PHYSICAL REVIEW D 96, 105004 (2017) Quantum estimation of parameters of classical spacetimes

T. G. Downes,1,* J. R. van Meter,2,3 E. Knill,3 G. J. Milburn,1 and C. M. Caves1,4 1Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St. Lucia, Queensland 4072, Australia 2Department of Mathematics, University of Colorado, Boulder, Colorado 80309, USA 3National Institute of Standards and Technology, Boulder, Colorado 80305, USA 4Center for Quantum Information and Control, University of New Mexico, MSC07–4220, Albuquerque, New Mexico 87131-0001, USA (Received 17 November 2016; published 6 November 2017) We describe a quantum limit to the measurement of classical spacetimes. Specifically, we formulate a quantum Cram´er-Rao lower bound for estimating the single parameter in any one-parameter family of spacetime metrics. We employ the locally covariant formulation of quantum field theory in curved spacetime, which allows for a manifestly background-independent derivation. The result is an uncertainty relation that applies to all globally hyperbolic spacetimes. Among other examples, we apply our method to the detection of gravitational waves with the electromagnetic field as a probe, as in laser-interferometric gravitational-wave detectors. Other applications are discussed, from terrestrial gravimetry to cosmology.

DOI: 10.1103/PhysRevD.96.105004

I. INTRODUCTION connection between local changes in the metric and relative changes in the states of quantum fields that live on the The geometry of spacetime can be inferred from physical spacetime. These changes, used to sense the spacetime measurements made with clocks and rulers or, more gen- parameters, can be characterized in terms of an evolution of erally, with quantum fields, sources, and detectors. We the state driven by a stress-energy integral with respect to assume that the ultimate precision achievable is determined ˆ by quantum mechanics. In this paper we obtain parameter- the change in the metric, which gives the operator P. based quantum uncertainty relations that bound the preci- For the universal connection between changes in states sion with which we can determine properties of spacetime in of quantum fields and the stress-energy integrals, we rely terms of stress-energy variances. Such uncertainty relations on the locally covariant formalism for quantum fields on might become increasingly relevant to empirical observation curved spacetime backgrounds [5]. For this purpose, we treat gravity classically as in general relativity, determined as, for example, laser-interferometric gravitational-wave − detectors are expected to approach quantum-limited sensi- by a metric with signature ð ; þ; þ; þÞ on a spacetime manifold. This is treated as a fixed background on which tivity across a wide bandwidth in the near future. — An informative, high-level way to quantify the precision the quantum fields used for measurements we call these “ ”— of a parameter measurement is by the inverse variance probe fields propagate. In particular, we do not con- sider back action from the quantum fields on the metric. In hðδθ~Þ2i of an estimator θ~. The best precision with which we any case, we expect that such back action would transfer can measure a parameter is determined by the quantum uncertainty in the quantum field being measured to the Fisher information [1,2]. For pure states, the Fisher infor- metric and reduce the achievable precision, for otherwise, mation reduces to a multiple of the variance hðΔPˆÞ2i of an ˆ by an argument given in Refs. [6,7], the uncertainty evolution operator P that describes how the quantum state principle for matter would be violated (see also Ref. [8] changes with changes in the parameter. This determines a for a study of the problem of back action when measuring parameter-based uncertainty relation [3,4], the structure of spacetime). We allow for the presence of classical fields that can ℏ2 hðδθ~Þ2ihðΔPˆÞ2i ≥ ; ð1:1Þ propagate on the spacetime background. We only consider 4 those fields that play a direct role as sources for the quantum fields used for measurement. These sources are whose form is reminiscent of the Heisenberg uncertainty determined by classical devices needed to implement the relations. measurement. For the types of measurement considered Such parameter-based uncertainty relations can be here, the parametrized changes of the metric are indepen- applied to parameters associated with local changes of dent of these equipment-related classical fields. In particu- the spacetime metric. They are derived from a universal lar, these classical fields contain no information about the parameter of interest, and for this reason we do not model *[email protected] them explicitly.

2470-0010=2017=96(10)=105004(23) 105004-1 © 2017 American Physical Society DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) Our work in this paper relies on the parameter-based This approach was used by Braunstein, Caves, and uncertainty relation (1.1). To put such uncertainty relations Milburn [4] to develop optimal quantum estimation for in context, we consider now how the familiar Heisenberg spacetime displacements in flat Minkowski spacetime. In uncertainty relations generalize to parameter-based uncer- spacetime, not only can one move a fixed proper distance or tainty relations. The Heisenberg uncertainty relation for time, one can also boost and rotate. Quantum parameter position and momentum states that the product of the estimation was thus also developed for the parameters uncertainties in position and momentum, for any quantum corresponding to these actions [4]. The results were state, must be greater than a constant. In terms of the developed with the quantized electromagnetic field as variances of position and momentum, the Heisenberg the probe. These results show that estimates of a spacetime uncertainty relation is written as translation can be made increasingly accurate as the uncertainty in the operator that generates the translation, ℏ2 boost, or rotation is made very large. Δ ˆ 2 Δ ˆ 2 ≥ hð xÞ ihð pÞ i 4 ; ð1:2Þ In this paper we are interested in limits on the precision of estimates of parameters of the classical gravitational field. In general relativity the gravitational field is a where ℏ is the reduced Planck’s constant. The Heisenberg manifestation of the geometry of spacetime, which is uncertainty relation is derived in standard quantum described by a metric. The metric determines the length mechanics, where position and momentum are both rep- of the invariant (proper) interval between two spacetime resented as Hermitian operators. events according to [9] A similar relation, albeit with a different interpretation, exists between time and energy: 2 μ ν ds ¼ gμνðxÞdx dx ; ð1:4Þ ℏ2 δ 2 Δ ˆ 2 ≥ μ ν 0 hð tÞ ihð HÞ i 4 : ð1:3Þ where gμνðxÞ is the metric tensor, with indices , ¼ ,1, 2, 3 for time and the three spatial coordinates. The dxμ are Unlike position, time in standard quantum mechanics is a infinitesimal coordinate differences. We assume the classical parameter. The time-energy uncertainty relation is Einstein summation convention, where repeated upper an example of a quantum limit on parameter estimation. and lower indices are summed over. One tries to estimate a classical parameter, in this case time, Before we describe the relevant quantum parameter by making measurements on a quantum system, a “clock,” estimation, we ask the following: can any insight be found whose evolution depends on time. In the time-energy by applying the Heisenberg uncertainty relation, in its 2 parameter-based form, directly to a proper distance? It was uncertainty relation (1.3), hðδtÞ i is the classical variance along these lines that Unruh [10] derived an uncertainty of the estimate of t; this classical variance arises ultimately relation for a component of the metric tensor. Once from quantum uncertainties in clock variables conjugate to coordinates are chosen, there should only be quantum the Hamiltonian H. uncertainty in the proper time and the proper distance. As The most common way to make quantum mechanics these are in turn related to the metric via Eq. (1.4), any compatible with classical relativity is to demote position to uncertainty in the proper distance is equivalent to uncer- a parameter, just like time in the previous example. Physical tainty in the metric. By applying the Heisenberg uncer- systems can then be thought of as living on, and intera- tainty relation to a proper distance, Unruh found a simple, cting with, the classical spacetime manifold. In this yet insightful uncertainty relation for a single component of relativistic context, the position-momentum uncertainty the metric. The Unruh uncertainty relation for the g11 relation naturally becomes a quantum limit on parameter component of the metric (assuming particular Cartesian- estimation [3]. like coordinates), in terms of variances, is This parameter-based approach is the natural, opera- tional way to think of uncertainty relations. Nothing in the ℏ2 traditional Heisenberg uncertainty relation (1.2) refers 2 ˆ 11 2 hðδg11Þ ihðΔT Þ i ≥ ; ð1:5Þ directly to a measurement of position or momentum. In V2 the parameter-based approach, one considers measure- ments of any sort whose results are used to estimate where here, and henceforth, we use units such that changes in a parameter; quantum mechanics then says, G ¼ c ¼ 1. The conjugate variable to g11 is the corre- via the quantum Fisher information, that the uncertainty of sponding component of the quantized stress-energy tensor, 11 the estimate is limited by the uncertainty in the operator that in this case the pressure Tˆ in the x1 direction. The key generates changes in the parameter, in a way that looks like feature of this uncertainty relation is the inverse propor- (but by being operational is more powerful than) a tradi- tionality to V2, the square of the four-volume of the tional Heisenberg uncertainty relation. measurement.

105004-2 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) We provide a general framework for deriving such an optimistic in the sense that they suggest a higher-than- uncertainty relation, by formulating it as a problem in achievable precision, we generally wish to make the quantum estimation theory. The metric gμνðxÞ is defined relationship between the measurement region, stress- for each point x on the manifold. If the quantum probe energy variance, and measurement precision tighter. For (measurement device) occupies some four-volume V, then this we show that the metric change can be localized, the probe’s state depends on the metric at every point in provided that the sensitivity of our measurement to the that region. If we consider the metric to be an arbitrary parameter of interest is not affected. Because the locally function on the manifold, then we need to estimate an covariant framework requires compactly supported regions, infinite number of parameters to define it completely. the localization is necessary when the parameter is a global Instead, we consider regions of spacetime that can be property of spacetime. described by metrics characterized by a single parameter θ. Another important issue—perhaps the most important For example, the Schwarzschild metric, which describes issue for the interpretation and relevance of our results—is the spacetime around a static nonrotating black hole, is that computations of the relevant stress-energy variances defined by the single parameter M, the mass of the black can be difficult. In most situations, however, the probe hole. The task is to estimate this mass parameter by devices introduce fields that have large mean values making measurements on physical systems living on the compared to a zero-mean reference state, which is typically spacetime manifold. the vacuum state. In these cases, the calculation can be There has recently been some promising work in this simplified by recognizing that mean-field-independent direction [11–13], focusing on quantum probes consisting contributions become negligible. of scalar fields in Gaussian states. Here we present a Once we have determined the general parameter-based general formalism for relativistic quantum metrology, using uncertainty relations connecting measurement precision to arbitrary fields and states. In so doing, we address several stress-energy variances, we apply them to several situations related issues, which have, we believe, not previously of interest. The first involves estimating constant metric received enough attention in this context. Ensuring that components and recovers the Unruh uncertainty relations. quantum observables in different spacetimes measure “the By considering specific metric components, we also obtain same” physical parameter is nontrivial. If the spacetimes the parametrized form of the Heisenberg uncertainty differ by a global perturbation, the positions of measure- relations. Next we study in detail the problem of interfero- ment devices therein might correspondingly differ, further metric gravitational-wave detection with light. This complicating the issue; more generally, spacetime points in requires the full power of our approach. Here we take two such spacetimes cannot unambiguously be identified advantage of localization by means of a “bump” function with each other. Care must also be taken to ensure supported in the region of the measurement devices and use coordinate independence. the large-mean-field property to enable the explicit calcu- Fortunately, a framework exists for comparing quantum lation of a bound on precision that depends on the observables in perturbed, classical spacetimes in a coor- amplitude of the light fields used. This recovers the dinate-independent manner [5]. This locally covariant well-known shot-noise limit, but goes beyond this limit framework, which was developed in the context of alge- in two ways. First, for the case of a wideband mean field on braic quantum field theory, serves as our starting point. As top of vacuum fluctuations, we obtain a general shot-noise the current work is geared more towards physical experi- bound that does not make an assumption of narrow ments than most literature invoking algebraic quantum field bandwidth detection of the probe field; in particular, we theory, it is worth a comment. The aim of algebraic find a wideband shot-noise limit in terms of a frequency- quantum field theory is to put quantum field theory on weighted integration over mean photon numbers. Second, rigorous mathematical footing, while the aim of what might we find that wideband squeezing gives the optimal, be termed pragmatic quantum field theory is to make sub-shot-noise sensitivity under the assumptions we are experimental predictions [14,15]. Strides toward connect- making. We briefly discuss other examples, including ing the two have been made recently [5,16–19], and this cosmological parameters and gravimetry. progress makes the current work possible. Our work is organized as follows. In Sec. II we review The locally covariant framework directly connects the quantum estimation theory including, in particular, the stress-energy to the change in state associated with a quantum Cram´er-Rao bound, which is the expression of compactly supported change in the metric. The connection parameter-based uncertainty principles. In Sec. III we is via the concept of relative Cauchy evolution developed in review the locally covariant approach to quantum field Ref. [5] and leads directly to our bounds on measurement theory in perturbed, classical spacetime developed by precision. An issue is that the bounds obtained are with Brunetti, Fredenhagen, and Verch [5]. In Sec. IV we show respect to the best observable supported in a region that can that the application of the Cram´er-Rao bound discussed in be much bigger than the region containing the probes. Sec. II to such a perturbed spacetime results in a coordinate- While this means that the bounds are guaranteed to be independent uncertainty relation between a local spacetime

105004-3 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) property and a quantum operator that depends on the probe The variance of an unbiased estimate of the parameter θ, field. In Sec. V we generalize this uncertainty relation to based on the distribution of measurement outcomes to be global spacetime properties. In Sec. VI we consider the observed, is bounded by the classical Cram´er-Rao lower application of the formalism to the estimation of metric bound [4], components, proper time, and proper distance in a certain class of perturbed spacetimes. In Sec. VII we derive a 2 1 hðδθ~Þ i ≥ ; ð2:3Þ quantum uncertainty bound on the detection of gravita- FðθÞ tional waves using the electromagnetic field as a probe, as in laser-interferometric gravitational-wave detectors. Then, where FðθÞ is the classical Fisher information for the in Sec. VIII, we consider additional applications, and measurement, given by finally we conclude in Sec. IX. Z 1 ∂pðωjθÞ 2 FðθÞ¼ dω : ð2:4Þ II. QUANTUM ESTIMATION THEORY pðωjθÞ ∂θ

