The black hole information paradox and macroscopic superpositions
aa r X i v : . [ g r- q c ] M a r The black hole information paradox and macroscopicsuperpositions
Stephen D.H. Hsu
Institute for Theoretical Science, University of Oregon, Eugene OR 97403-5203, USAE-mail: [email protected]
Abstract.
We investigate the experimental capabilities required to test whether black holesdestroy information. We show that an experiment capable of illuminating the informationpuzzle must necessarily be able to detect or manipulate macroscopic superpositions (i.e.,Everett branches). Hence, it could also address the fundamental question of decoherence versuswavefunction collapse.
1. Introduction: black hole information
In 1976 Hawking proposed that black holes destroy information: pure states which collapse toform black holes evaporate into mixed states described by density matrices [1]. The argumentin favor of information destruction can be pared to a few essential components; for reviews,see [2, 3, 4, 5, 6]. Hawking radiation, into which the hole evaporates, originates from outsidethe horizon and is causally disconnected from the interior: a spacelike slice can be constructedwhich intersects both the infalling matter and the outgoing Hawking radiation. The no-cloningtheorem [7] in quantum mechanics prevents information from residing in two places on the sameslice, so the outgoing radiation must be independent of the initial state. (See Figure 1.)It is safe to say that, over 30 years after Hawking’s paper, theoreticians remain divided as towhether Hawking was originally correct, or whether some locality-violating mechanism somehowallows the information to escape.In this discussion [8], we investigate the following questions: Is the black hole informationpuzzle simply philosophy, or is it subject to experimental test? If the latter, then whatcapabilities are required? We find connections to a different question, from the foundationsof quantum mechanics: do wavefunctions collapse, or is quantum evolution strictly unitary,leading inevitably to macroscopic superposition states? An experiment which sheds light onthe black hole information puzzle would also be capable of addressing fundamental issues inquantum mechanics. See Zeh [9] and also [10] for related discussion.To highlight the importance of macroscopic superpositions, we emphasize that the veryformulation of the information puzzle relies on the use of the semiclassical black hole spacetime– e.g., in the construction of the spacelike slice used in the no-cloning argument, or in theoriginal Hawking calculation. But, in any fully quantum mechanical treatment of the black holeformation process there exist branches of the wavefunction, possibly of very small amplitude, inwhich no apparent horizon is formed and the initial state particles all escape to infinity. In otherwords, in which the spacetime is radically different. This is most easily seen if one considersblack holes formed in the collision of two particles [11] (see Figure 2): there is always a non-zero igure 1.
Formulation of the black hole information puzzle on a semiclassical spacetime. Theno-cloning theorem prevents the same quantum information from existing in two different placeson the nice slice.
Figure 2.
Two particle scattering with sufficiently small impact parameter leads to black holeformation.amplitude for no scattering – i.e., no black hole creation – in which the particles simply passeach other. This remains the case even as the number of particles in the initial state becomeslarge, although for certain semiclassical initial data one can make the no-formation amplitudearbitrarily small. Nevertheless, the information puzzle cannot avoid the issue of macroscopicsuperpositions: could such small amplitude branches restore purity or unitarity? [12]
2. Decoherence: pure to mixed evolution
Quantum mechanics as conventionally formulated (the Copenhagen interpretation) allows fortwo kinds of time evolution: the usual Schr¨odinger evolution, which is unitary, and measurementcollapse, which is non-unitary and leads to von Neumann projection onto a particular eigenstateof the operator associated with the measurement. It is appealing to think that wavefunctioncollapse might only be an apparent phenomenon, which results from unitary evolution. Thisidea dates to Everett [13], but has been developed substantially in recent decades as the theoryof decoherence [14]. igure 3.
