Non-monogamy of spatio-temporal correlations and the black hole information loss paradox
C. Marletto, V.Vedral, S.Virzì, E.Rebufello, A.Avella, F.Piacentini, M.Gramegna, I.Degiovanni, M.Genovese
NNon-monogamy of spatio-temporal correlations and the black hole information loss paradox
Chiara Marletto and Vlatko Vedral
Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom andCentre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 andDepartment of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
Salvatore Virz`ı
Universit`a di Torino, via P. Giuria 1, 10125 Torino, Italy andIstituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy
Enrico Rebufello
Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, andIstituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy
Alessio Avella, Fabrizio Piacentini, Marco Gramegna, and Ivo Pietro Degiovanni
Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy
Marco Genovese
Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy, andINFN, sezione di Torino, via P. Giuria 1, 10125 Torino, Italy
Pseudo-density matrices are a generalisation of quantum states and do not obey monogamy of quantum cor-relations. Could this be the solution to the paradox of information loss during the evaporation of a black hole?In this paper we discuss this possibility, providing a theoretical proposal to extend quantum theory with thesepseudo-states to describe the statistics arising in black-hole evaporation. We also provide an experimentaldemonstration of this theoretical proposal, using a simulation in optical regime, that tomographically repro-duces the correlations of the pseudo-density matrix describing this physical phenomenon.
PACS numbers: 03.67.Mn, 03.65.Ud
INTRODUCTION
The possibility of black hole evaporation represents a prob-lem from the quantum mechanical perspective [1–4], as wellas other cosmological aspects [5–8]. In short, if the process isunitary as prescribed by quantum theory, then entanglementmust be created between the exterior and the interior of theblack hole as particle pairs are generated through the processof Hawking radiation [9–12]. If we provide an elementarymodel of evaporation based on a finite number of qubits, afterhalf of the qubits in black hole has evaporated, we should pre-sumably have a maximally entangled state between the qubitsin the interior and the qubits in the exterior of the black hole,assuming thermal radiation being emitted. As the black holecontinues to evaporate, Hawking radiation would imply thateven more entanglement is generated between the interior andthe exterior of the black hole. But this cannot be, since qubitsalready maximally entangled cannot be entangled to anythingelse. This fact, that a system cannot be maximally entangledto more than one other system, is known as the monogamy ofentanglement principle [16]. The claim therefore is that if weare trying to preserve unitarity of black hole evaporation, thenthe black hole evaporation itself ought to violate monogamyof entanglement [13].Here we will not discuss further on the issue of whetherthere is or is not such a paradox (which has been hotly de- bated, see e.g. [14] and references herein). We would likeinstead to suggest that, assuming that the paradox exists, asimple re-interpretation of the evaporation process could pro-vide a novel resolution. Informally, this is the rationale forour proposition. Following the Schwarzschild metric, that de-scribes space-time in the presence of a black hole, crossingthe horizon for a particle is tantamount to swapping the sig-natures of the spatial and temporal components of the met-ric [15]. Now, if we think of a typical quantum phase factor e i ( kx − ωt ) , the change of the sign of space and time simply cor-responds to complex conjugation of the phase factor. In thissense, the effect on a density matrix of an in-falling quantumsystem should be described by the operation of transposition(which swaps the off-diagonal elements and therefore imple-ments the complex conjugation).It is well known that transposition is a positive, but not com-pletely positive, operation. This means that, if we performtranspose on just one of two entangled systems, the overallstate may not end up being a valid density matrix. Here wewould like to use this fact to resolve the apparent violation ofmonogamy of entanglement during the evaporation of a blackhole, by suggesting to utilise an extended notion of quantumstate to describe it, which includes Hermitean operators thatare not positive. These generalised quantum states are calledpseudo-density