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Weak value analogue in classical stochastic process 2 1 Hiroyuki Tomita 0 2 Research Center for Quantum Computing, n Kinki University, a Kowakae 3-4-1, Higashi-Osaka, 577-8502, Japan J 0 1 ] h c Abstract e m The time evolution of the two-time conditional probability of the classical - stochastic process is described in an analogous form of the quantum mechan- t a ical wave equations. By using it, we emulate the same strange behaviors as t s those of the weak value in the quantum mechanics. A negative probability . t and abnormal expectations of some quantities remarkablely larger than their a m inherent norms are found in an example of a stochastic Ising spin system. - d Keywords: Weak value, Stochastic process, Two-time conditional n probability, Stochastic Ising model o c [ 1. Introduction 3 v A notion of the weak value proposed by Aharonov et al [1] has brought a 2 0 new understanding on the quantum observation, i.e. a weak measurement [2] 3 whichhardlydisturbsthequantumstate. Thereasonofthisstrangenatureof 4 . thequantum measurement isthat theweak value isdefined asanexpectation 9 0 with the condition of two-time observations of the initial and the final states 1 which differ from one another. This condition is very rare case with little 1 : probability and is far from the main behavior of a given quantum system. v i Then the observation of the weak value does not disturb the quantum system X not so fatally. As a result of this rather fictitious probability, the weak value r a happens to be abnormally enhanced from its inherent norm. The purpose of this letter is to make the mechanism of this abnormal behavior clearer by using a classical stochastic model, in which we can avoid the ambiguity of the complex probability in the quantum case [3]. Email address: [email protected](Hiroyuki Tomita) Preprint submitted to Physics Letters A January 11, 2012 We introduce a conventional transformation of the stochastic master equation to a self-adjoint form in the following section. A good analogy with the quantum mechanics is found by applying this tarnsformation to the two-time conditional probability (TTCP). This is shown in Sec.3. An example of the stochastic Ising model which shows an abnormal enhancement of the expectations of some quantities with respect to TTCP is given in Sec.4. In Sec.5 we discuss an extention of TTCP to a density matrix form to com- plete the analogy with the quantum mechanics. The last section is devoted to brief summary and discussions. 2. Self-adjoint form of stochastic master equation Firstlet ussurvey thewell-known transformation[4]toaself-adjointform of the stochastic master equation. Let x be a set of stochastic variables described by a time-dependent conditional probability, P(x,t x,t) for t t, which obeys the following sta- i i i | ≥ tionary, Markoffian master equation, i.e. the Chapman-Kolmogorov forward equation, ∂ P(x,t x,t) = W(x x′)P(x,t x,t)+ W(x′ x)P(x′,t x,t) i i i i i i ∂t | − → | → | Xx′ Xx′ = L(x,x′)P(x′,t x,t), (1) i i − | Xx′ where L(x,x′) =  W(x x′′)δ(x x′) W(x′ x). → − − → Xx′′   The matrix L has an eigenvalue λ = 0 correspondding to the steady state, 0 P (x) = lim P(x,t x,t). 0 i i t−ti→∞ | Let us introduce a wave function related to this forward conditional probability by ψ(x,t x,t) = φ (x)−1P(x,t x,t), (t t) (2) i i 0 i i i | | ≥ where φ (x) = P (x)1/2. This function ψ obeys the forward wave equation, 0 0 ∂ ψ(x,t) = H(x,x′)ψ(x′,t), (3) ∂t − Xx′ 2 where H is defined by H(x,x′) = φ (x)−1L(x,x′)φ (x′). (4) 0 0 For the time being the initial condition (x,t) in ψ is abbreviated. The i i function φ (x) is an eigenfunction of Eq.(3) for λ = 0. 0 0 The merit of this transformation is that the eigenvalue problem of a given master equation is simplified, if the matrix H is symmetric, i.e. H(x,x′) = H(x′,x). This situation is widely expected when the detailed balance codition, i.e. the time-reversal symmetry [5], P (x)W(x x′) = P (x′)W(x′ x), 0 0 → → or equivalently, L(x,x′)P (x′) = L(x′,x)P (x), (5) 0 0 is satisfied. In this case the eigenvalues of H are all real, and non-negative, if the steady state is stable. Therefore, φ (x) is the ground state. 0 A useful example is the Fokker-Planck equation for a single, continuous stochastic variable x, ∂ ∂ ǫ ∂ P(x,t) = [x]P(x,t), [x] = F′(x)+ , (6) ∂t −L L −∂x (cid:18) 2∂x(cid:19) which describes a one-dimensional Brownian motion in a potential F(x) with a small diffusion constant ǫ. By using its steady state solutions, P (x) exp[ 2F(x)/ǫ] and φ (x) exp[ F(x)/ǫ], 0 0 ∝ − ∝ − we find the continuous variable version of the above formulations, 1 ǫ2 ∂2 1 [x] = +V(x) , V(x) = F′(x)2 ǫF′′(x) . (7) H ǫ (cid:20)−2 ∂x2 (cid:21) 2 − (cid:2) (cid:3) Thus the Fokker-Planck equation is transformed into a self-adjoint form of an imaginary-time Schro¨dinger equation, ∂ ǫ2 ∂2 ǫ ψ(x,t) = +V(x) ψ(x,t), − ∂t (cid:20)−2 ∂x2 (cid:21) 3 ..........................................................................................................................................................................................................................................................................................................................................e...............................................−....................................................................................................λ..............................................................1....................................................t....................................../.................................................0........ǫ......................................................................................................................................................................................................................................................................................................................................................................................................................................................F...............................................................................................................