# Implementation of Classical Communication in a Quantum World

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Assumption: Quantum Theory is Universal

_{U}is a deterministic universal Hamiltonian. This assumption rules out any objective non-unitary “collapse” of ; it amounts to the adoption of what Landsman [10] calls “stance 1” regarding quantum theory, a stance that is realist about quantum states, and therefore demands an explanation for the appearance of classicality. All available experimental evidence is consistent with this universality assumption [11]. Alice, Bob, the systems that they observe and the systems that they employ to encode classical communications are, on this assumption, all collections of quantum degrees of freedom evolving under the action of the universal Hamiltonian H

_{U}.

_{U}. If the observed system S is regarded as a quantum information processor, this question of observational classicality becomes the question of how the behavior of S can be interpreted as computation. How, for example, do the unitary transformations of the quantum state of a quantum Turing machine (QTM, [12]) or Hamiltonian oracle [13] implement a computation on a classical data structure encoded by the system’s initial state? In what sense do the events that occur between measurements in a measurement-based quantum computer [14] implement computation? That these questions are both foundational to quantum computing and non-trivial has been emphasized by Aaronson [15].

#### 2.2. Consequence: Measurements are Actions by POVMs

_{i}} of positive-semidefinite Hilbert-space automorphisms that have been normalized so as to sum to unity; POVMs generalize traditional projective measurements (e.g., [17]) by dropping the requirement of orthogonality and hence the requirement that all elements of a measurement project onto the same Hilbert-space basis. If is a POVM representing a measurement of the state of some quantum system S, then each component is a Hilbert-space automorphism on , i.e., ; one can also write , where in general . Given the assumption of universality, it is clear that any such automorphism must be implemented by the unitary physical propagator acting on the universal Hilbert space , and hence on as a collection of components of some universal state . Hence a measurement can be thought of as a physical action by a POVM, as emphasized for example by Fuchs’ [18] depiction of a POVM as an observer’s prosthetic hand.

_{i}} defined on a Hilbert space . Each component Π

_{j}of a von Neumann projection {Π

_{i}} projects any state onto a basis vector of . If the set { } of images of the components of {Π

_{i}} is complete in the sense of spanning , one can write for states . In this case a general Hermitian observable M can be written where α

_{j}is the j

^{th}possible observable outcome of M acting on . Hence from an observer’s point of view, what a projection {Π

_{i}} produces is not just a new state vector, but a real outcome value α

_{j}; {Π

_{i}} is not just a Hilbert-space automorphism, but is also a mapping from to the set of real outcome values of some observable of interest.

^{S}is the set of bases of [3]. Indeed, any collection of mappings for which the probabilities P(α

_{j}) of obtaining real outcome values α

_{j}∈ sum to unity, and for which each of the components is implementable by the unitary physical propagator acting on the universal Hilbert space must be positive semi-definite (to yield real outcome values), normalized (to yield well-defined probabilities) and be a collection of Hilbert-space automorphisms (to be implementable by ); hence such a collection must be a POVM. The POVM formalism thus represents the extraction of classical information from quantum systems in the only way that it can be represented while maintaining consistency with the universality assumption.

#### 2.3. Consequence: Observers must Identify the Systems They Observe

Nothing is said in this definition, or in the surrounding discussion [4,5], about how observers are able to “access” a physical system “without prior knowledge” of such state variables as its location, size or shape, and without “prior agreement” about which item in their shared environment constitutes the system of interest. To find the identification of physical systems by observers treated explicitly, one must look to cybernetics, where unique identification of even classical finite-state machines (FSMs) by finite sequences of finite observations is shown to be impossible in principle [19,20], or to the cognitive neuroscience of perception, where the identification in practice of individual systems over extended periods of time is recognized as a computationally-intensive heuristic process [21,22,23].“A property of a physical system is objective when it is:

simultaneously accessible to many observers, who are able to find out what it is without prior knowledge about the system of interest, and who can arrive at a consensus about it without prior agreement.”

^{®}Ge(Li) detector” [3].