In this paper we consider estimating an individual In this paper we consider the special case where ρˆ is parameter of a spacetime metric, so we only need the differentiable at θ0, with differential generated by the self- theory of single-parameter quantum estimation. A general adjoint operator hˆ, as expressed by scheme for quantum parameter estimation is depicted in Fig. 1. A quantum state, represented by a density operator dρˆ i ˆ ˆ ¼ − ½h; ρˆ: ð2:5Þ ρˆ0, undergoes a unitary transformation UðθÞ that depends dθ ℏ on the parameter θ of interest, producing a one-parameter ˆ ˆ † family of states, ρˆðθÞ¼UðθÞρˆ0U ðθÞ. Measurements are Informally, we write made on the system, with results ω, which are fed into an ρˆ θ θ −idθh=ˆ ℏρˆ θ idθh=ˆ ℏ estimator θ~ðωÞ of the parameter. ð 0 þ d Þ¼e ð 0Þe : ð2:6Þ We consider generalized measurements, described by positive-operator-valued measures (POVMs). For simplicity It can then be shown that for any POVM, the classical ˆ 2 2 we consider such POVMs given by a positive-operator- Fisher information satisfies FðθÞ ≤ 4hðΔhÞ =ℏ [1–3], ˆ 2 valued density EˆðωÞ that satisfies the completeness property where hðΔhÞ i is the quantum variance of the generator ˆ Z h; moreover, there is a POVM that saturates this bound when ρˆðθ0Þ is pure [3]. Applying this bound to Eq. (2.3) dωEˆ ω 1ˆ; 2:1 ð Þ¼ ð Þ yields the quantum Cram´er-Rao lower bound [4],

1ˆ ℏ2 where is the identity operator. The outcomes of a particular hðδθ~Þ2ihðΔhˆÞ2i ≥ : ð2:7Þ measurement follow a probability distribution pðωjθÞ con- 4 ditional on the parameter θ. The probability distribution for the outcomes ω can be calculated as One can now construct examples by identifying parameters and their corresponding generators. For example, since the Hamiltonian is the generator of time translations, this gives pðωjθÞdω ¼ TrðEˆðωÞρˆðθÞdωÞ: ð2:2Þ the time-energy uncertainty relation (1.3) presented in the θ Introduction. Since the momentum operator generates The problem of estimating the parameter is essentially displacements, using it as the generator gives the θ~ θ that of choosing a value to make a good estimate of by parameter-based version of the Heisenberg uncertainty ω considering the observed in relation to the known relation. Another important example is provided by the ω θ ω probability distributions pð j Þd . A common example number operator and phase, which is the basis of is the maximum likelihood estimator, which is the choice of Heisenberg-limited phase estimation [20–22]. θ~ that retrospectively maximizes the probability of the observed measurement outcomes. III. RELATIVE CAUCHY EVOLUTION We employ the locally covariant formulation of quantum field theory in curved spacetime as developed by Brunetti, Fredenhagen, and Verch [5]. The approach has been used to develop a notion of “identical physics” on different space- times [23]. A key result is a method for calculating how quantum observables respond to local changes in the FIG. 1. Scheme for quantum parameter estimation. background spacetime. Say we believe some particular

105004-4 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017)

ðsÞ ðsÞ region of the universe to be well described by a metric gμν (b) Each ðM; g Þ is a globally hyperbolic that depends on a parameter s.Ifs is assumed to para- spacetime. ðsÞ metrize a compactly supported perturbation, the locally (c) C is also a Cauchy surface for each ðM; g Þ. covariant approach can be used to calculate how any The geometric assumptions listed here can be seen in observable Eˆ s responds to such a change. The response greater mathematical detail in Sec. 4.1 of Ref. [5]. ð Þ ˆ is evaluated as the rate of change of the observable with Consider now a Hilbert-space operator Að0Þ defined on 0 respect to the parameter. As we noted in the Introduction, ðM; gð ÞÞ; this is an operator acting on a representation of this is just what we need to calculate the quantum Cram´er- the algebra generated by the quantum fields on ðM; gð0ÞÞ. Rao lower bound. This operator could, for example, be a POVM element for a We emphasize, however, that only compactly supported particle detector and belong to the algebra of operators perturbations in spacetime were considered in Ref. [5]. The localized to the spatiotemporal extent where the detector is motivation for this restriction is similar to that for restrict- active. ing the domain of distributions to test functions and ensures It was shown in Ref. [5] that Aˆð0Þ unitarily transforms that relevant quantities are well defined. We consider how under an s-parametrized metric perturbation into a new to approach more general perturbations in Sec. IV. operator AˆðsÞ. The functional derivative of the action of this The locally covariant approach is formulated in a unitary transformation with respect to the metric is category-theory framework. It involves the category of defined as globally hyperbolic spacetimes and the mapping of each to Z an algebra of observables. By this formalism, which is δ ˆ d ˆ ∘ AðsÞ d ðsÞ summarized in Appendix A, the evolution of an observable AðsÞ¼ dμðxÞ gμν ðxÞ; ð3:1Þ ds 0 δgμνðxÞ ds 0 in response to a spacetime perturbation is made math- M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ematically well defined. ∘ ð0Þ 0 1 2 3 Note that here a spacetime is a pair ðM; gÞ, where M is a where dμðxÞ¼ j det g jdx dx dx dx is the proper four-manifold admitting a Lorentzian metric and g is a volume element for the metric gð0Þ. The interpretation of Lorentzian metric. The additional property of global hyper- this is as follows: fields are prepared in N−, they then scatter bolicity is a restriction on the causal structure on the off an intermediate region (the compact subset of geometric manifold. It removes the possibility of closed time-like assumption 3) and are measured in Nþ; s controls a localized curves and ensures the spacetime can be foliated into perturbation within this region, and Eq. (3.1) gives the Cauchy surfaces. This in turn ensures that any hyperbolic infinitesimal movement of the observables in the Hilbert- field equation (Klein-Gordon, Maxwell, etc.) has a well- space representation of AðM; gð0ÞÞ due to an infinitesimal posed initial-value formulation. perturbation ds around s ¼ 0. Further properties and inter- A one-parameter family of spacetimes fM; gðsÞg was pretations of this functional derivative and the relative considered in Ref. [5], all sharing “initial” and “final” Cauchy evolution can be found in Refs. [5,23,24]. Cauchy surfaces, as well as respective neighborhoods N− For the case of the Klein-Gordon field, with its corre- and Nþ of those Cauchy surfaces. These spacetimes differ sponding Weyl algebra of observables, it can be shown that only in their geometry, and only within a compact region both elements of this algebra and polynomials of field ðsÞ between N− and Nþ. We refer to the metric g as operators constructed from it obey the following relation perturbed when s ≠ 0 and fiducial when s ¼ 0. (Theorem 4.3 from Ref. [5]): To be more precise, we make the following geometric δ ˆ assumptions: AðsÞ i ˆ ˆ μν 0 ¼ ½Að0Þ; T ðxÞ: ð3:2Þ (1) ðM; gð ÞÞ is a globally hyperbolic spacetime. δ 2ℏ gμνðxÞ s¼0 (2) We choose a Cauchy surface C in ðM;gð0ÞÞ and two ð0Þ ˆ μν open subregions ðN ; g Þ with the following prop- Here T ðxÞ is the renormalized stress-energy tensor on the N erties: relevant Hilbert space as discussed in Ref. [5] and satisfy- ing Theorem 4.6.1 of Ref. [25]. We assume, more specifi- (a) Nþ is within the future and N− is within the past causal regions of the Cauchy surface C. cally, that it is normally ordered in accordance with the ð0Þ procedure advocated by Brown and Ottewill [26]. (b) ðN; gN Þ are globally hyperbolic spacetimes. Inserting Eq. (3.2) into Eq. (3.1), we have (c) N contain Cauchy surfaces for the whole ð0Þ Z spacetime ðM; g Þ. ðsÞ d ∘ i μν d (3) fg g ∈ −1 1 is a set of Lorentzian metrics on M with ˆ μ ˆ 0 ˆ ðsÞ s ½ ; AðsÞ¼ d ðxÞ 2ℏ ½Að Þ; T ðxÞ gμν ðxÞ; ð3:3Þ the following properties: ds 0 M ds 0 (a) Each gðsÞ deviates from gð0Þ only on a compact ˆ subset of the region in the past of N and the where we emphasize that the operator A does not depend on þ ˆ future of N−. x. By defining the operator P as

105004-5 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) Z 1 ˆ ∘ ˆ μν d ðsÞ from the measurement device. We can take advantage of a P ¼ dμðxÞT ðxÞ gμν ðxÞ; ð3:4Þ 2 M ds 0 flexibility built into quantum estimation theory whereby we can obtain an uncertainty bound from any parameter that we have our estimator θ~ is sensitive to. To see this, let Θ denote the global parameter of interest d ˆ i ˆ ˆ AðsÞj ¼ ½Að0Þ; P: ð3:5Þ for a spacetime with metric gμνðΘÞ, with Θ0 being the ds s¼0 ℏ fiducial value of the parameter and θ~ denoting its estimator. The above relative Cauchy evolution equation has also We can choose a smooth, compactly supported “bump” 1 – been shown to hold for spin-2 and spin-1 fields [16 18], function, 0 ≤ χ ≤ 1, with χðxÞ¼1 on the measurement with an appropriately defined stress-energy tensor. For region. We consider the localized perturbation example, for the electromagnetic field, which we consider gμνðθÞ ≡ gμνðθ0 þðθ − θ0ÞχÞ, parametrized in terms of θ, in Sec. VII, the Weyl algebra of the Klein-Gordon field is with θ0 ¼ Θ0.Ifχ has sufficiently large extent and simply replaced by the Weyl algebra of gauge-equivalence transitions to 0 sufficiently slowly (say, adiabatically), classes of the vector potential. Then by Theorem 3.2.9 in we can argue that the sensitivity of θ~ to θ, given by Ref. [17], the functional derivative with respect to the ~ ~ ðdhθθi=dθÞjθ θ approaches that of θ to Θ, which is perturbed metric of an operator Aˆ is again given by the ¼ 0 d θ~ =dΘ 1, where the latter identity follows commutator with the stress-energy tensor, as in Eq. (3.2). ð h iΘ ÞjΘ¼Θ0 ¼ from the assumption that θ~ is an unbiased estimator to first ~ IV. ESTIMATION OF SPACETIME order in Θ − Θ0. This means that θ is also an unbiased PERTURBATION estimator of θ, to first order in θ − θ0, so that the Cram´er- ~ Due to the dual nature of operators and states, a Rao bound applies to θ with gμνðθÞ, giving consequence of Eq. (3.5) is that we can write ℏ2 ~ 2 ˆ 2 ˆ ˆ hðδθÞ ihðΔPÞ i ≥ ; ð4:3Þ ρˆð0 þ dsÞ¼e−idsP=ℏρˆð0ÞeidsP=ℏ; ð4:1Þ 4 where ρˆðsÞ is a density operator in the Gelfand-Neimark- where – ð0Þ Z Segal [27 29] representation of the algebra on ðM;g Þ 1 ˆ ˆ ∘ ˆ μν d after the action of UðsÞ, the unitary in the Hilbert-space P ¼ dμT gμνðθÞ: ð4:4Þ 2 dθ representation corresponding to the relative Cauchy evo- M θ0 lution (more precisely to a unit-preserving automorphism β How the bump function transitions from 1 on the on the algebra of observables, called gðsÞ in Appendix A). measurement region to 0 is arbitrary. The choice affects Then, noting the equivalence of Eqs. (4.1) to (2.6),we the bound, however, through excess contributions to the obtain [30] variance hðΔPˆÞ2i in the bump function’s support outside ℏ2 the measurement region. To get the best bounds on δ~ 2 Δ ˆ 2 ≥ hð sÞ ihð PÞ i 4 ð4:2Þ precision, we choose bump functions that minimize this excess variance while achieving the desired sensitivity. In where s~ is the estimator for the perturbation parameter. the examples to be considered, this excess variance can be A limitation of the quantum Cram´er-Rao bound (4.2) is attributed to contributions from a reference state such as the that the spacetime perturbation is restricted to compact Minkowski vacuum. This is because the state associated support, whereas physically interesting perturbations are with the measurement and from which the bound is typically not so restricted. For example, a variation in the computed is a localized deviation from the reference state mass of the Earth would vary the metric at unbounded and the bump function necessarily extends beyond the distances away. To apply our formalism to this situation, we region of localization. We observe that the shape of the approximate global perturbations compactly. As a first transition of the bump function from 1 to 0 affects attempt, one might consider using compact perturbations the excess variance [31–33]. In particular, the excess can that approach the global one in some limit, but this might be reduced by ensuring that the transition from 1 to 0 is ˆ 2 lead to unbounded values of ðΔPÞ , which would trivialize slow. This is analogous to the adiabatic limit. For the case the bound (4.2). To avoid this we take advantage of the fact of the Minkowski vacuum, the reference-state contribution that the measurements that yield an estimate of θ~ are of can be made arbitrarily small by this method, as demon- observables that are accessible in a compact measurement strated for example in Ref. [31]. region determined by the measurement device. Given this, The use of slowly varying bump functions is expected the bounds above are necessarily conservative for pertur- to reduce excess variance from the reference state, but bations with large extent: they apply to all observables in does not necessarily lead to readily computable bounds. For the region of the perturbation, even those causally separated this, we observe that informative measurements rely on

