Quantum Enhanced Measurements Without Entanglement

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Article: Braun, Daniel, Adesso, Gerardo, Benatti, Fabio et al. (4 more authors) (2018) Quantum enhanced measurements without entanglement. Reviews of Modern Physics. 035006. ISSN 0034-6861 https://doi.org/10.1103/RevModPhys.90.035006

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[email protected] https://eprints.whiterose.ac.uk/ Quantum-enhanced measurements without entanglement

Daniel Braun1, Gerardo Adesso2, Fabio Benatti3,4, Roberto Floreanini4, Ugo Marzolino5,6, Morgan W. Mitchell7,8, Stefano Pirandola9 1Institute of Theoretical Physics, University Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany 2Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems (CQNE), School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 3Department of Physics, University of Trieste, 34151 Trieste, Italy 4Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34151 Trieste, Italy 5Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia 6Division of Theoretical Physics, Ruder BoˇskovićInstitute, 10000 Zagreb, Hrvatska 7ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 8ICREA – InstitucióCatalana de Recerca i Estudis Avan¸cats, 08015 Barcelona, Spain 9Computer Science and York Centre for Quantum Technologies, University of York, York YO10 5GH, United Kingdom

Quantum-enhanced measurements exploit quantum mechanical eﬀects for increasing the sensitivity of measurements of certain physical parameters and have great potential for both fundamental science and concrete applications. Most of the research has so far focused on using highly entangled states, which are, however, diﬃcult to produce and to stabilize for a large number of constituents. In the following we review alternative mechanisms, notably the use of more general quantum correlations such as quantum discord, identical particles, or non-trivial Hamiltonians; the estimation of thermodynamical parameters or parameters characterizing non-equilibrium states; and the use of quantum phase transitions. We describe both theoretically achievable enhancements and enhanced sensitivities, not primarily based on entanglement, that have already been demonstrated experimentally, and indicate some possible future research directions.

CONTENTS 1. Liftingspectrallimitations 28 2. Decoherence-enhancedmeasurements 29 I. Introduction 1 3.Coherentaveraging 30 A. Aim and scope 1 4.Quantumfeedbackschemes 32 B.Parameterestimationtheory 4 C. Quantumparameterestimationtheory 4 V. Thermodynamical and non-equilibrium steady states 34 A. Thermodynamical states and thermal phase II. Quantumcorrelationsbeyondentanglement 7 transitions 34 A. Parallel versus sequential strategies in unitary 1.Roleofquantumstatistics 35 quantum metrology 7 2. Interferometricthermometry 36 B. General results on the usefulness of entanglement 8 B. Thermodynamical states and quantum phase C. Role of quantum discord in parameter estimation with transitions 37 mixed probes 8 1. Quasi-freeFermionmodels 38 D. Black-box metrology and the interferometric power 10 2. Hubbard models 39 E. Quantumestimationofbosonicloss 10 3. Spin-1/2systems 39 F. Gaussianquantummetrology 11 4. Topologicalquantumphasetransitions 41 G. Quantumchanneldiscrimination 13 C. Non-equilibriumsteadystates 41 H. Averageprecisioninblack-boxsettings 15 D.Adaptivemeasurements 42

III. Identical particles 15 VI. Outlook 43 A. Particleandmodeentanglement 17 1.Particleentanglement 18 arXiv:1701.05152v2 [quant-ph] 16 Apr 2018 Acknowledgments 43 2.Modeentanglement 18 B. Modeentanglementandmetrology: bosons 19 References 44 C. Mode entanglement and metrology: fermions 23

IV.MoregeneralHamiltonians 24 A.Non-linearHamiltonians 24 I. INTRODUCTION B. Proposedexperimentalrealizations 25 1. Nonlinear optics 25 A. Aim and scope 2. Ultra-cold atoms 25 3. Nano-mechanicaloscillators 26 4. NonlinearFaradayrotation 26 Quantum-enhanced measurements aim at improving C. Observationsandcommentary 26 measurements of physical parameters by using quantum D. Nonlinear measurement under number-optimized effects. The improvement sought is an enhanced sensitiv- conditions 27 E. Signal amplification with nonlinear Hamiltonians 27 ity for a given amount of resources such as mean or maxi- F. OthermodificationsoftheHamiltonian 27 mum energy used, number of probes, number of measure- 2 ments, and integration time. Ideas in this direction go hi, respectively, taken for simplicity here as identical for back at least to the late 1960s when the effect of quantum all subsystems (Giovannetti et al., 2006). In fact, this noise on the estimation of classical parameters started to 1/√N scaling can be easily understood as a consequence be studied in a systematic way using appropriate math- of the central limit theorem in the simplest case that one ematical tools (Helstrom, 1969; Holevo, 1982). In the measures the systems independently. But since (1) is op- early 1980s first detailed proposals appeared on how to timized over all measurements of the full system, it also enhance the sensitivity of gravitational wave detectors implies that entangling measurements of all systems af- by using squeezed light (Caves, 1980, 1981). Nowadays, ter the parameter has been encoded in the state cannot squeezed light is routinely produced in many labs, and improve the 1/√N scaling. used for instance to enhance sensitivity in gravitational Unfortunately, there is no unique definition of the Stan- wave observatories (Aasi et al., 2013; Chua, 2015). dard Quantum Limit in the literature. Whereas in the Quantum-enhanced measurements have the potential described 1/√N scaling N refers to the number of dis- of enabling many important applications, both scientific tinguishable sub-systems, the term Standard Quantum and technological. Besides gravitational wave detection, Limit is used for example in quantum optics typically for there are proposals or demonstrations for the improve- a scaling as 1/√n¯ with the average number of photonsn ¯, ment of time- or frequency-standards, navigation, remote which in the same mode are to be considered as indistin- sensing, measurement of very small magnetic fields (with guishable (see Sec.III). In this context, the 1/√n¯ scaling applications to medical brain- and heart-imaging), mea- is also called “shot-noise limit”, referring to the quantum surement of the parameters of space-time, thermometry, noise that arises from the fact that the electromagnetic and many more. The literature on the topic of quantum energy is quantized in units of photons. Furthermore, metrology is vast and for a general introduction we refer the prefactor in these scaling behaviors is not fixed. We to the available reviews (Degen et al., 2016; Giovannetti therefore may define quite generally Standard Quantum et al., 2006; Paris, 2009; Pezzèand Smerzi, 2014; Pezzè Limit as the best scaling that can be achieved when em- et al., 2016; Tóth and Apellaniz, 2014; Wiseman and Mil- ploying only “classical” resources. burn, 2009). While this is not yet a mathematical definition either, From the theoretical side, the standard tool for evalu- it becomes precise once the classical resources are spec- ating a possible quantum enhancement has become the ified in the problem at hand. This may be achieved so-called quantum Cramér-Rao bound (Braunstein and adopting a resource-theory framework, in which classical Caves, 1994; Braunstein et al., 1996; Helstrom, 1969; states of some specific sort are identified and formalised Holevo, 1982). It provides a lower bound on the vari- as “free” states (i.e. given at no cost), and any other ance Var(θest) of any unbiased estimator function θest state is seen as possessing a resource content which may that maps observed data obtained from arbitrary quan- allow us to outperform free states in practical applica- tum measurements to an estimate of the parameter θ. tions, leading specifically to quantum-enhanced measure- The bound is optimized over all possible measurements ments beyond the Standard Quantum Limit scaling. For and data analysis schemes, in a sense made precise be- instance, separable states are the free (classical) states low. In the limit of an infinite number of measurements in the resource-theory of entanglement (Horodecki et al., the bound can be saturated. It thus represents a valu- 2009), while states diagonal in a reference basis are the able benchmark that can in principle be achieved once free (classical) states in the resource theory of quantum all technical noise problems have been solved, such that coherence (Streltsov et al., 2017). In quantum optics, only the unavoidable noise inherent in the quantum state Glauber’s coherent states and their mixtures are regarded itself remains. as the free (classical) states (Mandel and Wolf, 1965), A standard classical method of noise reduction is to av- and any other state can yield a nonclassical scaling. In erage measurement results from N independent, identi- the latter example, considering the mean photon num- cally prepared systems. In a quantum mechanical formu- bern ¯ as an additional resource, one can fix the prefactor lation with pure states, the situation corresponds to hav- of the Standard Quantum Limit scaling, so that quan- ing the N quantum systems in an initial product state, tum enhancements are possible not only by improving N ψ = φ i. Suppose that the parameter is encoded the scaling law, but also by changing the prefactor. | i ⊗i=1| i in the state through a unitary evolution with a Hamil- However, basing our review exclusively on a resource- N tonian H(θ) = θ hi, i.e. ψ(θ) = exp( iH(θ)) ψ . theory picture would be too restrictive, as cases of en- i=1 | i − | i Based on the quantum Cramér-Rao bound one can show hanced sensitivity are readily available for which no re- P that with M final measurements, the smallest achievable source theory has been worked out yet (see (Brandãoand variance Var(θest) of the estimation of θ is Gour, 2015) and references therein for a recent overview 1 of existing resource theories). Examples are the use of Var(θest)min = , (1) quantum phase transitions, for which one can compare NM(Λ λ)2 − the sensitivity at the phase transition with the sensi- where Λ and λ are the largest and smallest eigenvalue of tivities away from the phase transition, or instances of 3

