A macroscopic material consists of atoms, which are quantum particles. The starting point for their dynamics lies in quantum statistical mechanics. For a measurement, the apparatus must be macroscopic and have a macroscopic pointer so that the outcome of the measurement can be read off or processed automatically. Hereto, an operator formalism is required, with dynamics set by the Liouville–von Neumann equation, the generalization of the Schrödinger equation to mixed states.

Progress on solvable models for quantum measurement has been made in recent decades when we, together with A. Allahverdyan and R. Balian introduced and solved the so-called Curie–Weiss model for quantum measurement (Allahverdyan A. E. et al., 2003) in our “ABN” collaboration. Here, the classical Curie–Weiss model of a magnet is taken in its quantum version and applied to the measurement of a quantum spin 1 2 . Various further aspects were presented in Allahverdyan A. E. et al. (2003), Allahverdyan et al. (2005a), Allahverdyan et al. (2005b), Allahverdyan et al. (2007), and Allahverdyan et al. (2006). They were reviewed and greatly expanded in Allahverdyan et al. (2013). Lecture notes were presented by Nieuwenhuizen et al. (2014). A straightforward interpretation for a class of these measurement models was provided by Allahverdyan et al. (2017); it is a specified version of the statistical interpretation made popular by Ballentine (1970).

Simultaneous measurement of two noncommuting quantum variables was worked out (Perarnau-Llobet and Nieuwenhuizen, 2017a), as well as an application to Einstein-Podolsky-Rosen type of measurements (Perarnau-Llobet and Nieuwenhuizen, 2017b). A numerical test on a simplified version of the Curie–Weiss model reproduced nearly all of its properties (Donker et al., 2018).

Our ensuing insights, which are suitable for teachers of quantum theory (at the high school, bachelor’s, or master’s levels), are presented in Allahverdyan et al. (2024) and summarized in a feature article (Allahverdyan et al., 2025).

The mentioned Curie–Weiss model was recently generalized by us to measure a spin l > 1 2 (Nieuwenhuizen, 2022). This study will be termed “Models” henceforth. For spin l , the state of the magnet is described by 2 l order parameters. To assure an unbiased measurement, the Hamiltonian of the apparatus and the interaction Hamiltonian with the tested system have Z 2 l + 1 symmetry. The statics were solved for spin-1, 3 2 , 2, and 5 2 .

Here, the dynamics are worked out for spin-1, laying the groundwork for higher-spin dynamics. In the spin 1 2 Curie–Weiss model, it was found that Schrödinger cat terms disappear through two mechanisms: dephasing of the magnet, possibly followed by decoherence due to the thermal bath. Similar behavior is now investigated for spin-1.

The setup of the article is as follows. In Section 2, we recall the formulation of the Curie–Weiss model for general spin- l and discuss aspects of its physical implementation for spin 1 2 and spin-1. In Section 3, we revisit the spin- 1 2 case and cast its dynamics in a general form. In Section 4, we analyze the dynamics of the spin-1 situation. We close with a summary in Section 5.

A quantum measurement is often assumed to be “instantaneous.” In our idealized modeling, it will take a finite time, but the tested spin will not evolve in the meantime. In other words, the spin itself is “sitting still” and waiting to be measured. Neither should it evolve during the “fast” measurement. This is realized when its Hamiltonian H ̂ S commutes with s ̂ z ; we consider the simplest case: H ̂ S = 0 .

In order to have an unbiased apparatus, the Hamiltonian of the magnet should have degenerate minima and maximal symmetry. To construct such a functional, we consider, in the eigenvalue presentation, the form C 2 = ν 2 ∑ i , j = 1 N cos 2 π σ i − σ j 2 l + 1 , ν ≡ 1 N , (2.2) which is maximal in ferromagnetic states σ i = σ 1 ( i = 2 , … , N ) . In general, these interactions do not seem realistic, but here, the cosine rule allows expressing this as spin–spin interactions, C 2 = co l 2 + s i l 2 , (2.3)

which is bilinear in the single-spin sums co l = 1 N ∑ i = 1 N cos 2 π σ i 2 l + 1 ,   s i l = 1 N ∑ i = 1 N sin 2 π σ i 2 l + 1 . (2.4)

