Parameter estimation of wormholes beyond the Heisenberg limit
AArticle
Parameter estimation of wormholes beyond theHeisenberg limit
Carlos Sanchidrián-Vaca and Carlos Sabín * Departamento de Física Teórica, Universidad Complutense de Madrid, Plaza de Ciencias, 1 28040Madrid,Spain ; [email protected] Instituto de Física Fundamental, CSIC, Serrano, 113-bis, 28006 Madrid, Spain; [email protected] * Correspondence: [email protected] Editor: nameReceived: date; Accepted: date; Published: date
Abstract:
We propose to exploit the quantum properties of nonlinear media to estimate the parametersof massless wormholes. The spacetime curvature produces a change in length with respect toMinkowski spacetime that can be estimated in principle with an interferometer. We use quantummetrology techniques to show that the sensitivity is improved with nonlinear media and proposea nonlinear Mach-Zehnder interferometer to estimate the parameters of massless wormholes thatscales beyond the Heisenberg limit.
Keywords: quantum metrology, nonlinear media, quantum field theory in curved spacetime
0. Introduction
In last years, there have been many theoretical proposals to improve measurements throughquantum technologies in the context of quantum metrology [1], whose goal is to estimate unknownparameters of interest. Among many other developments, such as gravitational wave detection [2] ortimekeeping [3], there have also been more challenging ideas, such as the measurement of accelerationwith a Bose Einstein Condensate setup [4] or temperature through Berry’s phase in [5].Along these lines, it is natural to ask whether quantum metrology can be useful to test gravitationaleffects such as gravitational time delay (see, for instance, [6]). In particular, nonlinear media wereproposed in [7] to estimate the Schwarzschild radius of the Earth beyond the Heisenberg limit.Moreover, the detection of distant traversable wormhole spacetimes by means of quantum metrologywas considered in [8]. Following this path, we propose to use the nonlinear Kerr effect to estimate thesize of the throat of distant massless wormholes with super-Heisenberg scaling. Given the renewedinterest in the physics of wormholes coming from different fields [9–11] and the proposal of newmethods for their detection by classical means [12], our main goal is to explore the fundamentalsensitivity limits provided by quantum mechanics, which in principle overcome the classical ones.We consider the evolution of a coherent state under a nonlinear hamiltonian. If the propagationtakes place in the spacetime of a distant traversable wormhole, the phase of the coherent state acquiresa slight dependence on the relevant parameter of the metric, which in this case is the radius of thewormhole throat. The fundamental bound on the precision with respect to this parameter is givenby the Quantum Fisher Information (QFI). Using the QFI, we find that the scaling with respect to thenumber of photons surpasses the Heisenberg limit, as expected, due to the nonlinearity. Moreover,we show that a realistic measurement protocol can get close to this fundamental limit by proposing aparticular scheme of detection, consisting on an interferometer whose arms are stretched in a curvedspacetime. Since we consider an almost-flat spacetime, we can express the metric as Minkowski plus alittle perturbation, yielding a small correction to the length of the arm. This generates a phase shift inthe interferometer that in principle can be measured. We show that a standard homodyne detection
Universe , xx a r X i v : . [ g r- q c ] N ov niverse , xx , x 2 of 13 scheme with a nonlinear interferometer would also exhibit super-Heisenberg scaling with respect tothe number of photons.In the linear regime, gravitational bodies produce a stationary phase shift which is locallyindistinguishable from other sources. We find compulsory to rotate the interferometer in orderto get a controlled oscillating phase shift. The whole scheme resembles a laser interferometer setup forgravitational wave detection, such as LIGO. Indeed, we use experimental parameters from LIGO toshow that parameter estimation might be, in principle, possible. Thus we also show that LIGO couldin principle benefit from the use of nonlinear media.The structure of the paper is the following. In the first two sections we briefly recall some basicnotions of quantum metrology and traversable wormholes, respectively. Then in the next sectionwe present our results both for the QFI and the interferometric setup. Finally, we summarize ourconclusions in the last section.
