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Observations of quantum systems carried out by finite observers who subsequently communicate their results using classical data structures can be described as “local operations, classical communication” (LOCC) observations. The implementation of LOCC observations by the Hamiltonian dynamics prescribed by minimal quantum mechanics is investigated. It is shown that LOCC observations cannot be described using decoherence considerations alone, but rather require the

Suppose spatially-separated observers Alice and Bob each perform local measurements on a spatially-extended quantum system—for example, a pair of entangled qubits in an asymmetric Bell state—and afterwards communicate their experimental outcomes to each other. This “local operations, classical communication” (LOCC, e.g., [

Let us begin with classical communication. Any finite message from Bob to Alice can be represented as a finite sequence of classical bits. It must, moreover, be encoded in some physical medium [

The local operations performed by Alice and Bob have, therefore, two distinct targets. Alice and Bob must each operate locally on S to extract classical information, and must each operate locally on their shared communication medium to either encode (Bob) or decode (Alice, and Bob if he checks his encoding) the classical information contained in the transmitted message. Most discussions of LOCC acknowledge that the interactions with S involve quantum measurement; most neglect the fact that, if quantum theory is assumed to be universal, the encoding and decoding steps also involve interactions with a quantum system: the physical medium of communication. Most, moreover, neglect the fact that Alice and Bob are themselves quantum systems. The purpose of the present paper is to examine LOCC from a perspective that acknowledges these facts; it is, therefore, to ask what is required to implement LOCC in a quantum world.

The next section, “Preliminaries”, discusses the fundamental assumption that quantum theory is universal and two of its consequences: That the extraction of classical information from quantum systems can be represented by the actions of positive operator-valued measures (POVMs, reviewed by [

The first and most fundamental assumption made here is that quantum theory is universal: All _{U} is a deterministic universal Hamiltonian. This assumption rules out any objective non-unitary “collapse” of _{U}.

The assumption that all physical systems are quantum systems clearly does not entail that all

Under the assumption of universality, understanding the requirements of LOCC in the case of either Alice or Bob individually clearly requires understanding quantum measurement, and in particular understanding whether observational classicality can be supposed to “emerge” from the dynamics specified by _{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, [

What the LOCC concept adds to the quantum measurement problem as traditionally presented (e.g., [

If quantum theory is universal, measurements can be represented by POVMs. A POVM is a collection {_{i}

Treating a POVM as a collection of Hilbert-space automorphisms does not, however, capture the sense in which observations _{i}_{j}_{i}_{i}_{j}^{th}_{i}_{j}_{i}

A general POVM ^{S} is the set of bases of _{j}_{j}

The assumption that all measurements can be represented by POVMs clearly does not entail that an observer can explicitly write down the components of every POVM that he or she might deploy in the course of interacting with the world. Doing so in any particular case would require both a complete specification of the outcome values obtainable with that POVM and a complete specification of the Hilbert space upon which it acts, or as discussed below, a complete specification of the inverse image in

When a new graduate student enters a laboratory, he or she is introduced to the various items of apparatus that the laboratory employs. The reason for this ritual is obvious: The student cannot be expected to reliably report the state of a particular apparatus if he or she cannot

“A property of a physical system 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.”

In practice, observers identify items of laboratory apparatus by finite sets of classically-specified criteria: location, size, shape, color, overall appearance, laboratory-affixed labels, brand name. These criteria are encodable as finite binary strings. If quantum theory is universal, items of laboratory apparatus are quantum systems, and hence are characterizable by Hilbert spaces comprising their quantum degrees of freedom. Observing a laboratory apparatus, therefore, requires deploying an operator that maps a collection of quantum degrees of freedom to a finite set of finite binary strings; by the reasoning above, such operators can only be POVMs. ^{®} Ge(Li) detector” [

The formal definition of system-identifying POVMs is complicated by two related issues. First, the vast majority of systems identified by human observers are characterized, like laboratory apparatus are characterized, not by possible outcome values of their quantum degrees of freedom, but by possible outcome values of bulk degrees of freedom such as macroscopic size or shape. The exceptions—the systems that those who reject the universality of quantum theory consider to be the only

In recognition of the role of the intervening environment in the observation and hence identification of systems of interest, it has been proposed that system-identifying POVMs be defined, in general, over either the physically-implemented information channel with which an observer interacts (

Defining system-identifying POVMs over U as a whole does not render observations nonlocal. Any finite observer must expend finite energy to record the outcomes obtained by deploying a POVM; hence any observation requires finite time. Any finite observer can, moreover, deploy a POVM for only a finite time. A finite observer can, therefore, regard a system-identifying POVM—or any POVM—as extracting classical information from at most a local volume with a horizon at