(...........................................................x.................................................................................................................)...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................x.................... λλ1........0....................................................................................=.......................................................................................................................................................................................................................................................................................................................................................................0.........................................................................................................................................................................................................................................................................................................................................................................................................0............................................................................................................................................................................................................................................................................................................................................................................................................................................V..................................................................................................(.......................................x.........................................................................................).........................................................................................................................................................................................................................................................................................................................................................................................................................................................x.................... Figure 1: Stochastic decay process of the metastable state. anditseigenvalueproblemresultsinafamiliaroneofthequantummechanics. Figure.1 shows an early application [6] to the so-called Kramers escape problem [7]. The stochastic decay rate of the metastable state in a double- well potential F(x) is given by the first excited eigenvalue λ of the cor- 1 responding Schro¨dinger potential V(x). The first excited state is almost degenerated with the ground state for the small diffusion constant ǫ. 3. Two-time conditional probability So far the quantum mechanical reformulation merely helps us to simplify the eigenvalue problem of a given master equation. None of the remarkable quantum mechanical phenomena appears, until we are concerned with the TTCP, P(x,t x,t;x ,t), t t t . ( ; denoting ‘and’) (8) i i f f i f | ≤ ≤ Byusing theMarkoffianpropertyandthewell-known equality ofthesimplest Bayes’ theorem, P(A B) = P(A B)P(B) = P(B A)P(A), ∩ | | repeatedly, the TTCP can be written in the following form with a pair of the wave functions as 1 P(x,t x,t;x ,t ) = ψ(x,t x ,t)ψ(x,t x,t), (9) i i f f f f i i | ψ ψ | | f i h | i wheretheconjugatewavefunctionψ isrelatedtotheso-called posterior conditional probability, P(x,t x ,t) for t t , by f f f | ≤ ψ(x,t x ,t ) = φ (x)−1P(x,t x ,t ), (10) f f 0 f f | | 4 and obeys the backward wave equation, ∂ ψ(x,t) = H†(x,x′)ψ(x′,t). (11) ∂t Xx′ Here H† is the hermite conjugate of H, i.e. the transposed matrix in the present case. The eigensystem is common with the forward equation Eq.(3), when H is hermite, i.e. real and symmetric as has been assumed here. The denominator in Eq.(9) is the weight of overlap between the two wave functions defined by an inner product, ψ ψ = ψ(x′,t x ,t )ψ(x′,t x,t). (12) f i f f i i h | i | | Xx′ Of course this quantity is real, while the corresponding quantity in the quantum mechanics is complex in general. Let us define the ket- and the bra-vectors by ψ(t) = ψ(x,t x,t) † and ψ (t) = ψ(x,t x ,t ) . i i i f f f | i { | } h | { | } Then the wave equations Eqs.(3) and (11) are rewritten in the quantum mechanical form as ∂ ∂ ψ(t) = H ψ(t) and ψ (t) = ψ (t) H. (13) i i f f ∂t| i − | i ∂th | h | Henceforth, H is called the Hamitonian. ByusingthispairoftheSchro¨dingerequationsitisshownthattheoverlap integral, ψ ψ given by Eq.(12) does not depend on the current time t, i.e. f i h | i ∂ ψ ψ = ψ (t) H ψ(t) ψ (t) H ψ(t) = 0. f i f i f i ∂th | i h | | i−h | | i Furtherit canbeshown thatthisoverlap integral hasthefollowing properties in the two limits; (i) lim ψ ψ = 1, f i  tf−ti→∞h | i (14)    (ii) lim ψ ψ = [φ (x )φ (x)]−1δ(x x). f i 0 f 0 i f i tf−ti→0h | i −    5 Note that the two-time conditional expectation (TTCE) of a physical quantity Q with respect to TTCP defined by Q = Q(x′)P(x′,t x,t;x ,t ) (i;f) i i f f h i | Xx′ ψ (t) Q ψ(t) f i = h | | i, (15) ψ ψ f i h | i has just the analogous form of the weak value in the quantum mechanics [3]. Thus the TTCP is a non-linear quantity composed of a product of a pair of the forward and the backward wave functions, and cannot be described by a closed, linear evolution equation. Then it happens that the principle of the probability superposition is violated and the interference of wave functions may occur. However, its example is omitted here because none of nontrivial phenomenon from this view point has been found, yet. The reason may be that the wave functions are always real and possitive in the present case. Let us discuss only the weak value in the rest. 4. Stochastic model of classical Ising spins Anexample isapairoftheclassical Ising spinσ = 1having anexchange ± interaction, E(x) = Jσ σ , 1 2 − where x = (σ ,σ ). Let