_{j,k}^o〉$ are negligible if supports do not intersect. This implies an approximate decorrelation of the ensemble, which should mean that correlation between detail coefficients is predominantly concerned by inheritance along branches of the binary tree [9,10].

The wavelet-domain hidden Markov model, which is obtained in that manner, has been proven as tremendously useful in a variety of applications, including speech recognition and artificial intelligence [24]. Correlation between detail coefficients $\mathbf{D}=\left(D_{j,k}\right)$ is transmitted only through the Markovian tree of hidden variables $\mathbf{S}=\left(S_{j,k}\right)$, with one attributed to each node, and out of such an interrelation, the ensemble is considered decorrelated. The conditional distribution $\mathbf{D}|\mathbf{S}$ is supposed to be normal, which implies that $D_{j,k}|S_{j,k}$ are independent variables [9].

4.2. Psychophysical Parallelism

The projective measurement $\mathfrak{M}={\sum }_j{\mathfrak{M}}_j$ temporally decomposes into the sum of superprojectors ${\mathfrak{M}}_j={\sum }_k{\mathfrak{P}}_{j,k}$, whereby each ${\mathfrak{P}}_{j,k}\rho =P_{j,k}\rho P_{j,k}$ is a superprojection onto the orthogonal projector $P_{j,k}=\left|{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right|$. If one defines $\mathcal{U}\mathfrak{P}={\mathfrak{UPU}}^{†}$,

$${\mathcal{U}}^{†}{\mathfrak{P}}_{j,k}\rho =\left\{\begin{array}{c}2\left|{\psi }_{j-1,k}\right.〉\left.〈{\psi }_{j-1,k}\left(x\right)\right|{\int }_0^{1/2}F(x,t)F^{†}(t,y)dt\left|{\psi }_{j-1,k}\left(y\right)\right.〉\left.〈{\psi }_{j-1,k}\right|,\hspace{1em}\hspace{1em}\hspace{1em}k\le 2^{j-1}\\ 2\left|{\psi }_{j-1,k-2^{j-1}}\right.〉\left.〈{\psi }_{j-1,k-2^{j-1}}\left(x\right)\right|{\int }_{1/2}^{1}F(x,t)F^{†}(t,y)dt\left|{\psi }_{j-1,k-2^{j-1}}\left(y\right)\right.〉\left.〈{\psi }_{j-1,k-2^{j-1}}\right|,\hspace{1em}\hspace{1em}\hspace{1em}k>2^{j-1}\end{array}\right.$$

holds for $j\ge 1$. In that respect,

$${\mathcal{U}}^{†}{\mathfrak{M}}_j={\mathcal{U}}^{†}\sum _{k\le 2^{j-1}}{\mathfrak{P}}_{j,k}+{\mathcal{U}}^{†}\sum _{k>2^{j-1}}{\mathfrak{P}}_{j,k}=\sum _{k}2{\mathfrak{P}}_{j-1,k}=2{\mathfrak{M}}_{j-1}$$

and since $\mathcal{U}$ is unitary,

$${\mathfrak{M}}_j=2\mathcal{U}{\mathfrak{M}}_{j-1}=2^j{\mathcal{U}}^j{\mathfrak{M}}_0=2^j{\mathfrak{U}}^j{\mathfrak{M}}_0{\mathfrak{U}}^{j†}$$

(17)

which relates all superprojectors to the primary measurement ${\mathfrak{M}}_0\rho =P_0\rho P_0$.

The evolutionary operator U that maps a scale of the measurement hierarchy into the next one is extended onto the space of ensembles $\Delta \otimes \Sigma$ due to the baker map (15). It crosses information between coordinates of the domain in such a manner that the first digit of one, which has been lost by the Rényi map, becomes the first digit of another. The induced operator should cross spatial components, which is evident in the relation $U_{\chi }\left|\chi \right.〉\left.〈1\right|=\left|1\right.〉\left.〈\chi \right|$, and likewise for other wavelets. Considering identifications $\left|·\right.〉\hookrightarrow \left|·\right.〉\left.〈1\right|$ and $\left.〈·\right|\hookrightarrow \left|1\right.〉\left.〈·\right|$, the operator has crossed a measurement device into a state [6].