## 3. Decompositional Equivalence and Its Consequences

#### 3.1. Assumption: Our Universe Exhibits Decompositional Equivalence

_{U}is asumed to be independent of, and hence symmetric under arbitrary modifications of, boundaries drawn in by specifications of tensor product structures. Call this symmetry decompositional equivalence [3]. Stated formally, decompositional equivalence is the assumption that if a TPS S ⊗ E = S′ ⊗ E′ = U, then the dynamics H

_{U}= H

_{S}+ H

_{E}+ H

_{S−E}= H

_{S′}+ H

_{E′}+ H

_{S′ − E′}, where S and S′ are arbitrarily chosen collections of physical degrees of freedom, E and E′ are their respective “environments” and H

_{S − E}and H

_{S′ − E′}are, respectively, the S − E and S′ − E′ interaction Hamiltonians. Such equivalence of TPSs of can be alternatively expressed in terms of the linearity of H

_{U}: If H

_{U}=∑

_{ij}H

_{ij}where the indices i and j range without restriction over all quantum degrees of freedom within , decompositional equivalence is the assumption that the interaction matrix elements ( do not depend on the labels assigned to collections of degrees of freedom by specifications of TPSs. Decompositional equivalence is thus consistent with the general philosophical position of microphysicalism (for a recent review, see [29]), but involves no claims about explanatory reduction, and indeed no claims about explanation at all; it requires only that emergent properties of composite objects exactly supervene, as a matter of physical fact, on the fundamental interactions of the microscale components of those objects.

_{ij}are not just independent of where and when the degrees of freedom labeled by i and j interact, but are also independent of any other classical information that might be included in the specification of a reference frame from which the interaction of i and j might be observed. As such, it is similar in spirit to Tegmark’s “External Reality Hypothesis (ERH)” that “there exists an external physical reality completely independent of us humans” ([30] p. 101). If taken literally, however, the ERH violates energy conservation, as it allows human beings to behave arbitrarily without affecting “external physical reality” and vice-versa. The assumption of decompositional equivalence, on the other hand, does not involve, entail, or allow decoupling of observers or any other systems from their environments; any evidence that energy is not conserved, or evidence that energy is conserved but not additive would be evidence that decompositional equivalence is not satisfied in our universe. Were our universe to fail in fact to satisfy decompositional equivalence, any shift in specified system boundaries—any change in the TPS of —could be expected to alter fundamental physical laws or their dynamical outcomes; in such a universe, the notions of “fundamental physical laws” and “well-defined dynamics” would be effectively meaningless. It is, therefore, assumed in what follows that decompositional equivalence is in fact satisfied in our universe U, and hence that the dynamics H

_{U}is independent of system boundaries.

#### 3.2. Consequence: System-Environment Decoherence can have No Physical Consequences

_{S−E}. Such environmentally-mediated superselection or einselection [33,34] assures that observations of S that are mediated by information transfer through E will reveal eigenstates of H

_{S−E}; in the canonical example, observations of macroscopic objects mediated by information transfer through the ambient visible-spectrum photon field reveal eigenstates of position. From this perspective, it is the quantum mechanism of einselection that underlies the classical notion that the “environment” of a system—whether this refers to the ambient environment or to an experimental apparatus—objectively encodes the physical state of the system, where “objectively” has the sense given in the Ollivier–Poulin–Zurek definition [4,5] quoted in Section 2.3.

_{S−E}, where both S and E are specified completely independently of observers—allows decoherence to mimic “collapse” as a mechanism by which the world prepares or creates classical information about particular systems that observers can then detect. In this picture, as in the traditional Copenhagen picture, observers have nothing to do with what “systems” are available to observe: The world—in the decoherence picture, the environment—reveals some systems as “classical” and not others. The sense of “objectivity” defined by Ollivier, Poulin and Zurek [4,5] depends critically on this assumption; without it, the idea that observers can approach the world “without prior knowledge” of the systems it contains becomes uninterpretable. The second thing to note is that the formal mechanism of “tracing out the environment” in decoherence calculations [24,25,31,32] corresponds physically to an assumption that environmental degrees of freedom are irrelevant to the system-observer interaction, i.e., to an assumption that the physical interaction H

_{S−O}, where O is the observer, is independent of E. This assumption straightforwardly conflicts with the idea that observation—the S − O interaction—is mediated by E. This conflict between the formalism of decoherence and its model theory suggests that the trace operation is at best an approximate mathematical representation of the physics of decoherence.