Measurement of a single spin leads to entangled state in Eqn. (2).Consider a system prepared in a superposition state (see Figure 3) | ψ i = c | i + c | i . (1)Suppose that, due to interactions between the system and its environment (or, equivalently, ameasuring apparatus), the two evolve into an entangled state | ψ i ⊗ | E i → | Ψ i = c | i ⊗ | E i + c | i ⊗ | E i . (2)The states E , are referred to as pointer states of the environment or measuring device. Thesepointer states are determined by the dynamics – that is, the 1 , E , and can be read out by making simplemeasurements on subsets of the degrees of freedom – i.e., did the red light flash (look for redphotons), or did the green light flash?The environment is assumed to have a large number of degrees of freedom, so that aftera relatively short time (determined by the specific dynamics) the states E and E are nearlyorthogonal. The dimensionality of a Hilbert space describing N degrees of freedom is exponentialin N : for qubits, d = 2 N . Two randomly chosen vectors from this space will have overlap h E | E i ∼ /d , which is tremendously small for any macroscopic environment or measuringdevice (e.g., N ∼ Avogadro’s number). Consequently, interference phenomena between the two“branches” of system plus environment are highly suppressed (see below).Absent the capability to measure the environmental degrees of freedom, it is appropriate totrace over the degrees of freedom of E :ˆ ρ = Tr E | Ψ ih Ψ | = | c | | ih | + | c | | ih | ; (3)this density matrix, from which the outcome of all subsequent measurements on the systemalone can be predicted just as well as from the knowledge of the complete state | Ψ i , is diagonalif one neglects the exponentially small overlap h E | E i . This process then has the appearanceof measurement with fundamental wavefunction collapse and probabilistic outcomes, despitepurely unitary Schr¨odinger evolution. For All Practical Purposes – FAPP, as formulated by Bell[15] – decoherence leads to the usual Copenhagen interpretation.But a nagging issue remains – the presence of the other “branches” in the pure state | Ψ i . Arethey real? Can they ever be detected experimentally? The off-diagonal elements of the reduceddensity matrix are suppressed by the small overlap of typical environmental states E , E , butcan their effects be measured?mn`es [16] made detailed estimates of the capabilities required to distinguish betweendecoherence and fundamental collapse, which we will examine in the following section. Hefound that for macroscopic objects, e.g., containing Avogadro’s number of degrees of freedom,any beyond-FAPP device would have to be larger than the visible universe. Hence, Omn`esasserted, any distinction between the Copenhagen interpretation and unitary wavefunctionevolution (leading to Everett branches) for macroscopic objects is untestable, and beyond therealm of scientific inquiry.The foregoing discussion, in particular Omn`es’ estimate, assumes big environments (e.g., N ∼ Avogadro’s number). However, decoherence mechanisms are also at work if the numberof degrees of freedom N of the environment is smaller, or if the interaction between systemand environment is weaker; in these cases, however, decoherence effects are either not as strong(the off-diagonal elements in ˆ ρ cannot be completely neglected as in (3), i.e. FAPP does nothold in this case), or happen only over time scales longer than in the strongly-interacting case,respectively. In recent years, this gradual onset of decoherence in controlled environments (alsocalled the “Quantum-to-Classical transition”) has been the focus of quite a number of laboratoryexperiments. For example, in [17] the gradual loss of spatial coherence (interference pattern)of fullerene molecules in a slit experiment was observed with increasing pressure of the gas,i.e., with increasing interaction strength (cross section) between fullerenes and gas molecules(environment). Other experiments, e.g. [18], have verified the gradual loss of coherence in asystem (superposition of two states of a Rydberg atom) with increasing number of degrees offreedom of the interacting environment ( N ∼
10 photons in a cavity). On the other hand, oneof the big challenges in achieving useful quantum computing [19] is to build and control largeand scalable quantum systems ( N ∼
100 or more) in which coherence is maintained (possiblyvia quantum error correction) over the time of the computation.In the decoherence approach to measurement an initially pure state is later described bya reduced density matrix which, FAPP, represents a mixed state. The black hole informationpuzzle is often described in similar terms: infalling matter in a pure state is somehow transformedinto a mixed state of Hawking radiation. Or, equivalently, if the quantum information in theblack hole precursor is not to be found in the outgoing radiation, the radiation is surmised tobe in a mixed state. It has been claimed that pure to mixed evolution implies, necessarily,catastrophic consequences, such as violation of energy conservation [20] (see [21] for additionalarguments, for and against this point of view). Potential resolutions of the puzzle in which theinformation ends up somewhere outside our universe (e.g., involving baby universes or spacetimetopology change [22]) must still have an effective description in our universe in terms of pureto mixed state evolution. Decoherence provides an example of effective pure to mixed evolutionwithout catastrophic consequences. Tracing over the environmental states, one loses track ofthe total energy of the system, but without any resulting catastrophe.
