Source Information J
What is the maximum possible value of information I in a measurement? Obviously it must have something to do with the effect that is measured, in particular, its information “carrying capacity”. This is simply the level of information that is intrinsic to the effect. We call this the information level J. Hence, ideally
I = J (10)
The maximum possible information level I in the data equals the information carrying capacity of the phenomenon. (In many phenomena this is equivalently a principle of conservation of mean kinetic energy – but not in all phenomena!). From this we conclude that information J represents
(f) the amount of information that is “bound” to the effect (a concept originally due to Brillouin [1a,b]). Hence J is commonly called the “bound information”. It also has the significance of :
(g) representing the level of entanglement that is possible for the measured effect (see below eq. (14) ); and
(h) approximating the notion of an active information. In the EPR-Bohm experiment, a mother particle splits into two daughter particles, only one of which is observed. As postulated by Bohm and Hiley, the “active” information is information residing in the observed particle that is about the unobserved particle. This information is supposed to somehow flow from the unobserved particle to the observed one, “informing” it (or interacting with it, which is why it is called “active”) about some property of the unobserved particle (say, its spin state).
The difficulty with this premise of a physical “flow” is that under certain circumstances it would have to travel over a huge distance and, hence, faster than the speed of light. This is untenable. However, according to EPI, no such real-time “flow” or “activity” or “interaction” actually takes place. Therefore no “signal” has to travel faster than the speed of light. Rather, owing to an application of the EPI principle to the problem, the observer simply knows a priori the missing information about the unobserved particle. That is, information is received out of inference, and not out of the flow of a physical signal. An observation of the state of the observed particle results in an almost instantaneous observation of that of the unobserved one. See prediction (10) below.
Essential mathematical difference between I and J; property of unitarity
We mentioned before that the numbers I and J are evaluated as integrals. In fact these integrals are called functionals, i.e., single (scalar) numbers that arise out of integrals over all x of certain functions of p(x). Functional I always has the form eq. (4) or its equivalent. This is basically because acquired information is a universal concept, valid for describing the quality of any real measurement.
By comparison, J has generally a different functional form for each measured effect. It arises out of knowledge of an invariance principle that is appropriate to the effect (see alternatives (i), (j) below following eq. (13)). Examples of invariance principles are unitarity (invariance of length) in quantum mechanics or invariance of charge flow in electromagnetics. Because of its overriding importance, we next discuss the unitarity principle.
“Unitarity” simply means the preservation of lengths. Eq. (10) is an expression of unitarity in expressing the equality of “lengths” I and J. The concept of unitarity is the key to EPI derivations of quantum effects. Recall that the length I is generally expressed in the coordinate space of the observations. Correspondingly, J is expressed in a “unitary space” to coordinate space. For example, in describing quantum mechanics, so-called “momentum space” is the unitary space. Notice that when unitarity eq. (10) holds there is no loss of information during the information flow J→I. It follows that the resulting EPI theory is 100% correct. Hence, quantum theories are fully correct. By comparison, when unitarity does not hold, as where conservation of charge or some other invariance are chosen instead, I turns out to be less than J (the precise relation is eq. (16) below) and EPI gives output theories that are well-known to be approximations. These are the so-called “classical” electromagnetic or gravitational theories. Hence, EPI preducts that these are inaccurate in ignoring quantum effects since such ignorance lowers the information level in the observation.
However, surely all effects are ultimately quantum effects if the observer is only allowed to measure on a fine enough level. In fact, EPI corroborates this, in showing that all phenomena are capable of a quantum description, i.e. a scenario where eq. (10) holds. As before, the observer must simply know what space is physically unitary to his space of observation. This follows because the expression for Fisher information I has the precise form of a length that is generally capable of transforming by unitary transformation. This is a so-called “L2 length.” Any unitary transformation is equivalent to a rotation, and an L2 length remains invariant to a rotation. This is again the expression of eq. (10). The particular rotation that is chosen defines the particular quantum theory that is to be observed.
An L2 length has the form of a sum of squares. The Fisher information is simply a fine limit of such a sum, in the form of an integral of squares,
I = 4 ∫dx q’ 2(x). (11)
The definite integral ∫ is over all x, and by definition q’ = dq/dx where q(x) is a new function defined by p(x) = q2(x). [Note that (11) follows from eq. (4) simply by replacing p(x) with q2(x).] Hence q(x) is the square root of a probability, and is called a “probability amplitude.” It turns out to be easier to define laws of nature by the use of probability amplitudes than by use of the probabilities themselves. To those who know quantum mechanics, q(x) is a purely real component of the generally complex wave amplitude function ψ(x).
In real problems, where fully complex amplitude functions ψn(x), n=1,…,N/2 are to be found, these are formed in their successive real and imaginary parts by the EPI output solutions qm(x), m = 1,…,N (see eq. (14) for EPI principle).
In summary, the difference between I and J is the difference between the general and the specific – The information in general data is described by I, and these arise out of a specific effect at information level J. The effect is specified by an appropriate invariance principle (see above).
All effects are capable of a quantum representation, if the observer only observes on a fine enough scale (see also “coarse graining” discussion below). Correspondingly the L2 form (11) of I permits such an effect to be derived on the basis of an appropriate unitary space.
Vital role of the probe particle
How are data actually seen? Any observation requires the use of a probe particle, i.e., something that illuminates the subject and proceeds onward to the observer (see beginning of essay). Since Heisenberg (or, via eq. (7) ), it is known that the probe must interact with the subject while illuminating it. This necessarily changes its state, i.e. its probability law p(x), if only by a very slight amount. Then by Eq. (4) it must perturb information level I. Call the perturbation amount δI. Then, by Eq. (10) it perturbs the intrinsic information level J by the same amount, that is,
δI = δJ. (12)