operators (PDOs), [17].A density operator can be viewed as a collection of all pos- a r X i v : . [ g r- q c ] F e b sible statistics ensuing from measurements of observables ofa system of interest. For a d -qubit system, for instance, wecan write a general density operator as ρ d . = 12 d (cid:88) i =0 · · · (cid:88) i d =0 (cid:104) n (cid:79) j =1 σ i j (cid:105) n (cid:79) j =1 σ i j . PDOs generalise these operators into covering statistics thatpertain to the time domain. A general PDO for d qubits isdefined as: R d . = 12 d (cid:88) i =0 · · · (cid:88) i d =0 (cid:104){ σ i j } nj =1 (cid:105) n (cid:79) j =1 σ i j , where (cid:104){ σ i j } nj =1 (cid:105) denotes the expected value of a possibleset of Pauli measurements which could be either in space orin time, thus generalising standard density operators to coverboth space and time correlations. This is a Hermitian, trace-one but not necessarily positive operator.Let us understand how the PDO works with an exampledirectly relevant to our problem. Suppose that we want todescribe a physical process where a single qubit, initially ina maximally mixed state, is then measured at two differenttimes. Each measurement is performed in all three comple-mentary bases X, Y, Z (represented by the usual Pauli opera-tors). The evolution is trivial between the two measurements,i.e. the identity operator. Suppose now that we would like towrite the statistics of the measurement outcomes in the formof an operator, generalising the quantum density operator. Be-cause the whole state, as we said, is Hermitian and unit trace,but not positive, we refer to it as a pseudo-density operator[17].It would be represented as: R = 14 { I + X X + Y Y + Z Z } , (1)where subscripts 1 and 2 indicate, respectively, the two qubitsof a general bipartite state. This operator looks very much likethe density operator describing a singlet state of two qubits,however, the correlations all have a positive sign (whereas forthe singlet they are all negative, (cid:104) XX (cid:105) = (cid:104) Y Y (cid:105) = (cid:104) ZZ (cid:105) = − ). In fact, it is simple to show that R is not a density ma-trix, because it is not positive (i.e. it has at least one negativeeigenvalue). We can however, trace the label out and obtainone marginal, i.e. the “reduced” state of subsystem . In-terestingly, this itself is a valid density matrix (correspondingto the maximally mixed state I/ ). Likewise for subsystem . So, the marginals of this generalised operator are actuallyboth perfectly allowed physical states, but the overall state isnot.Interestingly, R = ( I ⊗ T ) | Σ (cid:105) (cid:104) Σ | , where | Σ (cid:105) = √ ( | (cid:105) + | (cid:105) ) and I ⊗ T denotes the partialtranspose operation. (This relation holds true up to a localbit flip and phase flip for any of the Bell states). Thus, giventhat the partial transposition can model what happens to a pairentangled qubits due to one of them falling into a black hole,we can use R as a candidate to describe the state of the pairof qubits, with one of them falling into the black hole.Based on this heuristic reasoning, we now proceed byproposing a PDO to model the situation where one of thequbits in the pair gets further entangled with a third parti-cle. Specifically, we show that a viable solution to the blackhole information problem can be achieved by postulating thata PDO (see eq. (2)), generalising the above pseudo-state R ,represents the state of two initially entangled qubits after oneof them has crossed the event horizon and fallen into the blackhole, getting entangled with a third qubit. As we shall explain,our proposal consists of introducing an extension of the den-sity matrix formalism, via pseudo-density operators, treatingequally temporal and spatial correlations. This proposed gen-eralised quantum state can describe perfectly the correlationsassociated with the black-hole evaporation scenario. RESULTS