The discrete values of the spin projections allow expressing these terms in the 2 l spin moments, m k = 1 N ∑ i = 1 N σ i k , k = 1 , … , 2 l , (2.5) while m 0 ≡ 1 . For l = 1 2 , the values s = ± 1 2 imply cos ⁡ π s = 0 , sin ⁡ π s = 2 s . (2.6)

Applying this for s → σ i and summing over i yields co 1 2 = 0 , s i 1 2 = 2 m 1 , m 1 = 1 N ∑ i = 1 N σ i . (2.7)

In the case l = 1 , one has s = 0 , ± 1 . The rule cos 2 π s 3 = 1 − 3 2 s 2 , sin 2 π s 3 = 3 2 s , (2.8) leads to s → σ i and summing over i leads to co 1 = 1 − 3 2 m 2 , s i 1 = 3 2 m 1 , (2.9) Here, m 2 ranges from 0 to 1 with steps of ν ≡ 1 / N , while m 1 ranges from − m 2 to m 2 with steps of 2 ν . At finite N , one can label the discrete m 1,2 as m 1 = 2 n 1 − n 2 ν , m 2 = n 2 ν , 0 ≤ n 2 ≤ N , 0 ≤ n 1 ≤ n 2 . (2.10)

The results for s = 3 2 , 2, and 5 2 are given in Models.

Let out of the N spins σ i , a number N σ = ∑ i δ σ i , σ take the value σ ∈ spec l and let x σ = N σ / N be their fraction. The sum rule ∑ σ N σ = N implies m 0 ≡ ∑ σ x σ = 1 . The moments read m k = ∑ σ = − l l x σ σ k , k = 1 , … , 2 l , (2.11)

Inversion of these relations determines the x σ as linear combinations of the m k . For l = 1 2 , one has m 1 = 1 2 x 1 2 − 1 2 x − 1 2 , x ± 1 2 = 1 2 ± m 1 . (2.12)

For spin-1 ( l = 1 ) , one has m 1 = − x − 1 + x 1 , m 2 = x − 1 + x 1 . (2.13) With x − 1 + x 0 + x 1 = 1 , their inversion reads x 0 = 1 − m 2 , x ± 1 = m 2 ± m 1 2 . (2.14)

In a quantum approach, one goes to operators and sets s → s ̂ z , σ i → σ ̂ z ( i ) , and m k → m ̂ k . For the Hamiltonian H ̂ M = N H ̂ , we follow Allahverdyan et al. (2003a) and Allahverdyan et al. (2003b) and adopt the spin–spin and four–spin interactions: H ̂ M = N H ̂ , H ̂ = − 1 2 J 2 C ̂ 2 − 1 4 J 4 C ̂ 2 2 . (2.15)

Multispin interaction terms like − 1 6 J 6 C ̂ 2 3 − 1 8 J 8 C ̂ 2 4 can be added without changing the overall picture.

The coupling between the tested spin S and the magnet M is chosen similar to Equation 2.2, H ̂ SA = N I ̂ ,   I ̂ = g N ∑ i = 1 N cos 2 π s ̂ z σ ̂ z ( i ) 2 l + 1 , (2.16) where g is the coupling constant. It takes the values I s σ i = − g N ∑ i = 1 N cos 2 π s 2 l + 1 cos 2 π σ i 2 l + 1 + sin 2 π s 2 l + 1 sin 2 π σ i 2 l + 1 , (2.17)

This can be expressed as a linear combination of the moments m 1 , ⋯   , m 2 l . For l = 1 2 , one has I s m 1 = − 4 g s m 1 , (2.18) and for l = 1 , denoting m = ( m 1 , m 2 ) , I s m = − g 1 − 3 2 s 2 1 − 3 2 m 2 + 3 4 s m 1 . (2.19)

The total spin Hamiltonian, H ̂ = H ̂ M + H ̂ SA = H ̂ M − N I ̂ , (2.20)

has Z 2 l + 1 symmetry: on the diagonal basis, a shift s → s + s ̄ with s ̄ = 1,2 , … , 2 l + 1 can be accompanied by a shift σ i → σ i + s ̄ for all i . This is evident in the cosine expressions and implies a somewhat hidden invariance in the formulation in terms of the moments m k , as discussed in Models.