105004-6 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) deviations from the reference state with relatively large two-point correlators [16,19,25]. Such a reference state localized mean fields. This is both out of necessity and to allows us to perform normal ordering via the point-splitting maximize the signal-to-noise. Typical measurements are approach. A corresponding stress-energy tensor can then be designed not to detect the reference-state contributions to constructed, with a well-defined expectation value [25], the variance, but rather to detect an effect in the presence of which is unique up to terms which cancel in the commu- a strong mean field, which greatly enhances the signal we tator (3.5) and in the variance (and is thus sufficient for our are looking for. Indeed, for arbitrary curved spacetimes, it is purposes). Moreover, while Hadamard reference states are not known how to calculate the reference-state contribu- generally not unique, their contribution to the relevant tions, nor is it known how to design a measurement on the variance is mean field independent. probe field that detects the corresponding mean-field- independent effects. The neglect of reference-state contri- V. COORDINATE INDEPENDENCE AND butions can then be regarded as a way of finding quantum COMPACT PERTURBATIONS limits on the kinds of measurements we know how to do, which involve large mean fields. The formulation given so far is generally covariant. For the case of free fields, the large-mean-field scenario Consider two compactly supported perturbations of the is formalized by considering the measurement state as a metric where one is obtained from the other by a local displacement by a local Weyl unitary of a reference state isometry, that is one acting as the identity except on a with mean field zero. In Minkowski space, these displace- compact region in the past of Nþ and the future N−. Then ments are enacted by conventional modal displacement both perturbations induce the same relative Cauchy evo- operators, where the displacement is by an amount deter- lution, as shown in Ref. [5] (see also Appendix B). mined by the mean field. We show in Sec. VII that the A complication to this obvious conclusion of general variance hðΔPˆÞ2i has terms that grow with the mean field as covariance arises, however, when the parameter of interest well as mean-field-independent terms. We identify the is expressed in a coordinate-dependent way and we require mean-field-independent terms as the reference-state con- the use of a bump function to localize the associated metric tribution to the variance. For a fixed bump function, but perturbation. This is the situation when estimating global large mean field, the reference-state contribution becomes parameters. An example is the invariant mass of a black negligible. This is the main strategy used for the analysis of hole, where there are a number of different standard gravitational-wave detection in Sec. VII. coordinate systems to choose from. When the bump For the above discussion, we assumed that the meas- function is expressed in the first coordinate system so as urement device is contained in a finite measurement region, to be independent of the invariant mass, in the second it can where an incoming reference state such as the vacuum state depend on the invariant mass. This means that in the second is temporarily modified for the measurement. This modi- coordinate system, the bump-function-modified metric fication is usually necessary to enhance the signal that we perturbation includes a term coming from the derivative are looking for. In the case of an interferometric measure- of the mass-dependent bump function with respect to mass, ment using light, the modification is accomplished by and this leads to discrepancies in the values of the variances introducing a large-amplitude light field confined between and Cram´er-Rao bounds depending on which coordinate mirrors. To accommodate these modifications in the gen- system is used to define the bump function. If our erally covariant formalism, the background includes exter- sensitivity argument for the choice of bump function is nally introduced classical sources with fixed relationships valid, we should obtain valid bounds regardless of coor- to the manifold, meaning that these relationships are dinate system. It is desirable, however, to choose bump unchanged by the perturbation of the metric under inves- functions for which the dependence on coordinate system is tigation. The formalism and relative Cauchy evolution still negligible. In this section, we show that such is the case for applies, as suggested in Ref. [5]. For the example of the variance of Pˆ in an arbitrary spacetime, assuming a gravitational-wave detection, the fixed relationship can sufficiently large mean field. To demonstrate the basic be justified by the observation that the classical sources mechanism by which coordinate independence is achieved, follow geodesics for the original as well as the perturbed we first consider as an illustrative example the expectation metric and are thus manifestly independent of the pertur- of Pˆ in Schwarzschild spacetime. bation in Gaussian normal coordinates. The fiducial metric in Schwarzschild coordinates is While our examples involving large mean-field devia- tions are on flat backgrounds, we expect that the justifi- 2 2 −1 S μ ν m0 2 m0 2 gμνdx dx ¼ − 1 − dt þ 1 − dr cation for neglecting reference-state contributions based on S S r r large mean-field deviations also applies to general curved 2 2 backgrounds. For any free field in a (globally hyperbolic) þ r dΩ ; ð5:1Þ spacetime, there exist zero-mean-field reference states, called Hadamard states, characterized by well-defined and in isotropic coordinates it is

105004-7 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) 2 μ 1 − m0=2ρ d d I ν − 2 S ≠ I gμνdxI dxI ¼ dt gμνðmÞ gμνðmÞ; ð5:6Þ 1 þ m0=2ρ dm dm 0 0 4 m0 þ 1 þ ðdρ2 þ ρ2dΩ2Þ: ð5:2Þ nor are these two tensor fields related by any coordinate 2ρ S transformation. To understand this, observe that gμνðmÞ − S I I gμνð0Þ and gμνðmÞ − gμνð0Þ also represent distinct tensor By our prescription for approximating a global pertur- fields; the first terms in the two tensor fields can be made bation by a compact perturbation, we add to every instance equal by an m-dependent coordinate transformation, but of the fiducial mass m0 a bump function of the form not without making the second terms unequal. It should ðm − m0Þχðt; r; θ; ϕÞ or ðm − m0Þχðt; ρ; θ; ϕÞ: ˆ ˆ come as no surprise, then, that h∶PS∶i and h∶PI∶i appear to 2 − χ be unequal: S μ ν ðm0 þðm m0Þ Þ 2 gμνdx dx ¼ − 1 − dt Z S S r 1 ∶ ˆ ∶ ∶ ˆ tt∶ ∶ ˆ rr∶ 2 ϑ ϑ φ −1 h PS i¼ ðh T iþh T iÞr sin dtdrd d ; 2ðm0 þðm − m0ÞχÞ K r þ 1 − dr2 þ r2dΩ2; r ð5:7Þ Z ð5:3Þ 1 h∶Pˆ ∶i¼ ðh∶Tˆ tt∶iþh∶Tˆ rr∶iþr2h∶Tˆ ϑϑ∶i I r 2 K μ 1 − m0 m − m0 χ =2ρ I ν − ½ þð Þ 2 2 2ϑ ∶ ˆ φφ∶ 2 ϑ ϑ φ gμνdxI dxI ¼ dt þ r sin h T iÞr sin dtdrd d : ð5:8Þ 1 þ½m0 þðm − m0Þχ=2ρ 4 m0 þðm − m0Þχ This appearance is deceptive, however, as we see from þ 1 þ ðdρ2 þ ρ2dΩ2Þ: 2ρ Z ˆ αr 2 ð5:4Þ h∶∇αT ∶ir sin ϑdtdrdϑdφ K Z Z S I ˆ αr ˆ ϑϑ The resulting metrics gμνðmÞ and gμνðmÞ are not related ¼ dλnαh∶T ∶i − ðrh∶T ∶i ∂ by a coordinate transformation and thus are no longer K K 2 ˆ φφ 2 physically equivalent. This reflects the fact that there is no þ rsin ϑh∶T ∶iÞr sin ϑdtdrdϑdφ; ð5:9Þ unique way to approximate a global perturbation with a λ compact perturbation. Our particular choice depends on our where d is the surface element induced on the boundary ∂ initial coordinates, out of convenience. These metrics are, K of K and nμ is the corresponding surface normal. Since ˆ μν ˆ μν however, locally related by a coordinate transformation on h∶T ∶i vanishes on ∂K and assuming ∇μT ¼ 0 (as a patch restricted to the region where χ ¼ 1. Therein, both required for a properly defined stress-energy tensor [25]), metrics are locally indistinguishable from that of a black which implies that the left-hand side of Eq. (5.9) vanishes hole with mass m and are related by the m-dependent identically, we conclude that coordinate transformation that relates Schwarzschild and isotropic coordinates. h∶Pˆ ∶i − h∶Pˆ ∶i I Z S Now letting m0 ¼ 0 for simplicity, consider the (nor- ˆ ϑϑ 2 ˆ φφ 2 mally ordered) expectation value of Pˆ, where the nonzero ¼ ðrh∶T ∶iþrsin ϑh∶T ∶iÞr sin ϑdtdrdϑdφ ¼ 0: expectation value of the stress-energy is assumed to be K ð5:10Þ confined to a region K in which χ ¼ 1, i.e., K ¼ ∶ ˆ μν∶ χ 1 ∈ suppðh T iÞ and ðxÞ¼ for x K. Note that K is Note the critical role played by the vanishing divergence of strictly contained in the interior of suppðχÞ, since as μν Tˆ in the above demonstration of coordinate independ- discussed in the previous section we assume the transition ence. This is no coincidence; for more discussion of the of the bump χ from 1 to 0 is both smooth and gradual. We relevance of the stress-energy tensor to diffeomorphism have invariance in the context of relative Cauchy evolution, Z see Ref. [5]. 1 ˆ ∘ ˆ μν d ∶P∶ dμ ∶T ∶ gμν m : 5:5 In the above example, we only considered the first h i¼2 h i ð Þ ð Þ ˆ K dm 0 moment of P, whereas the Cram´er-Rao bound involves the variance of Pˆ. We deal with this question now by S I Notice that gμνð0Þ¼gμνð0Þ, since both Schwarzschild and considering an arbitrary s-dependent coordinate transfor- isotropic coordinates reduce to standard spherical coordi- mation from unprimed coordinates to primed coordinates, nates in this limit. Yet on a coordinate patch that is assumed to cover the support

105004-8 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) ∂ ∂ K of the mean stress-energy (for a more general treatment ðsÞ ðsÞ 0 gαβ ðxÞ and gα0β0 ðx Þ; ð5:12Þ see Appendix B). In classic index notation, the metric ∂s 0 ∂s 0 components in the two systems are related by where we now write the s derivatives as partial derivatives ðsÞ 0 μ ν ðsÞ gα0β0 ðx Þ¼L α0 ðsÞL β0 ðsÞgμν ðxÞ; ð5:11Þ to emphasize that the respective coordinates are held fixed while taking the s derivative. Because the coordinate μ μ 0 α0 where L α0 ðsÞ¼∂x ðx ;sÞ=∂x . We are interested in the transformation is s dependent, these two tensors are not tensor fields the same, but are related by

γ γ ð0Þ ∂ ∂ ∂X 0 ∂X 0 ∂gμν ðsÞ 0 μ 0 ν 0 ðsÞ ð Þ ð Þ γ g 0 0 ðx Þ¼L α0 ð ÞL β0 ð Þ gμν ðxÞþ gγν þ gμγ þ X ∂s α β ∂s ∂xμ ∂xν ∂xγ 0 0 ∂ μ ν ðsÞ ¼ L α0 ð0ÞL β0 ð0Þ gμν ðxÞþ∇νXμ þ ∇μXν ∂s 0 ∂ μ ν ðsÞ ¼ L α0 ð0ÞL β0 ð0Þ gμν ðxÞþ∇β0 Xα0 þ ∇α0 Xβ0 ; ð5:13Þ ∂s 0 where ∂ γ γ X ¼ x ðx0;sÞ: ð5:14Þ ∂s 0

Now we find Z Z ∂ ∂ μ∘ 0 ˆ α0β0 ðsÞ μ∘ ˆ μν ðsÞ ∇ ∇ d T g 0 0 ¼ d T gμν þ νXμ þ μXν ∂s α β ∂s K 0 ZK 0 ∂ ∘ ˆ μν ðsÞ ˆ μν ˆ μν ¼ dμ T gμν þ 2∇μðT XνÞ − 2ð∇μT ÞXν ∂s ZK 0 Z ∂ ∘ ˆ μν ðsÞ ˆ μν ¼ dμT gμν þ 2 dλnμT Xν; ð5:15Þ K ∂s 0 ∂K

ˆ μν where in the last line we assume that ∇μT ¼ 0 and where we convert a volume integral over K to a surface integral over the boundary ∂K. ∶ ˆ μν∶ ∶ ˆ μν∶ 0 χ 1 As before, let K ¼ suppðh T iÞ, so that h T ij∂K ¼ , and we assume that jK ¼ [and thus K is strictly contained 0 within suppðχÞ]. In addition, consider a θ0-dependent coordinate transformation of gμνðθ0Þ, denoted gμνðθ0Þ, where we switch back from classic index notation to denoting a coordinate change with a prime on the tensor itself. As previously prescribed, to each instance of θ0 in the coordinate components of this new metric, add ðθ − θ0Þχ, denoting the result as 0 0 ˆ ˆ 0 gμνðθÞ¼gμνðθ0 þðθ − θ0ÞχÞ. It proves convenient to divide P and P into two parts, Z Z Z 1 1 1 ˆ ∘ ˆ μν d ∘ ˆ μν d ∘ ˆ μν d ˆ ˆ P ¼ dμT gμνðθÞ¼ dμT gμνðθÞþ dμT gμνðθÞ¼PK þ P ¯ ; ð5:16Þ 2 θ 2 θ 2 ¯ θ K M d θ0 K d θ0 K d θ0 Z Z Z 1 ∘ 1 ∘ 1 ∘ ˆ 0 0 ˆ 0μν d 0 0 ˆ 0μν d 0 0 ˆ 0μν d 0 ˆ 0 ˆ 0 P ¼ dμ T gμνðθÞ¼ dμ T gμνðθÞþ dμ T gμνðθÞ¼P þ P ¯ ; ð5:17Þ 2 θ 2 θ 2 ¯ θ K K M d θ0 K d θ0 K d θ0 where K¯ ¼ MnK. The integrals over K¯ are restricted to the neighborhood of K where the bump function is nonzero, but in which h∶Tˆ μν∶i¼0. The content of Eq. (5.15) is that Z ˆ 0 ˆ λ ˆ μν ˆ ˆ PK ¼ PK þ d nμT Xν ¼ PK þ B; ð5:18Þ ∂K