Hamiltonian engineering, for which one can evaluate the discord become possible. These become naturally effect of added terms in the Hamiltonian. Rather than important once we look at estimation of loss parameters, developing resource theories for all these examples, which quantum illumination problems, and other applications would be beyond the scope of this review, we point out that typically involve the loss of probes. Secondly, in the the enhancements achievable compared to the sensitivity derivation of the Standard Quantum Limit the quantum without the use of the mechanism under consideration. systems are distinguished by an index i, which supposes Based again on the quantum Cramér-Rao bound one that they are distinguishable. Cold atoms, on the other can show that initially entangled states can improve hand, have to be considered in general as indistinguish- the scaling to 1/N (see e.g. (Giovannetti et al., 2006)), able particles, and the same is true for photons, which known as the “Heisenberg-limit”. Similarly to the have been used for quantum-enhanced measurements Standard Quantum Limit, there is no unique definition from the very beginning. Hence, statements about of the Heisenberg-limit in the literature (see the remarks the necessity of entanglement have to be re-examined in sec.IV.A). Nevertheless, achieving “the Heisenberg- for indistinguishable particles. It turns out that the limit” has been the goal of large experimental and permutational symmetry of the quantum states required theoretical efforts over the last two decades. However, due to indistinguishability of the particles leads immedi- only few experiments achieved the 1/N scaling of the ately to the level of quantum-enhanced sensitivity that Heisenberg-limit and only for very small numbers of for distinguishable particles would require to entangle sub-systems, where the scaling advantage is still far them. Thirdly, the structure of the Hamiltonian is from allowing one to beat the best possible classical rather restrictive: a.) many hamitonians do not have a measurements. This has several reasons: first of all, it bound spectrum characterized by largest and smallest is already very difficult to achieve even the Standard eigenvalues Λ and λ as assumed in eq.(1). Indeed, one of Quantum Limit, as all non-intrinsic noise sources have to the most common systems used in quantum metrology, be eliminated. Secondly, resources such as photons are the harmonic oscillator that represents e.g. a single cheap, such that classically one can operate with very mode of an electro-magnetic field, has an unbound N large photon numbers, whereas entangled states with spectrum. And b.), the Hamiltonian H(θ) = θ i=1 hi large photon numbers are difficult to produce. Thirdly, does not allow for any interactions. Taking into account P and most fundamentally, quantum-enhanced measure- these freedoms opens the path to many new forms of ments schemes are plagued by decoherence. Indeed, enhanced sensitivity. Fourthly, unitary evolutions with a it has been shown that a small amount of Markovian Hamiltonian that depends on the parameter are not the decoherence brings the 1/N scaling for certain highly only way of coding a parameter in a state. In statistical entangled states back to the 1/√N scaling of the mechanics, for example, there are parameters that Standard Quantum Limit (Escher et al., 2011; Huelga describe the statistical ensemble, such as temperature or et al., 1997; Ko lodyński and Demkowicz-Dobrzański, chemical potential for systems in thermal equilibrium, 2010). The reduction to the Standard Quantum Limit but which are not of Hamiltonian origin. The same also affects the estimation of noise in programmable is true for non-equilibrium states. For many of these and teleportation-covariant channels (Laurenza et al., situations, the corresponding QCRs have been obtained 2017). Recent research has focussed on finding optimal only recently, and it often turned out that improvements states in the presence of decoherence, and at least for beyond the Standard Quantum Limit should be possible. non-Markovian noise, a certain improvement can still Furthermore, it is known even in classical statistical be obtained from entangled states (Chin et al., 2012; physics that phase transitions can lead to diverging Matsuzaki et al., 2011). Also, niche applications are pos- susceptibilities and hence greatly enhanced sensitivities. sible, for which the light-intensity must be very small, as The same is true for quantum phase transitions, and in some biological applications. Nevertheless, it appears we therefore review as well the use of quantum phase worthwhile to think about alternative possible quantum transitions for quantum-enhanced measurements. enhancement principles other than the use of highly entangled states, and this is the focus of the present survey. While a growing number of researchers are investi- gating possibilities of breaking the Standard Quantum Many results have been obtained over the last years for Limit without using entanglement (see e.g. (Tilma et al., such alternative schemes that are worth a comprehensive 2010)), these still appear to be a minority. The situa- and exhaustive review that compares their usefulness tion is comparable to other aspects and fields of quantum with respect to the main-stream research focused on information treatment, where previously it was thought highly entangled states. We structure the review by that entanglement is necessary. For example, for a long different ways of breaking the conditions that are known time entanglement has been considered as necessary for to lead to Standard Quantum Limit scaling of the sen- non-locality, until it was realized that certain aspects sitivity. Firstly, by going away from pure states, more of non-locality can arise without entanglement (Bennett general forms of quantum correlations such as quantum et al., 1999). Recent reviews of quantum-enhanced mea- 4 surements schemes using entanglement (Degen et al., of equalities, valid for an unbiased estimator: 2016; Giovannetti et al., 2011; Paris, 2009; Pezzèand ∂ ∂ Smerzi, 2014; Pezzè et al., 2016; Tóth and Apellaniz, 1= E(θ )= dM x p (x)θ (x) ∂θ est ∂θ θ est 2014) are available and we do not survey this vast lit- Z ∂ erature here, but focus rather exclusively on quantum- = dM x p (x) ln p (x) θ (x) θ ∂θ θ est enhanced measurements schemes that are not essentially Z based on the use of entanglement, hoping that our re- ∂ = dM x p (x) ln p (x) (θ (x) θ) view will stimulate research in these directions. Before θ ∂θ θ est − reviewing these schemes, we give a short introduction to Z ∂ ln pθ parameter estimation theory and the precise definition of = ,θest θ . (2) ∂θ − the quantum Cramér-Rao bound. A more elaborate ped- agogical introduction to classical and quantum parameter In the step before the last one we used that estimation theory can be found in (Fra¨ısse, 2017). θ(∂/∂θ)E(1) = 0 due to the normalization of the probability distribution valid for all values of θ. The scalar product in the last step is defined for any two real func- M tions a(x),b(x) as a,b = d x pθ(x)a(x)b(x). Using the Cauchy-Schwarzh inequalityi for this scalar product, we immediately arrive at the (classical)R Cramér-Rao (lower) bound for the variance of the estimator 1 Var(θest) , (3) ≥ J (M) B. Parameter estimation theory θ