For a general spin l , the magnet–bath coupling is taken as the spin–boson coupling of Opus Equation 3.10, H ̂ MB ≡ γ ∑ i = 1 N ∑ a = x , y , z σ ̂ a ( i ) B ̂ a ( i ) , (2.21) with γ ≪ 1 , where the bath operators read B ̂ a ( i ) = ∑ k c k b ̂ k , a ( i ) + b ̂ k , a † n , (2.22) for each i , a , there is a large set of oscillators labeled by k , having a common coupling parameter c k . These bosons have the Hamiltonian H ̂ B = ∑ i = 1 N ∑ a = x , y , z ∑ k ℏ ω k b ̂ k , a † ( i ) b ̂ k , a ( i ) , (2.23) with the ω k also identical for all n , a . The autocorrelation function of B defines a bath kernel K , which is identical for all i , a , t r B R ̂ B 0 B ̂ a ( i ) t B ̂ b j t ′ = δ i , j δ a , b   K t − t ′ , B ̂ a ( i ) t ≡ e i H ̂ B t B ̂ a ( i ) e − i H ̂ B t . (2.24) Writing c k = c ( ω k ) , this leads to K t = ∑ k c ω k e i ω k t e β ω k − 1 + e − i ω k t 1 − e − β ω k ≡ 1 2 π ∫ − ∞ + ∞ d ω   e i ω t K ̃ ω . (2.25)

The kernel K ̃ ( ω ) can be read off and expressed in the spectral density ρ c ( ω ) = ∑ k c ( ω k )   δ ( ω − ω k ) , K ̃ ω = 2 π | ω | ρ c ω ω e β ω − 1 . (2.26)

We adopt an Ohmic spectrum with a Debye cutoff, K ̃ ω = e − ω / Γ 4 ω e ω / T − 1 , (2.27) where T = 1 / β is the temperature of the phonon bath, and Γ the typical cutoff frequency. In Opus, we also consider a Lorentzian (power law) cutoff, for which the statics allows analytic results.

With the couplings in Equations 2.21, 2.22, and 2.23 independent of a , H ̂ MB is statistically invariant under Z 2 l + 1 . Combined with the invariance of H ̂ M and H ̂ SA , this ensures an unbiased measurement.

The evolution of the density matrix of the total system is given by the Liouville–von Neumann equation. On the eigenbasis of s ̂ z , its elements R ̂ s s ̄ evolve independently as given in Equation 4.8 of Opus; this involves the apparatus spins and the bath. The procedure of Opus for spin l = 1 2 appears to hold for general spin- l operators.

Let us consider the time evolution of R ̂ s s ̄ as given in Equation 4.8 of Opus (we now denote i → s , j → s ̄ ), where the action of the harmonic oscillator bath has been expressed in the bath kernel K ( t ) and which involves commutators of R ̂ s s ̄ with the spin operators σ ̂ a ( i ) , a = x , y , z ; i = 1 , … , N .

Formally, the initial state (Equation 5.4) is a constant function of the σ ̂ z ( i ) . In addition, R ̇ s s ̄ ( t i ) is a function of them, so it is consistent to assume that, at all t , R ̂ s s ̄ only depends on the σ ̂ z ( i ) . As a result, the a = z terms of Equation 4.8 in Opus have vanishing commutators for any spin l . Left with the x , y commutators, we define (using the index n rather than i to label the σ ̂ x , y ) σ ̂ ± ( n ) = σ ̂ x ( n ) ± i σ ̂ y ( n ) . (2.28) Because ∑ a x , y σ ̂ a ( n ) O ̂ σ ̂ a ( n ) = 1 2 ∑ α = ± 1 σ ̂ α ( n ) O ̂ σ ̂ − α ( n ) for any operator O ̂ , Equation 4.8 in Opus takes the form d R ̂ s s ̄ t d t = − i H ̂ s R ̂ s s ̄ t + i R ̂ s s ̄ t H ̂ s ̄ +   γ 2 ∑ α , β = ± 1 ∑ n = 1 N ∫ 0 t d u   K β u C ̂ s s ̄ , β α , n u , (2.29)

where C ̂ s s ̄ , + α , n u = e − i u H ̂ s σ ̂ − α ( n ) e i u H ̂ s R ̂ s s ̄ t ,   σ ̂ α ( n ) , C ̂ s s ̄ , − α , n u = σ ̂ − α ( n ) ,   R ̂ s s ̄ t e − i u H ̂ s ̄ σ ̂ α ( n ) e i u H ̂ s ̄ , (2.30)

are commutators involving the Hamiltonian of M coupled to S in state s , without the bath, viz. H ̂ s = H ̂ M + H ̂ SA s = N H m ̂ 1 + N I s m ̂ 1 . (2.31)

The action of the bath is expressed in the kernel K ( ± u ) , with the smallness of γ allowing truncation at its first order. Equations 2.29, 2.30 are valid for general spin l = 1 2 , 1 , 3 2 , ⋯   .