ˆ ∶ ˆ 0 ∶ ∶ ˆ ¯ ∶ ∶ ˆ∶ 0 where B is the boundary term. By construction, we have h PK¯ i¼h PK i¼h B i¼ ,so

105004-9 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) ∶ ˆ 0∶ ∶ ˆ 0 ∶ ∶ ˆ ∶ ∶ ˆ∶ ; h P i¼h PK i¼h PK i¼h P i ð5:19Þ this is the general version of what we showed for the particular case of Schwarzschild and isotropic coordinates. What we need for the Cram´er-Rao bound (4.3) are variances, not mean values, and it is in the variances that the problem with reference-state contributions arises. Equation (5.16) gives us

hðΔPˆÞ2i¼h∶Pˆ∶∶Pˆ∶i − h∶Pˆ∶i2 ∶ ˆ ˆ ∶∶ ˆ ˆ ∶ − ∶ ˆ ∶ 2 ¼h ðPK þ PK¯ Þ ðPK þ PK¯ Þ i h PK i Δ ˆ 2 ∶ ˆ ∶∶ ˆ ∶ ∶ ˆ ∶∶ ˆ ∶ ∶ ˆ ∶∶ ˆ ∶ ¼hð PKÞ iþh PK PK¯ iþh PK¯ PK iþh PK¯ PK¯ i; ð5:20Þ

ˆ 2 ˆ ˆ ˆ 2 ˆ where hðΔPKÞ i¼h∶PK∶∶PK∶i − h∶PK∶i . We now decompose ∶PK∶ into its expectation value and a correction, ∶ ˆ ∶ ∶ ˆ ∶ ∶Δ ˆ ∶ ∶ ˆ ∶∶ ˆ ∶ ∶ ˆ ∶ ∶ ˆ ∶ ∶Δ ˆ ∶∶ ˆ ∶ ∶Δ ˆ ∶∶ ˆ ∶ PK ¼h PK iþ PK , and use the fact that h PK PK¯ i¼h PK ih PK¯ iþh PK PK¯ i¼h PK PK¯ i to write the variance in the form

Δ ˆ 2 Δ ˆ 2 ∶Δ ˆ ∶∶ ˆ ∶ ∶ ˆ ∶∶Δ ˆ ∶ ∶ ˆ ∶∶ ˆ ∶ hð PÞ i¼hð PKÞ iþh PK PK¯ iþh PK¯ PK iþh PK¯ PK¯ i: ð5:21Þ

The final three terms are all reference-state contributions: the middle two terms express correlations between the reference state inside and outside of K; the third term is a reference-state contribution from outside the support of the probe’s mean stress-energy. ˆ 0 ˆ ˆ ˆ ˆ ˆ 0 Now, using Eqs. (5.17) and (5.18) to write P ¼ PK þ Q, with Q ¼ B þ PK¯ , the same considerations give us hðΔPˆ 0Þ2i¼h∶Pˆ 0∶∶Pˆ 0∶i − h∶Pˆ 0∶i2 ˆ ˆ ˆ ˆ ˆ 2 ¼h∶ðPK þ QÞ∶∶ðPK þ QÞ∶i − h∶PK∶i ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ ¼hðΔPKÞ iþh∶PK∶∶Q∶iþh∶Q∶∶PK∶iþh∶Q∶∶Q∶i ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ ¼hðΔPKÞ iþh∶ΔPK∶∶Q∶iþh∶Q∶∶ΔPK∶iþh∶Q∶∶Q∶i: ð5:22Þ

Again, the final three terms are reference-state contri- What we can say in the present context is that when we butions with the same sort of interpretation as that given assume that the mean-field terms predominate, we are above for hðΔPˆÞ2i, except that now there is a contribution finding quantum Cram´er-Rao bounds on measurements from the boundary ∂K. that we do know how to perform, which involve large mean If we take advantage of the improved signal-to-noise that fields. comes from a large mean field, we expect that our probe fields have a sufficiently substantial mean component that VI. SIMPLE EXAMPLES Δ ˆ 2 hð PKÞ i makes the dominant contribution to the variances A. Estimation of constant metric components and thus determines the quantum Cram´er-Rao bound. Under these assumptions we conclude that hðΔPˆ 0Þ2i and Consider now making measurements in a local inertial ˆ 2 ˆ 2 frame where the fiducial metric g(0) is flat or sufficiently hðΔPÞ i are both equal to hðΔPKÞ i up to subleading, mean-field-independent terms. flat for differences to be negligible in our calculations. In A complementary point of view acknowledges that for a this case we define a local inertial coordinate system where η real perturbation, not one that has been modified by a bump the fiducial metric is μν. Now further suppose that the μ0 ν0 function, the changes in the reference state do contribute to perturbed metric has variation gμνðθÞ¼ημν þ θδμ δν , for the quantum Cram´er-Rao bound. These changes in the some fixed μ0 and ν0. In other words, assume that the reference state (in some cases, the reference state could be parameter of interest is θ, and the fixed local coordinates δ θ vacuum, when that can be properly defined) could pre- are such that gμ0ν0 ¼ . Then the uncertainty relation (4.3) sumably be used to detect the perturbation in the absence of becomes a mean field. Yet we do not know how to calculate the  Z  ∘ 2 reference-state contributions in general curved spacetimes, δ 2 Δ μ ˆ μ0ν0 ≥ ℏ2 hð gμ0ν0 Þ i d T ; ð6:1Þ nor do we know how to measure the corresponding K modifications of the reference state. It would be desirable to determine how to do the necessary measurements, which where in this case there is no sum over the repeated indices might involve particle emission or Casimir-type effects. as we are dealing with a particular metric component

105004-10 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) Z  Z  −2 2 2 specified by the fixed values μ0 and ν0. This inequality is ℏ δτ 2 Δ 4 ˆ 00 ≥ reminiscent of the Unruh uncertainty relation (1.5). Indeed, dtaðxÞ hð Þ i d xT ðxÞaðxÞ 4 : the same restrictions were needed to derive Eq. (1.5) as were used to produce Eq. (6.1). ð6:5Þ It should be emphasized that the stress-energy tensor Tˆ μν Now we assume that the effective spatial volume of the in Eq. (6.1) is for the probe field. It is not the stress-energy confined probe field is small enough that the spacetime tensor for the matter distribution which gives rise to gμν via perturbation can be considered spatially uniform through- ’ Einstein s field equations. This is to be expected, as out. Then, recognizing that the integral of the energy Eq. (6.1) is essentially an uncertainty relation for the probe density over the spatial component of the four-volume is field: we are estimating the metric with field measurements, the Hamiltonian, we have so the uncertainty in the field in Eq. (6.1) has been replaced Z Z by uncertainty in our estimate of the metric, just as 4 ˆ 00 ˆ uncertainty in some “clock” variable is replaced by d xT ðxÞaðtÞ¼ dtHðtÞaðtÞ: ð6:6Þ uncertainty in a time estimate in the time-energy uncer- tainty relation (1.3). The uncertainty relation (6.1) can be Further assuming a time-independent Hamiltonian, the time thought of as the minimum uncertainty achievable when integrals cancel and the uncertainty relation reduces to attempting to verify with measurements that the metric takes the Minkowski form in the local coordinate system ℏ2 δτ 2 Δ ˆ 2 ≥ one has defined. hð Þ ihð HÞ i 4 : ð6:7Þ

B. Estimation of proper time and proper distance This demonstrates that the standard time-energy uncer- tainty relation is a special case of the metric uncertainty Suppose we are interested in the proper time as measured relation (4.2). by a stationary observer in a perturbed Minkowski space- Using a similar argument, one can derive a correspond- time. Assuming the metric perturbation is compactly ing uncertainty relation for proper distance X: supported in space, we might consider the passage of time as measured by an atomic clock at rest within the perturbed ℏ2 δ 2 Δ ˆ 2 ≥ region, relative to the passage of time as measured by an hð XÞ ihð PXÞ i 4 : ð6:8Þ atomic clock at rest in flat spacetime outside the perturbed region. Both can be considered proper time, but the latter is This is the parametric version of the Heisenberg uncertainty ˆ also equivalent to our Minkowskian coordinate time. The relation, where PX is the momentum in the direction of the proper time in our locally defined inertial frame is related to displacement. It demonstrates the consistency between the the coordinate time by metric uncertainty relation (4.2) and the earlier work on parameter-based uncertainty relations for the Lorentz group Z pffiffiffiffiffiffiffiffiffiffi in flat spacetime [4]. τ ¼ dt −g00: ð6:2Þ VII. QUANTUM-LIMITED GRAVITATIONAL-WAVE DETECTION In these coordinates, then, we are interested in a metric We now consider estimating the amplitude of a gravi- ðsÞ η perturbation of the form g00 ¼ 00 þ saðxÞ, where a is a tational wave. To a good approximation, a gravitational smooth function of the coordinate four-position x. Using wave can be modeled by a small perturbation of Minkowski the approximation spacetime satisfying the linearized Einstein equations. Thus we write 2 δ 2 ≃ df δ 2 h½ fðXÞ i hð XÞ i; ð6:3Þ gμν ¼ ημν þ hμν; ð7:1Þ dX X¼hXi where for a plane-fronted, parallel-propagating wave, the uncertainty in the metric is related to the uncertainty in linearly polarized along the x and y axes and cast in the the proper time by transverse-traceless gauge [9], the nonvanishing compo- nents of the metric perturbation are Z −2 δ 2 4 δτ 2 − − hð sÞ i¼ dtaðxÞ hð Þ i: ð6:4Þ hxx ¼ hyy ¼ Aþfðz tÞ: ð7:2Þ

Physical solutions have a suitably localized envelope along Then the relation (4.2) becomes the propagation direction, which can be approximated by a

105004-11 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) compactly supported function of z − t. Physical solutions the probe electromagnetic field as being confined within a also are not exactly plane-fronted, but rather are confined in laser-powered interferometer, as in the LIGO detectors the directions transverse to the propagation direction or, as [34–36], but there is no need to specialize to this particular in the case of astrophysical sources, have spherical wave field configuration. Instead, we let the probe be a free fronts. We assume for our analysis that any physical electromagnetic field: the field is turned on, receives an deviations from a plane-fronted wave are negligible on imprint from the gravitational wave as it propagates freely the spatial scales of our probe field. Assuming that the through the gravitational wave, and is then turned off. gravitational-wave detector, which might be the electro- Recall that the Cram´er-Rao bound optimizes over all magnetic field confined within a laser-powered interfer- measurements we could make on the probe field, so we ometer, is compactly supported in space, the intersection do not have to specify what measurement is used to read out with the detector’s support is compactly supported in the imprint of the gravitational wave on the electromagnetic spacetime. The volume integrals of concern to us are field, although we will have something to say about this as therefore well defined. we proceed. This approach allows us to use the free For simplicity, however, we analyze broadband detection electromagnetic field and the free-field commutators. In of a gravitational wave, i.e., detection that is essentially this approach, it is clear that we do not find a “standard instantaneous compared to the scale of variation (period or quantum limit” that is enforced by back-action forces that wavelength) of the gravitational wave. What this means is act on masses that confine the field, because there are no that the probe field’s support is sufficiently confined such masses. spatially and temporally relative to the gravitational wave’s Notice that if we did regard the field as being in an envelope and wavelength that within the probe’s window of interferometric configuration, there would need to be beam observation, the gravitational wave is well approximated by splitters and mirrors to split, confine, and recombine the a constant: field. To neglect back-action and thus to be consistent with the present calculation, we could make these optical hxx ¼ −hyy ≃ A: ð7:3Þ elements sufficiently massive that they are unaffected by the field’s back-action radiation-pressure noise and thus For consistency with this assumption and to ensure that move on geodesics. All of this is consistent with the now the relevant volume integrals remain well defined, we well-established result that there is no back-action-enforced assume a finite duration of detection. Since our perturbed “standard quantum limit” that fundamentally limits inter- metric happens to be in Gaussian normal coordinates (i.e., ferometric gravitational-wave detectors. The absence g00 ¼ −1 and g0i ¼ 0), the resulting coordinate bounds of of a back-action-enforced fundamental limit for interfero- integration are independent of the perturbation. Our metric detectors follows from a substantial body of work assumptions amount to saying that the probe field is to on specialized, back-action-evading designs for laser- be turned on and off, i.e., emitted and absorbed, within a interferometer gravitational-wave detectors [37–39] and compact spatial region in such a way that it senses an from general analyses of quantum limits on the detection essentially instantaneous amplitude of the gravitational of waveforms [40,41]. wave over this compact spatial region. For the electromagnetic field, the diagonal components The generator (4.4) of changes in the probe field is of the stress tensor are Z 1 1 2 2 2 2 2 2 1 2 2 ˆ 4 ˆ μν d Tˆ jj Eˆ Eˆ Eˆ Bˆ Bˆ Bˆ − Eˆ Bˆ ; P ¼ d xT gμνðAÞ ¼ ð x þ y þ z þ x þ y þ z Þ ð j þ j Þ 2 dA 8π 4π Z 0 1 ð7:5Þ 4 ˆ xx d ˆ yy d ¼ d x T g ðAÞþT g ðAÞ 2 dA xx dA yy Z 0 0 where Eˆ 2 and Bˆ 2 are normally ordered and we use cgs 1 j j ¼ d4xðTˆ xx − Tˆ yyÞ; ð7:4Þ Gaussian units with c ¼ 1. The local generator of changes 2 in the field due to the gravitational wave is 1 1 where we have assumed the fiducial amplitude is zero (i.e., ˆ xx ˆ yy ˆ 2 ˆ 2 ˆ 2 ˆ 2 ðT − T Þ¼ ðEy − Ex þ By − BxÞ perturbation about flat spacetime). The domain of integra- 2 8π 1 X tion encloses the finite extent of the probe mean field. ∶ ˆ2∶ ¼ 8π rσ fσ : ð7:6Þ We now take the probe field to be the electromagnetic σ field, having a large mean field that is turned on and off, as ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ we have discussed. We assume that the domain of inte- Here we let f1 ¼ Ey, f2 ¼ Ex, f3 ¼ By, f4 ¼ Bx and gration in Eq. (7.4) is large, both temporally and spatially, r1 ¼ r3 ¼ 1, r2 ¼ r4 ¼ −1. Beginning with the last form, compared to the scales of variation (periods and wave- we indicate normal ordering explicitly where it is needed. lengths) of the mean electromagnetic field. We could regard The generator (7.4) becomes

