Spacelike Separated Regions Research Articles

Localization in quantum field theory

We review the issue of localization in quantum field theory and detail the nonrelativistic limit. Three distinct localization schemes are examined: the Newton–Wigner, the algebraic quantum field theory, and the modal scheme. Among these, the algebraic quantum field theory provides a fundamental concept of localization, rooted in its axiomatic formulation. In contrast, the Newton–Wigner scheme draws inspiration from the Born interpretation, applying mainly to the nonrelativistic regime. The modal scheme, relying on the representation of single particles as positive frequency modes of the Klein–Gordon equation, is found to be incompatible with the algebraic quantum field theory localization.This review delves into the distinctive features of each scheme, offering a comparative analysis. A specific focus is placed on the property of independence between state preparations and observable measurements in spacelike separated regions. Notably, the notion of localization in algebraic quantum field theory violates this independence due to the Reeh–Schlieder theorem. Drawing parallels with the quantum teleportation protocol, it is argued that causality remains unviolated. Additionally, we consider the nonrelativistic limit of quantum field theory, revealing the emergence of the Born scheme as the fundamental concept of localization. Consequently, the nonlocality associated with the Reeh–Schlieder theorem is shown to be suppressed under nonrelativistic conditions.

Subregion independence in gravity

In gravity, spacelike separated regions can be dependent on each other due to the constraint equations. In this paper, we give a natural definition of subsystem independence and gravitational dressing of perturbations in classical gravity. We find that extremal surfaces, non-perturbative lumps of matter, and generic trapped surfaces are structures that enable dressing and subregion independence. This leads to a simple intuitive picture for why extremal surfaces tend to separate independent subsystems. The underlying reason is that localized perturbations on one side of an extremal surface contribute negatively to the mass on the other side, making the gravitational constraints behave as if there exist both negative and positive charges. Our results support the consistency of islands in massless gravity, shed light on the Python’s lunch, and provide hints on the nature of the split property in perturbatively quantized general relativity. We also prove a theorem bounding the area of certain surfaces in spherically symmetric asymptotically de Sitter spacetimes from above and below in terms of the horizon areas of de Sitter and Nariai. This theorem implies that it is impossible to deform a single static patch without also deforming the opposite patch, provided we assume spherical symmetry and an energy condition.

Towards a measurement theory in QFT: "Impossible" quantum measurements are possible but not ideal

Naive attempts to put together relativity and quantum measurements lead to signaling between space-like separated regions. In QFT, these are known as impossible measurements. We show that the same problem arises in non-relativistic quantum physics, where joint nonlocal measurements (i.e., between systems kept spatially separated) in general lead to signaling, while one would expect no-signaling (based for instance on the principle of no-nonphysical communication). This raises the question: Which nonlocal quantum measurements are physically possible? We review and develop further a non-relativistic quantum information approach developed independently of the impossible measurements in QFT, and show that these two have been addressing virtually the same problem. The non-relativistic solution shows that all nonlocal measurements are localizable (i.e., they can be carried out at a distance without violating no-signaling) but they (i) may require arbitrarily large entangled resources and (ii) cannot in general be ideal, i.e., are not immediately reproducible. These considerations could help guide the development of a complete theory of measurement in QFT.

On separable states in relativistic quantum field theory

We initiate an investigation into separable, but physically reasonable, states in relativistic quantum field theory. In particular we will consider the minimum amount of energy density needed to ensure the existence of separable states between given spacelike separated regions. This is a first step towards improving our understanding of the balance between entanglement entropy and energy (density), which is of great physical interest in its own right and also in the context of black hole thermodynamics. We will focus concretely on a linear scalar quantum field in a topologically trivial, four-dimensional globally hyperbolic spacetime. For rather general spacelike separated regions A and B we prove the existence of a separable quasi-free Hadamard state. In Minkowski spacetime we provide a tighter construction for massive free scalar fields: given any R > 0 we construct a quasi-free Hadamard state which is stationary, homogeneous, spatially isotropic and separable between any two regions in an inertial time slice all of whose points have a distance . We also show that the normal ordered energy density of these states can be made (in Planck units). To achieve these results we use a rather explicit construction of test-functions f of positive type for which we can get sufficient control on lower bounds on .

Channel Divergences and Complexity in Algebraic QFT

We consider a notion of divergence between quantum channels in relativistic continuum quantum field theory (QFT) that is derived from the Belavkin–Staszewski relative entropy and the concept of bimodules for general von Neumann algebras. Key concepts of the divergence that we shall prove based on a new variational formulation of that relative entropy are the subadditivity under composition and additivity under the tensor product between channels. Based on these properties, we propose to use the channel divergence relative to the trivial (identity-) channel as a novel measure of complexity. Using the properties of our channel divergence, we prove in the prerequisite generality necessary for the algebras in QFT that the corresponding complexity has several reasonable properties: (i) the complexity of a composite channel is not larger than the sum of its parts, (ii) it is additive for channels localized in spacelike separated regions, (iii) it is convex, (iv) for an N-ary measurement channel it is log N, (v) for a conditional expectation associated with an inclusion of QFTs with finite Jones index it is given by log (text {Jones Index}).