1. Quantum Metrology
Quantum metrology is the branch of physics which attempts to improve the estimation ofparameters of interest through quantum resources. To infer the value of a parameter, θ from the datacollected by n measurements ( x , x , ..., x n ) , we must build an estimator ˆ θ that is a function of thepossible outcomes. The statistical error is bounded by the classical Fisher information (FI), whichrepresents the amount of information that a random variable x carries about an unknown parameter θ of a distribution that models x. It can be written as F ( θ ) = − (cid:28) ∂ log p ( x ; θ ) ∂θ (cid:29) (1)where p ( x ; θ ) represents the probability that the outcome of a measurement is x when the parameter is θ . Fisher information can be extended to a metric on a statistical manifold and it obeys the Cramer-Raoinequality for estimating the variance of θ . ∆ θ ≥ (cid:112) F ( θ ) (2)In the quantum realm, p ( x ; θ ) = Tr ( ρ θ Π x ) . The ultimate bound on the sensitivity ∆ θ ρ of a state ρ withrespect to the parameter θ is obtained maximizing the FI over all possible measurements: ∆ θ ρ ≥ (cid:112) H ( θ ) (3)where H ( θ ) is the Quantum Fisher Information (QFI), which represents the maximum informationthat can be obtained. For instance, in the case of pure states, we can estimate the QFI as H = ∆ H ,where H is the Hamiltonian. A possible loophole of this analysis is the assumption that the observedmodel is the predicted, so in case there is a different model underneath, we are estimating a falseparameter. However, Bayesian theory is probably the best way to deduce the model that fits the data.In quantum metrology, a typical parameter is the phase θ and a typical experimental set-up isan interferometer [1]. It is a well-known result that the phase estimation for a coherent state with N photons per pulse goes as ∆ θ ∝ √ N , which is the Standard Quantum Limit (SQL) and is a consequenceof the central limit theorem. This resolution can be improved using quantum states such as squeezedlight, achieving the so-called Heisenberg limit which set bounds to the sensitivity in the linear caseas ∆ θ ∝ N . As it has been shown by different authors [13–16], the precision can be improved withnonlinear media as ∆ θ ∝ √ N . This limit has been confirmed experimentally [17]. niverse , xx , x 3 of 13
2. Traversable wormholes
Wormholes are a class of spacetime predicted by Einstein’s equations with interesting properties.They allow for spacetime travel and closed timelike curves, rising paradoxes about causality, whichare not so when quantum mechanics is considered [18]. The creation of a traversable wormholewould require some exotic source of negative energy, whose existence is bounded by quantum energyinequalities (QIs) and is dubious in classical physics. The QIs entail that the more negative the energydensity is in some time interval, the shorter its duration. These constraints apply only to free quantumfields [19] and they extend to curved spacetime. Intuitively, light rays converge while coming into thewormhole throat but they will diverge once they leave so negative energy is compulsory for a timelikeobserver. Moreover, it has been argued that a traversable wormhole must be only a little larger thanPlanck size, otherwise, its negative energy must be confined in a shell many orders of magnitudesmaller than the throat size [20]. However, there might be unexplored possibilities to avoid this bound,such as consider interacting quantum fields [21]. It remains an open question whether wormholes areforbidden by the laws of physics.Interestingly, wormholes can mimic black-hole metrics at first approximation, which in principlecould question the identity of objects at the center of the galaxies as well as the observed gravitationalwaves [9],[10]. Besides, a recent conjecture ER = EPR , involves wormholes and entanglement. Itstates that there might be difference kind of bridges for each kind of entanglement [11].The estimation of parameters of these objects, as well as a general curved spacetime can beimproved, in principle, by quantum metrology [8], which surpasses classical techniques such asgravitational lensing [12].Let us consider a field propagating along the radial direction in the presence of a traversablewormhole whose general metric is given by [22] ds = − c e Φ dt + − b ( r ) r dr + r ( d θ + sin θ d φ ) , (4)where Φ , the redshift function and b ( r ) , the shape function are arbitrary functions of the radius r. Inthe case of an Ellis massless wormhole, Φ = b ( r ) = b r . Ignoring the angular part results in: ds = − c dt + − b r dr . (5)As we will see below, for a far-away observer, the metric can be expressed as flat with a perturbation onthe spatial part. Then, there will be a phase shift in the presence of a wormhole in the radial direction.