The superprojector (17) is factorized into measurement operators ${\mathfrak{M}}_j\rho =M_j\rho M_j^{†}$, in which $M_{j}F=2^{j/2}{U}^j{P}_0{U}^{j†}F$. First of all, it concerns the evolution by $U^{†}$ crossing states into devices. Thereafter, $P_0$ projects the ensemble onto a primary device, which annihilates all devices out of the measurement display. Finally, the evolution by U crosses devices back into states. Supposing the measurement hierarchy that is realized by the Haar base (9), the primary device has corresponded to the ensemble ${\chi }_0=\left|\chi \right.〉\left.〈1\right|$ which produces the base of ensembles ${\prod }_{j\in (j_1<\cdots <j_n)}{\chi }_j$ by the evolution ${\chi }_j=U_{\chi }^j{\chi }_0$ [17]. Each element ${\chi }_{\overrightarrow{j}}$ is specified by an increasing sequence of integers $\overrightarrow{j}=(j_1<\cdots <j_n)$ and it evolves by $U_{\chi }{\chi }_{\overrightarrow{j}}={\chi }_{\overrightarrow{j}+1}$, wherein $\overrightarrow{j}+1=(j_1+1<\cdots <j_n+1)$. The measurement operator $M_j=2^{j/2}{U}_{\chi }^j{P}_0{U}_{\chi }^{j†}$ implies the process $U_{\chi }^{j†}{\chi }_{\overrightarrow{j}}={\chi }_{\overrightarrow{j}-j}$, due to which some states have become devices. The projector $P_0$ should fix an element ${\chi }_{\overrightarrow{j}}={\chi }_{j_1}\cdots {\chi }_{j_n}$ if it is started by the primary device ${\chi }_{j_1}={\chi }_0$ and annihilate it if not, which means that all devices out of the measurement display come to be annihilated. The terminal step concerns the evolution $U_{\chi }^j{\chi }_{\overrightarrow{j}}={\chi }_{\overrightarrow{j}+j}$, wherein some devices have become states. In that respect, crossing between them due to an evolution in the temporal domain is substantial for a hierarchy [6].

The measurement display defines a boundary between states and devices, which is arbitrary to a very large extent. Self-duality of the signal space representing both states and devices concerns the principle of psychophysical parallelism, as has been noted by von Neumann [12]. A problem occurs in that the principle is violated so long as it is not demonstrated that the display has been placed in an arbitrary manner, which is achieved by crossing due to the evolutionary operator. In that regard, the evolution of measurement operators corresponds to its displacement by designating another ${\chi }_j$ to be a primary device. Crossing devices into states elucidates the term psychophysics that is used in order to transcend any separation between the two [6].

Von Neumann made a reference to Bohr, who “was the first to have pointed out that the dual description of quantum theory relates to the principle of psychophysical parallelism [25]”. Although he has never mentioned it, Bohr adopted Fechner’s psychophysics as he had been taught by Høffding [26]. The most significant source for psychophysical parallelism by Fechner is the foreword and the introduction from the Elements of Psychophysics [27]. His attitude is termed the identity view, since the observer is not considered to be a conglomeration of two substances but one single entity. Fechner primarily discerns the outer psychophysics, which is a link between sensation and stimulation. It is realized through the neuroesthetical computation that relates sensation to neural activity, which is termed inner psychophysics [28].

A significant repercussion of von Neumann’s solution to the measurement problem is that irreversibility takes place in the presence of the observer’s mind, which seems to play an active role in the process. The only manner to make such an insight compatible to psychophysical parallelism concerns switching to inner psychophysics by a change in representation [6]. In that regard, inner psychophysics should correspond to a Markovian tree of the wavelet-domain hidden Markov model [28].

