- D. Accelerated detectorsIn quantum field theory, the vacuum is defined as thelowest energy state of a field. A free field with linearequations of motion can be resolved into normal modes,such as standing waves. Each mode has a fixed frequencyand behaves as a harmonic oscillator. The zero point motionof all these harmonic oscillators is called “vacuumfluctuations”and the latter, under suitable conditions,may excite a localized detector that follows a trajectoryxν (τ) parametrized by its proper time τ. The internalstructure of the detector is described by non-relativisticquantum mechanics, so that we can indeed assume that itis approximately localized, and it has discrete energy levelsEn. Furthermore, we assume the existence of a linearcoupling of an internal degree of freedom, μ, of the detector,with the scalar field φ(x(τ)) at the position of thedetector. First-order perturbation theory gives the followingexpression for the transition probability per unitproper timeFor
*inertial detectors*(that is, x^ν = v^ντ with a constantfour-velocity vν) the transition probability is zero,as one should expect. However, the response rate doesnot vanish for more complicated trajectories. Considerin particular one with*constant proper acceleration*a.With an appropriate choice of initial conditions, it correspondsto the hyperbola t^2 +x^2 = 1/a^2, shown in Fig. 3.Then the transition rate between levels appears to bethe same as for an inertial detector in equilibrium withthermal radiation at temperature T ~ ha/2πckB. Thisphenomenon is called the Unruh (1976) effect. It wasalso discussed by Davies (1975) and it is related to thefluctuation-dissipation theorem (Candelas and Sciama,1977) and to the Hawking effect that will be discussed inthe next section.17 A rigorous proof of the Unruh effectin Minkowski spacetime was given by Bisognano andWichmann (1976) in the context of axiomatic quantumfield theory, thus establishing that the Unruh effect is notlimited to free field theory.*For any reasonable acceleration, the Unruh temperature**is incomparably smaller that the black-body temperature**of the cosmic background, or any temperature ever**attained in a laboratory,*and is not observable. Levin,Peleg, and Peres (1992) considered the effect of shieldinga hypothetical experiment from any parasitic sources.This, however, creates a radically new situation, becausethe presence of a boundary affects the dynamical propertiesof the quantum field by altering the frequencies ofits normal modes. Finite-size effects on fields have beenknown for a long time, both theoretically (Casimir, 1948)and experimentally (Spaarnay, 1958). Levin, Peleg, andPeres showed that if the detector is accelerated togetherwith the cavity that shields it, it will not be excited bythe vacuum fluctuations of the field. On the other hand,an inertial detector freely falling within such an acceler-ated cavity will be excited. The relevant property in allthese cases is the relative acceleration of the detector andthe field normal modes.17 Properties of detectors undergoing circular acceleration, as inhigh energy accelerators, were investigated by Bell and Leinaas(1983), Levin, Peleg, and Peres (1993), and by Davies, Dray, andManogue (1996).We now consider the evolution of an arbitrary quantumsystem. An*observer at rest (Alice)*can describe thequantum evolution on consecutive parallel slices of spacetime,t = const. What can*Bob, the accelerated observer*,do? From Fig. 3, one sees that there is no communicationwhatsoever between him and the region of spacetimethat lies beyond both horizons. Where Alice sees a purestate, Bob has only a mixed state. Some information islost. We shall return to this subject in the next, finalsection.VI. BEYOND SPECIAL RELATIVITYIt took Einstein more than ten years of intensive workto progress from special relativity to general relativity.Despite its name, the latter is not a generalization ofthe special theory, but a radically different construct:spacetime is not only a passive arena where dynamicalprocesses take place, but has itself a dynamical nature.At this time, there is no satisfactory quantum theory ofgravitation (after seventy years of efforts by leading theoreticalphysicists).In the present review on quantum information theory,we shall not attempt to use the full machinery of generalrelativity, with Einstein’s equations.18 We still considerspacetime as a passive arena, endowed with a