3. Parameter estimation of wormholes through nonlinear effects
The Kerr effect is produced when the polarization of the medium is proportional to | E | and iswell-known in nonlinear optics. In quantum optics, it is usually used to generate non-classical states oflight such as Schrodinger’s cats.Basically, there is a coupling between the intensity I and the proper time τ in the phase [23]: φ = χτ I , (6)where χ is the nonlinearity. It has been shown that in the presence of a nonlinear medium thephase can be detected with higher precision than the Heisenberg limit, replacing the vacuum by anonlinear medium [13–17]. Classically, nonlinear effects can be interpreted as anharmonic terms in theHamiltonian. Then, we can consider the following quantum Hamiltonian:ˆ H = χ ( ˆ a † ˆ a ) + ω ˆ a † ˆ a = χ ˆ n ( ˆ n + ) + ˆ nkc . (7) niverse , xx , x 4 of 13 where a and a † are the standard creation and annihilation operators of a single electromagnetic fieldmode. As we will see, the effective nonlinearity becomes χτ , which essentially implies a coupling withthe curvature of spacetime.The more general form of a Gaussian state is a squeezed coherent state for a quantum harmonicoscillator, which is given by: | α , ζ (cid:105) = D ( α ) S ( ζ ) | (cid:105) . (8)A laser is the typical coherent source of the electromagnetic field, which is represented by a coherentstate and is obtained by letting the unitary displacement operator D( α ) act on the vacuum, | α (cid:105) = e α ˆ a † − α ∗ ˆ a | (cid:105) = D ( α ) | (cid:105) = e − | α | ∞ ∑ n = α n √ n ! | n (cid:105) . (9)The evolution of the coherent state with frequency ω propagating through a nonlinear medium, isgiven by: | α NL ( τ ) (cid:105) = ˆ U | α (cid:105) = e − i τ ˆ n ( χ ( ˆ n + )+ kc ) | α (cid:105) . (10)It is impossible to give an analytic expression of the phase due to the nonlinear terms whichproduce a superposition of coherent states. However, we can give a bound for the variance of thephase. As explained before, the bound for the variance of an estimator is given by the Cramer-Raoinequality. The Quantum Fisher Information for estimating the parameter τ , H ( τ ) , is given by [7], H ( τ ) = lim d τ → [ − (cid:112) F ( ρ ( τ ) , ρ ( τ + d τ )))] d τ = N { [ ω + ( N + ) χ ] + N χ } , (11)where F is the fidelity, F ( ρ , σ ) = [ Tr (cid:112) √ ρσ √ ρ ] which is a distance between two states ρ , σ in theHilbert space. When N, the number of photons per pulse, is large N → ∞ , the Heisenberg limit issurpassed ∆ τ ∝ N . (12)The proper time for a free-falling observer has a dependence on b , so the quantum Fisher informationmay be rewritten as H ( b ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂τ∂ b (cid:12)(cid:12)(cid:12)(cid:12) H ( τ ) (13)In 2D , the metric (5) becomes flat by means of a change of coordinates: ds = − c dt + dl , (14)where l = (cid:113) r − b . (15)Nevertheless, there is a phase shift for propagation in the radial direction. To see this, notice thatfor a free-falling observer, the proper length equals the proper time times c , so it can be expressed inlaboratory coordinates as: c τ = L (cid:48) = | l − l | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:113) r − b − (cid:113) r − b (cid:12)(cid:12)(cid:12)(cid:12) . (16)We assume that the separation L (cid:48) = c τ is small as compared to the distance to the wormhole; L << r , r , and for simplicity we consider that r > r , thus L = r − r . If the propagation takes place far niverse , xx , x 5 of 13 away from the wormhole throat, r , r >> b , the proper length can be approximated using a Taylorexpansion up to second order in b r , giving τ ≈ L / c (cid:32) + b r − b r Lr (cid:33) . (17)So the proper length as