Defining system-identifying POVMs over U as a whole does not, moreover, resolve the question of how such POVMs—or how any POVMs—can yield outcome values for bulk degrees of freedom such as macroscopic size or shape. This question is, clearly, the question of quantum measurement itself; in particular, it is the question of the “emergence of classicality” that is taken up in

A fundamental requirement of observational objectivity, and hence of science as practiced, is that reality is independent of the language chosen to describe it. This fundamental assumption that reality is independent of the descriptive terms and hence the semantics chosen by observers—in particular, human observers—underlies the assumption in scientific practice that any arbitrary collection of physical degrees of freedom can be stipulated to be a “system of interest” and named with a symbol such as “S” without this choice of language affecting either fundamental physical laws or their outcomes as expressed by the dynamical behavior of the degrees of freedom contained within S. It similarly underlies the assumption that, given the technological means, an experimental apparatus to investigate the behavior of S can be designed and constructed without altering either fundamental physical laws or the dynamical behavior of the degrees of freedom contained within S. These assumptions operate prior to apparatus-dependent experimental interventions into the behavior of S, and hence prior to

This fundamental assumption that reality is independent of semantics can be generalized to state an assumed dynamical symmetry: The universal dynamics _{U} is asumed to be independent of, and hence symmetric under arbitrary modifications of, boundaries drawn in _{U} = _{S} + _{E} + _{S−E} = _{S′} + _{E′} + _{S′ − E′}, where S and S′ are arbitrarily chosen collections of physical degrees of freedom, E and E′ are their respective “environments” and _{S − E} and _{S′ − E′} are, respectively, the S − E and S′ − E′ interaction Hamiltonians. Such equivalence of TPSs of _{U}: If _{U} =∑_{ij} H_{ij}

As is the assumption that quantum theory is universal, the assumption that the universe satisfies decompositional equivalence is an empirical assumption. Its empirical content is most obvious in its formulation as the assumption that interaction matrix elements (_{ij}_{U} is independent of system boundaries.

The assumption of decompositional equivalence has immediate, but largely unremarked, consequences in two areas: The characterization of system-environment decoherence and the characterization of system identification by observers. Let us consider decoherence first. The usual understanding of system-environment decoherence (e.g., [_{S−E}. Such environmentally-mediated superselection or _{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

Two features of this standard account of decoherence deserve emphasis. First, the idea that the environment einselects particular eigenstates of S in an observer-independent way—that environmental einselection depends only on _{S−E}, where both S and E are specified completely independently of observers—allows decoherence to mimic “collapse” as a mechanism by which the _{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

By definition, einselection depends on the Hamiltonian _{S−E}, which is defined at the boundary, in Hilbert space, between S and E [_{ij}_{S−E} and _{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 _{U} = ∑_{ij} H_{ij}

It has been proposed, under the rubric of “quantum Darwinism” [

The physics of continuous fluid flow provides a simple example of decompositional equivalence and its consequences for einselection. It is commonplace to describe fluid flow in terms of deformable voxels, stipulated to be cubic at some initial time t0, that contain some particular collection of molecules. The stipulation of such a voxel has no effect on the intermolecular interactions between the molecules composing the fluid, whether these molecules are within, outside, or on opposite sides of the boundary of the voxel. Stipulation of a voxel boundary immediately defines, however, a Hamiltonian _{in} _{−out}^{−20} s [

The situation with bulk material objects appears, intuitively, to be different from the fluid-flow situation just described. When viewed in terms of pairwise interactions between the quantum degrees of freedom of individual atoms, however, the intuitive difference vanishes. Consider a uniform sphere of Pb embedded in a solid mass of Plexiglas^{®} 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 _{in−out}

As a final example, consider observers of the experimental apparatus employed by Brune

If decoherence has no physical consequences for interaction matrix elements, it can have no physical consequences for entanglement. The total entanglement in a quantum universe satisfying decompositional equivalence is, therefore, strictly conserved. Measurements, in particular, cannot

While they cannot, without violating decompositional equivalence, physically destroy entanglement, observations nonetheless have real-valued outcomes that can be recorded in classical data structures and reported by one observer to another using classical communication. If the “systems” that these outcome values describe cannot be assumed to be specified for observers by decoherence and environmental witnessing, they must be specified by observers themselves, by the deployment of system-identifying POVMs. It was argued in