Bipartite entanglement distillation by unilateral and bilateral local filters using polarizing Mach–Zehnder interferometers

The extraordinary correlation seen in entangled states in space-like separated regions is one of the most intriguing aspects of quantum states. Practical utility of entanglement as a resource for quantum key distribution, dense coding, or teleportation generally requires maximally entangled states. In practice, entanglement quality degrades substantially due to channel noise. The problem may be mitigated by entanglement distillation. The simplest distillation protocol is enforced by local filtering operations and classical communication. The local filtering operations are merely generalized positive operator valued measures utilizing additional degrees of freedom (DoFs) as ancillary qubits. In this work, we experimentally show that filtering on a single channel (unilateral) is equally effective as filtering on both channels (bilateral) for distillation of pure non-maximally entangled bipartite states. This result holds for a non-maximally entangled multi-qubit Greenberger-Horne-Zeilinger (GHZ) like states as well, as they show a straightforward extension of the Bell state structure. Further, we provide a theoretical comparison of the efficacy of unilateral and bilateral filtering for the case of mixed states resulting from local depolarizing noise introduced either in one or both of the non-maximally entangled pairs. Surprisingly, when noise is introduced in either one of the channels, we find that unilateral filtering on the noise-free channel outperforms the filtering on the noisy channel and bilateral filtering on both channels. We also find that bilateral filtering is more effective when both channels are noisy. A reduction in the number of local operations, in general, has the advantage of reducing the complexity of the experimental apparatus and the reduction of measurement related errors.

Bell’s nonlocality and gravity

The experimental results that test Bell’s inequality have found strong evidence suggesting that there are nonlocal aspects in nature. Evidently, these nonlocal effects, which concern spacelike separated regions, create an enormous tension between general relativity and quantum mechanics. In addition, by avoiding the coincidence limit, semiclassical gravity can also accommodate nonlocal aspects. Motivated by these results, we study if it is possible to construct geometrical theories of gravitation that are nonlocal in the sense of Bell. We propose three constructions of such theories, which could constitute an important step towards our understanding of the interplay between quantum mechanics and gravitation.

Photonic realization of erasure-based nonlocal measurements

Abstract Relativity theory severely restricts the ability to perform nonlocal measurements in quantum mechanics. Studying such nonlocal schemes may thus reveal insights regarding the relations between these two fundamental theories. Therefore, for the last several decades, nonlocal measurements have stimulated considerable interest. However, the experimental implementation of nonlocal measurements imposes profound restrictions because the interaction Hamiltonian cannot contain, in general, nonlocal observables such as the product of local observables belonging to different particles at spacelike-separated regions. In this work, we experimentally realize a scheme for nonlocal measurements with the aid of probabilistic quantum erasure. We apply this scheme to the tasks of performing high-accuracy nonlocal measurements of the parity, as well as measurements in the Bell basis, which do not necessitate classical communication between the parties. Unlike other techniques, the nonlocal measurement outcomes are available locally (upon successful postselection). The state reconstructed via performing quantum tomography on the system after the nonlocal measurement indicates the success of the scheme in retrieving nonlocal information while erasing any local data previously acquired by the parties. This measurement scheme allows to realize any controlled-controlled-gate with any coupling strength. Hence, our results are expected to have conceptual and practical applications to quantum communication and quantum computation.

Quantum reality via late-time photodetection

We further investigate postulates for realist versions of relativistic quantum theory and quantum field theory in Minkowski space and other background space-times. According to these postulates, quantum theory is supplemented by local variables that depend on possible outcomes of hypothetical measurements on the late time electromagnetic field in spacelike separated regions. We illustrate the implications in simple examples using photon wave mechanics, and discuss possible extensions to quantum field theory.

The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields

Conditions for the appearance of topological charges are studied in the framework of the universal C*-algebra of the electromagnetic field, which is represented in any theory describing electromagnetism. It is shown that non-trivial topological charges, described by pairs of fields localised in certain topologically non-trivial spacelike separated regions, can appear in regular representations of the algebra only if the fields depend non-linearly on the mollifying test functions. On the other hand, examples of regular vacuum representations with non-trivial topological charges are constructed, where the underlying field still satisfies a weakened form of "spacelike linearity". Such representations also appear in the presence of electric currents. The status of topological charges in theories with several types of electromagnetic fields, which appear in the short distance (scaling) limit of asymptotically free non-abelian gauge theories, is also briefly discussed.

The split property for quantum field theories in flat and curved spacetimes

The split property expresses a strong form of independence of spacelike separated regions in algebraic quantum field theory. In Minkowski spacetime, it can be proved under hypotheses of nuclearity. An expository account is given of nuclearity and the split property, and connections are drawn to the theory of quantum energy inequalities. In addition, a recent proof of the split property for quantum field theory in curved spacetimes is outlined, emphasising the essential ideas.