measured in the radial direction of the Ellis metric is larger than the Minkowskione. Now, we calculate the Jacobian for estimating the QFI as a function of b in (13): ∂τ∂ b = L (cid:32) b cr − b Lcr (cid:33) . (18)Finally, putting everything together the sensitivity is given by: ∆ b b = b (cid:112) H ( b ) = b (cid:12)(cid:12)(cid:12)(cid:12) ∂ b ∂τ (cid:12)(cid:12)(cid:12)(cid:12) (cid:112) H ( τ ) = (cid:39) L ( b cr − Lb cr ) (cid:112) N { [ ω + ( N + ) χ ] + N χ }(cid:39) L ( b r − Lb r ) (cid:112) N { [ ω + ( N + ) χ ] + N χ }(cid:39) cr Lb (cid:112) N { [ ω + ( N + ) χ ] + N χ } . (19)In the last step we have neglected the term b r Lr and expanded in Taylor series around 0 in x andy the ratio 1/ ( x + xy ) ≈ x , where x = b r and y = − Lr . Therefore, when N goes to infinity we find apolynomial enhancement in the sensitivity which goes beyond the Heisenberg limit; ∆ b b ∝ N . (20)Moreover, repeating the experiment M times, we would reduce the noise as a Gaussian variance: ∆ b b = cr Lb (cid:112) N M { [ ω + ( N + ) χ ] + N χ } . (21)Note that in the limit χ = Let us consider a more realistic scenario in which the measurement is carried out by homodynedetection. This consists in estimating an unknown signal comparing it with a known local oscillator.The unknown signal is one of the output modes b k , where k makes reference to the wave number incase there are several frequencies. As we assume a monochromatic source this will be suppressedlater. We follow the nonlinear Mach-Zehnder interferometer proposed in [7] (see Figure 1). Metrics arequalitatively different so the set up is not obtained just by replacing the black hole with a wormhole.We have: b k = [ a k ( e − ik ( φ + φ ) − e − ik ( φ + φ ) )] + v k ( e − ik ( φ + φ ) + e − ik ( φ + φ ) ) , (22) niverse , xx , x 6 of 13 where a k is an input coherent state and v k an input vacuum state. We must introduce only a non linearmaterial in one intereferometer arm and an adjustable linear phase β in the other arm. Figure 1.
A Mach-Zehnder interferometer with a non linear medium placed along the radial direction.
As we will explain in the next section, we should rotate the interferometer in order to determinethe position of the wormhole. The proper distance is only modified basically in the radial axis, thiswill produce an oscillating signal that allows to identify the radial axis of the wormhole where thephase shift is maximum. In fact, the proper distance on the top and bottom arms will differ but wecan neglect this term with respect to the main correction in the radial direction as will be explained indetail below.We assume that the length of the top and bottom arms, h , is small with respect to L ( h << L ) , sowe can neglect curvature effects and take the value φ = φ =
0. Using that c τ = L (cid:48) , it follows that φ = b k = ( e − ik φ + i χτ a † k a k − e i β ) a k + ( e − ik φ + i χτ a † k a k + e i β ) v k . (23)The phase shift is induced in one arm of the interferometer through a nonlinear medium such that φ = φ = L (cid:48) − cn (cid:48) τ (cid:39) L (cid:32) + b r (cid:33) (cid:18) − n (cid:48) (cid:19) , (24) niverse , xx , x 7 of 13 where τ ≈ Lc (cid:32) + b r (cid:33) (25)is the proper time, L (cid:48) the proper distance, n (cid:48) is the first-order refractive index of the material, (weconsider the generic value n (cid:48) = r is the distance to the wormhole previously called r .We use the approximation of linearized Gaussian regime given in [7] valid when τχ √ N << e − i χτ a † k a k ≈ e − i χ | α | τ [ − i χτ ( α ∗ δ a + αδ a † )] α + e − i χ | α | τ δα (26)The quadrature X b = b ( τ ) e i θ + b † ( τ ) e − i θ can be measured by homodyne detection as can be seen in Figure 2.
