3.5 Inexhaustible uncertainty

In 1927 Werner Heisenberg discovered a fundamental limitation of quantum mechanics: a bound on the accuracies with which certain complementary pairs of observables can be measured.


Figure 3: Werner Heisenberg
The “canonical” understanding of complementarity is expressed in Messiah [100, p. 154]
The description of properties of microscopic objects in classical terms requires pairs of complementary variables; the accuracy in one member of the pair cannot be improved without a corresponding loss in the accuracy of the other member.

...

It is impossible to perform measurements of position and momentum with uncertainties (defined by the root-mean square deviations) and such that the product of is smaller than a constant unit of action .
In Prigogine’s words [111, p. 51],
the world is richer than it is possible to express in any single language.
Many other instances of complementarity are well known. A simple example is offered by the so-called two-slit experiment (see Firgure 3.5, a snapshot of an animation posted at http://www.colorado.edu/physics/2000/index.pl). A source is “shooting" electrons towards a wall which has two tiny holes (slits), each of them just enough for one electron to get through at a time. A second wall has a detector, that can be moved up and down, with the aim to count the number of electrons reaching a given position of the second wall. Experimentally one can determine the probabilities that electrons reach some positions on the second wall, depending upon the number of open slits, one or two. Contrary to common intuition, due to interference, there are places where one counts fewer electrons in the case when both slits are open than in the case when only one slit is open! Detecting through which slit an electron went (a particle measurement) or recording the interference pattern (a wave measurement) cannot be done in the same experiment.

Moore [104] has used simple automata “Gedanken" experiments to illustrate uncertainty (see more in Conway [46], Brauer [23], Svozil [128], Calude [41]). A (simple) Moore experiment can be described as follows: a copy of the machine will be experimentally observed, i.e. the experimenter will input a string of input symbols to the machine and will observe the sequence of output symbols. The correspondence between input and output symbols depends on the particular chosen machine and on its initial state. The experimenter will study the sequences of input and output symbols and will try to conclude that “the machine being experimented on was in state at the beginning of the experiment". This is often referred to as a state identification experiment.

In what follows we will work with finite deterministic automata with a finite set of states, an input alphabet , and a transition function . Instead of final states we will consider an output function . At each time the automaton is in a given state and is continuously emitting the output . The automaton remains in state until it receives an input signal , when it assumes the state and starts emitting . As we will discuss only the simplest case when the alphabet , an automaton will be just a triple .

The transition function can be extended to a function as follows: and for all .

The output produced by an experiment started in state with input sequence is described by , where is the function defined by the following equations:

_


Figure 4: Two-slit experiment


                                                                 Table 3: Moore’s automaton transition.

for all ; recall that is the output function.

Consider, for example, Moore’s automaton, in which the transition is given by the following tables:

and the output function is defined by , . A graphical display appears in Figure 3.5.


The experiment starting in state with input sequence leads to the output Indeed,



















Consider now an automaton , and following Moore [104] say that a state is “indistinguishable" from a state (with respect to ) if every experiment performed on starting in state produces the same outcome as it would starting in state . Formally, for all strings .

An equivalent way to express the indistinguishability of the states and is to require, following Conway [46, p. 3], that for all ,

Indeed,

_
           
                                                                          Figure 5: Moore’s automaton.

for all

A pair of states will be said to be “distinguishable" if they are not “indistinguishable", i.e. if there exists a string , such that

Moore [104] has proven the following important theorem:
There exists an automaton such that any pair of its distinct states are distinguishable, but there is no experiment which can determine what state the machine was in at the beginning of the experiment.
Moore used the automaton displayed in Figure3.5. His result can be thought of as being a discrete analogue of the Heisenberg uncertainty principle. The state of an electron is considered specified if both its velocity and its position are known. Experiments can be performed with the aim of answering either of the following:
1. What was the position of at the beginning of the experiment?
2. What was the velocity of at the beginning of the experiment?
For an automaton, experiments can be performed with the aim of answering either of the following:
1. Was the automaton in state at the beginning of the experiment?
2. Was the automaton in state at the beginning of the experiment?
In either case, performing the experiment to answer question 1 changes the state of the system, so that the answer to question 2 cannot be obtained. This means that it is only possible to gain partial information about the previous history of the system, since performing experiments causes the system to “forget" about its past.

An exact quantum mechanical analogue has been given by Foulis and Randall [115, Example III]: Consider a device which, from time to time, emits a particle and projects it along a linear scale. We perform two experiments. In experiment A, the observer determines if there is a particle present. If there is no particle, the observer records the outcome of A as the outcome If there is a particle, the observer measures its position coordinate . If , the observer records the outcome , otherwise A similar procedure applies for experiment B: If there is no particle, the observer records the outcome of B as If there is, the observer measures the -component of the particle’s momentum. If the observer records the outcome , otherwise the outcome

Moore’s automaton is a simple model featuring an “uncertainty principle” (cf. Conway [46, p. 21]), later termed “computational complementarity” by Finkelstein and Finkelstein [62]; for a detailed analysis see Svozil [128], Calude, Calude, Svozil and Yu [28], Calude and Lipponen [38], Jurvanen and Lipponen [77].