Building the Observer into the System: Toward a Realistic Description of Human Interaction with the World

Human beings do not observe the world from the outside, but rather are fully embedded in it. The sciences, however, often give the observer both a"god's eye"perspective and substantial a~priori knowledge. Motivated by W. Ross Ashby's statement,"the theory of the Black Box is merely the theory of real objects or systems, when close attention is given to the question, relating object and observer, about what information comes from the object, and how it is obtained"(Introduction to Cybernetics, 1956, p. 110), I develop here an alternate picture of the world as a black box to which the observer is coupled. Within this framework I prove purely-classical analogs of the"no-go"theorems of quantum theory. Focussing on the question of identifying macroscopic objects, such as laboratory apparatus or even other observers, I show that the standard quantum formalism of superposition is required to adequately represent the classical information that an observer can obtain. I relate these results to supporting considerations from evolutionary biology, cognitive and developmental psychology, and artificial intelligence.


Introduction
Like its quantum-theoretic counterpart, the classical Hamiltonian is additive: given any arbitrary partitioning of a closed system U into subsystems S  ...S n , to second order the classical Hamiltonian H U = i H i + i =j H ij . The terms in the sum can, moreover, be grouped in any arbitrary way: for a three-way partition of U, for example, one is free to group two of the subsystems and write H U = (H  + H  + H  ) + (H  + H  ) + H  . In general, whether or how U is partitioned into subsystems can have no effect on H U ; H U is completely invariant under such partitionings and re-partitionings of U.
Consider some arbitrary two-way partition of U into subsystems O and B O : these will be referred to as "the observer" and the observer's "box" respectively. Let  Under these conditions, well-known results of the classical theory of the black box apply. In particular, O cannot determine, by finite observations conducted at finite resolution, anything beyond lower limits on the numbers of degrees of freedom, accessible states, or allowed state transitions of B O [1,2,3]. No finite set of finite observations justifies anything other than a uniform prior probability distribution over the finite set of finitely-encoded (without loss of generality, binary) observational outcomes recordable by O, and nothing justifies any assumption that the observational outcomes O can record exhaust the available behavioral complexity of B O .

No-signalling and its corollaries
Consider now a further arbitrary partition of B O into subsystems O ′ and B; these will be referred to as the "second observer" and the "shared box" respectively. Assume the O-O ′ interaction is negligible, i.e. H OO ′ ∼ 0. The Hamiltonian H U can then be written:  In particular, O cannot observationally distinguish between the three situations shown in Fig. 1. Physically, O cannot determine whether outcomes are being received from "inside" or "outside" and hence cannot distinguish between the two.  That observers have no access to memory is obviously counter-intuitive. Zurek, for example, considers the ability of observers to "readily consult the content of their memory" as distinguishing them from apparatus ( [6] p. 759). The no-memory corollary shows that observers defined in this way do not exist in any universe in which observers interact only with black boxes, i.e. in any universe characterized by an additive Hamiltonian. It explains, moreover, why such observers do not exist. Assuming specific access by O to the particular degrees of freedom of a memory M requires assuming a specific interaction H OM . Any such assumption violates the invariance of H U under arbitrary partitionings of U that is required by additivity. As any time-persistent object effectively functions as a memory, Corollary 2.3 shows that O cannot determine by finite observation that any object is time-persistent. Proof : If O cannot determine whether either S or S ′ is time-persistent, O clearly cannot determine that either as a "copy" of the other (cf. [7]).