The irreversibility is actually manifested by the fact that a state before the measurement process results in the sum of diverse states thereafter. The primary measurement designed by an operator $M_0=P_0$ is not irreversible in that respect, since it corresponds to the projector onto a single state. A problem occurs considering that the measurement operator $M_j=2^{j/2}{\mathfrak{U}}^j{P}_0={\sum }_k{2}^{j/2}{P}_{j,k}$ evolves into a combination of diverse projectors. It concerns the evolution represented by $\mathfrak{U}$, whose irreversibility comes to prominence due to a change in representation (16). The evolution $M_{j+1}=\sqrt{2}\mathfrak{U}{M}_j$ in terms of the Markov process $\mathfrak{W}$ becomes $M_{j+1}=\sqrt{2}\mathfrak{U}{\sum }_k{2}^{j/2}{P}_{j,k}=\sqrt{2}{\sum }_k{\Lambda }^{†}\mathfrak{W}{\Lambda }^{-1†}{2}^{j/2}{P}_{j,k}$ and one denotes $S_{j,k}={\Lambda }^{-1†}{2}^{j/2}{P}_{j,k}F=2^{j/2}{\Lambda }^{-1†}\left|{\psi }_{j,k}\right.〉\left.〈D_{j,k}\right|$ which indicates an irreversible evolution of hidden variables $\sqrt{2}\mathfrak{W}{\sum }_k{S}_{j,k}={\sum }_k{S}_{j+1,k}$. In that manner, the change in representation should transfigure detail coefficients $\mathbf{D}=\left(D_{j,k}\right)$ into a Markovian tree $\mathbf{S}=\left(S_{j,k}\right)$.

The outer psychophysical information of an ensemble is independent of orthonormal wavelets, considering that $\mathrm{H}\left(C\mathbf{D}\right)=\mathrm{H}\left(\mathbf{D}\right)+log|detC|=\mathrm{H}\left(\mathbf{D}\right)$ for any operator C which should be unitary since it represents a base substitution [28]. The canonical relation $\mathrm{H}\left(\mathbf{D}\right)=\mathrm{H}\left(\mathbf{S}\right)+\mathrm{H}\left(\mathbf{D}\right|\mathbf{S})$ separates the inner psychophysical information $\mathrm{H}\left(\mathbf{S}\right)$ from irreducible randomness $\mathrm{H}\left(\mathbf{D}\right|\mathbf{S})$. The global entropy $\mathrm{H}\left(\mathbf{S}\right)$ is related to an increase of the local entropy $\mathrm{H}(S_{j,k})$ in the temporal domain, corresponding to the scale of the measurement hierarchy [29]. The optimal decomposition concerns the most significant increase of the information entropy, which is the measurement process characterized by [12].

An innate component of the wavelet-domain hidden Markov model is a denoising procedure that has proven to be advantageous over other methods [24]. It is performed in a superior manner using the optimal base [29], which presents an active strategy to cope with the measurement problem. In that respect, the optimal measurement is related to maximization of the inner psychophysical information which remains unaltered by the procedure [28].

5. Frame Wavelets and General Measurements

5.1. Duality in Frame Theory

The concept of frame refers to elements ${\psi }_{j,k}$ such that

$$A\le \sum _{j\ge 0}\sum _{k=1}^{2^j}\left|{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right|\le B$$

for positive numbers A and B which are termed frame bounds [13]. If $A=B=1$, i.e., $1={\sum }_{j,k}\left|{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right|$, such a frame is the Parseval one. It is termed frame wavelets on the unit interval if axioms (10)–(13) hold.