_{S−E}, which is defined at the boundary, in Hilbert space, between S and E [33,34]. In a universe that satisfies decompositional equivalence, this boundary can be shifted arbitrarily without affecting the interactions between quantum degrees of freedom, i.e., without affecting the interaction H

_{ij}, and hence without affecting the matrix element , between any pair of degrees of freedom i and j within U. An arbitrary boundary shift, in other words, has no physical consequences. In particular, a boundary shift that transforms S ⊗ E into an alternative TPS S′ ⊗ E′ has no physical consequences for the values of matrix elements ( where i and j are degrees of freedom within the intersection E ∩ E′, and hence has no physical consequences for states of E ∩ E′ or for the classical information that such states encode. The encodings within E ∩ E′ of arbitrary states of S and S′, and hence of einselected pointer states of S and S′ are, therefore, entirely independent of the boundaries of these systems, and hence entirely independent of the Hamiltonians H

_{S−E}and H

_{S′−E′}defined at those boundaries. The encoding of information about S in E is, in other words, entirely a result of the action of H

_{U}= ∑

_{ij}H

_{ij}, and is entirely independent of specified system boundaries or “emergent” system-environment interactions definable at such specified boundaries.

_{in}

_{−out}that describes the bulk interaction between the molecules within the voxel and those outside. This bulk interaction can be viewed as decohering the collective quantum state of the molecules within the voxel, with a decoherence time at room temperature and pressure of substantially less than 10

^{−20}s [35], and as einselecting as an eigenstate of position within the fluid at all subsequent times. Such einselection prevents the wavefunction from spreading into a macroscopically-extended spatial superposition, just as decoherence and einselection by interplanetary dust, gasses and radiation prevent the wavefunction of Hyperion from doing so [36]. Does the state of the fluid outside the stipulated voxel objectively encode the position of the continuously-deforming voxel boundary at which this einselection takes place? Could observers with no prior knowledge of the stipulated voxel boundary determine its position by observing the state of the fluid? Obviously they could not.

^{®}plastic. The interatomic interactions between Pb, C, O and H atoms are completely independent of whether the Pb sphere, the Pb sphere together with a surrounding spherical shell of plastic, a voxel of Pb entirely within the Pb sphere, or a voxel containing only plastic is considered the “system of interest.” The boundary of the system stipulated, in each of these cases, is the site of action of a Hamiltonian H

_{in−out}that describes the bulk interaction between the atoms within the stipulated boundary and those outside; the action of this Hamiltonian einselects positional eigenstates of the collective quantum state of the atoms inside the boundary just as it does in the case of a voxel boundary in a fluid. Observers of the states of some arbitrary sample of the atoms in the plastic part of this combined system would, however, be no more capable of determining the site of a stipulated boundary than observers of some arbitrary sample of the fluid molecules in the previous example.

#### 3.3. Consequence: Identification of Systems by Observers is Intrinsically Ambiguous

_{i}} with a nonlinear function such that:

_{i}} over all of as in (1) renders the definition of “system” implicit: A system S is whatever returns finite outcome values when acted upon by some POVM composed with . The detectable degrees of freedom of such a system are, at some time t, the degrees of freedom in the inverse images of the components for which at t.

## 4. Decoherence as Semantics

#### 4.1. Decoherence as Implemented by a POVM

_{S−E}, a POVM can be defined as a mapping:

_{i}} defined over U, such that the inverse image Im

^{−1}E

_{k}is outside O for all components E

_{k}for which . In this case, O can be considered the “system” and ∪

_{k}(Im

^{−1}E

_{k}) ⊂ U where can be considered the “environment” in (2); the Hamiltonian H

_{ik}then characterizes the observer-environment interaction, and encodes classical information—the outcome values α

_{k}—about ∪

_{k}(Im

^{−1}E

_{k}) into . Hence (2) provides a general definition of decoherence as the deployment of a POVM by an observer. For observers embedded in a relatively static environment, for which the total observer-environment interaction ∑