where the (classical) Fisher information Jθ is defined as x 2 Consider the following task in classical statistical anal- (M) ∂ ln pθ( ) J = dM xp (x) ysis: Given a probability distribution p (x) of a random θ θ ∂θ θ Z variable x that continuously varies as function of a sin- 2 1 ∂pθ(x) gle real parameter θ, estimate θ as precisely as possi- = dM x . (4) p (x) ∂θ ble from M samples drawn, i.e. a set of random values Z θ x , i = 1, 2,...,M. We denote this M-sample as x The bound can be saturated iff the two vectors in { i} for short, and denote the probability to find the drawn the scalar product are parallel, i.e. for ∂ ln pθ(x)/∂θ = M samples in the intervals xi ...xi +dxi as pθ(x)d x, with A(θ)(θest(x) θ), where A(θ) is a possibly θ dependent M − − d x = dx1 ...dxM . For independently drawn, identi- proportionality factor. If one differentiates this condi- cally distributed samples, p (x) = p (x ) ... p (x ), tion once more and then integrates it over with p (x), θ θ 1 · · θ M θ but the formalism allows for arbitrary joint-probability one finds that A(θ)= J (M). Hence, an unbiased estima- x θ distributions pθ( ) i.e. also correlations between differ- tor exists iff there is a function f(x) independent of θ ent samplings of the distribution. For simplicity we take (M) such that ∂ ln pθ/∂θ = J (f(x) θ). In that case one the support of x to be the real numbers. θ − can choose θest(x)= f(x). One can show that for many (but not all) members of the family of exponential proba- The task is accomplished by using an estimator func- bility distributions, i.e. distributions that can be written tion θest(x1,...,xM ) that takes as input the drawn ran- in the form pθ(x)= a(x)exp(b(θ)c(x)+ d(θ)) with some dom values and nothing else, and outputs an estimate functions a(x),b(θ),c(x),d(θ), this condition is satisfied, of the parameter θ. Many different estimator functions meaning that in such cases the Cramér-Rao bound can are possible, some more useful than others. Through its be saturated even for finite M. For M and iden- random arguments the estimator will itself fluctuate from tically, independently distributed samples,→ ∞the so-called one sample to another. One would like to have an estima- maximum-likelihood estimator saturates the bound. One tor that on average gives the true value of θ, E(θest)= θ, easily shows from eq.(4) that the Fisher information is M x x where E(...) = d pθ( )(...) is the mean value of a additive, such that for independently drawn, identically quantity over the distribution. This should hold at least (M) (1) distributed samples J = MJθ with Jθ J . in an infinitesimalR interval about the true value of θ; such θ ≡ θ an estimator is called “unbiased”. Secondly, one would like the estimator to fluctuate as little as possible. The C. Quantum parameter estimation theory latter request makes only sense together with the first one, as otherwise we could just choose a constant esti- In quantum mechanics the state of a system is given mator, which of course would not reproduce the correct by a density matrix ρθ, i.e. a positive hermitian operator value of θ in most cases. Now consider the following chain with trace equal one that can depend on the parameter 5

θ, which we assume to be a classical parameter. Ran- based on the Cauchy-Schwarz inequality that leads to the dom data are created when measuring some observable bound of the system whose statistics will depend on θ through 2 Jθ Iθ tr(ρθL ) , (7) the quantum state ρθ. Again we would like to estimate θ ≤ ≡ ρθ as precisely as possible based on the measurement data where Iθ is known as the “quantum Fisher information”. (see FIG.1). Similarly as for the classical Fisher information, the quantum Fisher information of uncorrelated states is additive, (Fujiwara and Hashizum´e, 2002):

I (ρ(θ) σ(θ)) = I (ρ(θ)) + I (σ(θ)) , (8) θ ⊗ θ θ such that for M independent identical POVM measurements of the same system, prepared always in the same Figure 1 (Color online) General setup of quantum parameter estimation. A quantum measurement of a system in quantum state, the total quantum Fisher information satisfies (M) (1) state ρθ that depends on a classical parameter θ is performed I = MI with I I . Inequalities (7) and (3) θ θ θ ≡ θ (general POVM measurement) and produces data x1,...,xM . then lead to the so-called quantum Cramér-Rao bound, The data are analysed with an estimator function that outputs an estimate θest of θ. The goal is to obtain an unbiased 1 Var(θest) . (9) estimate with as small as possible statistical fluctuations. ≥ MIθ Additivity of the quantum Fisher information also immediately implies the 1/√N scaling in eq.(1), as the The most general measurements are so-called POVM quantum Fisher information of the N uncorrelated sub- measurements (POVM=positive-operator-valued mea- systems is just N times the quantum Fisher information sure). These are measurements that generalize and in- of a single subsystem. Inequality (7) can be saturated clude projective von Neumann measurements and are rel- with a POVM that consists of projectors onto eigenstates evant in particular when the system is measured through of L (Braunstein and Caves, 1994; Helstrom, 1969; an ancilla system to which it is coupled (Peres, 1993). θ Holevo, 1982). As the quantum Cramér-Rao bound They consist of a set of positive operators M , where x x is already optimized, no measurement of the whole labels possible measurement outcomes that we take again system, even if entangling the individual systems, can as x R for simplicity. They obey a completeness rela- improve the sensitivity when the parameter was already tion, ∈ M dx = I, where I is the identity operator on R x imprinted on a product state. the Hilbert-space of the system. The probability-density to findR outcome x is given by p (x) = tr(ρ M ), and it is θ θ x The quantum Cramér-Rao bound has become the most through this equation that the contact with the classical widely used quantity for establishing the ultimate sensi- parameter estimation theory can be made: Plugging in tivity of measurement schemes. It derives its power from p (x) in eq. (4) with M = 1, we are led to the Fisher θ the facts that firstly it is already optimized over all possi- information ble data analysis schemes (unbiased estimator functions) 2 1 ∂ρθ and all possible (POVM-) measurements, and that sec- Jθ = dx tr Mx ondly it can be saturated at least in the limit of infinitely tr(ρθMx) ∂θ Z many measurements and using the optimal POVM con- 1 1 2 = dx tr (ρ L + L ρ )M (5), sisting of projectors onto the eigenstates of Lρθ . tr(ρ M ) 2 θ ρθ ρθ θ x Z θ x In (Braunstein and Caves, 1994) it was shown that Iθ is a geometric measure on how much ρ(θ) and ρ(θ + dθ) where in the last step we have introduced the so-called differ, where dθ is an infinitesimal increment of θ. The symmetric logarithmic derivative L , defined indirectly ρθ geometric measure is given by the Bures-distance, through 2 ∂ρ 1 dsBures(ρ, σ) 2 1 F (ρ, σ) , (10) θ = (ρ L + L ρ ) , (6) ≡ − ∂ 2 θ ρθ ρθ θ p θ where the fidelity F (ρ, σ) is defined as in analogy to the classical logarithmic derivative 1/2 1/2 2 F (ρ, σ)= ρ σ 1 , (11) ∂ ln ρθ/∂θ. Compared to the classical case, one has in || || the quantum mechanical setting the additional freedom and A 1 tr√AA† denotes the trace norm (Miszczak to choose a suitable measurement in order to obtain a et al.||, 2009).|| ≡ With this (Braunstein and Caves, 1994) distribution pθ(x) that contains as much information as showed that possible on the parameter θ. Based on eq. (5), one can 2 2 find a similar chain of inequalities as in the classical case Iθ = 4dsBures(ρ(θ), ρ(θ + dθ))/dθ , (12) 6 unless the rank of ρ(θ) changes with θ and thus produces Trees, 2001), Bayesian approaches (Macieszczak et al., removable singularities (Banchi et al., 2015; Safránek,ˇ 2014; Rivas and Luis, 2012), adaptive measurements (Ar- 2017), a situation which we do not consider in this re- men et al., 2002; Berry and Wiseman, 2000, 2002, 2006, view. The quantum Cramér-Rao bound thus offers the 2013; Fujiwara, 2006; Higgins et al., 2009; Okamoto et al., physically intuitive picture that the parameter θ can be 2012; Serafini, 2012; Wheatley et al., 2010; Wiseman, measured the more precisely the more strongly the state 1995), and approaches specialized to particular param- ρ(θ) depends on it. For pure states ρ(θ) = ψ(θ) ψ(θ) , eter estimation problems such as phase estimation (Hall the quantum Fisher information reduces to| theih overlap| et al., 2012). Another point to be kept in mind is that of the derivative of the state ∂θψ(θ) with itself and the the quantum Cramér-Rao bound can be reached asymp- original state (Braunstein et| al., 1996;i Paris, 2009), totically for a large number of measurements, but not necessarily for a finite number of measurements. The I ( ψ ψ = 4( ∂ ψ ∂ ψ + ∂ ψ ψ 2) . (13) θ | θi|h θ| h θ θ| θ θi h θ θ| θi latter case is clearly relevant for experiments and subject of active current research (see e.g. (Liu and Yuan, 2016) If the parameter is imprinted on a pure state via a ). unitary transformation with hermitian generator G as These limitations not withstanding, we base this review ψ = exp(iθG) ψ , eq.(13) gives I = 4Var(G) θ θ almost exclusively on the quantum Cramér-Rao bound |4( ψi G2 ψ ψ | Gi ψ 2). With a maximally entangled≡ θ θ θ θ (with the exception of Sec.II.G on quantum channel dis- stateh | of the| i−hN subsystems| | i and a suitable measurement, crimination and parts of Sec.IV.B.1, where a signal-to- one can reach a scaling of the quantum Fisher informa- noise ratio is used), given that the overwhelming major- tion proportional to N 2 (Giovannetti et al., 2006), the ity of results have been obtained for it and allow an in- mentioned Heisenberg-limit. This can be seen most eas- depth comparison of different strategies. A certain num- ily for a pure state of the form ψ =( Λ N + λ N )/√2, ⊗ ⊗ ber of results have been obtained as well for the quan- where Λ and λ are two eigenstates| i | ofiG to two| i different tum Fisher information optimized over all input states eigenvalues| i Λ,λ| .i (Fujiwara, 2001b; Fujiwara and Imai, 2003), a quantity For mixed states, the Bures-distance is in general diffi- sometimes called channel quantum Fisher information. cult to calculate, but I (ρ(θ)) is a convex function of ρ(θ), θ We do not review this literature here, as in this type of i.e. for two density matrices ρ(θ) and σ(θ) and 0 λ 1 work sensitivity is typically not separately optimized over we have (Fujiwara, 2001a) ≤ ≤ entangled or non-entangled initial states.