3. Black hole information experiments
Below we describe two categories of experiments which test different aspects of theoretical ideasabout black hole information. Despite their differences, both require the ability to detect ormanipulate macroscopic superpositions.
A. Test purity vs decohered mixed state.
A basic goal would be to differentiate betweenpure and mixed states of the type produced by decoherence. Without this minimal capabilityone can hardly investigate whether pure states evolve to mixed states, as proposed originally byHawking. (Was the initial state pure? Is the final state pure or mixed?)In the black hole context, one could imagine forming the hole from an initial state with atleast some degrees of freedom in a superposition (e.g., two spin states). The remaining degreesf freedom can be considered the environment E from our earlier discussion, assuming thedynamics are such that the environment evolves into two different pointer states correspondingto the superposition. This would be the case if, for example, the two spin states had slightlydifferent energy due to a magnetic field provided by the other degrees of freedom.If, for each i = 1 ,
2, the initial state | i i of the system leads to the final state | Ψ i i = | i i ⊗ | E i i for system plus environment, then, starting from the initial state c | i + c | i , two candidatesfor the final state would be: on the one hand a pure state superposition | Ψ i = c | Ψ i + c | Ψ i (4)as predicted by unitary evolution, corresponding to the pure density matrixˆ ρ P = | Ψ i h Ψ | , (5)and, on the other hand, a mixed state density matrixˆ ρ M = | c | | Ψ i h Ψ | + | c | | Ψ i h Ψ | , (6)predicted by wavefunction collapse. What is required to differentiate between these alternatives?Consider a measurement operator M , which, without loss of generality, we might take tobe a projector onto some subspace of the Hilbert space. The pure and mixed states can bedistinguished if we can detect the differenceTr[ˆ ρ P M ] − Tr[ˆ ρ M M ] = c ∗ c h Ψ | M | Ψ i + c ∗ c h Ψ | M | Ψ i . (7)Let us consider two possibilities. If M is a generic operator – for example, only couples to alimited subset of the degrees of freedom – then the matrix elements in (7) will be exponentiallysmall. In particular, h Ψ | M | Ψ i ∼ H S ⊗ E ∼ exp( − N ) , (8)where N is the number of degrees of freedom of system plus environment. For a black hole, N is of order its entropy, or area in Planck units. Omn`es [16] has argued that when N is of orderAvogadro’s number (e.g., for one gram of ordinary matter), a measurement of this accuracyis impossible in principle . Based on this, Omn`es concludes that questions about decoherentbranches other than the one observed as an outcome are not scientific. In effect, his calculationspurport to extend Bell’s FAPP to a statement of principle. Roughly speaking, he argues thatthe sensitivity of a classical device with N ′ degrees of freedom only improves as a power of N ′ (not exponentially with N ′ ). Since the precision needed to detect a decohered macroscopicsuperposition involving N degrees is ∼ exp( − N ), as in (8), the required N ′ grows exponentiallywith N . To detect a macroscopic superposition with N ∼ , Omn`es concludes, would require N ′ larger than the number of particles in the visible universe.In AdS space a concrete proposal [23] has been made for how to decide between unitary (pureto pure) and non-unitary (pure to mixed) evaporation of a black hole, by measuring whethera certain correlation function drops, over time, either to zero or to the finite value exp( − N ).Interestingly, the required measurement accuracy is