_{ik}H

_{ik}is nearly constant, (2) is reasonably interpreted as defining a single, continuously-deployed POVM. For observers embedded in highly-variable environments that nonetheless exhibit some periodicity, as most human observers are, it is reasonable to view (2) as describing the deployment of not one but a periodic sequence of POVMs, each normalized over a subset of the environmental degrees of freedom with which O interacts. As such a sequence must be finite for a finite observer, a finite observer can only be viewed as decohering his, her or its environment in a finite number of ways. Hence unlike the “environment as witness”, a finite observer as witness can physically encode the states of at most a finite number of distinct “systems”. Because the POVMs encoded by finite observers are limited in their resolution by , each of the distinct “systems” representable by a finite observer is in fact an equivalence class under observable-dependent exchange symmetry.

^{µ}can be represented by a collection of distinct POVMs . The detectable outcome values produced by these POVMs have non-overlapping inverse images; hence they all mutually commute. If these POVMs are regarded as all acting at each of a sequence of times t

_{i}, their outcomes at those times can be considered to be a sequence of real vectors . These vectors form a consistent decoherent history of the S

^{µ}at the t

_{i}, in the sense defined by Griffiths [8]. In a universe in which decoherence is an informational process, the number of such consistent decoherent histories and hence the number of “classical realms” [40] is limited only by the number of distinct sets of subspaces of , i.e., is combinatorial in the number of degrees of freedom of . Each of these histories, as a discrete time sequence of real vectors, can be regarded as a sequential sample of the state transitions of a classical finite state machine (FSM; [19]). As shown by Moore [20], no finite sequence of observations of an FSM is sufficient to uniquely identify the FSM; hence no finite sample of any decoherent history is sufficient to identify the TPS boundaries at which the POVMs contributing to the history are defined, confirming the observable-dependent exchange symmetry of observations in a universe satisfying decompositional equivalence.

#### 4.2. Decoherence Defines a Virtual Machine

_{k}, it is clear that any consistent decoherent history of U can be represented as an execution trace of a classical virtual machine. Hence decoherence can, in general, be represented as a mapping of to the space of classical virtual machines, i.e., by a diagram such Figure 1; as such a mapping takes quantum states to classical information, it can be represented as a POVM {E

_{i}}. The requirement that this diagram commutes is the requirement that the action of the physical propagator eacting from t

_{n}to t

_{n}

_{+1}is represented, by the mapping {E

_{i}}, as a classical state transition from the n

^{th}to the (n + 1)

^{th}state of some virtual machine V. This commutativity requirement is fully equivalent to the commutativity requirement that defines consistency of observational histories of U (e.g., [8] Equation 10.20). Hence an evolution H

_{U}is consistent under a decoherence mapping {E

_{i}} if it can be interpreted as an implementation of a classical virtual machine.

**Figure 1.**Semantic relationship between physical states of U and einselected virtual states of a virtual machine V implemented by U. Commutativity of this diagram assures that the decoherence mapping {E

_{i}} is consistent.

_{i}} deployed by a finite observer must be collected within a finite time, any such mapping interprets only some local sample of the time evolution of U as computation. This perspective on decoherence is consistent with the cybernetic intuition—the intuition expressed by the Church–Turing thesis—that any classical dynamical process, and in particular any classical communicative process can be represented algorithmically.

## 5. Observation as Entanglement

#### 5.1. Classical Communication is Regressive

_{1}and its use in directing observations at t

_{2}; they differ only in the source of the information received at t

_{1}. As noted earlier, however, the only means of obtaining classical information provided by quantum theory is the deployment of a POVM. The two processes differ, therefore, only in which POVM the observers deploy at t

_{1}: In (A) they each deploy a POVM that identifies and determines the state of the “classical source,” while in (B) they each deploy a POVM that identifies and determines the state of S. Hence the coordination question asked at t

_{2}can also be asked at t

_{1}; even if the intrinsic ambiguity of observations with POVMs is ignored, the LOCC scenario cannot get off the ground without an agreement between the observers about which POVM to deploy at t

_{1}.