105004-12 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) Z 1 X ∶ ˆ∶ 4 ∶ ˆ2∶ zero-mean-field reference states, except where noted oth- P ¼ 8π rσ d x fσ : ð7:7Þ σ erwise, and we eventually get to the case of a squeezed- vacuum state as the reference state that provides optimal Now we express the electric and magnetic fields as a sum sensitivity under the assumptions we make. Our main of a mean field and field fluctuations, defined as the conclusions are aimed at the case where the displacement deviation from the mean: is large, in which case we only keep the terms that are leading order in the displacement. ˆ ˆ ˆ fσðx;tÞ¼hfσðx;tÞi þ Δfσðx;tÞ: ð7:8Þ The expectation value of the generator (7.9) is ˆ ˆ This puts the generator in the form h∶P∶i¼P þh∶F∶i: ð7:14Þ ˆ ˆ ˆ ∶P∶ ¼ P þ ΔX 1 þ ∶F∶; ð7:9Þ Using

ˆ ˆ 2 ˆ ˆ 2 ˆ ˆ where ∶P∶∶P∶ ¼ P þ 2P∶F∶ þðΔX 1Þ þ ∶F∶∶F∶ Z ˆ ˆ ˆ ˆ ˆ 1 X þ 2PΔX 1 þ ΔX 1∶F∶ þ ∶F∶ΔX 1; ð7:15Þ 4 ˆ 2 P ¼ 8π rσ d xhfσi ; ð7:10Þ σ we have Z 1 X Xˆ 4 ˆ ˆ ∶ ˆ∶∶ ˆ∶ 2 2 ∶ ˆ∶ ΔXˆ 2 ∶ ˆ∶∶ ˆ∶ 1 ¼ 4π rσ d xhfσifσ; ð7:11Þ h P P i¼P þ Ph F iþhð 1Þ iþh F F i σ ˆ 2 ˆ 2 ˆ ˆ ˆ 2 Z ¼h∶P∶i þhðΔX 1Þ iþh∶F∶∶F∶i − h∶F∶i : 1 X ∶ ˆ∶ 4 ∶ Δ ˆ 2∶ ð7:16Þ F ¼ 8π rσ d x ð fσÞ : ð7:12Þ σ Here we assume that the odd moments of the reference ˆ Notice that we do not need to normal order X 1 because it is (zero-mean-field) state are zero, which is the case if the linear in field operators. reference state is Gaussian or is invariant under parity and Note that by our formalism, the quantum fields in time reversal. Eqs. (7.5)–(7.12) need only be evaluated in the fiducial Rewriting Eq. (7.16) in terms of the variance, we get spacetime, which in the present case is flat. Therefore, the ˆ 2 ˆ ˆ ˆ 2 ˆ 2 ˆ 2 vacuum state is unambiguous, and the splitting of the field hðΔPÞ i¼h∶P∶∶P∶i − h∶P∶i ¼hðΔX 1Þ iþhðΔFÞ i: operators into positive- and negative-frequency parts and the use of normal ordering are appropriate and well defined. ð7:17Þ (For a discussion of the issues arising in curved spacetime, see Sec. 1 of Ref. [5].) Thus we have We subsume the normal ordering into the definition of the definition of the variance ðΔFˆÞ2 when using this notation. ˆ2 ˆðþÞ ˆð−Þ 2 Δ ˆ 2 ∶Δfσ∶ ¼ ∶½Δfσ þ Δfσ ∶ The term hð FÞ i is mean field independent, so for large mean field, we can drop it. Before doing so, however, it is 2Δ ˆð−ÞΔ ˆðþÞ Δ ˆðþÞ2 Δ ˆð−Þ2 ¼ fσ fσ þ fσ þ fσ : ð7:13Þ worth taking a closer look at the mean-field-independent contributions. When we put the right-hand side of The free-field commutators and vacuum correlators that Eq. (7.13) into the spacetime integral (7.12) to get ∶Fˆ∶, we need are summarized in Appendix C. we can expand the field operators in the last two terms of Our separation of the field operators into a mean field Eq. (7.13) into integrals over the wave vectors of free-field plus field fluctuations is different from our treatment in plane-wave modes, as in Appendix C. Performing the Sec. V, where we separated the stress-energy, which is spacetime integral first, the amplitudes for a pair of wave generally quadratic in field operators, into its mean and its vectors, k and k0, average to nearly zero, except for field fluctuation about the mean. To identify the mean-field- modes whose period and wavelength are as large or larger independent contributions to the variance, we view the state than the temporal and spatial extent of the region of as being obtained from a zero-mean-field state by a integration. Realistic measurement devices such as laser displacement operator D, which is generated by a linear interferometers are neither designed for nor capable of function of the fields. This operator is determined by † detecting such low-frequency photons. If we neglect these requiring that D fσD ¼ fσ þhfσi. The displacement essentially dc contributions, we are left with parameter is the mean field. For the present purposes, Z the zero-mean-field state is the reference state and is 1 X ∶ ˆ∶ 4 Δ ˆð−ÞΔ ˆðþÞ considered fixed. One example is where this reference F ¼ 4π rσ d x fσ fσ ; ð7:18Þ state is the vacuum state. Our initial arguments apply to all σ

105004-13 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) where the equals sign now assumes that we have omitted the dc contributions. The corresponding variance is

hðΔFˆÞ2i¼h∶Fˆ∶∶Fˆ∶i − h∶Fˆ∶i2 Z 1 X 4 4 0 ˆð−Þ ˆðþÞ ˆð−Þ 0 0 ˆðþÞ 0 0 rσrσ0 d xd x Δfσ x;t Δfσ x;t Δf x ;t Δf x ;t ¼ 16π2 ðh ð Þ ð Þ σ0 ð Þ σ0 ð Þi σ;σ0 − Δ ˆð−Þ x Δ ˆðþÞ x Δ ˆð−Þ x0 0 Δ ˆðþÞ x0 0 h fσ ð ;tÞ fσ ð ;tÞih fσ0 ð ;tÞ fσ0 ð ;tÞiÞ: ð7:19Þ

If the electromagnetic field is excited into a coherent commutator implies a Heisenberg uncertainty relation, state, where the field fluctuations are those of vacuum, both precisely analogous to the position-momentum uncertainty h∶Fˆ∶i and hðΔFˆÞ2i, as calculated from Eqs. (7.18) and relation (1.2), (7.19) vanish. For coherent states, the only mean-field- ˆ ℏ2 independent contributions to the total variance of F come ˆ 2 ˆ 2 2 hðΔX 1Þ ihðΔX 2Þ i ≥ C : ð7:23Þ from the dc terms discarded in going from Eqs. (7.13) 4 to (7.18). If the field fluctuations are redistributed relative to To put these two observables on the same footing relative to vacuum, as in the squeezed state discussed below, the vacuum, we require that terms of hðΔFˆÞ2i given in Eq. (7.19) make the dominant ℏ ˆ 2 ˆ 2 mean-field-independent contribution, expressing the fact h0jX 1j0i¼h0jX 2j0i¼ C: ð7:24Þ that these nonvacuum field fluctuations are affected by the 2 presence of a gravitational wave and can be used to detect ˆ ˆ the wave. For sufficiently large mean field, these mean- The observables X 1 and X 2 are generalized quadrature field-independent contributions are small compared to components [42–44] for the single field mode that is ΔXˆ 2 determined with respect to the Minkowski vacuum by hð Þ1i, leaving us with ˆ the mean field according to the definition of X 1, with their ˆ 2 ˆ 2 ℏ 2 Xˆ hðΔPÞ i¼hðΔX 1Þ i; ð7:20Þ vacuum level of noise given by C= . We calculate 2 explicitly below after restricting to the case of a plane wave. ˆ as we assume henceforth. Equation (3.5) specifies the response of X 2 to the We can now summarize our results by saying that the gravitational wave, Cram´er-Rao bound (4.2) on the estimate of the gravita- ˆ tional-wave amplitude A is dX 2 i ˆ ˆ ¼ ½X 2; X 1¼C: ð7:25Þ dA ℏ ℏ2 ~ 2 ˆ 2 δA ΔX 1 ≥ : 7:21 hð Þ ihð Þ i 4 ð Þ Linear-response analysis gives the variance of an estimate ˆ of A based on a measurement of X 2, We could stop here, having confirmed the valuable lesson, generic to Cram´er-Rao bounds, that the precise determi- ΔXˆ 2 ΔXˆ 2 ℏ2 1 δ ~ 2 hð 2Þ i hð 2Þ i ≥ nation of the gravitational-wave amplitude requires that the hð AÞ i¼ ˆ 2 ¼ 2 ˆ 2 ; ˆ jdhX 2i=dAj C 4 hðΔX 1Þ i observable X 1, which in the presence of a large mean field generates the change in the probe-field state, be as ð7:26Þ uncertain as possible. In this case, however, we can say considerably more. matching the Cram´er-Rao bound (7.21). If the probe field is ˆ Since X 1 is linear in the fields, one can find an placed in a minimum-uncertainty state relative to the ˆ ˆ observable X 2, conjugate to X 1 and also linear in the uncertainty relation (7.23), the bound (7.26) is saturated, fields, which is the observable one should measure to effect and it is particularly useful to write the variance of the ˆ estimate as the precise determination of A. The commutator of X 1 and ˆ X 2 is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔXˆ 2 ℏ ΔXˆ 2 δ ~ 2 hð 2Þ i hð 2Þ i ˆ ˆ hð AÞ i¼ 2 ¼ ˆ 2 : ð7:27Þ ½X 1; X 2¼iℏC; ð7:22Þ C 2C hðΔX 1Þ i where the real constant C is to be determined (we can We stress that Eq. (7.27) is not a general expression for the ˆ make C positive by, say, changing the sign of X 2). The Cram´er-Rao bound, but rather is the form the bound