Suppose that a maximally entangled state is created justabove the event horizon of a black hole as in the process ofHawking radiation. One of the particles, e.g. particle 1, nowfalls into the black hole. According to our proposal, we con-jecture that time-like correlations are created between the twoparticles (out of what used to be spatial correlations). Sothe pair is now described by a pseudo-density operator like R , defined above. Now, when another particle, i.e. particle3, becomes entangled with particle 1, this leads to a three-qubit entangled pseudo-state. In this state, qubits 1 and 2are maximally temporally correlated, while qubits 1 and 3 aremaximally spatially correlated. The total three qubit pseudo-density operator can be written as: R = 18 { I + Σ − Σ − Σ } , (2)where Σ ij = X i X j I k + Y i Y j I k + Z i Z j I k . The reducedstates are R = { I + Σ } , R = { I − Σ } and R = I { I − Σ } . Now we can see that qubits 1 and 2 canbe maximally entangled (in time), while qubits 1 and 3 canalso be maximally entangled (in space), as well as qubits 2and 3. Therefore, correlations described by pseudo-densitiesneed not obey the principle of monogamy of entanglement.We conjecture this pseudo-density operator could be used todescribe the elementary step involved in the Black-Hole evap-oration.The usual entanglement monogamy of three qubits can beencapsulated in the following inequality: E + E ≤ . (3)This is violated by the state described by R .Therefore, the proposed PDO description for the BlackHole evaporation incorporates the monogamy violation. Thisis because PDOs, unlike density operators, can be used to de-scribe a situation with two qubits (1 and 2) maximally tem-porally correlated and one of them forming an entangled state(maximally spatially correlated) with qubit 3. In this frame-work, the violation of monogamy in Eq. (4) is allowed andpredicted. Or course, this requires to modify quantum the-ory by generalising quantum states from density operators toPDOs. This paper only offers a first exploration of applyingthis idea to the specific scenario of black-hole evaporation,leaving a more general theory to be developed in the future,should this approach prove useful.To provide an experimental demonstration of this situa-tion, we performed a quantum optical simulation of suchframework. Note that this is not an experimental test, but anillustration of our theoretical proposal within a qubit simula-tion. In this experiment, initially, we generate a maximallyentangled pair of photons (A and B) in a singlet state. Thecorrelations between the particle fallen inside the black holeand the one that remained outside, originally belonging to thesame maximally entangled state, are observed by measuringphoton A at two different times ( t and t ). Correlationsbetween the two (spatially) entangled particles inside theblack hole, instead, are sensed by measuring photons A and Bat the same time t . The simulation consists in reconstructingall the relevant statistics contained in the PDO R , byconstructing different ensembles of particles. In our setup
404 nm Type-IIBBO 808 nm 808 nmSecond HarmonicGenerator Ti:Sapphire 76 MHz mode-locked laserLens Si-SPADSi-SPADCoincidenceelectronicsQuarter waveplate Half waveplate Polarizingbeam splitter Interferencefilter
FIG. 1: Experimental setup. A maximally entangled singlet stateis generated by pumping a type-II BBO crystal. Two polarisationmeasurements, M1 and M2 (at times t and t , respectively) are per-formed in sequence on photon A, while a single measurement (M3) iscarried on photon B. Correlations among them certify entanglementmonogamy violation for the whole PDO R in Eq. (2), describingthe scenario of the spatio-temporal multi-partite entanglement (out-side and inside the black hole) considered. (see Fig. 1), a CW laser at nm pumps a Ti:Sapphire crystal in an optical cavity, generating a mode-locked pumpat 808 nm (repetition rate: 76 MHz) whose second harmonicgeneration (SHG) is injected into a . mm thick β -bariumborate (BBO) crystal to generate type-II parametric down-conversion (PDC) [18]. The maximally entangled singletstate | ψ − (cid:105) = √ ( | HV (cid:105) − | V H (cid:105) ) (being H and V thehorizontal and vertical polarization components, respectively)is obtained by spatially selecting the photons belonging to theintersections of the two degenerate PDC cones and properlycompensating the temporal and phase walk-off [19].In photon A path, two polarization measurements occur incascade (M1 and M2), each carried by a quarter-wave plate(QWP) followed by a half-wave plate (HWP) and a polarizingbeam splitter (PBS). Between the two measurements, a HWPand a QWP are put in order to compensate the polarizationprojection occurred in M1. Photon B, instead, undergoes asingle polarization measurement (M3) performed by the sameQWP+HWP+PBS unit used for M1 and M2. After thesemeasurements, photons A and B are filtered by bandpassinterference filters (centered at λ = 808 nm and with a nm full width at half-maximum) and coupled to multi-modeoptical fibers connected to silicon single-photon avalanchediodes (Si-SPADs), whose outputs are sent to coincidenceelectronics.Initially, we perform an