essentially that necessary to detect Everettbranches of a system with a similar number N of degrees of freedom, as described in the previousparagraph.On the other hand, for a carefully engineered operator M , the amplitude h Ψ | M | Ψ i in (7)can be of order 1/2, even if h Ψ | Ψ i = 0. Indeed, the maximum value is obtained when M is a projector onto a macroscopic superposition of the type ∼ | Ψ i + | Ψ i , like | Ψ i itself. Itis a formidable challenge to perform a measurement of such an operator M , presumably evenarder than preparing | Ψ i itself, and we give a concrete, albeit idealized, example below in thecontext of the Coleman-Hepp model of measurement [24] to illustrate the difficulties arising evenin simple cases.Coleman and Hepp proposed an explicit model of a quantum measurement which results fromthe interaction of spin states. In their model the interaction between the system (itself a spin)and measuring device causes evolution as in (2), with | Ψ i = | i ⊗ | E i = | + i ⊗ | + + + · · · + i (9) | Ψ i = | i ⊗ | E i = |−i ⊗ | − − − · · · −i , where ± are spin-up and spin-down states along the z axis, and the measuring device has N degrees of freedom (qubits). That is, the interaction between system and measuring device leadsto a correlation between the initial spin state and the (macroscopic, when N is large) state ofthe device. The state of the spin can be read out by measuring some subset of the N degrees offreedom in the device.In this context it is straightforward to design an operator M for which h Ψ | M | Ψ i is large:we simply take the tensor product operator M = O i σ ix , (10)where σ ix |±i = |∓i measures the spin of environmental qubit i in the x -direction. Anexperimental realization of M would be able to distinguish the two considered post-measurementstates of the system plus device: pure ˆ ρ P vs. mixed ˆ ρ M . Note, though, that the simple M considered in this toy example is not a projection (in particular, not the projector intothe one-dimensional subspace | Ψ i ) and so cannot, e.g., distinguish the considered macroscopicsuperposition pure state (4) from the (seemingly simpler) pure state | + x i ⊗ | + x + x · · · + x i inwhich all spins are aligned in the + x -direction.One might object that in this simple example the pointer states E , of the device in (9)are mutually orthogonal, but not thermalized. In this sense the model does not represent arealistic measurement (the “environment” is highly constrained). This could easily be remediedby allowing some interactions between the spins in the device, which leads to some (presumably)ergodic but unitary evolution. If one keeps the spins isolated from the rest of the universe, theseinteractions evolve Ψ , into something more random, at least in appearance: | Ψ i → | Ψ ′ i = U ( c | Ψ i + c | Ψ i ) (11)or, for the proposed mixed state (6) after the measurement,ˆ ρ M → ˆ ρ ′ M = U ˆ ρ M U † . (12)It would still be the case that the operator M ′ = U M U † can distinguish between pure andmixed states of the post-measurement device. However, if the N spins are separated in space(e.g., correspond to isolated qubits in a quantum device), then the operator M ′ would itselfhave to be realized out of macroscopic superpositions of spatially separated objects, unlike theoriginal M which acted independently on each of the spins.To summarize, a measurement which can differentiate between pure and mixed states eitherhas to rely on extreme precision to detect very small matrix elements as in (8), or on the abilityto prepare a very special, typically non-local, operator like M ′ . B. Test Hawking mixed state vs typical pure state.