Most importantly, the R ̂ s s ̃ are decoupled in the separate s , s ̄ sectors, a property of ideal measurement but absent in general. Examples of these non-idealities are a spin S having nontrivial dynamics during the measurement and a biased measurement, in which the Hamiltonian of the magnet and/or the bath depends on the state of S.

The spin- 1 2 Curie–Weiss model for quantum measurement (Allahverdyan A. et al., 2003) was initially conceived as a tool to understand the dominant physical aspects of idealized quantum measurements. It has served this purpose well. Let us look here at possible realizations of the model.

Curie–Weiss models are mean-field types of spin models. Their distance-independent couplings apply to a small magnetic grain. The grain need not be very large. From studies of spin glasses and cluster glasses, it is known that “fat spins,” clusters of hundreds or thousands of coherent spins, are easily detectable (Mydosh, 1993).

The Ising nature of the couplings refers to fairly anisotropic spin–spin interactions. For spin 1 2 , Equation 2.15 expresses the pair and quartet couplings between the z -components of the spins. Multispin interactions are a natural result of the overlap of electronic orbits; here, they are approximated as not decaying with the distance between the spins in the grain. How reasonable this approximation is must be considered in each separate application. The main feature of our modeling, a first-order phase transition in the magnet, suggests that it represents a large class of short-range systems. This is underlined by the model’s support of the Copenhagen postulates of collapse and Born probabilities.

These features also hold for the spin-1 Curie–Weiss model. However, on top of this, Equation 2.8 produces the combination Σ ̂ i ≡ σ ̂ z ( i ) 2 − 2 / 3 , which takes the values 1 / 3 for the “out-of-plane” cases σ i = ± 1 and − 2 / 3 for the “in-plane” case σ i = 0 . Separate-spin terms of the form ∑ i D σ ̂ z ( i ) 2 are well known, stemming from crystal fields. For the apparatus, the co 1 2 term of Equation 2.3 relates to the interaction ∑ i j Σ ̂ i Σ ̂ j between the Σ ̂ i , so it involves both the aforementioned D -term and also the terms σ ̂ z ( i ) 2 σ ̂ z ( j ) 2 . How to implement these crystal-field-type spin–spin interactions in practice is an open question.

Concerning numerical implementations, Donker et al. (2018)’s approximation of the Curie–Weiss model can be generalized to higher spin.

We set the stage by considering the spin- 1 2 situation, the original Curie–Weiss model for quantum measurement in slightly adapted notation ³ . The spin operators are σ ̂ x , y , z , with σ ̂ z = d i a g ( 1 2 , − 1 2 ) . It holds that [ σ ̂ a , σ ̂ b ] = i ε abc σ ̂ c and σ ̂ x 2 + σ ̂ y 2 + σ ̂ z 2 = 3 4 σ ̂ 0 with σ ̂ 0 = d i a g ( 1,1 ) .

The magnet has N these spins σ ̂ x , y , z ( i ) , i = 1,2 , … , N . They have magnetization operator M ̂ 1 = N m ̂ 1 , m ̂ 1 = 1 N ∑ i = 1 N σ ̂ z ( i ) , (3.1)

taking eigenvalues − 1 2 ≤ m 1 ≤ 1 2 . In the paramagnetic state, m 1 = 0 . The Hamiltonian is taken as pair and quartet interactions, H ̂ M = N H ̂ , H ̂ = − 2 J 2 m ̂ 1 2 − 4 J 4 m ̂ 1 4 . (3.2) With x ̂ σ = N ̂ σ / N , it holds that m ̂ 1 = 1 2 x ̂ 1 / 2 − 1 2 x ̂ − 1 / 2 , x ̂ ± 1 / 2 = 1 2 I ̂ ± m ̂ 1 . (3.3)

The spins have eigenvalues σ i = ± 1 2 , so that m ̂ 1 has eigenvalues m 1 = ν ∑ i σ 1 ranging from − 1 2 to 1 2 with steps of ν .