Recall that any finite observer is restricted to a finite encoding of the outcomes obtained with any POVM; any POVM can be considered a mapping to binary codes of some finite length _{i}_{i}

In general, many TPSs of

By imposing observable-dependent exchange symmetry on observers, the assumption of decompositional equivalence removes the final sense in which observational classicality might be regarded as objective classicality: Two observers who record the same outcomes can no longer infer that their respective POVMs have detected the same collection of quantum degrees of freedom. As observable-dependent exchange symmetry applies, in principle, to

If decoherence is not a _{S−E}, a POVM

In a universe that satisfies decompositional equivalence, the meanings of “S” and “E” in (2) can be shifted arbitrarily provided S ⊗ E = U. Suppose an observer O deploys a POVM {_{i}^{−1}_{k}_{k}_{k}^{−1}_{k}_{ik}_{k}_{k}^{−1}_{k}_{ik} H_{ik}

Using (2), any collection of Hilbert-subspace boundaries that enclose disjoint collections of degrees of freedom and hence define distinct “systems” S^{µ}_{i}^{µ}_{i}

A classical _{k}_{i}_{n}_{n}_{+1} is represented, by the mapping {_{i}^{th}^{th}_{U} is consistent under a decoherence mapping {_{i}

Semantic relationship between physical states of U and einselected virtual states _{i}

The semantic relationship shown in _{i}

We can now return to Alice and Bob, who each perform local observations of a quantum system and then exchange their results by classical communication. If the dynamics in U exhibit decompositional equivalence, Alice and Bob cannot rely on decoherence by their shared environment to uniquely identify the system of interest; instead they must each rely on their own POVM to identify it. Observable-dependent exchange symmetry prevents them, moreover, from determining by observation that they have identified the

The first thing to note is that any answer to this question that relies on prior agreements between Alice and Bob is straightforwardly regressive, and hence incapable of explaining anything. How, for example, do Alice and Bob know which POVM to deploy in order to perform a joint observation? How, in other words, do observers coordinate their observations, independently of whether they manage to observe a single, shared system? There are two possibilities, as illustrated in

From the perspective of the observers, the two processes illustrated in _{1} and its use in directing observations at _{2}; they differ only in the source of the information received at _{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 _{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 _{2} can also be asked at _{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 _{1}.

In order to reach an agreement about which POVMs to deploy at _{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 “_{0}. Clearly the same question can be asked at _{0} as at _{2} and _{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

Two options for coordinating the selection of POVMs

The regress of classical communications encountered here is equivalent to the regress of the von Neumann chain that motivates the adoption of “collapse” as a postulate of quantum mechanics [

The second thing to note regarding LOCC is that the physical implementation of any classical memory, whether it comprises words written on a page or neural excitation patterns in someone’s brain, is a quantum system. Physically accessing a classical memory requires extracting classical information from this quantum system, and hence requires deploying a POVM. Observable-dependent exchange symmetry assures that an observer cannot be confident that the physical degrees of freedom accessed with a “memory-accessing” POVM are the same physical degrees of freedom that were accessed when a memory was encoded, or on any previous occasion when the memory was read. Hence Bob’s predicament when accessing his own memory of an observation is no different from Alice’s predicament when accessing a report from Bob; in both cases, all the usual caveats pertaining to quantum measurement apply.

The requirement that classical memories be observed in order to function as memories renders the LOCC scenario descriptive of all

Let us suppose that Alice obtains a report from Bob simply by observing his state _{U}. The action of _{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 [

To say of any observer O that “O deploys _{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”—_{U}—whether it appears classical to third parties is virtual, _{U} and cannot be assumed without circularity. Any publicly-communicable classical description of the world is, therefore, intrinsically logically circular.

The intrinsic circularity of public classical communication renders an explanation of a shared classical world in terms of fundamental physics unattainable. The shared classical world of ordinary experience cannot, therefore, be regarded as “emergent” from fundamental physics alone; instead it must be thought of as _{U} that implements any particular instance of classical communication.

As Bohr [_{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

The dependence of physics on model-theoretic or semantic assumptions explored here ties physics explicitly to classical computer science: the selection of a shared POVM that enables quantum theory to get off the ground as a description of a shared observable world is fully equivalent to the selection of a virtual-machine description that enables the description of a physical process as the instantiation of a classical algorithm to get off the ground. All physical descriptions are, from this point of view, specifications of classical virtual machines. What distinguishes “quantum” from “classical” computation is the choice of a POVM. The increased efficiency of quantum computation is, therefore, not the result of a different kind of device executing a different kind of behavior, but rather the result of a different choice of description. Castagnoli [