Iθ(λρ(θ) + (1 λ)σ(θ)) λIθ(ρ(θ)) + (1 λ)Iθ(σ(θ)) . The Fisher information can be generalized to multi- − ≤ − (14) parameter estimation (Helstrom, 1969; Paris, 2009), θ = This can be used to obtain an upper bound for the (θ1,θ2,... ). The Bures distance between two infinitesi- quantum Fisher information. Convexity also implies mal close states then reads that the precision of measurements cannot be increased 2 ds (ρθ, ρθ θ) = 2 1 tr √ρθ ρθ θ√ρθ . (15) by classically mixing states with mixing probabilities Bures +d − +d independent of the parameter (Braun, 2010). q 2 An expansion of dsBures(ρθ, ρθ+dθ) leads to the quantum In principle, the optimal measurement that saturates Fisher information matrix (Paris, 2009; Sommers and Zy- the quantum Cramér-Rao bound can be constructed by czkowski, 2003), diagonalizing L(θ). The projectors onto its eigenstates 2 Iθk,θk′ form a POVM that yields the optimal measurement. dsBures(ρθ, ρθ+dθ)= dθkdθk′ , (16) However, such a construction requires that the precise 4 value of the parameter θ is already known. If that was where I = trρθ(L L + L ′ L )/2, and L is the θk,θk′ θk θk′ θk θk θk the case, one could skip the measurement altogether and symmetric logarithmic derivative with respect to param- choose the estimator as θest = θ, with vanishing uncer- eter θk. The quantum Cramér-Rao bound generalizes to tainty, i.e. apparently violating the quantum Cramér-Rao a lower bound on the co-variance matrix Cov[θ] of the bound in most cases (Chapeau-Blondeau, 2015) (note, parameters θi (Helstrom, 1969, 1976; Paris, 2009), however, that for a state that depends on θ, the condi- 1 1 tion θest = θ for an unbiased estimator cannot be fulfilled Cov[θ] (I(θ))− , (17) in a whole ǫ-interval about θ, such that there is no for- ≥ M mal contradiction). If θ is not known, the more common where Cov[θ]ij = θiθj θi θj , and A B means approach is therefore to use the quantum Cramér-Rao that A B is a positive-semidefiniteh i−h ih i matrix.≥ Contrary to bound as a benchmark as function of θ, and then check the single− parameter quantum Cramér-Rao bound, the whether physically motivated measurements can achieve bound (17) can in general not be saturated, even in the it. More general schemes have been proposed to mitigate limit of infinitely many measurements. The Bures metric the problem of prior knowledge of the parameter. This in- has also been called fidelity susceptibility in the frame- cludes the van Trees inequality (Gill and Levit, 1995; van work of quantum phase transitions (Gu, 2010). 7

II. QUANTUM CORRELATIONS BEYOND gratz et al., 2014; Marvian and Spekkens, 2016; Streltsov ENTANGLEMENT et al., 2017) in sequential schemes further extends to certain schemes of quantum metrology in the presence of A. Parallel versus sequential strategies in unitary quantum noise, namely when the unitary encoding the parame- metrology ter to be estimated and the noisy channel commute with each other (e.g. in the case of phase estimation affected One of the most typical applications of quantum by dephasing) (Boixo and Heunen, 2012; Demkowicz- metrology is the task of unitary parameter estimation, Dobrzański and Maccone, 2014), although in more gen- exemplified in particular by phase estimation (Giovan- eral instances entanglement is shown to provide an ad- netti et al., 2006, 2011). Let U = exp( iθH) be a θ − vantage (Demkowicz-Dobrzański and Maccone, 2014; Es- unitary transformation, with θ the unknown parameter cher et al., 2011; Huelga et al., 1997). In general, se- to be estimated, and H a selfadjoint Hamiltonian opera- quential schemes such that individual probes are ini- tor which represents the generator of the transformation. tially correlated with an ancilla (on which the param- The typical estimation procedure then consists of the fol- eter is not imprinted) and assisted by feedback control lowing steps: a) preparing an input probe in a state ρ; (see Sec. IV.F.4) can match or outperform any paral- b) propagating the state with the unitary transformation lel scheme for estimation of single or multiple param- Uθ; c) measuring the output state ρθ = UθρUθ†; d) per- eters encoded in unitary transformations even in the forming classical data analysis to infer an estimator θest presence of noise (Demkowicz-Dobrzański and Maccone, for the parameter θ. 2014; Huang et al., 2016; Nichols et al., 2016; Sekatski Let us now assume one has the availability of N uti- et al., 2017; Yousefjani et al., 2017; Yuan, 2016; Yuan lizations of the transformation Uθ. Then, the use of N and Fung, 2015). While probe and ancilla typically need N uncorrelated probes in a global initial state ρ⊗ , each to be entangled for such sequential schemes to achieve of which is undergoing the transformation Uθ in paral- maximum quantum Fisher information, this observation lel, yields an estimator whose minimum variance scales removes the need for large-scale multiparticle entangled as 1/N (Standard Quantum Limit). On the other hand, probes in the first place. by using an initial entangled state ρ of the N probes, and propagating each with the unitary Uθ in parallel, Similarly, in continuous variable optical interferometry one can in principle achieve the Heisenberg-limit, mean- (Caves, 1981), equivalent performances can be reached ing that an optimal estimator θest can be constructed (for unitary phase estimation) by using either a two- whose asymptotic variance, in the limit N 1, scales as mode entangled probe, such as a N00N state, or a 1/N 2. However, it is not difficult to realize≫ that the very single-mode non-classical state, such as a squeezed state. same precision can be reached without the use of entan- These are elementary examples of quantum-enhanced glement, by simply preparing a single input probe in a measurements achievable without entanglement, yet ex- superposition state with respect to the eigenbasis of the ploiting genuinely quantum effects such as nonclassical- generator H, and letting the probe undergo N sequential ity and superposition. Such features can be understood iterations of the transformation Uθ. by observing that both optical nonclassicality in infinite- For instance, thinking of each probe as a qubit for sim- dimensional systems and coherence (superposition) in plicity, and fixing the generator H to be the Pauli matrix finite-dimensional systems can be converted to entangle- σz, one can either consider a parallel scheme with N in- ment within a well-defined resource-theoretic framework put probes in the Greenberger-Horne-Zeilinger (GHZ, or (Asbóth et al., 2005; Killoran et al., 2016; Streltsov et al., cat-like) maximally entangled state Ψ = ( 00 ... 0 + 2015; Vogel and Sperling, 2014), and can be thought-of as 11 ... 1 )/√2, or a sequential scheme| withi a single| probei equivalent resources to entanglement for certain practical in| the superpositioni ψ = ( 0 + 1 )/√2. In the first purposes, as is evidently the case for unitary metrology. 1 case, the state after| imprintingi | i the| i parameter reads . N iNθ iNθ Uθ⊗ Ψ = (e− 00 ... 0 + e 11 ... 1 )/√2, while | i | N i iNθ| i iNθ in the second case Uθ ψ = (e− 0 + e 1 )/√2. Hence, in both schemes| onei achieves| ani N-fold| increasei of the phase between two orthogonal states, and both schemes reach therefore the Heisenberg-limit scaling in the estimation of the phase shift θ, meaning that the quantum Cramér-Rao bound can be asymptotically sat- 1 An additional scenario in which the quantum limit can be urated in both cases by means of an optimal measure- reached without entanglement is when a multipartite state is ment, associated to a quantum Fisher information scal- used to measure multiple parameters, where each parameter is encoded locally onto only one subsystem — it has recently been ing quadratically with N. The equivalence between en- shown that entanglement between the subsystems is not advanta- tanglement in parallel schemes and coherence (namely, geous, and can even be detrimental, in this setting (Knott et al., superposition in the eigenbasis of the generator) (Baum- 2016; Proctor et al., 2018) 8