$\widetilde{{\psi }_{j,k}}$ is a dual frame of ${\psi }_{j,k}$ if the resolution of identity applies in the manner

$$1=\sum _{j,k}\left|\widetilde{{\psi }_{j,k}}\right.〉\left.〈{\psi }_{j,k}\right|$$

If there is an operator $\left[\right]$ such that $\widetilde{{\psi }_{j,k}}=\left[\right]{\psi }_{j,k}$, the frame is canonical dual. Let ] be an invertible operator such that $]{\psi }_{j,k}$ is the Parseval frame and [ is its ajoint. In that regard,

$$1=\left[\left]{\psi }_{j,k}\right.〉\right.\left.〈{\psi }_{j,k}\right|=\left[\right]\sum _{j,k}{{]}^{-1}\left]{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right[{{[}^{-1}=\left[\right]]}^{-1}\sum _{j,k}\left]{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right[{{[}^{-1}=\left[\right]]}^{-1}[}^{-1}$$

wherefrom it follows that $\left[\right]$ is factorized into the product of [ and ].

The general measurement $\mathfrak{M}\rho ={\sum }_{j,k}{M}_{j,k}\rho M_{j,k}^{†}$ is characterized by operators $M_{j,k}$ satisfying $1={\sum }_{j,k}{M}_{j,k}^{†}{M}_{j,k}$, which means that a density is mapped into a density

$$Tr\mathfrak{M}\rho =\sum _{j,k}\parallel M_{j,k}{F\parallel }^2=〈F|\sum _{j,k}{M}_{j,k}^{†}{M}_{j,k}F〉={\parallel F\parallel }^2=Tr\rho$$

In order to elucidate how it relates to the frame concept, one should consider operators $Q_{j,k}=\left|\widetilde{{\psi }_{j,k}}\right.〉\left.〈{\psi }_{j,k}\right|$ that meet the resolution of identity $1={\sum }_{j,k}{Q}_{j,k}$. Under the term $\parallel \widetilde{{\psi }_{j,k}}\parallel =1$, it follows that $1={\sum }_{j,k}\left]{\psi }_{j,k}\right.〉\left.〈{\psi }_{j,k}\right[={\sum }_{j,k}\left]Q_{j,k}^{†}{Q}_{j,k}\right[$ which implies that $Q_{j,k}[=\left|\widetilde{{\psi }_{j,k}}\right.〉\left.〈{\psi }_{j,k}\right[$ is the measurement operator $M_{j,k}$. Its evolution requires the Parseval frame $]{\psi }_{j,k}$ and the dual one $\widetilde{{\psi }_{j,k}}$ to be wavelets satisfying (10)–(13).

The evolutionary operator $U=C{U}_{\chi }{C}^{†}$ on the space of ensembles is obtained through conjugation of the natural extension $U_{\chi }$ by $C:\left|{\chi }_{j,k}\right.〉\left.〈{\chi }_{j,k}\right|\mapsto \left|\widetilde{{\psi }_{j,k}}\right.〉\left.〈{\psi }_{j,k}\right[$, which transforms the Haar base to the Parseval frame and the dual one. Crossing devices into states due to the evolution by U concerns a duality relation $\Sigma =\tilde{\Delta }$ [14]. The signal space of the general measurement might not be self-dual, but it separates into dual spaces generated by ${\psi }_{j,k}$ and $\widetilde{{\psi }_{j,k}}$, respectively, of which the first one should correspond to states and the second one to devices [13].