Local and global thermodynamics
If O undergoes a physical state transition on receipt of each outcome from B O in compliance with Landauer's principle [8,9], then O's physical state transitions can be regarded, from an external, theoretical perspective, as counting the outcomes received. As this count increases monotonically, O can be regarded, from this external perspective, as a clock with period ∆t O = 1 by definition. Let a coordinate t count ticks of this clock; with respect to this t, the interaction H OB O is clearly irreversible. The acquisition of classical information by O can, therefore, be regarded as costing at least 0.7 kT per bit, where the temperature T is taken to be defined by the integral: Acquiring an N-bit outcome then requires an action of at least 0.7 NkT ∆t O .
If one assumes that the irreversible receipt of one photon can, under ideal conditions, transmit one bit of information from B O to O, the minimal action required to transmit one bit can be estimated from the ∼ 200 fs required, at physiological temperature, for absorption of a photon by rhodopsin [10]. Taking T = 310 K (i.e. 37 C), kT ∼ 4.3 · 10 −21 J, one obtains an action of ∼ 6.0 · 10 −34 J·s, a value remarkably close to h ∼ 6.6 · 10 −34 J·s. In what follows, therefore, h will be regarded as the "quantum of action" required to transmit one bit of classical information.
This local action is defined, however, only relative to the O-B O partition. Because this partition is arbitrary, it can have no effect on the overall state of U. From a physical perspective, no clock characterizes U as a whole, and no observer observes U's overall state. Observations and hence transfers of classical information are only defined relative to partitions of U. The overall state of U can, therefore, be regarded as fixed without loss of generality; differences between observational outcomes obtained by different observers can be attributed entirely to differences between observer-box partitions. As no observer within U can place an upper limit on the complexity of U, an arbitrarily large "multiverse" of observer-box partitions is consistent with the information about U obtainable by any observer.
at a common temperature T.
Physically, detailed balance requires the classical information content of O's "question" to B O to equal the classical information content of B O 's "answer." The no-memory corollary shows that such information, once transferred, becomes inaccessible to its source, thus justifying the assumption of Landauer's principle. In particular, O can never determine that B O has returned to the "same" state or vice-versa.

Hilbert-space representation
where H B O is an abstract space representing the physical states of B O .   Intuitively, a reference frame such as a meter stick acts on only a limited collection of the degrees of freedom of U; when one measures a length with a meter stick, one interprets it as the length of something. No single measurement, however, reveals which particular collection of degrees of freedom is being measured; an outcome of 0.6 m, for example, indicates that something is 0.6 m long, but does not reveal which something is 0.6 m long. Any given outcome could, in principle, have been obtained from any of an arbitrarily large number of 0.6 m long somethings. If we give the measured something a name, e.g. 'S' on the basis of its measured 0.6 m length, that name is arbitrarily referentially ambiguous [12] We can, similarly, choose to regard the something S as occupying a state |S(t) at t, bearing in mind that for consistency with Corollary 2.3, the physical meaning of |S(t) is simply a state of B O , one of possibly arbitrarily many, that yields some α ij as an outcome when queried with ζ i at t. Proof : Assume that S is treated as a classical ensemble. It is then a specification, for O, of classical information, namely the prior probability distribution of the α ij over S. If S picks out a proper subsystem S of B O , then this S must be regarded as the source of this classical information for O. By Theorem 2, no subsystem of B O can be a source of classical information for O; hence S cannot be a source of classical information. In this case specifying S cannot specify a prior probability distribution of the α ij , so S cannot be treated as a classical ensemble.
Physically, S being a classical ensemble corresponds to ζ i being a reference frame that rules out, a priori, any potential effects of the degrees of freedom of B O that are not in S on the outcomes of observations made with ζ i . It corresponds, in other words, to ζ i perfectly isolating S from the rest of B O . Any such isolation clearly violates the additivity of H U . Proof : By Theorem 6, substituting the label 'E' for 'S'.
In the terminology of decoherence, E is the "environment" of S (e.g. [6]). Corollary 6.1 shows that the environment cannot be treated as a classical ensemble in any theory in which H U is additive. Assuming a classical environment for the purposes of decoherence calculations, or equivalently, assuming that O does not interact with and hence is isolated from E, is therefore formally inconsistent with quantum theory, not merely an example of circular reasoning as has been shown previously [13,14]. It remains now only to characterize the abstract state space H B O on which the reference frames ζ i act. Over the course of some experiment S employing ζ i , a particular outcome α ik will occur, if it occurs at all, for some observations but not, in general, for all observations. We can therefore think of H OS as an interaction that on some occasions "picks out" degrees of freedom of B O that yield α ik as an outcome and that on other occasions picks out degrees of freedom that yield other outcomes. In this case we can write: where e −(ϕ j t) is the periodic function that picks out a collection of degrees of freedom of B O that yields the j th outcome when acted upon by ζ i and b j measures how many such collections there are. Making the action of H OS (t) on |S explicit, (7) becomes: which can equally well be written: where physically |S j is, as above, just a state of B O that yield α ij as an outcome when acted upon by ζ i .
Physically, the superposition (9) expresses O's uncertainty about which of the potentially arbitrarily many collections of degrees of freedom of B O that could yield α ij as an outcome when acted upon by ζ i are actually being acted upon by ζ i and yielding the outcome α ij at t. It expresses, in other words, O's uncertainty about which 0.6 m long something is currently being measured. Bell's theorem tells us, very explicitly, that O cannot resolve this uncertainty: the phase ϕ j that determines when α ij is produced as an outcome is unmeasurable in principle. This phase cannot, therefore, be real-valued; it must be the case that ϕ j = iφ j for some real value φ j . The abstract space H B O must, therefore, be a Hilbert space, not a real configuration space. As the maps ζ i are POVMs, for each ζ i there must be some collection of states |S j such that j |S j S j | = Id. Hence j b 2 j = 1, allowing the b 2 j to be interpreted as probabilities. As the |S j and hence the b j are defined only relative to H OB O and hence relative to the O-B O partitioning of U, these probabilities are observer-relative and hence "subjective" in the Bayesian sense.