_{1}, the observers must exchange classical information. Each observer must, therefore, deploy a POVM that enables the acquisition of classical information from the other; call Alice’s POVM for acquiring information from Bob “ ” and Bob’s POVM for acquiring information from Alice “ ”, and suppose that these POVMs are deployed at some time t

_{0}. Clearly the same question can be asked at t

_{0}as at t

_{2}and t

_{1}, and clearly it cannot be answered by postulating yet another agreement, another classical communication, and another deployment of POVMs. The same kind of regress infects any simple joint assumption by Alice and Bob that they are observing the same system, an assumption that must be communicated to be effective. Any instance of measurement under LOCC conditions, in other words, requires the postulation of a priori classical communication between the observers, and hence requires that the observers themselves be regarded as classically objective a priori. Minimal quantum mechanics with decompositional equivalence provides no mechanism by which such a priori classical objectivity can be achieved; hence minimal quantum mechanics with decompositional equivalence does not support LOCC. At best, minimal quantum mechanics with decompositional equivalence supports the appearance of LOCC in cases in which observers agree to treat their observations as observations of the same system.

**Figure 2.**Two options for coordinating the selection of POVMs and Bob by Alice and Bob, respectively. (A) Alice and Bob receive POVM selection instructions from a classical source. (B) Alice and Bob jointly observe the production of S and agree that their selected POVMs identify it.

#### 5.2. Memory is Communication

#### 5.3. Implementation of POVMs by H_{U}

_{U}. The action of H

_{U}maintains a counterfactual-supporting classical correlation between states of S and B just in case S and B are entangled; if the correlation that is maintained is perfect, S and B must be monogamously entangled. Whether joint states of two identified systems appear to be entangled is, however, dependent on the choice of basis and hence the POVM deployed to determine their joint states [46,47,48,49,50]. Bob’s state is, therefore, a classical encoding of for Alice only if she deploys a POVM that projects onto a Hilbert-space bases in which is entangled, and is a perfectly classical encoding if this apparent entanglement is monogamous.

_{U}on the quantum degrees of freedom that implement O and S: Observation is entanglement. The existence of such entanglement is an objective fact that is, in a universe satisfying decompositional equivalence, independent of the boundaries of S and O. Whether S and O appear to be entangled to a third-party observer, however, is not an objective fact; it rather depends on the POVM employed by that observer to extract classical information from the degrees of freedom implementing S and O. Hence while the classical correlation between S and O is “real”—i.e., physical, a result of the action of H

_{U}—whether it appears classical to third parties is virtual, i.e., dependent on semantic interpretation. All public communication is, therefore, nonfungible or “unspeakable” in the sense defined in [51]: The information communicated is always strictly relative to a POVM—a “reference frame” in the language of [51]—that is not specified by H

_{U}and cannot be assumed without circularity. Any publicly-communicable classical description of the world is, therefore, intrinsically logically circular.

_{U}that implements any particular instance of classical communication.

## 6. Conclusions

_{U}. Such entanglement is not publicly accessible to multiple observers without the further specification of a POVM. Any such specification is, however, itself an item of classical information; hence any claim that classical communication “emerges” from quantum entanglement involves logical circularity. The idea that quantum theory can produce a shared classicality—can be an “ultimate theory that needs no modifications to account for the emergence of the classical” ([53] p. 1)—therefore cannot be maintained. This loss of “emergent classicality” is, however, balanced by a powerful gain: The possibility that a POVM can be discovered that will reveal, in particular cases, the entanglement by which the transfer of classical information from system to observer is implemented.

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**MDPI and ACS Style**

Fields, C.
Implementation of Classical Communication in a Quantum World. *Information* **2012**, *3*, 809-831.
https://doi.org/10.3390/info3040809

**AMA Style**

Fields C.
Implementation of Classical Communication in a Quantum World. *Information*. 2012; 3(4):809-831.
https://doi.org/10.3390/info3040809

**Chicago/Turabian Style**

Fields, Chris.
2012. "Implementation of Classical Communication in a Quantum World" *Information* 3, no. 4: 809-831.
https://doi.org/10.3390/info3040809