Homodyne detection: The unknown signal, b , is introduced into a beam splitter with a localoscillator, a , whose phase θ is controlled. The difference of photons is proportional to the quadrature X b . niverse , xx , x 8 of 13 Figure 2. This can be written as X b = | α | [ cos ( θ + ξ ) − cos ( θ + β )] − χ | α | τ [ sin ( θ + ξ ) − sin ( θ + β )] X + ( X θ + ξ − X θ + β ) + ( X v θ + ξ − X v θ + β ) (27)where we have defined ξ = k φ − τχ | α | (cid:39) L (cid:32) + b r (cid:33) (cid:18) k (cid:18) − n (cid:48) (cid:19) − χ | α | c (cid:19) (28)Then, the expectation value turns out to be (cid:104) X b (cid:105) = | α | ( cos ( θ + ξ ) − cos ( θ + β )) (29)and (cid:68) ∆ X b (cid:69) = χ | α | τ ( sin ( θ + ξ ) − sin ( θ + β )) − χ | α | τ ( sin ( θ + ξ ) − sin ( θ + β )) · [ cos ( θ + ξ ) − cos ( θ + β )] + (cid:104) X (cid:105) = (cid:10) X (cid:11) = and consequently (cid:10) ∆ X (cid:11) = . Now,defining γ = b r we find that the derivative of X with respect the parameter is d (cid:104) X b (cid:105) db = | α | (cid:18) kc (cid:18) − n (cid:48) (cid:19) + | α | χ (cid:19) Lcb γ sin ( θ + ξ ) . (31)The nonlinearity creates noise from antisqueezing in the axis of rotation, which can be removed making β = ξ , making the variance shot noise ( (cid:10) ∆ X b (cid:11) = b we use the sensitivity of the quadrature through thefollowing equation ∆ b b = ∆ X b b (cid:12)(cid:12)(cid:12) d (cid:104) X b (cid:105) db (cid:12)(cid:12)(cid:12) . (32)We can optimize our choice of θ in (31) making θ = π /2 − ξ which gives d (cid:104) X b (cid:105) db = γ Lcb | α | (cid:18) kc (cid:18) − n (cid:48) (cid:19) + | α | χ (cid:19) . (33)Finally, the sensitivity is: ∆ b b = c γ L (cid:114) N [ ω (cid:16) − n (cid:48) (cid:17) + N χ ] = (cid:32) cr L b (cid:33) (cid:114) N [ ω (cid:16) − n (cid:48) (cid:17) + N χ ] . (34)This expression is quite different from (19), however, it still scales as N (see Figure 3). We compareboth expressions in figure 4. We now discuss the theoretical prospects for wormhole parameter estimation with thesetechniques. The nonlinear parameter χ depends on the third -order susceptibility χ ( ) because thesecond-order susceptibility χ ( ) vanishes due to inversion symmetry of the lattice. Let us considertypical values of the LIGO interferometer (L=1 km), with frequency ω ∼ Hz, and a typical value niverse , xx , x 9 of 13 of the nonlinear parameter χ = ¯ h ω χ ( ) (cid:101) V ≈ − − − ≈ χ = − − Hz. For the ratio of rb = and χ = − Hz, sensitivity can be improvedwith a nonlinear interferometer by 2 orders of magnitude, as shown in Figure 3,with respect to thetheoretical SQL, which is obtained by making χ = Figure 3.
Sensitivity of the Quadrature Measurement with a nonlinear medium (Eq.(34)) togetherwith theoretical SQL vs. number of photons N for L = ω = Hz, rb = , n (cid:48) = χ = − Hz. We see that in the high-number regime shown in the plot, the sensitivity goes inprinciple significantly beyond the standard quantum limit. This range for the number of photonswould correspond to a laser with a wavelength around a thousand of nm and a few W of power, as inLIGO.