Riemannianmetric, instead of the Minkowski metric of specialrelativity. The difference between them is essential: itis necessary to introduce notions of topology, because itmay be impossible to find a single coordinate system thatcovers all of spacetime. To achieve that result, it may benecessary to use several coordinate patches, sewed to eachother at their boundaries. Then in each patch, the metricis not geodesically complete: a geodesic line stops after afinite length, although there is no singularity there. Thepresence of singularities (points of infinite curvature) isanother consequence of Einstein’s equations. It is likelythat these equations, which were derived and tested forthe case of moderate curvature, are no longer valid undersuch extreme conditions. We shall not speculate on thisissue, and we shall restrict our attention to the behaviorof quantum systems in the presence of horizons, inparticular of black holes. Before we examine the latter,let us first return to entanglement, now in curved spacetime,and to the Unruh effect, still in flat spacetime, butdescribed now in an accelerated coordinate system.18 Concepts of quantum information were recently invoked in severalproblems of quantum gravity and quantum cosmology, butwe restrict ourselves to conventional black hole physics.A. Entanglement revisitedCalculations on EPRB correlations require a commonreference frame. Only then can statements such as“if m1z = 1/2 , then m2z = – 1/2”have an operational meaning. In a curved spacewe can choose an arbitrary frame at onespacetime point and then translate it parallel to itselfalong a geodesic path. For example, spin-1/2 particles maybe sent to Alice and Bob, far away. After a referenceframe is chosen at the emission point, local frames areestablished for them by parallel transport along the particles’trajectories. However, particles only approximatelyfollow classical geodesic trajectories, and this inevitablyintroduces uncertainties in the definition of directions.Using path integral methods, von Borzeszkowski andMensky (2000) have shown that if certain conditions aremet, approximate EPR correlations still exist, but “thelonger the propagation and the stronger the gravitationalfield, the poorer is the correlation”.One of the difficulties of quantum field theory in curvedspacetimes is the absence of a unique (or preferred)Hilbert space, the reason being that different representationsof canonical commutation or anticommutationrelations lead to unitarily inequivalent representations(Emch, 1972; Bogolubov et al., 1990; Haag 1996). Forthe Minkowski spacetime, the existence of a preferredvacuum state enables us to define a unique Hilbert spacerepresentation. A similar construction is also possible instationary curved spacetimes (Fulling, 1989;Wald, 1994).However, in a general globally hyperbolic spacetime thisis impossible, and one is faced with multiple inequivalentrepresentations.Genuinely different Hilbert spaces with different densityoperators and POVMs apparently lead to predictionsthat depend on the specific choice of the method of calculation.The algebraic approach to field theory can resolvethis difficulty for PVMs. The essential ingredientis the notion of physical equivalence (Emch, 1972; Araki,1999; Wald, 1994), which allows to extend the formalismof POVMs and CP maps to general globally hyperbolicspacetimes (Terno, 2002).As a consequence of the Reeh-Schlieder theorem, itfollows that a Minkowski vacuum corresponds to amixed state in the Rindler spacetime. To relate theMinkowski and Rindler Hilbert spaces, fields in*both**wedges are required*. The relation between the standardMinkowski Fock space and a tensor product of RindlerFock spaces is given by a formally unitary operator U,whose action on the Minkowski vacuum isthe previous discussion, since the energy of such a systemis also infinite.The relationship between Minkowski and Rindler wavepackets was analyzed by Audretsch and M¨uller (1994a).These authors also discussed local detection by Rindlerobservers and EPR-like correlations (Audretsch andM¨uller, 1994a, b).Alsing and Milburn (2002, 2003) examined the fidelityof teleportation from Alice in an inertial frame to Bobwho is uniformly accelerated.............................................Jack: in the special case that Alice and Bob are momentarily spatially coincident,that is a tetrad mapping. Not so when they are separated.................................................