optimized quantum tomographicstate reconstruction [20] on branch A, extracting the temporalcorrelations allowing to estimate the reduced pseudo-density R = ( I + Σ ) , describing the correlations betweenparticle 1, fallen into the black hole, and particle 2, formingthe initial maximally entangled state. To do this, we sum theresults obtained in two different acquisitions obtained choos-ing for M3 orthogonal projectors, e.g. | H (cid:105) (cid:104) H | and | V (cid:105) (cid:104) V | ,erasing this way the information on such measurement. Then,on the spatial side, we measure correlations between M1 andM3 (by having M2 performing the same polarization pro-jection as M1), tomographically reconstructing the reducedpseudo-density R = ( I − Σ ) , corresponding to thegenerated singlet state | ψ − (cid:105) , i.e. the one formed by particle 1with particle 3 within the black hole. DISCUSSION
The results of these two reconstructions are reported inFig.s 2 and 3, respectively. In both cases, the experimen-tal results are in excellent agreement with the theoretical ex-pectations, as stated by the Uhlmann’s fidelity computed forpseudo-density marginal R (the only reconstruction corre-sponding to a physical density matrix), i.e. F = 96 . .These two reconstructed PDOs would be enough to state theviolation of the entanglement monogamy relation reported inEq. (3), but, as a further test, we experimentally demonstratethe violation of such relation by evaluating the Clauser-Horne-Shimony-Holt (CHSH) inequality [19] between qubits 1 and2 ( CHSH ) and qubits 1 and 3 ( CHSH ). With the same (d)(b)(c)(a) FIG. 2: Tomographic reconstruction of the real (panel a) and imagi-nary (panel b) part of the reduced pseudo-density operator R = ( I + Σ ) , describing the temporal correlations between qubits1 and 2, compared with the corresponding theoretical expectations(panels c and d, respectively). (d)(b)(c)(a) FIG. 3: Tomographic reconstruction of the real (panel a) and imag-inary (panel b) part of the reduced pseudo-density operator R = ( I − Σ ) , related to the spatially maximally entangled state withinthe black hole, compared with the corresponding theoretically-expected counterparts (panels c and d, respectively). methodology followed for the quantum tomographic recon-structions of R and R , we select the proper polarizationprojections allowing to reach the maximal violation of theCHSH inequality in the temporal domain (M1 and M2) as wellas in the spatial one (M1 and M3), obtaining the experimentalvalues CHSH = 2 . ± . and CHSH = 2 . ± . ,respectively. These values grant for the left side of inequality(3) the value: E (CHSH)12 + E (CHSH)13 = 1 . ± . , (4) being E (CHSH) ij = CHSH ij / the amount of entanglementshared between qubits i and j , demonstrating a standarddeviations violation of the entanglement monogamy bound. CONCLUSIONS
In summary, in this paper we propose an alternative reso-lution of the entanglement paradox in black hole evaporationbased on the pseudo-density matrix formalism. We conjec-tured that the phenomenology of black hole evaporation,as described by Hawking’s radiation, could be describedby a pseudo-density operator instead of a standard densityoperator. We propose a specific form for the PDO that coulddescribe a pair of qubits, one falling into a black and the othermaximally entangled with a third qubit, after evaporating. Theusual paradoxes due to violations of entanglement monogamydo not arise in this formalism, as the PDO can accommodateand describe correlations that violate monogamy. In order toillustrate the temporal and spatial correlations in the proposedPDO, we use a quantum optical demonstration, buildingan experiment that simulates this physical phenomenon asdescribed by the same pseudo-density matrix. We reconstructexperimentally the correlations in the pseudo-density matrixproposed , demonstrating how violation of entanglementmonogamy can emerge in the proposed framework.
ACKNOWLEDGEMENTS
CM’s research was supported by the Templeton WorldCharity Foundation and by the Eutopia Foundation. VVthanks the Oxford Martin School, the John Templeton Foun-dation, the EPSRC (UK). This research is also supported bythe National Research Foundation, Prime Ministers Office,Singapore, under its Competitive Research Programme (CRPAward No. NRF- CRP14-2014-02) and administered by Cen-tre for Quantum Technologies, National University of Singa-pore. This research has also been developed in the context ofthe European Union’s Horizon 2020 project “Pathos”.
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