In this scenario we compareHawking radiation in a mixed state ρ ∗ to radiation in a pure state ψ .t is widely believed that large scale violation of locality at the black hole horizon is necessaryfor unitary evolution. This might lead to significant deviations from Hawking’s results describingwhat is emitted from the hole. If such deviations were observed, they would undermine theusual semiclassical reasoning which leads to the information puzzle, although deviations of theradiation from Hawking’s mixed state description do not by themselves imply that evolution isunitary or purity-conserving.Perhaps a more plausible scenario, assumed in what follows, is that the radiation, althoughdescribed by a pure state ψ , only deviates in subtle ways from the Hawking mixed state. Thatis, information is encoded in correlations (phases or superpositions) between particle states inthe radiation, but the overall distribution appears to be thermal and the temperature evolutionis as predicted by Hawking.It is extremely difficult to differentiate between a pure state ψ of this type and the Hawkingmixed state ρ ∗ . Local measurements on the radiation (i.e., over length scales much smaller thanits full spatial extent) can only exclude tiny subsets of the ψ Hilbert space. Moreover, it can beshown that for almost all states ψ these local measurements are governed by the same (thermal)probability distribution as the one obtained from ρ ∗ .A simple way to understand this is to recall that maximizing the entropy of a system subjectto an energy constraint leads to a thermal distribution. Pure states which conform, at leastmacroscopically, to the Hawking predictions are constrained to describe the same total energyemission over any particular interval of time. To be specific, consider a time interval ∆ t i overwhich the Hawking temperature is close to constant, T = T i , but during which many particlesare emitted. Let the Hawking calculation predict that a total amount of energy E i be emittedduring the interval (see Figure 4). Ordinary statistical mechanics tells us that the overwhelmingmajority of states (of the system in the volume corresponding to ∆ t i ) with energies close to E i will be approximately thermal – i.e., maximizing the entropy leads to a Boltzmann distributionfor energy occupation numbers, with temperature equal to the Hawking temperature T i . Theprobability distribution governing measurements of the energy distribution of individual emittedquanta over the interval ∆ t i will then coincide with that given by ρ ∗ , except for an exponentiallyrare subset of states satisfying the constraint (i.e., configurations with much lower entropies thanthe Boltzmannian, or thermal, ones). Figure 4.
Hawking radiation emitted in a pure state ψ HR which conforms to the semiclassicalprediction for temperature and energy radiated in each time interval i .A more explicit computation follows. Consider the subset of pure states ψ which conformto the Hawking predictions governing the amount of energy radiated in each time interval ∆ t i .Specifically, require that, for every i , the reduction (by taking the partial trace) of ψ to theegrees of freedom emitted in the interval ∆ t i be a mixture of superpositions of the energyeigenstates (of the theory in that volume) with eigenvalues close to E i . That is, consider region i and only the degrees of freedom within it (neglecting boundary effects, which are negligiblefor large regions). The reduction of ψ to these degrees of freedom, when expanded in an energyeigenstate basis for the region ∆ t i , must have support only on states with energies close to E i . The superposition of two pure states ψ and ψ satisfying this condition will also satisfythe condition as superposition does not extend the region of support; therefore the conditiondefines a subspace of the larger Hilbert space. Denote by H R this restricted (“energeticallyconforming”) subspace of the overall radiation Hilbert space H .Further, divide the radiation into a subsystem S , to be measured, and the remaining degreesof freedom which constitute an environment E , so H = H S ⊗ H E and ρ S ≡ ρ S ( ψ ) = Tr E | ψ ih ψ | (13)is the density matrix which governs measurements on S for a given pure state ψ . Note theassumption that these measurements are local to S , hence the trace over E .Then, a recent theorem [25] on entangled states, which exploits properties of Hilbert spacesof very high dimension, shows that for almost all ψ ∈ H R , ρ S ( ψ ) ≈ Tr E ( ρ ∗ ) as long as d E ≫ d S , where d E,S are the dimensionalities of the H E and H S Hilbert spaces. In the theorem, ρ ∗ = 1 R /d R is the equiprobable mixed state on the restricted Hilbert space H R (1 R is the identityprojection on H R and d R the dimensionality of H R ), so Tr E ( ρ ∗ ) is the corresponding canonicalstate of the subsystem S . In other words, ρ ∗ describes a perfectly thermalized radiation systemwith temperature profile equal to that of Hawking radiation from an evaporating black hole.To