B. General results on the usefulness of entanglement information, a measure of correlations between two (or more) variables described by a joint probability distri- More generally, for unitary metrology with multipar- bution (Ollivier and Zurek, 2001). A direct generaliza- tite probes in a parallel setting, a quite general formalism tion leads to the so-called quantum mutual information has been developed to identify the metrologically useful I(ρ) = S(TrAρ) + S(TrBρ) S(ρ), that quantifies to- correlations in the probes in order to achieve quantum- tal correlations in the state ρ−of a bipartite system AB, enhanced measurements (Pezzèand Smerzi, 2009) (see with S(ρ) = Tr(ρ log ρ) being the von Neumann en- (Pezzèand Smerzi, 2014; Tóth and Apellaniz, 2014) for tropy. An alternative− generalization leads instead to (A) A recent reviews). Specifically, let us consider an input J (ρ) = sup ΠA I(Π [ρ]), a measure of one-sided clas- state ρ of N qubits and a linear interferometer with sical correlations{ that} quantifies how much the marginal 1 N (i) Hamiltonian generator given by H = Jl = 2 i=1 σl , entropy of, say, subsystem B is decreased (i.e., how much i.e. a component of the collective (pseudo-)angular mo- additional information is acquired) by performing a mini- mentum of the N probes in the direction l = x,y,zP , with mally disturbing measurement on subsystem A described A A σ(i) denoting the lth Pauli matrix for qubit i. If ρ is by a POVM Π , with Π [ρ] being the conditional state l { } k-producible, i.e., it is a convex mixture of pure states of the system AB after such measurement (Henderson which are tensor products of at most k-qubit states, then and Vedral, 2001). The difference between the former the quantum Fisher information is bounded above as fol- and the latter quantity is precisely the quantum discord, lows (Hyllus et al., 2012; Tóth, 2012), D(A)(ρ)= I(ρ) J(ρ) , (19) 2 2 − Iθ(ρ, Jl) nk +(N nk) , (18) ≤ − that quantifies therefore just the quantum portion of the where n is the integer part of N/k. This means that total correlations in the state ρ from the perspective of genuine multipartite entangled probes (k = N) are re- subsystem A. It is clear from the definition above that quired to reach the maximum sensitivity, given by the the state ρ of a bipartite system AB has nonzero discord 2 Heisenberg-limit Iθ N , even though partially entan- (from the point of view of A) if and only if it is altered by ∝ gled states can still result in quantum-enhanced measure- all possible local measurements performed on subsystem ments beyond the Standard Quantum Limit. A: disturbance by measurement is a genuine quantum A similar conclusion has been reached in (Augusiak feature which is captured by the concept of discord, see et al., 2016) considering the geometric measure of entan- (Adesso et al., 2016; Modi et al., 2012) for more details. glement, which quantifies how far ρ is from the set of fully Every entangled state is also discordant, but the converse separable (1-producible) states according to the fidelity. is not true; in fact, almost all separable states still exhibit Namely, for unitary metrology with N parallel probes nonzero discord (Ferraro et al., 2010). The only bipartite initialized in the mixed state ρ, in the limit N a states with zero discord, from the point of view of sub- → ∞ nonvanishing value of the geometric measure of entangle- system A, are so-called classical-quantum states, which ment of ρ is necessary for the exact achievement of the take the form Heisenberg-limit. However, a sensitivity arbitrarily close 2 ǫ (A) A B to the Heisenberg-limit, I N for any ǫ > 0, can χ = pi i i τ , (20) θ − | ih | ⊗ i ∝ i still be attained even if the geometric measure of entan- X glement of ρ vanishes asymptotically for N . In where the states i A form an orthonormal basis for deriving these results, the authors proved an→ important ∞ subsystem A, and{| τiB} denote a set of arbitrary states continuity relation for the quantum Fisher information i for subsystem B, while{ } p stands for a probability dis- in unitary dynamics (Augusiak et al., 2016). i tribution. These states are{ } left invariant by measuring A in the basis i A , which entails that D(A)(χ(A)) = 0. {| i } C. Role of quantum discord in parameter estimation with In a multipartite setting, one can define fully classi- mixed probes cal states as the states with zero discord with respect to all possible subsystems, or alternatively as the states Here we will focus our attention on possible advantages which are left invariant by a set of local measurements stemming from the use of quantum correlations more gen- performed on all subsystems. Such states take the form A1 AN eral than entanglement in the (generally mixed) state of χ = pi ,...,i i1 i1 iN iN for an i1,...,iN 1 N | ih | ⊗···⊗| ih | the input probes for a metrological task. Such correla- N-particle system A1 ...AN ; i.e., they are diagonal in tions are usually referred to under the collective name a localP product basis. One can think of these states as of quantum discord (Henderson and Vedral, 2001; Ol- the only ones which are completely classically correlated, livier and Zurek, 2001), see also (Adesso et al., 2016; that is, completely described by a classical multivariate