105004-14 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) assumes for minimum-uncertainty states relative to the where α1;kσ ¼hakσi. The assumption of a nearly plane uncertainty relation (7.23). wave propagating in the x direction is that α1;kσ has The physical content here is that if the field is excited substantial support only for k pointing nearly along the into a coherent state, the uncertainties in the quadrature x direction, with linear polarization nearly along the y components are equal, and the variance of the estimate of A, direction, in which case we drop the polarization index and ˆ equal to ℏ=2C, is set by the vacuum-level noise in X 1 and write it as α1;k (formally, we might write α1;kσ ¼ δσyα1;k); ˆ X 2. To achieve a sensitivity better than ℏ=2C, one should this leads to the final form in Eq. (7.31). ˆ ˆ ˆ squeeze the vacuum so that the uncertainty in X 2 decreases We assume that X 2 looks the same as X 1, ˆ and the uncertainty in X 1 increases, as in the original Z proposal for decreasing shot noise in a laser-interferometer 1 gravitational-wave detector by using squeezed light [45],a Xˆ 4 x ˆ x 2 ¼ 4π d xE2ð ;tÞEyð ;tÞ; ð7:32Þ proposal that has been implemented in large-scale laser- interferometer detectors [46,47] and might be incorporated into Advanced LIGO [48]. but with a different (real) waveform E2, which is also a There is one task remaining, quite an important one, and nearly plane wave propagating in the x direction, with that is to evaluate the constant C. To do that, we specialize a linear polarization nearly along the y direction, bit, to the case where the mean probe field is that of a nearly plane wave propagating in the x direction and linearly polarized along the y axis. We do not need to assume that ðþÞ ð−Þ E2ðx;tÞ¼E ðx;tÞþE ðx;tÞ; ð7:33Þ this wave is close to monochromatic, but we do assume that 2 2 the transverse extent of the wave is much larger than the wave’s typical wavelengths. We neglect the small correc- tions to a plane wave due to the finite transverse extent. ðþÞ x ð−Þ x ˆ ˆ ˆ E2 ð ;tÞ¼E2 ð ;tÞ With these assumptions, we have hExi¼hEzi¼hBxi¼ Z ˆ X d3k pffiffiffiffiffiffiffiffiffiffiffiffi hB i¼0 and 2πℏωα e e iðk·x−ωtÞ y ¼ i 3 2;kσ kσ · ye σ ð2πÞ ˆ ˆ x Z 3 hEyi¼hBzi¼E1ð ;tÞ; ð7:28Þ pffiffiffiffiffiffiffiffiffiffiffiffi d k iðk·x−ωtÞ ¼ i 2πℏωα2 ke : ð7:34Þ ð2πÞ3 ; where the (real) waveform E1ðx;tÞ is mainly a function of x − t and only a weak function of y and z. With these assumptions, we have Again we understand that α2;k has support only for k Z 1 close to the x direction and corresponds to linear polari- ˆ 4 X x ˆ x zation nearly along the y direction (α2;kσ ¼ δσ α2 k). 1 ¼ 4π d xE1ð ;tÞEyð ;tÞ: ð7:29Þ y ; The following calculations show that the above assumption is warranted. Specifically, we find in Eq. (7.43) that E2 can It is useful to divide E1 into positive- and negative- frequency parts and to write these in terms of the Fourier be obtained from E1 by a 90° phase shift of every transform, monochromatic mode that contributes to E1, as one might expect for a broadband version of conjugate quadrature x ðþÞ x ð−Þ x components. E1ð ;tÞ¼E1 ð ;tÞþE1 ð ;tÞ; ð7:30Þ ˆ Notice that in the expressions (7.29) and (7.32) for X 1 ˆ ðþÞ ð−Þ and X 2, we can extend the spatial integrals over all of space E1 ðx;tÞ¼E1 ðx;tÞ Z because the waveforms E1ðx;tÞ and E2ðx;tÞ are zero X d3k pffiffiffiffiffiffiffiffiffiffiffiffi 2πℏωα e e iðk·x−ωtÞ outside the original domain of spatial integration. ¼ i 3 1;kσ kσ · ye σ ð2πÞ To determine C, we use the field commutators and Z vacuum correlators of Appendix C [see Eqs. (C12) and d3k pffiffiffiffiffiffiffiffiffiffiffiffi 2πℏωα iðk·x−ωtÞ ¼ i 3 1;ke ; ð7:31Þ (C14)] to find the commutator (7.22) and the second ð2πÞ moments (7.24):

Z 1 Xˆ ; Xˆ d4xd4x0E x;t E x0;t0 Eˆ x;t ; Eˆ x0;t0 ½ 1 2¼16π2 1ð Þ 2ð Þ½ yð Þ yð Þ Z ℏ ∂2 ∂2 i 4 4 0 0 0 0 0 ¼ d xd x E1ðx;tÞE2ðx ;tÞ − Gðx − x ;t− t Þ; ð7:35Þ 16π2 ∂t2 ∂y2

105004-15 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) Z 1 1 0 Xˆ 2 0 d4xd4x0E x;t E x0;t0 0 Eˆ x;t Eˆ x0;t0 Eˆ x0;t0 Eˆ x;t 0 h j aj i¼16π2 að Þ að Þ 2 h j½ yð Þ yð Þþ yð Þ yð Þj i Z ℏ ∂2 ∂2 ¼ d4xd4x0E ðx;tÞE ðx0;t0Þ − þ Dðx − x0;t− t0Þ;a¼ 1; 2: ð7:36Þ 16π3 a a ∂t2 ∂y2

Here Gðx;tÞ, the difference between retarded and advanced Green functions, is defined in Eq. (C10), and Dðx;tÞ, the principal value of the inverse of the invariant interval, is defined in Eq. (C11). In Eqs. (7.35) and (7.36), we can integrate by parts twice on the y derivatives. The boundary terms vanish because we can take the boundary of the region of integration to be outside the spatial extent of the waveforms E1 and E2, and we can neglect the resulting integrals because E1 and E2 are weak functions of y. The upshot is that we can omit the y derivatives in Eqs. (7.35) and (7.36).UsingEqs.(C10) and (C11) to start getting back into the Fourier domain, we have Z Z Z Z Z ℏ 3 ˆ ˆ i d k −iωt 3 ik·x 0 iωt0 3 0 0 0 −ik·x0 ½X 1; X 2¼ Im ω dte d xE1ðx;tÞe dt e d x E2ðx ;tÞe ; ð7:37Þ 4π ð2πÞ3 Z 3 Z Z Z Z 2 ℏ d k 0 0 h0jXˆ j0i¼ Re ω dte−iωt d3xE ðx;tÞeik·x dt0eiωt d3x0E ðx0;t0Þe−ik·x ;a¼ 1; 2: ð7:38Þ a 8π ð2πÞ3 a a

The spatial Fourier transforms are

Z pffiffiffiffiffiffiffiffiffiffiffiffi −iωt 3 x ik·x 2πℏω − α α −2iωt e d xEað ;tÞe ¼ ð i a;k þ i a;−ke Þ: ð7:39Þ

Z The counter-rotating terms average to nearly zero in the 1 3 ℏ d k ωτ 2 α 2 temporal integrals, so we discard them and obtain C ¼ 3 ð Þ j 1;kj : ð7:44Þ 2 ð2πÞ Z Z pffiffiffiffiffiffiffiffiffiffiffiffi −iωt 3 x ik·x − τ 2πℏωα dte d xEað ;tÞe ¼ i a;k; ð7:40Þ Consider now a nearly monochromatic mean field with wave vector k ¼ ωex. The vacuum level of noise in the where τ is the time interval over which the mean field is estimate of the gravitational-wave amplitude is turned on. Our final results for the commutator and vacuum ℏ=2C ¼ 1=ðωτÞ2n¯, where n¯ is the number of photons second moment are ðþÞ iωðx−tÞ carried by the mean field. We have E1 ∝ α2e and Z ðþÞ ∝ α iωðx−tÞ α α iϕ ∝ 1 d3k E2 i 2e ; writing 2 ¼j 2je , we have E1 Xˆ Xˆ ℏ ℏ ωτ 2 α α ½ 1; 2¼i 3 ð Þ Imð 1;k 2;kÞ ; ð7:41Þ α2 cos½ωðt − xÞ − ϕ and E2 ∝ α2 sin½ωðt − xÞ − ϕ. Thus 2 ð2πÞ ˆ ˆ X 1 and X 2 are proportional to the standard quadrature Z ℏ 1 3 components for a monochromatic field mode. ˆ 2 d k 2 2 0 X 0 ℏ ωτ α k ;a1; 2: h j aj i¼2 2 2π 3 ð Þ j a; j ¼ In the case of a nearly monochromatic mean field, the ð Þ quantum-limited sensitivity (7.27) for detecting a gravita- ð7:42Þ tional-wave amplitude becomes

A glance at Eqs. (7.22) and (7.24) shows that the quantities sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in large parentheses are all equal to C. Since we want this to 1 ΔXˆ 2 δ ~ 2 hð 2Þ i be true whatever the probe waveform is, we must have hð AÞ i¼ 2 ˆ 2 : ð7:45Þ ðωτÞ n¯ hðΔX 1Þ i

α2;k ¼ iα1;k; ð7:43Þ If the field is excited into a nearly monochromatic coherent i.e., as promised, E2 is obtained from E1 by a 90° phase state, the quadrature components have equal, vacuum- 2 shift of every monochromatic mode that contributes to E1. level uncertainties, n¯ ¼hnˆi¼hðΔnˆÞ i is the expectation Finally, we obtain value and the variance in the number of photons, and

105004-16 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) ℏC the sensitivity is shot-noise-limited, i.e., hðδA~ Þ2i1=2 ¼ ΔXˆ 2 2r hð 1Þ i¼ 2 e ; 1=ωτ nˆ 1=2. This result has a physically intuitive interpre- h i ℏ ˆ 2 C −2r tation. The gravitational wave changes the coordinate hðΔX 2Þ i¼ e : ð7:50Þ speed of light in the x direction by −A=2, leading to a 2 phase shift δϕ ¼ðωτÞA=2; the shot-noise limit on detec- It thus beats shot-noise-limited sensitivity by a factor ting the gravitational-wave amplitude translates to of e−2r. 2 2 1 hðδϕÞ ihðΔnˆÞ i¼4, which is the conventional uncer- tainty-principle bound on phase and photon number. To VIII. OTHER APPLICATIONS ˆ do better than shot noise, one can squeeze the X 2 quad- We now consider briefly a few of the many other rature, reducing its uncertainty while increasing the uncer- applications of our formalism. Xˆ tainty in the 2 quadrature. Cosmology is one field where accurate measurement of A bonus of our approach is that Eq. (7.27) gives us the gravitational parameters is of obvious interest. For a simple quantum limit on detecting a gravitational wave using a example, consider the spatially closed Friedmann- nearly plane-wave, but broadband probe field: Lemaître-Robertson-Walker spacetime. This is a universe filled with a uniform density of matter, e.g., “galaxies,” and Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 −1 ΔXˆ 2 radiation. At any instant in time, in the comoving frame of ~ 2 d k 2 2 hð 2Þ i hðδAÞ i¼ ðωτÞ jα1 kj : ð7:46Þ the galaxies, the universe looks the same everywhere 2π 3 ; ΔXˆ 2 ð Þ hð 1Þ i (homogeneous) and in all directions (isotropic). The metric for this universe is given by [9] Comparison to the monochromatic sensitivity (7.45) shows 2 that the way to generalize ðωτÞ n¯ to a broadband mean field ds2 ¼ −dt2 þa2ðtÞ½dχ2 þsin2 χðdθ2 þsin2 θdϕ2Þ; ð8:1Þ is to integrate over contributions from all the monochro- matic modes. If the field is excited into a broadband where t is the proper time of an observer comoving with coherent state, the quantum-limited sensitivity is given any of the galaxies. The spatial coordinates χ; θ; ϕ describe by a sort of generalized shot noise quantified by this homogeneous and isotropic three-spheres of constant frequency-weighted integration over mean photon numbers proper time t. The function aðtÞ, known as the expansion in the monochromatic modes. parameter, is the ratio of the proper distance between any To do better than shot-noise-limited sensitivity, one two galaxies at the initial time t ¼ 0 and the time t. should put the appropriate field mode into a squeezed During an infinitesimal duration of proper time dt a state. We can write down the required squeezed state by photon travels the distance dη ¼ dt=aðtÞ. It is convenient noting that for a nearly planepffiffiffiffiffiffiffiffiffiffiffi wave, the field quadra- to use η, known as the conformal time coordinate, as the Xˆ ℏ 2 ˆ ˆ† Xˆ pturesffiffiffiffiffiffiffiffiffiffiffi take the form 1 ¼ C= ðb þ b Þ and 2 ¼ time parameter. Transforming to conformal time has the † ℏC=2ð−ibˆ þ ibˆ Þ (see Appendix C), where effect of shunting the time dependence into a conformal factor. We furthermore consider a universe dominated by Z η 1 − η τ d3k matter, in which case að Þ¼amaxð cos Þ and the ˆ ffiffiffiffiffiffiffiffiffi ℏωα ˆ b ¼ p 3 1 kaky ð7:47Þ metric becomes 2ℏC ð2πÞ ; 2 ˆ† 2 amax 1 − η 2 − η2 χ2 and b satisfy the canonical bosonic commutation ds ¼ 4 ð cos Þ ½ d þ d ˆ ˆ † relation, ½b;b ¼1. This means that the desired minimum- 2 2 2 2 þ sin χðdθ þ sin θdϕ Þ; ð8:2Þ uncertainty state is the squeezed state η 2π 1 where runs between 0 at the beginning of expansion to μbˆ†−μbˆ ˆ† 2 − ˆ2 0 at the end of recontraction. We wish to estimate the e exp 2 r½ðb Þ b j i; ð7:48Þ parameter amax which controls the maximum size the universe reaches before contraction commences. Since where μ and r are real, with μ chosen to give the assumed 2 dgμν=damax ¼ð =amaxÞgμν, the operator (4.4) becomes mean field E1ðx;tÞ. Indeed, one can see that this state has Z μτ 3 ffiffiffiffiffiffiffiffiffi ℏωα ˆ amax 4 2 ˆ μν hakyi¼p 1;k; ð7:49Þ Pða Þ¼ dηdχdθdϕð1 − cos ηÞ sin χ sin θgμνT 2ℏC max 16 ZM ffiffiffiffiffiffiffiffiffi 5 p amax 6 2 ˆ ηη so consistency requires that μ 2ℏC=ℏωτ. The sque- ¼ dηdχdθdϕð1 − cos ηÞ sin χ sin θ½−T ¼ pffiffiffiffiffiffiffiffiffi 64 ˆ μ Xˆ 2ℏ μ 2 ωτ M ezed state (7.48) has hbi¼ , h 1i¼ C ¼ C= , ˆ χχ 2 ˆ θθ ˆ ϕϕ 2 ˆ þ T þ sin χðT þ T sin θÞ: ð8:3Þ hX 2i¼0, and