us number the stochastic variable x in the order, 1 2 (1,1), (1, 1), ( 1,1), ( 1, 1) and choose the following transition matrices, − − − − 0 1 1 0 2p2 1 1 0 − −  p2 0 0 p2   p2 2 0 p2  W = or L = − − , (16) p2 0 0 p2 p2 0 2 p2    − −   0 1 1 0   0 1 1 2p2     − −  where p = e−βJ. Evidently this transition matrix W satisfies the detailed balance condition, x x′ e−βE( ) W(x x′) = e−βE( ) W(x′ x), → → at the steady state, i.e. the thermal equilibrium of a temperature parameter, β = 1/k T. With use of the equilibrium distribution function, B 1 1 P (x) = (1, p2, p2, 1) and φ (x) = (1, p, p, 1), 0 2(1+p2) 0 2(1+p2) p 6 we find the corresponding hermite Hamiltonian, 2p2 p p 0 − −  p 2 0 p  H = − − p 0 2 p  − −   0 p p 2p2   − −  = (1+p2) σ σ (1 p2) σ σ p (σ σ +σ σ ),(17) 0 0 z z 0 x x 0 ⊗ − − ⊗ − ⊗ ⊗ where σ and σ are the usual Pauli matrices and σ denotes the two dimen- x z 0 sional unit matrix I . This is the Hamiltonian of a pair of quantum Ising 2 spins with the exchange interaction in a transverse magnetic field. The eigenvalues and the eigenstates of this Hamiltonian H, λ = 0, λ = 2p2, λ = 2, λ = 2(1+p2), 0 1 2 3   1  0 = [ + p + p + ],   | i 2(1+p2) |↑↑i |↑↓i |↓↑i |↓↓i   p1  1 = [ ],   | i √2 |↑↑i − |↓↓i 1 2 = [ ],  | i √2 |↑↓i − |↓↑i   1   3 = [ p + p ],   | i 2(1+p2) |↑↑i − |↑↓i− |↓↑i |↓↓i     p  can be easily obtained, where 0 = φ , the ground state. Here the familiar 0 | i | i notations , are used for σ = 1. Note that the first excited state is almost ↑ ↓ ± degenerated with the ground state for a small transition probability p2. By using this eigensystem we can calculate the state vectors, ψ(t) and i | i ψ (t) for arbitrary initial and final states in just the same manner of the f h | elementary quantum mechanics except for the fact that the time t is imaginary. An interesting example is the case where the initial and the final states differ from each other, just like the case of the weak value. For example, let x = ( ) at t = 0 and x = ( ) at t = t , i f f ↑↑ ↓↓ that is, P(x,0) = (1,0,0,0) and P(x,t) = (0,0,0,1), f 7 xP,t|↑↑,↓↓,t(0;)f1.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................↑...........................................................................................................................↓..........................................................................................................................................................................................+.......................................................................................................................................................................↑............................................................................↓.............................................↑............................................................................↑..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................p.........................................................................................................................................................≪..................................................................................................................................................................................................................................................................1...........................................................................................................................................................0.....................................................................................................,........................................................................................................................................................t.......................................................................f...........................................................................................................=................................................................................................................................................................................................................2.........................................................................................................................................................................................................................................................................................................................................................................................................................................↓..........................................................................................................................↓..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 0 t tf Figure 2: Two-time conditional probability or equivalently, ψ(0) = 2(1+p2) and ψ (t) = 2(1+p2) . i f f | i |↑↑i h | h↓↓| p p By using the eigenvector expansion we obtain, ψ(t) = 0 + 1+p2 e−λ1t 1 +p e−λ3t 3 , i | i | i | i | i p (18) ψ (t) = 0 1+p2 e−λ1(tf−t) 1 +p e−λ3(tf−t) 3 , f h | h |− h | h | p and ψ ψ = 1 (1+p2) e−λ1tf +p2 e−λ3tf (> 0). (19) f i h | i − The TTCP is shown in Figure.2. This result itself is very natural and well-expected, all probabilities being always non-negative. Astrangebehaviorappearswhenweusethebasis k , k = 0,1,2,3 ,the {| i } eigenstates of the Hamiltonian H instead of the spin states x = σ σ . 1 2 {| i | i} We can calculate the probability, i.e. the TTCE of the projection operator k k onto each eigenstate k in the same manner. The result is given by | ih | ψ (t) 0 0 ψ(t) 1 f i P(0,t) = h | ih | i = , ψ ψ ψ ψ f i f i ψ (t)h1| 1iψ(t) h (1| +ip2)e−λ1tf f i P(1,t) = h | ih | i = (< 0 ) , ψ ψ − ψ ψ h f| ii h f| ii (20) ψ (t) 2 2 ψ(t) f i P(2,t) = h | ih | i = 0 , ψ ψ f i ψ (t)h3| 3iψ(t) p2e−λ3tf f i P(3,t) = h | ih | i = . ψ ψ ψ ψ f i f i h | i h | i 8 1234560.