Modi et al., 2012) for recent reviews. The name quan- probability distribution pi1,...,iN , embedded into a den- tum discord originates from a mismatch between two sity matrix formalism.{ An alternative} way to quantify possible quantum generalizations of the classical mutual discord in a (generally multipartite) state ρ is then by 9 taking the distance between ρ and the set of classically analysis in (Modi et al., 2011) was performed at fixed correlated states, according to a suitable (quasi)distance spectrum (and thus degree of purity) of the input probes, function. For instance, the relative entropy of discord a constraint which allowed the authors to still identify (Modi et al., 2010) is defined as an advantage in using correlations weaker than entanglement, as opposed to no correlations. However, it is DR(ρ) = inf S(ρ χ) , (21) presently unclear whether these conclusions are special to χ k the selected classes of states, or can be further extended where the minimization is over all classically correlated to more general settings, including noisy metrology. states χ, and S(ρ χ) = Tr(ρ log ρ ρ log χ) denotes the quantum relative entropy.k For a dedicated− review on different measures of discord-type quantum correlations we refer the reader to (Adesso et al., 2016). Let us now discuss the role of quantum discord in In a more recent work, (Cable et al., 2016) analyzed metrological contexts. (Modi et al., 2011) investigated a model of unitary quantum metrology inspired by the the estimation of a unitary phase θ applied to each of N computational algorithm known as deterministic quan- qubit probes, initially prepared in mixed states with ei- tum computation with one quantum bit (DQC1) or ther (a) no correlations; (b) only classical correlations; “power of one-qubit” (Knill and Laflamme, 1998). Using or (c) quantum correlations (discord and/or entangle- only one pure qubit supplemented by a register of l maxi- ment). All the considered families of N-qubit probe mally mixed qubits, all individually subject to a local uni- states were chosen with the same spectrum, i.e. in partic- tary phase shift Uθ, their model was shown to achieve the ular the same degree of mixedness (which is a meaning- Standard Quantum Limit for the estimation of θ, which ful assumption if one is performing a metrology experi- can be conventionally obtained using the same number of ment in an environment with a fixed common tempera- qubits in pure uncorrelated states. They found that the ture), and were selected due to their relevance in recent Standard Quantum Limit can be exceeded by using one nuclear magnetic resonance (NMR) experiments (Jones additional qubit, which only contributes a small degree et al., 2009). In particular, given an initial thermal state of extra purity, which, however, for any finite amount 1+p 1 p of extra purity leads to an entangled state at the stage ρ0(p) = 2 0 0 + −2 1 1 for each single qubit (with purity parameter| ih | 0 | ihp | 1), the product states of parameter encoding. In this model, incidentally, the (a) N ≤ ≤ output state after the unitary encoding was found to be ρ (p)=[Hρ0(p)H]⊗ were considered for case (a), and N always separable but discordant, with its discord vanish- the GHZ-diagonal states ρ(c)(p)= CH Cρ (p) N CH C N 1 0 ⊗ 1 ing only in the limit of vanishing variance of the estima- were considered for case (c), with H denoting the single- tor for the parameter θ. It is not quite clear if and how qubit Hadamard gate (acting on each qubit in the first the discord in the final state can be interpreted in terms case, and only on the first qubit in the second case), and of a resource for metrology, but the achievement of the C = N C-Not a series of Control-Not operations j=2 1j Standard Quantum Limit with all but one probes in a acting⊗ on pairs of qubits 1 and j. These two classes of (a) fully mixed state was identified as a quantum enhance- states give rise to quantum Fisher information Iθ = ment without the use of entanglement. This suggests 2 (c) 2 2 p N and Iθ ' p N , respectively. By comparing the that further investigation on the role of quantum discord two cases, the authors of (Modi et al., 2011) concluded (as well as state purity) in metrological algorithms with (c) (a) that a quantum enhancement, scaling as Iθ /Iθ N, vanishing entanglement may be in order. A protocol for is possible using pairs of mixed probe states with≈ arbi- multiparameter estimation using DQC1 was studied in trary (even infinitesimally small) degree of purity. This (Boixo and Somma, 2008), although the resource role of advantage persists even when the states in strategy (c) correlations was not discussed there. In (MacCormick are fully separable, which occurs for p . a + b/N (with et al., 2016) a detailed investigation of a DQC1-based a and b determined numerically for each value of N), in protocol was made based on coherently controlled Ryd- which case both strategies are unable to beat the Stan- berg interactions between a single atom and an atomic dard Quantum Limit, yet the quadratic enhancement of ensemble containing N atoms. The protocol allows one (c) over (a) is maintained, being independent of p. The to estimate a phase shift assumed identical for all atoms authors then argue that multipartite quantum discord in the atomic ensemble with a sensitivity that interpo- — which increases with N according to the relative en- lates smoothly between Standard Quantum Limit and tropy measure of eq. (21) and vanishes only at p = 0 Heisenberg-limit when the purity of the atomic ensemble — may be responsible for this enhancement. Let us re- increases from a fully mixed state to pure states. It leads mark that, even though the quantum Fisher information to a cumulative phase shift proportional to N, and the is convex (which means that for every separable but dis- scheme can in fact also be seen as an implementation of cordant mixed state there exists a pure product state “coherent averaging”, with the control qubit playing the with a higher or equal quantum Fisher information), the role of the “quantum bus” (see Sec. IV.F.3). 10