105004-17 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) It is interesting to note that since we are estimating a IX. CONCLUSION scale factor, the above integrand is proportional to the trace In this paper we presented the quantum Cram´er-Rao of the stress-energy tensor. Thus, for any field with a lower bound for the uncertainty in estimating parameters traceless stress-energy tensor, such as the free electromag- describing a spacetime metric. Our specific derivation Δ ˆ 2 netic field, hð PÞ i vanishes, and we get no information applies for any quantum state on an arbitrary globally about the scale factor. This is an expression of the well- hyperbolic manifold. To demonstrate the utility of our known scale invariance of the electromagnetic field. formalism, we applied it to the estimation of metric To give another cosmological example, suppose we are components and found uncertainty principles akin to those Λ interested in measuring the cosmological constant in a de found by Unruh [10] using heuristic arguments. We also Sitter universe. Again using the conformal time coordinate, considered the quantum estimation of gravitational-wave the line element can be written as amplitudes and obtained generalizations of known quantum limits for laser interferometers such as LIGO. 3 ds2 ¼ sec2ηð−dη2 þ dχ2 þ sin2χðdθ2 þ sin2θdϕ2ÞÞ; Λ ACKNOWLEDGMENTS ð8:4Þ T. G. D. thanks K. Fredenhagen and support from DESY, as well as R. Verch and support from the University of which gives us Leipzig. We also thank A. Kempf, T. C. Ralph, J. Doukas, Z C. Fewster, L. H. Ford, and S. Carlip for useful discussions. 9 ˆ Λ − η χ θ ϕ 4η 2χ θ ˆ μν This work was supported in part by US National Science Pð Þ¼ 3 d d d d sec sin sin gμνT : ð8:5Þ 2Λ M Foundation Grant Nos. PHY-1314763 and PHY-1521016. This work includes contributions of the National Institute Again, because we are estimating a scale factor, the of Standards and Technology, which are not subject to U.S. integrand is proportional to the trace of the stress-energy copyright. tensor, which vanishes for the free electromagnetic field. These and other cosmological parameters represent APPENDIX A: CATEGORY THEORY overall scale factors of the universe. As such, the con- FRAMEWORK formally invariant electromagnetic field alone is not an In this appendix we briefly review category theory as adequate probe. For example, it is only possible to measure employed by Brunetti, Fredenhagen, and Verch in Ref. [5]. the cosmological redshift of light if the atomic emission A primary motivation is that it provides a convenient way to spectrum of its source is also known. Indeed, Penrose has rigorously define the change in a quantum observable due essentially argued that should all matter in the universe to a spacetime perturbation, which we outline here. To decay into photons, the scale of the universe would become begin with, a category (or more precisely, a concrete unobservable and thus physically irrelevant [49]. So a more category) consists of objects and functions between objects useful calculation should include fermionic fields, which called morphisms. A category of particular relevance for break this scale invariance. Alternatively, a massive scalar our purposes, given in Ref. [5], is the following: or boson field would also yield a finite Cram´er-Rao bound, Definition. Man is the category whose objects are since its stress-energy has nonvanishing trace. globally hyperbolic spacetimes and whose morphisms are A more down-to-earth application is that of a gravimeter. isometric embeddings (or in other words inclusion maps). One way to deal with this case is to approximate the near- We next consider quantum fields in those spacetimes, earth spacetime by a Schwarzschild metric, and assume the formulated in terms of C-algebras. The relevant category floor of the laboratory has constant Schwarzschild coor- of C-algebras is given in Ref. [5] as follows: dinate radius; then the problem of gravimetry becomes one Definition. Alg is the category whose objects of estimating the Schwarzschild mass of the Earth. This are C-algebras and whose morphisms are injective problem certainly lends itself to our method, provided the *-homomorphisms. appropriate stress-energy correlations in a Schwarzschild To associate C-algebras to our spacetimes, we use a background are calculated. Of course, the same calculation covariant functor from the spacetimes to C-algebras. would also be applicable to a black hole. A functor is a function between categories which maps objects Our method is also applicable to the estimation of to objects and morphisms to morphisms, such that the identity dynamical quantities of a probe in flat spacetime, such maps to the identity and compositions map to compositions. as acceleration, rotation, etc. One approach is to assume A covariant functor is a functor that maps domains to domains such dynamics are due to coupling with a nongravitational, and images to images (pictorially, it preserves the directions of classical field. The locally covariant approach then morphism arrows). Thus we arrive at the following [5]: “applies, mutatis mutandis, also to this case” [5], and so Definition. A locally covariant quantum field theory is a also does our Cram´er-Rao bound. covariant functor,

105004-18 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017)

FIG. 2. Diagram of the perturbed spacetime, along with the embedded subregions N, the locally covariant quantum field theory ∘ thereon, and the relevant morphisms, where g ¼ gð0Þ and g ¼ gðsÞ.

A∶ → ˆ π β Man Alg: into AðsÞ¼ ð gðsÞ AÞ. This process is termed relative Cauchy evolution [5]. Note that a locally covariant quantum field theory is local in the sense that it maps submanifolds of manifolds to APPENDIX B: COORDINATE INDEPENDENCE subalgebras of the corresponding algebras. We further require that any causal, locally covariant In this appendix we derive Eq. (5.15), and thus the quantum field theory obey the following axiom [5]: coordinate independence of ðΔPˆÞ2, without the assumption Axiom A . (Time-slice axiom) If is a locally covariant of a single coordinate patch covering K. This assumption is ∈ ψ ∈ quantum field theory, ðN;gÞ; ðM; gÞ Man, and not valid, for example, when the interior of K is not ψ homððN;gÞ; ðM; gÞÞ such that ðN;gÞ contains a Cauchy homeomorphic to R4. More generally, it is often simply A ψ A A surface of ðM; gÞ, then ð Þð ðN;gÞÞ ¼ ðM;gÞ. more convenient to use multiple coordinate patches. In other words, the algebra associated with a Cauchy For the purposes of this proof, however, we avoid surface of a manifold determines the algebra associated explicitly juggling multiple coordinate transition maps with the entire manifold. by considering each to be locally induced by a global The perturbed spacetime discussed above, along with the diffeomorphism φðsÞ∶M → M that depends continuously embedded subregions N , the locally covariant quantum ∈ 0 1 φðsÞ field theory thereon, and the relevant morphisms, can all be on the parameter s ½ ; . This induces a pushforward ∘ ð0Þ of a contravariant tensor field or (in the same direction) a represented by the diagram in Fig. 2, where g ¼ g and φðsÞ −1 ðsÞ pullback of the inverse diffeomorphism ðð Þ Þ of a g ¼ g . Note that the assumption of the time-slice axiom ðsÞ A → A covariant field, which we will also denote by φ . Note that implies the morphisms ðN; gN Þ ðM; gÞ are bijec- ðsÞ tive. Therefore, the arrows on the right-hand side of the if it is restricted to a coordinate patch φ is related to the μ φðsÞ μ diagram are invertible. This allows these morphisms to be transformation Lα0 ðsÞ of Sec. V by ð vÞα0 ¼ Lα0 ðsÞvμ. composed in such a way as to construct an automorphism (And notice that in the classic index analysis of Sec. V, the 0 on AðM; gð ÞÞ: prime serves double duty, denoting both this s-dependent coordinate transformation, and its s ¼ 0 instance.) −1 −1 − − To further obviate the need for explicit coordinate charts, β ¼ αψ ∘ αψ ∘ αψ þ ∘ α þ : ðA1Þ g ∘ g g ψ ∘ we use here Penrose’s abstract index notation [50], denoted by latin indices. Like coordinate component indices, the number of such subscripted/superscripted indices indicate By the Gelfand-Neimark-Segal construction [27–29], tensor ranks, and repeated indices indicate tensor contrac- every C-algebra admits a linear *-representation π by tions. But Penrose’s abstract indices do not index coor- β bounded operators on a Hilbert space. Thus g induces an dinate components; rather they signify entire tensors. automorphism on Hilbert-space operators. Thus any oper- For example, gμν ∈ R while gab ∈ T M ⊗ T M. Thus ator Aˆ ¼ πðAÞ, where A ∈ AðM; gð0ÞÞ, i.e., any operator Penrose’s abstract indices provide all the convenience of associated with our fiducial spacetime, can be said to indices without any of the commitment to coordinates. In evolve under our s-parametrized spacetime perturbation these terms, our objective is to show

105004-19 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) Z Z d d μ φðsÞ ab φðsÞ ðsÞ μ ab ðsÞ d φðsÞ ðsÞ T g ¼ d ðsÞ T g : ðB1Þ g ab g ab K ds s¼0 K ds s¼0

ˆ ab Assuming that ∇aT ¼ 0,wehave

Z Z ∘ ∘ 0 ˆ 0ab d ðsÞ ðsÞ 0 ˆ 0ab d ðsÞ ðsÞ0 dμ T φ g ¼ dμ T ϕ g ds ab ds ab K 0 ZK 0  ∘ 0 0 ˆ 0ab d ðsÞ d ðsÞ ð0Þ0 ¼ dμ T g þ ϕ g ds ab ds ab ZK  0 0  ∘ 0 0 ˆ ab d ðsÞ ¼ dμ T0 g þ ∇0 X0 þ ∇0 X0 ds ab a b b a ZK  0  ∘ ˆ ab d ðsÞ ¼ dμT g þ ∇ X þ ∇ X ds ab a b b a ZK  0  ∘ ˆ ab d ðsÞ ˆ ab ˆ ab ¼ dμ T g þ 2ð∇ ðT X Þ − ð∇ T X ÞÞ ds ab a b a b ZK 0 Z ∘ ∘ ˆ ab d ðsÞ ˆ ab ¼ dμT g þ 2 dμ∇ ðT X Þ ds ab a b ZK 0 ZK ∘ d μ ˆ ab ðsÞ 2 λ ˆ ab ¼ d T gab þ d naT Xb; ðB2Þ K ds 0 ∂K

Z where ϕðsÞ ¼ φðsÞ∘ðφð0ÞÞ−1, X0 generates ϕðsÞ, dλ is the X d3k pffiffiffiffiffiffiffiffiffiffiffiffi a Eˆ ðþÞ Eˆ ð−Þ† 2πℏω ˆ e iωðn·x−tÞ ¼ ¼ i 3 akσ kσe ; surface element induced on the boundary of K, na is the 2π ð0Þ σ ð Þ corresponding surface normal, and primes denote ϕ . Then neglecting the boundary term, as explained in Sec. V, ðC3Þ we achieve the desired diffeomorphism independence. Z Note that ϕðsÞ above corresponds to ϕðsÞ in the proof of X d3k pffiffiffiffiffiffiffiffiffiffiffiffi Bˆ ðþÞ Bˆ ð−Þ† 2πℏω ˆ n e iωðn·x−tÞ ∇ ˆ ab ¼ ¼ i 3 akσ × kσe : Theorem 4.2 in Ref. [5]. That theorem implies that aT σ ð2πÞ must vanish if the above integral is invariant with respect to ðsÞ ðC4Þ the transformation ϕ of the metric. The above result is a generalization of the converse. Here k ¼ ωn is the wave vector (ω ¼jkj is the angular frequency and n is a unit vector), and aˆkσ and ekσ are the APPENDIX C: COMMUTATORS AND VACUUM annihilation operator and unit (transverse) polarization k CORRELATION FUNCTIONS FOR THE vector for the plane-wave mode with wave vector and σ ELECTROMAGNETIC FIELD polarization . The creation and annihilation operators satisfy the canonical commutator, The field operators for the free electric and magnetic fields can be written as (we use cgs Gaussian units with ˆ ˆ† 2π 3δ k − k0 δ ½akσ; ak0σ0 ¼ð Þ ð Þ σσ0 : ðC5Þ c ¼ 1) The Hamiltonian for the electromagnetic field is ˆ ˆ ðþÞ ˆ ð−Þ Z Z Eðx;tÞ¼E ðx;tÞþE ðx;tÞ; ðC1Þ 1 ˆ 3 ∶ ˆ 00∶ 3 ∶Eˆ Eˆ Bˆ Bˆ ∶ H ¼ d x T ¼ 8π d x · þ · Z Z 1 X d3k 3 Eˆ ð−Þ Eˆ ðþÞ ℏω † Bˆ x;t Bˆ ðþÞ x;t Bˆ ð−Þ x;t ; ¼ 2π d x · ¼ 3 akσakσ ð Þ¼ ð Þþ ð Þ ðC2Þ σ ð2πÞ ðC6Þ where the positive- and negative-frequency parts of the fields are given by where the integral extends over all space.