Spacetime-constrained oblivious transfer

In 1-out-of-2 oblivious transfer (OT), Alice inputs numbers x_0, x_1, Bob inputs a bit b and outputs x_b. Secure OT requires that Alice and Bob learn nothing about b and x_{\bar{b}}, respectively. We define spacetime-constrained oblivious transfer (SCOT) as OT in Minkowski spacetime in which Bob must output x_b within R_b, where R_0 and R_1 are fixed spacelike separated spacetime regions. We show that unconditionally secure SCOT is impossible with classical protocols in Minkowski (or Galilean) spacetime, or with quantum protocols in Galilean spacetime. We describe a quantum SCOT protocol in Minkowski spacetime, and we show it unconditionally secure.

The locality axiom in quantum field theory and tensor products of C*-algebras

The prototypes of mutually independent systems are systems which are localized in spacelike separated regions. In the framework of locally covariant quantum field theory, we show that the commutativity of observables in spacelike separated regions can be encoded in the tensorial structure of the functor which associates unital C*-algebras (the local observable algebras) to globally hyperbolic spacetimes. This holds under the assumption that the local algebras satisfy the split property and involves the minimal tensor product of C*-algebras.

EPR States and Bell Correlated States in Algebraic Quantum Field Theory

A mathematical rigorous definition of EPR states has been introduced by Arens and Varadarajan for finite dimensional systems, and extended by Werner to general systems. In the present paper we follow a definition of EPR states due to Werner. Then we show that an EPR state for incommensurable pairs is Bell correlated, and that the set of EPR states for incommensurable pairs is norm dense between two strictly space-like separated regions.

Noncommutative Common Cause Principles in algebraic quantum field theory

States in algebraic quantum field theory “typically” establish correlation between spacelike separated events. Reichenbach's Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions VA and VB, respectively, there is a local projection C not necessarily commuting with A and B such that C is supported within the union of the backward light cones of VA and VB and the set {C, C⊥} screens off the correlation between A and B.

Time Asymmetry of Probabilities Versus Relativistic Causal Structure: An Arrow of Time

There is an incompatibility between the symmetries of causal structure in relativity theory and the signaling abilities of probabilistic devices with inputs and outputs: while time reversal in relativity will not introduce the ability to signal between spacelike separated regions, this is not the case for probabilistic devices with spacelike separated input-output pairs. We explicitly describe a nonsignaling device which becomes a perfect signaling device under time reversal, where time reversal can be conceptualized as playing backwards a videotape of an agent manipulating the device. This leads to an arrow of time that is identifiable when studying the correlations of events for spacelike separated regions. Somewhat surprisingly, although the time reversal of Popescu-Rohrlich boxes also allows agents to signal, it does not yield a perfect signaling device. Finally, we realize time reversal using postselection, which could to lead experimental implementation.

Extraction of timelike entanglement from the quantum vacuum

An intriguing property of the massless quantum vacuum state is that it contains entanglement between both spacelike and timelike separated regions of space-time. The implications of timelike entanglement and its connection to standard entanglement, however, are unexplored. Here we show that timelike entanglement can be extracted from the massless Minkowski vacuum and converted into standard entanglement ``at a given time'' between two inertial, two-state detectors at the same spatial location: one coupled to the field in the past and the other coupled to the field in the future. The procedure used here demonstrates a time correlation as a requirement for extraction; e.g., if the past detector was active at a quarter to 12:00, then the future detector must wait to become active at precisely a quarter past 12:00 in order to achieve entanglement.

Simplified instantaneous non-local quantum computation with applications to position-based cryptography

Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially with the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given bound and is restricted to classical communication.

Einstein's Dissatisfaction with Nonrelativistic Quantum Mechanics and Relativistic Quantum Field Theory

It is argued that in his critique of standard nonrelativistic quantum mechanics Einstein formulated three requirements as necessary for a physical theory to be compatible with the field-theorectical paradigm, and it is shown that local, relativistic, algebraic quantum field theory typically satisfies those criteria—although, there are still open questions concerning the status of operational separability of quantum systems localized in spacelike separated space-time regions. It is concluded that local algebraic quantum field theory can be viewed as a research program that Einstein suggested informally in 1948 and that was realized only later in mathematical physics.

How local are local operations in local quantum field theory?

A notion called operational C⁎- separability of local C*-algebras ( A ( V 1 ) and A ( V 2 ) ) associated with spacelike separated spacetime regions V 1 and V 2 in a net of local observable algebras satisfying the standard axioms of local, algebraic relativistic quantum field theory is defined in terms of operations (completely positive unit preserving linear maps) on the local algebras A ( V 1 ) and A ( V 2 ) . Operational C*-separability is interpreted as a “no-signaling” condition formulated for general operations, for which a straightforward no-signaling theorem is shown not to hold. By linking operational C*-separability of ( A ( V 1 ) , A ( V 2 ) ) to the recently introduced ( Rédei & Summers, forthcoming) operational C*-independence of ( A ( V 1 ) , A ( V 2 ) ) it is shown that operational C*-separability typically holds for the pair ( A ( V 1 ) , A ( V 2 ) ) if V 1 and V 2 are strictly spacelike separated double cone regions. The status in local, algebraic relativistic quantum field theory of a natural strengthening of operational C*-separability, i.e. operational W*-separability, is discussed and open problems about the relation of operational separability and operational independence are formulated.