QFI in (19) and the quadrature measurement in the nonlinear case are practically indistinguishablebeyond 10 photons as can be seen in figure 4.To analyze these results, we must set the range of parameters in which a wormhole can beexpected in principle. Considering that the maximum tolerance is ∆ b b = rb = . The size of a wormhole in the centre of the Milky way ( 10 Pc), taking into accountthat 1 Pc = · m, would be 10 m, which seems too large. In the case of a black-hole mimicker,the throat is of the order of the Schwarzschild radius of the black hole, which in the case of LIGO is200 km, detection would be possible from r (cid:39) ∆ b b ∝ L χ , we can keep thesame sensitivity for the same ratio rb if the distance is reduced while the non linear coupling is raised.In fact, χ is expected to be bigger than 10 − Hz so the sensitivity could be improved significantly asshown in [24]. niverse , xx , x 10 of 13 Figure 4.
Theoretical sensitivity for the Quantum Fisher Information and Quadrature Measurement inthe nonlinear cases, with the same parameters as in Figure 3. We see that in the high-number regimediscussed in Figure 3 our realistic measurement protocol achieves a precision extremely close to thefundamental limit, exhibiting super-Heisenberg scaling.
We can consider a modest value for the length of the fiber approximately given by 1 m. Inthis case it would be needed a larger nonlinear parameter ( χ (cid:39) − Hz) in order to obtain a ratio rb = . Including repetitive measurements ( M = = Hz), the ratio rb can be as high as10 . Considering χ (cid:39) rb = with repetitivemeasurements and a value of L = η in the QFI [25] while for a thermal state with an averagenumber of excitations n T , the QFI will be reduced by a factor ( + n T ) − [26]: ∆ b b = η + n T cr Lb (cid:114) N [ ω (cid:16) − n (cid:48) (cid:17) + N χ ] (35)Further analysis of errors would depend on the particular experimental set up. niverse , xx , x 11 of 13 Figure 5.
Improvement of sensitivity with respect to the SQL vs. number of photons N for severalvalues of the nonlinear parameter, L = M = = Hz and rb = , with the rest of theparameters as in Figure 3. The nonlinear case becomes relevant for χ < − Hz in the high-numberregime.
We have considered to be in an almost-flat space time with a tiny perturbation which does notdepend on time. This might rise the question of how to actually determine the Minkoswki length, L . For this, we need to rotate the interferometer two angles θ and φ around two independent axis.This would change the proper length producing an oscillating phase-shift, necessary for deducing theposition of the wormhole.The interferometer must be translated along the radial axis so we can repeat the experiment andcheck the distance to the wormhole.Assuming that the interferometer is in the plane φ =
0, and in the radial direction as desired, wecan rotate the interferometer an angle θ , resulting in the following change of coordinates: (cid:126) e r = (cid:126) e x → cos θ (cid:126) e x + sin θ (cid:126) e y . (36) niverse , xx , x 12 of 13 Notice that the module of the proper length would not change in the Minkowski metric due to Lorentzcovariance, but this only apply locally. When rotating, the laboratory stops being an inertial referenceframe. The proper length changes at first order as: L θ = (cid:113) ( L (cid:48) cos θ ) + ( L sin θ ) (cid:39) (cid:115) L + L b r cos θ = L ( + b r cos θ ) . (37)
4. Conclusions
We have applied quantum metrology techniques to estimate gravitational parameters. Inparticular, we propose to use nonlinear media for the detection of Ellis wormholes. The tinyperturbation generated by a distant wormhole on a flat spacetime induces a slight phase shift ina coherent state which propagates along the radial direction. We show that, in principle, this effect canbe estimated with a precision which not only goes beyond the classical limits but can also overcomethe Heisenberg limit. We show that the fundamental sensitivity limit provided by the QFI is notrestricted by the Heisenberg limit when a nonlinear medium is considered. While this is an abstractresult, we also show that a particular measurement protocol can get extremely close to the theoreticalbound. In particular, we find that an standard homodyne detection protocol with a nonlinearityin one interferometer arm also exhibits super-Heisenberg scaling. We consider parameters of laserinterferometers such as LIGO and realistic values of the nonlinearity, and discuss the theoreticalprospects for parameter estimation. Of course, a complete experimental proposal would require anextremely careful characterization of all the sources of error -as in LIGO- and in any case wouldpresumably be highly challenging. However, our goal is only to show that, in principle, an extremelysensitive estimation could be achieved by exploiting quantum resources. Along this vein, our resultscan motivate further exploration of this research path.