state the theorem in [25] more precisely, the (measurement-theoretic) notion of the trace-norm is required, which can be used to characterize the distance between two mixed states ρ S and Ω S : k ρ S − Ω S k ≡ Tr q ( ρ S − Ω S ) . (14)This sensibly quantifies how easily the two states can be distinguished by measurements,according to the identity k ρ S − Ω S k = sup k O k≤ Tr ( ρ S O − Ω S O ) , (15)where the supremum runs over all observables O with operator norm k O k smaller than 1(projectors P = O are in some sense the best observables, all other observables can be composedout of them, and they have k P k = 1). Note that the trace on the right-hand side of (15) is thedifference of the observable averages h O i evaluated on the two states ρ S and Ω S , and thereforespecifies the experimental accuracy necessary to distinguish these states in measurements of O .A special form of the theorem then states that the probability that k ρ S ( ψ ) − Tr E ( ρ ∗ ) k ≥ d − / R + s d S d R (16)is less than 2 exp( − d / R / π ). In words: let ψ be chosen randomly (according to thenatural Hilbert space measure) out of the space of allowed states H R ; the probability thata measurement on the subsystem S only , with measurement accuracy of d − / R + q d S /d R ,will be able to tell the pure state ψ (of the entire system) apart from the mixed state ρ ∗ isexponentially small ( ∼ exp − d / R ) in the dimension of the space H R of allowed states. Thatis, the overwhelming majority of energetically conforming pure Hawking evaporation states ψ ∈ H R cannot be distinguished from Hawking’s predicted density matrix ρ ∗ by measurementsn a small subsystem S of the radiation, even if the experimental error in measurements ofprojectors is only d − / R + q d S /d R . This is an incredible precision, considering the estimates d R ∼ exp S BH ∼ exp M and d S ∼ exp S S , where the entropy S S ∼ V T ≪ S BH of the(energetically conforming) subsystem S can be computed from its volume V and the Hawkingtemperature T of the particle excitation distribution inside.Thus, as long as individual measurements are localized in spacetime, so that S is smallrelative to E , one cannot distinguish a typical state ψ ∈ H R from ρ ∗ without exponentialsensitivity in the measurement on S . This is true even if one performs many measurements ondistinct subsystems S i , in particular even if the union of S i covers all of the radiation. Thisis because, even if each subsystem were in a pure state that could in principle be measuredexactly (as opposed to merely measuring all the single-particle excitations in it separately), atthe very least the phase relations between the states of the different subsystems S i are lost.Only complete measurements (including phase relations) on very big subsystems S ( ∼ half ofthe degrees of freedom of E ) have a non-infinitesimal chance of distinguishing ψ from ρ ∗ .In analogy to what we learned in the previous cases, measurements that can distinguishgeneric ψ ∈ H R from ρ ∗ with reasonable probability, or without exquisite sensitivity, must behighly non-local, covering a spacetime region which includes most of the radiation emitted bythe black hole. They must, in a sense, measure it all “at once”. But measurements of thissophistication could, again, also differentiate between macroscopic superpositions and mixedstates.
4. Conclusions
Black hole information experiments are at least as hard as experiments which test decoherencevs fundamental collapse. One has to create a semiclassical black hole (in fact, many of themin identical states) and then make very challenging measurements on the relativistic decayproducts, which include gravitons. (Note, it appears difficult to determine the quantum state ofgravitons with physically realizable detectors that obey the positive energy condition [26].) Inparticular, the experiment must be sensitive to the phase relations in coherent superpositions ofFock states rather than simply counting occupation numbers as ordinary particle detectors do.By comparison, the most accessible tests of decoherence would be in the context of an artificialtoy system like the Coleman-Hepp model, or other controlled quantum computing environment.Our results can be summarized rather simply. Hawking suggested black holes cause purestates to evolve to mixed states. But, for all practical purposes (FAPP), decoherence does thesame thing – or at least appears to . In order to test Hawking’s proposal one therefore has to gobeyond FAPP and beyond decoherence. Such capability allows fundamental tests of quantummeasurement.If fundamental questions about measurement, decoherence and wavefunction collapse arephilosophy rather than science – i.e., cannot be tested by experiments – then so is the black holeinformation puzzle.
Acknowledgments —
The author thanks the organizers of the MCCQG for a stimulating andpleasant meeting. This work was supported by the Department of Energy under DE-FG02-96ER40969.
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