D. Black-box metrology and the interferometric power malization constant) the interferometeric power of the bipartite state ρ with respect to the probing system A, As explicitly discussed in Sec. II.A, for unitary param- (A) 1 eter estimation, if one has full prior information on the P (ρ)= min Iθ(ρ, H) . (22) 4 H generator H of the unitary transformation Uθ imprinting the parameter θ, then no correlations are required what- Remarkably, as proven in (Girolami et al., 2014), the in- soever, and probe states with coherence in the eigenbasis terferometric power turns out to be a measure of discord- of H suffice to achieve quantum-enhanced measurements type correlations in the input state ρ. In particular, it in a sequential scheme. Recently, (Adesso, 2014; Giro- vanishes if and only if ρ takes the form of a classical- lami et al., 2014, 2013) investigated quantum metrology quantum state, eq. (20). This entails that states with in a so-called black-box paradigm, according to which the zero discord cannot guarantee a precision in parameter generator H is assumed not fully known a priori. In such estimation in the worst case scenario, while any other bi- a case, suppose one selects a fixed (but arbitrary) input partite state (entangled or separable) with nonzero dis- single-particle probe ρ, then it is impossible to guarantee cord is suitable for estimating parameters encoded by a precision in the estimation of θ for all possible non- a unitary transformation (acting on one subsystem) no trivial choices of H. This is because, in the worst case matter the generator, with minimum guaranteed preci- scenario, the black-box unitary transformation may be sion quantified by the interferometric power of the state. generated by a H which commutes with the input state This conclusion holds both for parameter estimation in ρ, resulting in no information imprinted on the probe, finite-dimensional systems (Girolami et al., 2014), and hence in a vanishing quantum Fisher information. It is for continuous-variable optical interferometry (Adesso, clear then that, to be able to estimate parameters inde- 2014). Recently, it has been shown more formally that pendently of the choice of the generator, one needs an entanglement accounts only for a portion of the quantum ancillary system correlated with the probe. But what correlations relevant for bipartite quantum interferome- type of correlations are needed? It is in this context try. In particular, the interferometric power, which is that discord-type correlations, rather than entanglement by definition a lower bound to the quantum Fisher in- or classical correlations, are found to play a key resource formation (for any fixed generator H), is itself bounded role. from below in bipartite systems of any dimension by a Consider a standard two-arm interferometric configu- measure of entanglement aptly named the interferometric ration, and let us retrace the steps of parameter esti- entanglement, which simply reduces to the squared con- mation in the black-box scenario (Girolami et al., 2014): currence for two-qubit states (Bromley et al., 2017). The a) an input state ρ of two particles, the probe A and interferometric power can be evaluated in closed form, the ancilla B, is prepared; b) particle B is transmit- solving analytically the minimization in eq. (22), for all ted with no interaction, while particle A enters a black- finite-dimensional states such that subsystem A is a qubit box where it undergoes a unitary transformation Uθ = (Girolami et al., 2014), and for all two-mode Gaussian exp( iθH) generated by a Hamiltonian H, whose spec- − states when the minimization is restricted to Gaussian trum is known but whose eigenbasis is unknown at this unitaries (Adesso, 2014). An experimental demonstra- stage; c) the agent controlling the black-box announces tion of black-box quantum-enhanced measurements rely- the full specifics of the generator H, so that parties A and ing on discordant states as opposed to classically corre- B can jointly perform the best possible measurement on lated states has been reported using a two-qubit NMR the two-particle output state ρ = (U I)ρ(U I)†; θ θ ⊗ θ ⊗ ensemble realized in chloroform (Girolami et al., 2014). d) the whole process is iterated N times, and an opti- We finally notice that, while (quantum) correlations mal unbiased estimator θest is eventually constructed for with an ancilla are required to achieve a nonzero worst the parameter θ. In the limit N 1, for any specific ≫ case precision when minimizing the quantum Fisher in- black-box setting H, the corresponding quantum Fisher formation over the choice of the generator H within a information Iθ(ρ, H) determines the maximal precision fixed spectral class, as in the scenario considered here, enabled by the input state ρ in estimating the param- single-probe (non-maximally mixed) states may however eter θ generated by H, as prescribed by the quantum suffice to be useful resources in the arguably more prac- Cramér-Rao bound. tical case in which the average precision, rather than the One can then introduce a figure of merit quantifying minimal, is considered instead as a figure of merit. This the worst case precision guaranteed by the state ρ for scenario is further discussed in Sec. II.H. the estimation of θ in this black-box protocol. This is done naturally by minimizing the quantum Fisher information over all generators H within the given spectral E. Quantum estimation of bosonic loss class (the spectrum is assumed nondegenerate, with a canonical choice being that of equispaced eigenvalues) Any quantum optical communication, from fibre-based (Girolami et al., 2014, 2013). This defines (up to a nor- to free-space implementations, is inevitably affected by 11 energy dissipation. The fundamental model to describe ing the separation of two incoherent optical point-like this scenario is the lossy channel. This attenuates an in- sources (Lupo and Pirandola, 2016). In this con- coming bosonic mode by transmitting a fraction η 1 text (Tsang et al., 2016) showed that a pair of weak of the input photons, while sending the other fraction≤ thermal sources can be resolved independently from their 1 η into the environment. The maximum number of separation if one adopts quantum measurements based bits− per channel use at which we can transmit quan- on photon counting, instead of standard intensity mea- tum information, distribute entanglement or generate surements. Thus, quantum detection strategies enables secret keys through such a lossy channel are all equal one to beat the so-called “Rayleigh’s curse” which af- to log(1 η) (Pirandola et al., 2017), a fundamental fects classical imaging (Tsang et al., 2016). This curse rate-loss− tradeoff− that only quantum repeaters may sur- is reinstated in the classical limit of very bright ther- pass (Pirandola, 2016). For these and other implications mal sources (Lupo and Pirandola, 2016; Nair, R. and to quantum communication, it is of paramount impor- Tsang, T., 2016). On the other hand, (Lupo and Pi- tance to estimate the transmissivity of a lossy channel in randola, 2016) showed that quantum-correlated sources the best possible way. can be super-resolved at the sub-Rayleigh scale. In fact, Quantum estimation of bosonic loss was first studied it is possible to engineer quantum-correlated point-like in (Monras and Paris, 2007) by using single-mode pure sources that are not entangled (but discordant) which Gaussian states (see also (Pinel et al., 2013)). In this displays super-resolution, so that the closer the sources setting, the performance of the coherent state probes at are the better their distance can be estimated. fixed input energy provides the shot-noise limit or classi- The estimation of loss becomes complicated in the cal benchmark, which has to be beaten by truly quantum presence of decoherence, such as thermal noise in the en- probes. Let us denote byn ¯ the mean number of photons, vironment and non-unit efficiency of the detectors. From 1 then the shot-noise limit is equal to Iη η− n¯ (Monras this point of view, (Spedalieri et al., 2016) considered a and Paris, 2007; Pinel et al., 2013). The≃ use of squeez- very general model of Gaussian decoherence which also ing can beat this limit, following the original intuition includes the potential presence of non-Markovian mem- for phase estimation of (Caves, 1981). In fact, (Mon- ory effects. In such a scenario, (Spedalieri et al., 2016) ras and Paris, 2007) showed that, in the regime of small showed the utility of asymmetrically correlated thermal loss η 1 and small energyn ¯ 0, a squeezed vac- states (i.e., with largely different photon numbers in the uum state≃ can beat the Standard≃ Quantum Limit. The two modes), fully based on discord and void of entangle- use of squeezing for estimating the interaction parameter ment. These states can be used to estimate bosonic loss in bilinear bosonic Hamiltonians (including beam-splitter with a sensitivity that approaches the shot noise limit and interactions) was also discussed in (Gaiba and Paris, may also surpass it in the presence of correlated noise and 2009), showing that unentangled single-mode squeezed memory effects in the environment. This kind of thermal probes offer equivalent performance to entangled two- quantum metrology has potential applications for practi- mode squeezed probes for practical purposes. cal optical instruments (e.g., photometers) or at different 1 The optimal scaling Iη [η(1 η)]− n¯ can be achieved wavelengths (e.g., far infrared, microwave or X-ray) for by using Fock states at the≃ input− (Adesso et al., 2009). which the generation of quantum features, such as coher- Note that, because Fock states can only be used when the ence, number states, squeezing or entanglement, may be input energy corresponds to integer photon numbers, in challenging. all the other cases one needs to engineer superpositions, e.g., between the vacuum and the one-photon Fock state if we want to explore the low-energy regimen ¯ . 1. Non- F. Gaussian quantum metrology Gaussian qutrit and quartet states can be designed to beat the best Gaussian probes (Adesso et al., 2009). It Clearly we may also consider the estimation of other is still an open question to determine the optimal probes parameters beyond loss. In general, Gaussian quantum for estimating loss at any energy regime. It is certainly metrology aims at estimating any parameter or mul- 1 known that the bound I [η(1 η)]− n¯ holds for any tiple parameters encoded in a bosonic Gaussian chan- η ≤ − n¯, as it can be proven by dilating the lossy channel into nel. As shown in (Pirandola and Lupo, 2017), the most a beam-splitter unitary and then performing parameter general adaptive estimation of noise parameters (such estimation (Monras and Paris, 2007). Note that this as thermal or additive noise) cannot beat the Standard bound is computed by considering N uncorrelated probes Quantum Limit. This is because Gaussian channels are in parallel. It is therefore an open question to find the teleportation-covariant, i.e., they suitably commute with best performance that is achievable by the most general the random operations induced by quantum teleporta- (adaptive) strategies. tion, a property which is shared by large class of quantum Interestingly, the problem of estimating the loss pa- channels at any dimension (Pirandola et al., 2017). The rameters of a pair of lossy bosonic channels has been joint estimation of specific combinations of parameters, proven formally equivalent to the problem of estimat- such as loss and thermal noise, or the two real compo- 12 nents of a displacement, has been widely studied in the form, valid for any multi-mode Gaussian state. For an literature (Bellomo et al., 2009, 2010a,b; Duivenvoorden explicit parametrization via multiple parameters θ = et al., 2017; Gagatsos et al., 2016; Gao and Lee, 2014; (θ1,θ2,... ), one can expand the differential and write