To use the magnet coupled to its bath as an apparatus for a quantum measurement, a system–apparatus (SA) coupling is needed. According to Equation 2.18, it is chosen as a spin–spin coupling, H ̂ SA = − 4 g ∑ i = 1 N s ̂ σ ̂ z ( i ) = − 4 g N s ̂ m ̂ 1 , (3.4)

and takes the values H SA s ( m 1 ) = − 4 g s N m 1 . The full Hamiltonian of S + A in the sector s thus reads H ̂ s = − 2 J 2 N m ̂ 1 2 − 4 J 4 N m ̂ 1 4 − 4 g s N m ̂ 1 . (3.5)

The eigenvalues of s ̂ z are s = ± 1 2 and those of σ ̂ z ( i ) are σ i = ± 1 2 , so that m ̂ 1 has the eigenvalues ν ∑ i = 1 N σ i . The degeneracy of a state with magnetization m 1 is G N = N ! N − 1 2 !   N 1 2 ! = N ! N x − 1 2 ! N x 1 2 ! , (3.6)

and entropy S N = log G N . At large N , we get the standard result for the entropy S N = N S with S = 1 − 2 m 1 2 log 1 − 2 m 1 2 − 1 + 2 m 1 2 log 1 + 2 m 1 2 . (3.7)

Combining Equation 3.5 and Equation 3.6, the free energy in the s -sector reads F s m 1 = − 2 J 2 N m 1 2 − 4 J 4 N m 1 4 − 4 g s N m 1 − T ⁡ log G N m 1 , (3.8)

which yields, for large N , F s N = − 2 J 2 m 1 2 − 4 J 4 m 1 4 − 4 g s m 1 − T S m 1 . (3.9)

At the initial time t i of the measurement, the state of the tested system, S, here s ̂ , a spin- 1 2 operator, is described by its 2 × 2 density matrix r ̂ ( t i ) with elements r s s ̄ ( t i ) for s , s ̄ = ± 1 2 . The magnet M has N ≫ 1 quantum spins- 1 2 σ ̂ ( i ) ( i = 1 , … , N ) . In each s , s ̄ sector, S + M lie in the state R ̂ s s ̄ ( t ) = R ̂ s ̄ s † ( t ) , which is an operator that can be represented by a 2 N × 2 N matrix. At t i , M is assumed to lie in the paramagnetic state wherein the spins are fully disordered and uncorrelated. Multiplying by the respective element of r ̂ ( t i ) leads to the elements of the initial density matrix of S + M R ̂ s s ̄ t i = r s s ̄ t i σ ̂ 0 ( 1 ) 2 ⊗ σ ̂ 0 ( 2 ) 2 ⊗ ⋯ ⊗ σ ̂ 0 ( N ) 2 . (4.1)

The dynamics of the off-diagonal elements (cat terms) were worked out in Opus. In the relevant short-time domain, the spin–spin couplings are ineffective; therefore, it suffices to study independent spins coupled by the interaction Hamiltonian and the bath. These elements vanish dynamically, truncating the density matrix R ̂ to a form diagonal on the eigenbasis of s ̂ z . There is no reason to repeat that here; for spin-1, this will be worked out in Section 4.1.

Registration of the measurement is described by the evolution of the diagonal elements of the density matrix of the full system. For the situation of higher spin, it is instructive to reconsider and slightly reformulate the spin 1 2 situation.

For s ̄ = s , the Hamiltonian terms drop out of Equation 2.29; hence, the dynamics are a relaxation set by d R ̂ s s t d t = γ 2 ∑ α , β = ± 1 ∑ n = 1 N ∫ 0 t d u   K β u C ̂ s s , β α , n u . (4.2)