105004-20 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) The positive- and negative-frequency parts of the fields have the free-field commutators, 2 2 Z 3 − − ∂ ∂ d k 1 ˆ ðþÞ x ˆ ð Þ x0 0 ˆ ðþÞ x ˆ ð Þ x0 0 2πℏ −δ iω½n·ðx−x0Þ−ðt−t0Þ ½Ej ð ;tÞ; Ek ð ;tÞ ¼ ½Bj ð ;tÞ; Bk ð ;tÞ ¼ jk 2 þ 3 e ; ðC7Þ ∂t ∂xj∂xk ð2πÞ ω

2 Z 3 − − ∂ d k 1 ˆ ðþÞ x ˆ ð Þ x0 0 − ˆ ðþÞ x ˆ ð Þ x0 0 2πℏϵ iω½n·ðx−x0Þ−ðt−t0Þ ½Ej ð ;tÞ; Bk ð ;tÞ ¼ ½Bj ð ;tÞ; Ek ð ;tÞ ¼ jkl 3 e : ðC8Þ ∂t∂xl ð2πÞ ω

The integral on the right evaluates to Z d3k 1 i 1 eiω½n·ðx−x0Þ−ðt−t0Þ ¼ − Gðx − x0;t− t0Þþ Dðx − x0;t− t0Þ; ðC9Þ ð2πÞ3 ω 4π 2π2 where Z d3k 1 δðt − jxjÞ − δðt þjxjÞ Gðx;tÞ¼−4πIm eiωðn·x−tÞ ¼ ðC10Þ ð2πÞ3 ω jxj is the difference between retarded and advanced Green functions, i.e., the solution of the homogeneous wave equation for an incoming spherical wave that reflects off the origin and becomes an outgoing spherical wave, and Z d3k 1 1 1 Dðx;tÞ¼2π2Re eiωðn·x−tÞ ¼ p:v: ¼ p:v: ðC11Þ ð2πÞ3 ω −t2 þjxj2 ðΔsÞ2 is the principal value of the inverse of the invariant interval. From these follow the free-field commutators and vacuum correlators [51]:

ˆ ˆ 0 0 ˆ ˆ 0 0 ½Ejðx;tÞ; Ekðx ;tÞ ¼ ½Bjðx;tÞ; Bkðx ;tÞ − ¼ 2iImð½Eˆ ðþÞðx;tÞ; Eˆ ð Þðx0;t0ÞÞ j k ∂2 ∂2 ℏ δ − x − x0 − 0 ¼ i jk 2 Gð ;t t Þ; ðC12Þ ∂t ∂xj∂xk

ˆ ˆ 0 0 ˆ ˆ 0 0 ½Ejðx;tÞ; Bkðx ;tÞ ¼ −½Bjðx;tÞ; Ekðx ;tÞ 2 ˆ ðþÞ x ˆ ð−Þ x0 0 ¼ iImð½Ej ð ;tÞ; Bk ð ;tÞÞ ∂2 0 0 ¼ −iℏϵjkl Gðx − x ;t− t Þ; ðC13Þ ∂t∂xl 1 1 0 ˆ x ˆ x0 0 ˆ x0 0 ˆ x 0 0 ˆ x ˆ x0 0 ˆ x0 0 ˆ x 0 2 h j½Ejð ;tÞEkð ;tÞþEkð ;tÞEjð ;tÞj i¼2 h j½Bjð ;tÞBkð ;tÞþBkð ;tÞBjð ;tÞj i − ¼ Reð½Eˆ ðþÞðx;tÞ; Eˆ ð Þðx0;t0ÞÞ j k ℏ ∂2 ∂2 −δ x − x0 − 0 ¼ jk 2 þ Dð ;t t Þ; ðC14Þ π ∂t ∂xj∂xk 1 1 0 ˆ x ˆ x0 0 ˆ x0 0 ˆ x 0 − 0 ˆ x ˆ x0 0 ˆ x0 0 ˆ x 0 2 h j½Ejð ;tÞBkð ;tÞþBkð ;tÞEjð ;tÞj i¼ 2 h j½Bjð ;tÞEkð ;tÞþEkð ;tÞBjð ;tÞj i ˆ ðþÞ x ˆ ð−Þ x0 0 ¼ Reð½Ej ð ;tÞ; Bk ð ;tÞÞ ℏ ∂2 0 0 ¼ ϵjkl Dðx − x ;t− t Þ: ðC15Þ π ∂t∂xl

105004-21 DOWNES, VAN METER, KNILL, MILBURN, and CAVES PHYSICAL REVIEW D 96, 105004 (2017) The field quadratures (7.29) and (7.32) for a nearly plane-wave mean field can be evaluated in the Fourier domain as follows (a ¼ 1, 2):

Z Z 1 1 Xˆ 4 x ˆ x 3 ð−Þ x ˆ ðþÞ x a ¼ 4π d xEað ;tÞEyð ;tÞ¼4π dtd xEa ð ;tÞEy ð ;tÞþH:c: Z 1 3 X d k τ ℏω α ;kσaˆkσ0 e e e ekσ0 : : ¼ 2 2π 3 a ð y · kσÞð y · ÞþH c ð Þ σ;σ0 Z 1 d3k ¼ τ ℏωα kaˆk þ H:c:: ðC16Þ 2 ð2πÞ3 a; y

ffiffiffiffiffiffiffiffiffi ˆ ˆ ˆ p In the first line we discard counter-rotating terms that i.e., b ¼ðX 1 þ iX 2Þ= 2ℏC, where average to nearly zero over the temporal integral; in the second line, we insert the Fourier transforms of the field operators and the wave forms and do the temporal integral Z over the duration τ for which the mean fields are turned on; τ d3k ˆ ffiffiffiffiffiffiffiffiffi ℏωα ˆ α b ¼ p 3 1;kaky ðC18Þ in the third line, we use the fact that a;kσ has support only 2ℏC ð2πÞ for k pointing nearly in the x direction, with polarization nearly along the y direction, to restrict the two sums over polarization to y linear polarization. Using Eq. (7.43), we can write [the constant C is given in Eq. (7.44)]. One can verify that rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi the quadrature components obey the commutation relation ˆ ˆ† ˆ ℏC ˆ ˆ† ˆ ℏC ˆ ˆ† (7.22) or, equivalently, that b and b satisfy the canonical X 1 ¼ ðbþb Þ; X 2 ¼ ð−ibþib Þ; ðC17Þ † 2 2 bosonic commutation relation, ½b;ˆ bˆ ¼1.

[1] C. W. Helstrom, Quantum Detection and Estimation Theory [11] M. Ahmadi, D. Bruschi, C. Sabin, G. Adesso, and I. (Academic Press, New York, 1976). Fuentes, Relativistic quantum metrology, exploiting rela- [2] A. S. Holevo, Probabilistic and Statistical Aspects of tivity to improve quantum measurement technologies, Sci. Quantum Theory (North-Holland, Amsterdam, 1982). Rep. 4, 4996 (2014). [3] S. L. Braunstein and C. M. Caves, Statistical Distance and the [12] M. Ahmadi, D. Bruschi, and I. Fuentes, Quantum metrology Geometry of Quantum States, Phys.Rev.Lett.72, 3439 (1994). for relativistic quantum fields, Phys. Rev. A 89, 065028 [4] S. L. Braunstein, C. M. Caves, and G. J. Milburn, General- (2014). ized uncertainty relations: Theory, examples, and Lorentz [13] J. Doukas, L. Westwood, D. Faccio, A. Di Falco, and I. invariance, Ann. Phys. (N.Y.) 247, 135 (1996). Fuentes, Gravitational parameter estimation in a waveguide, [5] R. Brunetti, K. Fredenhagen, and R. Verch, The generally Phys. Rev. D 90, 024022 (2014). covariant locality principle—A new paradigm for local [14] F. Kronz and T. Lupher, Quantum theory: von Neumann vs. quantum field theory, Commun. Math. Phys. 237, 31 (2003). Dirac, in The Stanford Encyclopedia of Philosophy (Summer [6] K. Eppley and E. Hannah, The necessity of quantizing the 2012 Edition), edited by E. N. Zalta, available online at plato gravitational field, Found. Phys. 7, 51 (1977). .stanford.edu/archives/sum2012/entries/qt-nvd. [7] J. Adelman, The necessity of quantizing gravity, arXiv: [15] G. Fleming, Comments on Paul Teller’s book “An Inter- 1510.07195. pretive Introduction to Quantum Field Theory”,inOnto- [8] Y. J. Ng and H. van Dam, Limit to space-time measurement, logical Aspects of Quantum Field Theory, edited by M. Mod. Phys. Lett. A 09, 335 (1994). Kuhlmann, H. Lyre, and A. Wayne (World Scientific, [9] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation Singapore, 2002), p. 135. (W. H. Freeman, San Francisco, 1973). [16] K. Sanders, The locally covariant Dirac field, Rev. Math. [10] W. G. Unruh, Why study quantum gravity?, Can. J. Phys. Phys. 22, 381 (2010). 64, 128 (1986). [17] M. Benini, MSc thesis, University of Pavia, Italy, 2011.

105004-22 QUANTUM ESTIMATION OF PARAMETERS OF CLASSICAL … PHYSICAL REVIEW D 96, 105004 (2017) [18] M. Benini, Relative Cauchy evolution for the vector a 22-Solar-Mass Binary Black Hole Coalescence, Phys. potential on globally hyperbolic spacetimes, Mathematics Rev. Lett. 116, 241103 (2016). and Mechanics of Complex Systems 3, 177 (2015). [37] W. G. Unruh, Readout state preparation and quantum non- [19] C. J. Fewster and M. J. Pfenning, A quantum weak energy demolition, in Quantum Optics, Experimental Gravitation, inequality for spin-one fields in curved space-time, J. Math. and Measurement Theory, edited by P. Meystre and M. O. Phys. (N.Y.) 44, 4480 (2003). Scully (Plenum, New York, 1983), p. 647. [20] M. J. Holland and K. Burnett, Interferometric detection of [38] H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. optical phase shifts at the Heisenberg limit, Phys. Rev. Lett. P. Vyatchanin, Conversion of conventional gravitational- 71, 1355 (1993). wave interferometers into quantum nondemolition interfer- [21] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum ometers by modifying their input and/or output optics, Phys. Metrology, Phys. Rev. Lett. 96, 010401 (2006). Rev. D 65, 022002 (2002). [22] M. D. Lang and C. M. Caves, Optimal quantum-enhanced [39] F. Ya. Khalili, Increasing future gravitational-wave interferometry, Phys. Rev. A 90, 025802 (2014). detectors’ sensitivity by means of amplitude filter cavities [23] C. J. Fewster and R. Verch, Dynamical locality and covari- and quantum entanglement, Phys. Rev. D 81, 122002 ance: What makes a physical theory the same in all space- (2010). times?, Ann. Inst. Henri Poincar´e 13, 1613 (2012). [40] M. Tsang and C. M. Caves, Coherent Quantum-Noise [24] C. J. Fewster, On the notion of, the same physics in all Cancellation for Optomechanical Sensors, Phys. Rev. Lett. spacetimes, arXiv:1105.6202. 105, 123601 (2010). [25] R. M. Wald, Quantum Field Theory in Curved Spacetime [41] M. Tsang, H. M. Wiseman, and C. M. Caves, Fundamental and Black Hole Thermodynamics (University of Chicago Quantum Limit to Waveform Estimation, Phys. Rev. Lett. Press, Chicago, 1994). 106, 090401 (2011). [26] M. R. Brown and A. C. Ottewill, The energy-momentum [42] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, operator in curved spacetime, Proc. R. Soc. A 389, 379 and M. Zimmermann, On the measurement of a weak (1983). classical force coupled to a quantum-mechanical oscillator. [27] I. M. Gelfand and M. A. Naimark, On the imbedding of I. Issues of principle, Rev. Mod. Phys. 52, 341 (1980). normed rings into the ring of operators on a Hilbert space, [43] C. M. Caves and B. L. Schumaker, New formalism for two- Mat. Sb. 12, 197 (1943). photon quantum optics. I. Quadrature phases and squeezed [28] I. E. Segal, Irreducible representations of operator algebras, states, Phys. Rev. A 31, 3068 (1985). Bull. Am. Math. Soc. 53, 73 (1947). [44] B. L. Schumaker and C. M. Caves, New formalism for two- [29] R. Haag, Local Quantum Physics: Fields, Particles, photon quantum optics. II. Mathematical foundation and Algebras, 2nd ed. (Springer, Berlin, 1996). compact notation, Phys. Rev. A 31, 3093 (1985). [30] T. G. Downes, PhD thesis, School of Mathematics & [45] C. M. Caves, Quantum-mechanical noise in an interferom- Physics, The University of Queensland, 2011. eter, Phys. Rev. D 23, 1693 (1981). [31] J. Borgman and L. H. Ford, The effects of stress tensor [46] LIGO Scientific Collaboration, A gravitational-wave fluctuations upon focusing, Phys. Rev. D 70, 064032 observatory operating beyond the quantum shot-noise limit, (2004). Nat. Phys. 7, 962 (2011). [32] A. Satz, F. D. Mazzitelli, and E. Alvarez, Vacuum polari- [47] LIGO Scientific Collaboration, Enhanced sensitivity of the zation around stars: Nonlocal approximation, Phys. Rev. D LIGO gravitational wave detector by using squeezed states 71, 064001 (2005). of light, Nat. Photonics 7, 613 (2013). [33] C. J. Fewster and L. H. Ford, Probability distributions for [48] J. Miller, L. Barsotti, S. Vitale, P. Fritschel, M. Evans, and quantum stress tensors measured in a finite time interval, D. Sigg, Prospects for doubling the range of Advanced Phys. Rev. D 92, 105008 (2015). LIGO, Phys. Rev. D 91, 062005 (2015). [34] LIGO Scientific Collaboration, Advanced LIGO, Classical [49] R. Penrose, Cycles of Time: An Extraordinary New View of Quantum Gravity 32, 074001 (2015). the Universe (The Bodley Head, London, 2010). [35] LIGO Scientific Collaboration and Virgo Collaboration, [50] R. Penrose, Structure of space-time, in Battelle Rencontres, Observation of Gravitational Waves from a Binary Black edited by C. DeWitt and J. Wheeler (Benjamin, New York, Hole Merger, Phys. Rev. Lett. 116, 061102 (2016). 1968). [36] LIGO Scientific Collaboration and Virgo Collaboration, [51] W. Greiner and J. Reinhardt, Field Quantization (Springer, GW151226: Observation of Gravitational Waves from New York, 1996).

105004-23