Conclusion
Assuming an additive Hamiltonian and hence the invariance of the Hamiltonian under decompositions of the state space is equivalent, clearly, to assuming a commutative and associative product to represent state space decomposition. The in-principle non-uniqueness of state space decompositions within quantum theory has been widely noted, particularly under the rubric of "entanglement relativity" [15,16,17,18,19,20,21,22]. What has been shown here is that the invariance of the Hamiltonian under decompositions of the state space is not merely an interesting feature of quantum theory, but is rather the source of quantum theory. As even the classical Hamiltonian is invariant under state-space decompositions, any physics of the classical world that is consistent with Landauer's principle must be quantum theory.
If merely assuming an additive Hamiltonian and Landauer's principle yields quantum theory as shown here, what makes classical physics "classical"? The answer, clearly, is that classical physics standardly assumes that observers can obtain outcome information both specifically and exclusively from particular, identified subsystems of B O , i.e. from particular identified subsystems of the universe U minus the observer. This is not a formal, axiomatic assumption, but is rather an implicit metaphysical assumption. This implicit assumption contradicts Theorem 2, and it takes various forms. For example, it may be assumed that O interacts so quickly with a particular, identified subsystem S that the measurement resolution is effectively infinite (i.e. c → ∞), or that O interacts so subtly with S that no energy is transferred and hence no finite action is required (i.e. h → 0). Alternatively, it may be assumed that S is a classical ensemble, i.e. that S is completely isolated from all other systems. As this is a fortiori an assumption that S is isolated from any physical O, it amounts to the assumption that O is not a physical system and hence that observation is not a physical interaction.
Common "interpretations" of quantum theory make similar assumptions. Decoherence theory, for example, typically assumes that O is "not observing" the environment E and hence is unaffected by it. This is effectively an assumption that no state change of E transfers information to O that affects the outcome of O's observations of S, an assumption that is readily falsified by walking into any laboratory and turning off the lights. The "environment as witness" formulation of decoherence (e.g. [23]) assumes that interactions with E transfer information specifically about S to O, i.e. that E is a classical information channel from S to O, contradicting Theorem 2. While the Copenhagen interpretation in its purest form concerns only the observer's knowledge, it is often presented to both students and the public as requiring quantum states to physically collapse, a requirement that clearly contradicts additivity. Even QBism (e.g. [24]) assumes that observers who are ignorant of a system's physical state -who might not, in principle, know where the system is or what it looks like -can nonetheless re-identify that very same system for subsequent observations, again contradicting Theorem 2.
The spectacular empirical success of quantum theory tells us that assuming an additive Hamiltonian yields excellent and enormously useful predictions about the world. What has been shown here is that the formalism of quantum theory also presents us with a clear and simple message: we are physical systems, and the world that we observe is, to us, a black box.