Bidirectional classical stochastic processes with measurements and fee - PDF document

1 Bidirectional classical stochastic proce
Bidirectional classical stochastic processes with measurements and feedback G. E. Hahne“ NASA, Arms Etese2rch ‘3er;ter Moffett Field, California, 94035 USA - - - ----be 2~~2005- - .- - - - an operator/observable. address another aspect aspect (mentioned in Sec. 4 therein), that is, is there a measurement the input This problem problem for an extension of quantum mechanics that can describe physical processes certain measurements represen

2 t a physical system by point moving tran
t a physical system by point moving transverse space, a function motion along the time direction, jump fiom motion, in presume a given set transition probabilities associated incomina signal have simplifying terference phenomena closed channels-domains a measurement the 6rst probabili9- distribmion sible measurement probability and processes, e.g. e.g. [6], [7], [8], and specifically on Markov processes, e.g. e.g. Ch. XV, [lo], [ll], [12] Ch. 7, [1

3 3], [14] Ch. 6. We shall keep to the ana
3], [14] Ch. 6. We shall keep to the analysis of state spaces that are discrete, and adhere to the nomenclature recommended recommended p. 188, Table 6.1: the entities to be studied wdl be called Markov chains; ivlarkov processes deal with continuous chains can a either a discrete chains with a nonnegative real real Ch. 6.1), and is adopted in order to facilitate the eventual com- parision to formulas in nonnegative components The product stochast

4 ic matrices matrices Ch. VI, Th. l.l(d))
ic matrices matrices Ch. VI, Th. l.l(d)). The present work is based on a result derived below: there is another way to combine two stochastic matrices, output, and combining process converges. in that int,erval in a measurement the input, i.e., non-Markovian, mapping input into pletes the paper with grandfather paradox. Bidirectional Markov deconstruct stochastic same order fashion a another stochastic matrix matrix product can partition elements,

5 with . , to output left in the off-diag
with . , to output left in the off-diagonal measurement process components each, current vectors. feedback loop causal sequence entry points loop are marked on exit points transverse directions: conventional physical only one such systems can that the feedback loop either zero, matrices with both matrices, e.g., overall probability is conserved to the apply similarly (9c) thereby obtaining latter set product matrices exist. Then Then + AFBMBBCBB

6 VB yB = LBB[cBFMFFAFF~F + cBBvB] We now
VB yB = LBB[cBFMFFAFF~F + cBBvB] We now represent the complete mapping Given convergence, be represented infinite series Each sumniand in matrices with overdl input remark further entry points all elements will converge: and therefore majorized by each element in converges. Since . . in (17a) be arranged no element dimensionality, a all possible 1)-dimensional domain yields divergent In this section, both the and presume each crossing point throug

7 h is made which coarse shall also presum
h is made which coarse shall also presume, analogous to the the Eq. (7)-see also [lG], Eq. (43), (43), Eq. (2)), that within a coarse grain process does currents in The latter assumption entails least for individual a coarse-grained conditional probabilities. probabilities. p. 75), that is, a second current vector yield the same result as . . . . certain probabilities fact, to associated with an apparatus with labels and let vectors be ~ ~~~ a nor

8 malization factor. is governed and by pr
malization factor. is governed and by probability current coarse-grained channel a detection (cf. Fig. be continuous. consider, for both the coarse-grained chan- large number in practice measurements and physical sequence is, therefore, consider below This coiistruction entails measurenlenm in time interval associated sequence occurring is zero. built into that the that the undergoes more nontrivial example, from which one can infer scenarios list

9 ed conditional probability time interval
ed conditional probability time interval the trajectory recursively as complete feedback loop stop the and say cannot happen, hive rise measurements, therefore, same normalization The output four measurements, fine-grained probabilities is not is in outcome comprises Our claim (32j) with ()= N(AFF,UF). . .N(ABB,X1,) x U~(~l~,Y,Uo’,Vo;UF) (34a) - ABBIBBcBFIFF ~FBIBBCBFIFFAFFD-F I/, I v1 ~ ”0‘ ”0 - (34b) We reemphasize that, in

10 (34), the notation implies that the outp
(34), the notation implies that the output on the lhs is not conditioned on the the output given sequence measurements, together with modes, allow sequence of system trajectory the apparatuses and not that the the have the same ratios have in enhanced by a common factor such net output current is equal (32h), and trajectory’s first passage across a in (32c), the device rule for just the given by by (UF) = IEFg[, with no multiplying factor. A s

11 sidar argument obtains for (32e), (32g),
sidar argument obtains for (32e), (32g), Therefore, (34) with exactly feedback loop, This result all possible all possible measurement outcomes (lGc), and when no as in measurements on now consider in the that it to infer Given only outcome occurs no detailed measurements are still be disjoint subsets unordered measurements. set with information the infer a collapsed probability distribution otherwise. p(CIS2) =def (39) We obtain the usual condit

12 ional probability law I in an obvious ob
ional probability law I in an obvious obvious )and p(ClS2) can also be described as the distributions before and after the limiting case of a non-interventional there are intermediate measurements with sub- sub- passim), and show that in this feedback loop than it us take take 01 p” 1, (41a) c=[; ;jl (41b) 0 0 1-CY--p 1-7 0 0 0 (- or) so that (434 (43b) Pr/P -Q7) 1 (1 -?)/(I - ar) 0 SFF = 1 [ SBF = [y(1- a- P)/(l -cry) 01. Given that the input

13 s are (UFj1 = 1, (UF)2 = 0, and VB = 0,
s are (UFj1 = 1, (UF)2 = 0, and VB = 0, the outputs to stand backwards from ~~ ~ feedback loop plausible, therefore, nonzero feedback loop can only the F-type value when only quantum-mechanical survival probability within a feedback sysyem Two recent online online [20], and references given therein) study time travel from the viewpoint of conventional quantum mechanics. I believe that the quantum-mechanical time-travel phenomena in [a]; [a]; G. E.

14 Hahne. J. Phys. A36, 7149 (2003). Onlin
Hahne. J. Phys. A36, 7149 (2003). Online at arXiv:quant- ph/04040 12. [a] G. E. Hahne. J. Phys. A35, 7101 (2002). Online at arxiv-quant- ph/0404103. [3] mi. Fa&. General Principles Springer, Berlin, Berlin, S. Karlin and H. M. Taylor. A Erst course in stochastic processes. Acadenlic Press, New York, 1975, 2nd edition. [5] A. Papoulis. Probability, random variables, and stochastic stochastic A. Leon-Garcia. Probabilzty and MA; 1994. 1994. R. v. Mis

15 es. Mathematical theory of probability a
es. Mathematical theory of probability and statistics. Edited and P. E. Pfeiffer. Concepts of probability theory. Dover, New York, 1978. [13] A. T. Bharucha-Reid. Elements of the theory of Adarkov processes and their upplacations. McGraw-Hill, New York, 1960. [14] H. Minc. Nonnegatzwe matrices. matrices. G. Luders. Ann. Physik (Lezpzig) 8, 322 (1951). [16] W. H. Furry. in Lectures in Theoretical Physics, Vol. VIIIA, edited by W. E. Brittin, Gordon