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................h...........................................................................M.................................................................................................................................................................................x.................................................................................i......................................................................(...........................................i................................;..............................f..........................................).....................................................................................................................................................................................................................................................................................................................................................................||................................................................↑↑......................................................................................↑↑.....................................................................................ii.........................................................................................................................tt..........................................t..........................oo.......................................................................................................................................||...................................................................↑↓....................................................................................↑↓.....................................................................................ii....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................t................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... f Figure 3: Abnormal and normal TTCE of M for p = 0.2 and p2t = 0.01. x f The fictitious negative probability is found in P(1,t). Of course the com- pleteness of the probability, 3 P(k,t) = 1, X k=0 is satisfied evidently because of Eq.(19). A related unusual behavior to this fact is the abnormal enhancement of some observables. For example, if we calculate the TTCE of a quantity, 1 M = (σ σ +σ σ ), (21) x x 0 0 x 2 ⊗ ⊗ an abnormal behavior 1 2p 1 p2 M = 1 p2e−λ3tf − e−λ3t +e−λ3(tf−t) > 1, h xi(i;f) ψ ψ (cid:20)1+p2 − − 1+p2 (cid:21) h f| ii (cid:0) (cid:1) (cid:0) (cid:1) isfoundforsufficiently small p andt . Anexampleisshown inFigure.3. Note f that the natural norm of M must be less than 1, because the eigenvalues of x M are 1,0,0,1 . When the transition rate is very small, i.e. p2t 1, x f {− } ≪ we find M 1. x (i;f) h i ≫ A plain reason of this singular behavior is that the overlap integral ψ ψ in f i h | i the denominator may be expected to be very small owing to (ii) of Eq.(14), 9 0123456..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................h.........................................................................................................A......................................................................................................................................................................i..........................................................................(................................................i.................................;.................................f............................................)..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................||..............................................................2.....↑↑.....................................................................................↑↑......................................................................................ii...........................................................................................................................tt.................................................................oo..........................................................................................................................................||...................................................................↓.↑......................................................................................↓↑.......................................................................................ii.......................................................................................................................................................................................................................................................................................................................................................................................................................................2....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... p t f Figure 4: Abnormal and normal TTCE of A= σ σ for p = 0.2. x x ⊗ whenever the initial and the final states differ from each other, i.e. x = x . i f 6 This means that to reach x = ( ) starting from x = ( ) in a given time f i ↓↓ ↑↑ occurs scarcely and is far from the main flow of the conditional probability. On the contrary none of such strange behaviors are found when x = x , e.g. i f x = x = ( ). The result for the latter case for the same parameters as the i f ↑↑ upper abnormal case is shown by the lower curve in Figure.3, its maximum being 0.09 at t = t /2 and minimum 0.05 at t = 0 and t . f f ∼ ∼ In Figure.4 the TTCE of another quantity A = σ σ are shown also. x x ⊗ Note that A is commutative with H and a conserved quantity. Then the horizontal axis in this figure shows a parameter of the transition probability, not the time. 5. Extension of TTCP to a density matrix Itshouldbenotedthatthephysical quantitiesM andAarenon-diagonal x in the spin-state representation and have no corresponding quantities in the classical Ising spin system. They are related to the transition rate of the stochastic Ising spin. In order to calculate the expectations of such non- diagonal quantities we need an extension of the TTCP to the two-time conditional (TTC) density matrix defined by 1 ρ (t) = ψ(t) ψ (t) (i;f) i f ψ ψ | ih | f i h | i 10