5.2. Measuring an Open System

According to the Naimark theorem, states of the measurement extend to a direct sum ${\Sigma }^{*}=\Sigma \oplus {\Sigma }^{\prime }$, wherein the Parseval frame $\left.〈{\psi }_{j,k}\right[$ corresponds to the projection of an orthonormal base $\left\{{\psi }_{j,k}\right[$ onto the subspace $\Sigma$. Likewise, the dual frame concerns measurement devices which are extended to ${\Delta }^{*}=\Delta \oplus {\Delta }^{\prime }$. The measurement operator $M_{j,k}$ is a restriction of the projector $M_{j,k}^{*}{=[}^{-1}\left.\widetilde{{\psi }_{j,k}}\right\}\left\{{\psi }_{j,k}\right[$ onto $\Delta \otimes \Sigma$, and, in that manner, the projective measurement ${\mathfrak{M}}^{*}$ restricts to the general one $\mathfrak{M}$ by neglecting an environment which has remained out of scope [30]. The general measurement is therefore related to an open system that has been partially described by the stochastic process. Devices and states might be some subspaces of signals, respecting the duality between them. In that regard, frames $\left.〈{\psi }_{j,k}\right|$ and $\left|\widetilde{{\psi }_{j,k}}\right.〉$ are projections of the Riesz base $\left\{{\psi }_{j,k}\right|$ and its dual $\left|\widetilde{{\psi }_{j,k}}\right\}=\left[\left]{\psi }_{j,k}\right\}\right.$, which are biorthonormal [13].

A practical realization of the Naimark theorem implies a method that is analogous to heterodyne detection in communication engineering: the ensemble to be observed combines with another one, which is termed ancilla [31]. Thereafter, the von Neumann measurement corresponding to projectors $M_{j,k}^{*}$ has been performed on the combined space ${\Delta }^{*}\otimes {\Sigma }^{*}$ that is the tensor product of states and devices which are extended by the environment. The amount of information which is obtained in that manner might be larger than if the observer is restricted to the von Neumann measurements without ancilla. Optimal measurements are therefore not even close to being just projective ones which correspond to orthonormal wavelets in statistical signal processing [30].

A frame ${\psi }_{j,k}$ should be optimal for the ensemble from $\Delta \otimes \Sigma$ if the orthonormal base $\left]{\psi }_{j,k}\right\}$ is optimal for an ensemble in the combined space ${\Delta }^{*}\otimes {\Sigma }^{*}$. One assumes $F={\sum }_{j,k}\left|\widetilde{{\psi }_{j,k}^o}\right.〉\left.〈D_{j,k}^o\right|$, wherein ${\psi }_{j,k}^o$ is the optimal frame. Detail coefficients correspond to those of $F^{*}={\sum }_{j,k}\left[{}^{-1}\widetilde{{\psi }_{j,k}^o}\right\}\left.〈D_{j,k}^o\right|$ in the base $\left[{}^{-1}\widetilde{{\psi }_{j,k}^o}\right\}=\left]{\psi }_{j,k}^o\right\}$ which is orthonormal, though it might not imply any hierarchy (10)–(13). In that respect, general measurements spread the optimal decomposition to some ensembles which cannot be decorrelated in a hierarchical base but which has been restricted to the frame providing a hierarchy of devices and states.

6. Conclusions

The measurement problem is formulated in terms of mathematical physics, notwithstanding any interpretation of physical theories. The significance of time for its elaboration has explicated a substantial relation between signals and stochastic processes, which is the definition of statistical signal processing. A paradigmatic measurement concerns commensuration of magnitudes by the Euclidean algorithm, producing a time series of binary digits. It constitutes the hierarchy of the binary tree whose nodes correspond to both states and devices of the measurement process.

The time operator formalism of complex systems, which has proposed a unification of reversible and irreversible processes, relates the problem respecting its definition that was postulated by von Neumann. He indicated two fundamentally diverse types of interventions in a system, the first of which corresponds to a temporal evolution that is reversible and the second one to an irreversible measurement. The main advancement concerns a formulation of the measurement process in terms of a temporal evolution, wherein irreversibility has occurred due to the change in representation switching from outer to inner psychophysics. The principle of psychophysical parallelism that was pointed out by Bohr and von Neumann should be consistently realized in such a manner.

The optimal measurement corresponds to the most significant increase of the information entropy in the temporal domain. This implies a decorrelation of the ensemble, which is a consequence of its invariance under the process. Generalization to an open system is performed by the use of duality in frame theory, spreading the optimal decomposition to some ensembles which cannot be decorrelated in a hierarchical base. The denoising procedure, which is an innate component of the model, presents an active strategy to cope with the measurement problem in statistical signal processing.