For l = 1 2 , the spin operators σ ̂ x , y , z anticommute; hence, for any function f of the σ ̂ z ( i ) , it holds that σ ̂ α ( n ) f σ ̂ z ( i ) = f ̂ − 1 δ i , n σ ̂ z ( i ) σ ̂ α ( n ) ≡ f ( n ) σ ̂ z ( i ) σ ̂ α ( n ) . (4.3) This brings the σ ̂ α ( n ) and σ ̂ − α ( n ) next to each other, which allows to eliminate them using the sum ∑ α = ± 1 σ ̂ α ( n ) σ ̂ − α ( n ) = σ ̂ 0 ( n ) . With only functions of the σ ̂ z ( i ) ( i = 1 , … , N ) remaining, we can go to their diagonal bases to work with scalar functions of their eigenvalues σ i = ± 1 2 (see also Opus, Section 4.4). This expresses Equation 2.30 as C s s ̄ , + ( n ) u ≡ ∑ α = ± 1 C s s ̄ , + α   n u = e − i u H s e i u H s ( n ) R s s ̄ ( n ) t − e − i u H s ( n ) e i u H s R s s ̄ t , C s s ̄ , − ( n ) u ≡ ∑ α = ± 1 C s s ̄ , − α   n u = R s s ̄ ( n ) t e − i u H s ̄ ( n ) e i u H s ̄ − R s s ̄ t e − i u H s ̄ e i u H s ̄ ( n ) , (4.4)

where for any function f ( { σ i } ) , f ( n ) has the sign of σ n reversed, f ( n ) σ i = f − 1 δ i , n σ i . (4.5) We employed the obvious rules ( f g ) ( n ) = f ( n ) g ( n ) and [ f ( g ) ] ( n ) = f ( g ( n ) ) . The terms in Equation 4.4, being scalars, yield the relation C s s ̄ , − ( n ) ( u ) = C s s ̄ , + ( n ) ( − u ) , which allows combining the integrals of Equations 2.29 and 2.30 as a single one from u = − t to t . Because γ ≪ 1 , the typical scale of t , the registration time 1 / γ T is much larger than the bath equilibration time 1 / T . Hence, we may now take the integral over the entire real axis to arrive at the Fourier-transformed kernel K ̄ ( ω ) at specific frequencies.

The next step is to reduce the 2 N × 2 N matrix problem to a problem of N + 1 variables by considering R s s ( { σ i } ) = R s s ( m 1 ) to be functions of the order parameter m 1 = ν ∑ σ i . This is formally true at t i and valid for R ̇ s s ̄ ( t i ) ; hence, it remains valid over time. Denoting P s ( m 1 ) as the probability that R s s ( { σ i } ) / r s s ( t i ) involves m 1 = ν ∑ i σ i , it picks up the degeneracy number G N in Equation 3.6 of realizations { σ i } with the same m 1 , P s m 1 = G N m 1 R s s m 1 r s s t i . (4.6)

To obtain the evolution of P ̇ s , we multiply Equation 4.2 by G N ( m 1 ) / r s s ( t i ) . At given m 1 , one has m 1 ( n ) = m 1 − 2 ν σ n , so we can split the terms with σ n = 1 2 (and − 1 2 ) and perform the sum over n . The fraction of terms that flips an up spin σ n = 1 2 is x 1 2 ( m 1 ) , which multiplies P ( m 1 − ν ) ; flipping a down spin σ n = − 1 2 happens with probability x − 1 2 ( m 1 ) , which multiplies P ( m 1 + ν ) . Due to Equation 4.6, these P s involve the ratios G N m 1 G N m 1 − ν = x − 1 2 m 1 − ν x 1 2 m 1 , G N m 1 G N m 1 + ν = x 1 2 m 1 + ν x − 1 2 m 1 , (4.7)

which has the effect of eliminating the x ± 1 2 ( m 1 ) . Introducing the operators E ± and Δ ± = E ± − 1 by E ± f m 1 = f m 1 ± ν , Δ ± f m 1 = f m 1 ± ν − f m 1 , (4.8)

the evolution of P s gets condensed as P ̇ s m 1 = γ N 2 ∑ α = ± 1 Δ α x 1 2 α m 1 K ̃ Ω s α m 1 P s m 1 . (4.9)

where Ω s ± m 1 = Δ ∓ H s = H s m 1 ∓ ν − H s m 1 . (4.10) This is now a problem for N + 1 functions P ( m 1 ; t ) subject to the normalization ∑ m 1 P ( m 1 ; t ) = 1 .

In Figure 1, the distribution of the magnetization m 1 is depicted at various times. In Figure 2, this evolution is represented in a 3 d plot.

The dynamical entropy of the distribution P s ( m 1 ; t ) = G N ( m 1 ) R s s ( { σ i } ; t ) / r s s ( t i ) is defined as S s t = − T r R ̂ s s t r s s t i log R ̂ s s t r s s t i = − ∑ m 1 P s m 1 ; t log P s m 1 ; t G N m 1 . (4.11)