Acknowledgments:
C. S. has received financial support through the Junior Leader Postdoctoral FellowshipProgramme from “la Caixa” Banking Foundation and Fundación General CSIC (ComFuturo Programme)
Author Contributions:
C. S. proposed the general idea and supervised the work and the writing of the mansucript.C.S-V. developed the ideas, made all the computations and wrote the mansucript.
Conflicts of Interest:
The authors declare no conflict of interest.1. Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone. Advances in Quantum Metrology. Nature Photonics , 222(2011).2. Roman Schnabel, Squeezed states of light and their applications in laser interferometers. Phys. Rep. , (2017).3. P. Kómar et al. A quantum network of clocks Nature Phys. 10, 582 (2014).4. M. Ahmadi, David Edward Bruschi, Nicolai Friis, Carlos Sabín, Gerardo Adesso, and Ivette Fuentes, RelativisticQuantum Metrology: Exploiting relativity to improve quantum measurement technologies. Sci. Rep. , 4996(2014).5. Eduardo Martín-Martínez, Andrzej Dragan, Robert B. Mann, and Ivette Fuentes, Berry Phase QuantumThermometer. New Journal of Phys. (2013) 053036 (11pp).6. Magdalena Zych, Fabio Costa, Igor Pikovski, Timothy C. Ralph, and Caslav Brukner. General relativistic effectsin quantum interference of photons. Class. and Quant. Gravity , 22. (2012).7. S. P. Kish and T. C. Ralph, Quantum limited measurement of space-time curvature with scaling beyond theconventional Heisenberg limit. Phys. Rev. A , 041801(R) (2017).8. Carlos Sabín, Quantum detection of wormholes. Sci.Rep. , 716 (2017).9. Vitor Cardoso, Edgardo Franzin, Paolo Pani. Is the gravitational-wave ringdown a probe of the event horizon?Phys. Rev. Lett. 116, 171101 (2016).10. R. A. Konoplya, A. Zhidenko. Wormholes versus black holes: quasinormal ringing at early and late times.JCAP
043 (2016).11. Juan Maldacena, Leonard Susskind. Cool horizons for entangled black holes. Fortschr. Phys.
781 (2013). niverse , xx , x 13 of 13
12. Abe, F. Gravitational Microlensing by the Ellis Wormhole. Astrophys. J. 725, 787 (2010).13. S. Boixo, S.T. Flammia, C.M. Caves, et al. Generalized limits for single-parameter quantum estimation Phys.Rev. Lett,
469 (2018).18. D. Deutsch, Quantum mechanics near closed timelike lines. Phys. Rev. D , 3197 (1991).19. T. A. Roman Some thoughts on energy conditions and wormholes. In The Tenth Marcel Grossmann Meeting:On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic FieldTheories (In 3 Volumes) (pp. 1909-1924) (2005).20. L.H. Ford and Thomas A. Roman, Quantum field theory constrains trasversable wormholes geometries.Phys.Rev. D , 5496 (1996).21. Christopher J. Fewster. Lectures on quantum energy inequalities. arXiv e-print:1208.539922. Michael S. Morris and Kip S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool forteaching general relativity. American Journal of Physics , 395 (1988).23. Barry C. Sanders, Gerard J. Milburn. Quantum limits to all-optical phase shifts in a Kerr nonlinear medium.Physical Review, A , (1992).24. N. Matsuda, R. Shimizu, Y. Mitsumori, H. Kosaka, and K. Edamatsu, Observation of optical-fibre Kerrnonlinearity at the single-photon. level, Nat. Photonics , 95 (2009).25. N. Spagnolo et al. Phase Estimation via Quantum Interferometry for Noisy Detectors. Phys. Rev. Lett. 108,233602 (2012).26. M. Aspachs, J. Calsamiglia, R. Muñoz-Tapia. and E. Bagan, Phase estimation for thermal Gaussian states.Phys. Rev. A 79, 033834 (2009)c (cid:13)(cid:13)