Genoni et al., 2013; Monras and Illuminati, 2011), but dV = k ∂θk V dθk, and similarly for du. In this way, 2 1 ′ ′ the ultimate performance achievable by adaptive (i.e., dsBures = k,k 4 Iθk,θk′ dθkdθk as in Eq. (16), with feedback-assisted) schemes is still unknown. P If we employ Gaussian states at the input of a Gaus- P T 1 Iθk,θk′ =(∂θk u )V − (∂θk′ u) sian channel, then we have Gaussian states at the out- 1 + 2Tr[(∂ V )(4 + )− (∂ V )] . (26) put and we may exploit closed formulas for the quantum θk LV LΩ θk′ Fisher information. These formulas can be derived by Eqs. (25) and (26) have been derived following eq. (12), direct evaluation of the symmetric logarithmic deriva- namely explicitly computing the fidelity function for two tive (Jiang, 2014; Monras, 2013; Nichols et al., 2017; ˇ most general multi-mode Gaussian states, and then tak- Safránek et al., 2015) or by considering the infinitesi- ing the limit of two infinitesimally close states. A sim- mal expression of the quantum fidelity (Banchi et al., ilar approach was used for fermionic Gaussian states 2015; Pinel et al., 2012, 2013). The latter approach may in (Banchi et al., 2014). exploit general and handy formulas. In fact, for two arbi- An alternative derivation of the bosonic quantum trary multi-mode Gaussian states, ρ1 and ρ2, with mean Fisher information for multi-mode Gaussian states, based values u1 and u2, and covariance matrices V1 and V2, on the use of the symmetric logarithmic derivative, et al. we may write the Uhlmann-Jozsa fidelity (Banchi , has been recently obtained in (Nichols et al., 2017). 2015) Furthermore, (Nichols et al., 2017) derived a neces- 1 T −1 (u2 u1) (V1+V2) (u2 u1) sary and sufficient compatibility condition such that the Ftote− 2 − − F (ρ1, ρ2)= , (23) quantum Cramér-Rao bound eq. (17) is asymptotically det(V1 + V2) achievable in multiparameter Gaussian quantum metrol- 2 2 p (VauxΩ)− ogy, meaning that a single optimal measurement exists Ftot = det 2 11+ + 11 Vaux ,(24) which is able to extract the maximal information on " 4 ! # r all the parameters simultaneously. For any pair of pa- T 1 rameters θ ,θ ′ θ, in terms of the symmetric log- where we set Vaux := Ω (V1 + V2)− (Ω/4+ V2ΩV1) k k ∈ arithmic derivatives Lρθ and Lρθ , the correspond- with Ω being the symplectic form (Banchi et al., 2015). k k′ Specific expressions for the fidelity were previously ing quantum Fisher information matrix element is de- given for single-mode Gaussian states (Scutaru, 1998), fined as I ′ Re Tr ρθL L , while the mea- θk,θk ρθk ρθ ′ two-mode Gaussian states (Marian and Marian, 2012), ≡ k surement compatibilityh condition amountsi to Yθk,θ ′ multi-mode Gaussian states assuming that one of the k ≡ Im Tr ρθLρθ Lρθ = 0 (Ragy et al., 2016). In terms states is pure (Spedalieri et al., 2013), and multi-mode k k′ squeezed thermal Gaussian states with vanishing first of theh first and secondi statistical moments u and V of a moments (Paraoanu and Scutaru, 2000). m-mode Gaussian state ρθ, we have then (Nichols et al., 2 2017): From eq. (23) we may derive the Bures metric dsBures. In fact, consider two infinitesimally-close Gaussian states T 1 I =(∂ u )V − (∂ u) + 2Tr(∂ VK ), (27) ρ, with statistical moments u and V , and ρ + dρ, with θk,θk′ θk θk′ θk′ θk statistical moments u + du and V + dV . Expanding at 1 T 1 1 Yθ ,θ ′ = (∂θ u )V − ΩV − (∂θ ′ u) + 16Tr ΩKθ ′ VKθ , the second order in du and dV , one finds (Banchi et al., k k 2 k k k k 2015) (28)

ij T 1 m 3 (aθ ) 1 ij du V − du δ l T − 1 2 with Kθ = i,j=1 l=0 ν ν ( 1)l S Ml S− , where dsBures = 2[1 F (ρ, ρ + dρ)] = + , (25) i j − − − 4 8 ij 1 ij (a ) = Tr S 1∂ VST − M , ν are the symplec- p θ l P − θP l i 1 { } where δ := 4Tr[dV (4 + )− dV ], X := AXA, 1 LV LΩ LA tic eigenvalues of the covariance matrix V , S− is the and the inverse of the superoperator 4 V + Ω refers to symplectic transformation that brings V into its diago- L L 1 m the pseudo-inverse. A similar expression was also com- 1 T − nal form, S− VS = i=1 νi11, and the set of ma- puted by (Monras, 2013) using the symmetric logarithmic ij ˇ trices Ml have all zero entries except for the 2 2 derivative, with further refinements in (Safránek et al., L ij × block in position ij which is given by (M)l l 0,...,3 = 2015). From the Bures metric in eq. (25) we may derive ∈{ } 1 iσ , σ , 11, σ . Note that eq. (27) can also be the quantum Fisher information (see eq. (12)) for the √2 y z x estimation of any parameter encoded in a multi-mode obtained from (25) by explicitly writing all the opera- (pure or mixed) Gaussian state directly in terms of the tors in the basis in which V is diagonal, observing that 1 statistical moments. Eq. (25) is written in a com- (4 V + Ω)− (∂θV )= Kθ. On the other hand, Eq. (28) pact basis-independent and parametrization-independent cannotL beL obtained from the limit of the fidelity formula. 13

In the context of this review, (Pinel et al., 2012) separable probes can achieve exactly the same precision studied in particular the quantum Cramér-Rao bound as entangled probes, leading the authors of (Safránekˇ and for estimating a parameter θ which is encoded in a Fuentes, 2016) to remark how entanglement does not play pure multi-mode Gaussian state. It was realized that, any significant role in achieving the Heisenberg-limit for in the limit of large photon number, no entanglement unitary Gaussian quantum metrology. nor correlations between different modes are necessary The same conclusion has been reached by consider- for obtaining the optimal sensitivity. Rather, a de- ing the estimation of any small parameter θ encoded in tection mode can be used based on the derivative of Bogoliubov transformations, i.e., Gaussian unitary chan- the mean photon field with respect to the parameter nels corresponding to arbitrary linear transformations of θ, into which all the resources in terms of photons and a set of n canonical mode operators (Friis et al., 2015). squeezing should be put. The mean photon field is de- In the limit of infinitesimal transformations (θ 1), and fined asa ¯ (r,t) = ψ a(r,t) ψ , with all parameter considering an arbitrary (Gaussian or not) pure≪n-mode θ h θ| | θi dependence in the pure Gaussian quantum state ψ θ, probe state with input mean photon number Nθ, (Friis r r r | i a( ,t) = i aivi( ,t), where vi( ,t) are orthonormal et al., 2015) showed by means of a perturbative analysis mode functions found from solving Maxwell’s equation that the maximal achievable quantum Fisher informa- P 2 with appropriate boundary conditions, ai is the annihi- tion scales as Iθ Nθ , that is, at the Heisenberg-limit. lation operators of mode i, and the sum is over all modes. Remarkably, such∝ a quantum-enhanced scaling requires The mean field can be normalized, u =a ¯ (r,t)/ a¯ , nonclassical (e.g., squeezed) but not necessarily entan- θ θ || θ|| where the norm f(r,t) = ( f(r,t) 2d2rdt)1/2 con- gled states. || || | | tains spatial integration over a surface perpendicular to Further results on the use of bosonic probes and the R the light beam propagation and temporal integration over role of mode entanglement in Gaussian and non-Gaussian the detection time. The detection mode is then defined ′ r quantum metrology are presented in Sec. III.B. r a¯θ ( ,t) asv ˜1( ,t) = ′ , where ′ means derivative with re- a¯θ spect to θ. The|| detection|| mode can be complemented by other, orthonormal modes to obtain a full basis, but G. Quantum channel discrimination these other modes need not be excited for achieving maximum quantum Fisher information. The quantum Fisher A fundamental protocol which is closely related to information reads then quantum metrology is quantum channel discrimina- 2 tion (Acin, 2001; Childs et al., 2000; Invernizzi et al., 2 Nθ′ 1 I = N 4 u′ + V − , (29) 2011; Lloyd, 2008; Pirandola, 2011; Sacchi, 2005; Tan θ θ || θ|| N θ,[1,1] θ ! et al., 2008), which may be seen as a sort of digitalized 1 version of quantum metrology. Its basic formulation is where N is the mean photon number, and V − the ma- θ θ,[1,1] binary and involves the task of distinguishing between trix element of the inverse covariance matrix of the Gaus- two quantum channels, or , associated with two a sian state corresponding to the detection modev ˜ (r,t). 0 1 1 priori probabilities π :=Eπ andE π = 1 π. During the All other modes are chosen orthonormal to it. The Stan- 0 1 encoding phase, one of such channels is− picked by Alice dard Quantum Limit corresponds to a quantum Fisher 1 and stored in a box which is then passed to Bob. In the information of a coherent state, in which case V − = 1. θ,[1,1] decoding phase, Bob uses a suitable state at the input Hence, an improvement over the Standard Quantum of the box and performs a quantum measurement of its Limit is possible with pure Gaussian states by squeezing output. Bob may also use ancillary systems which are the detection mode. The scaling with Nθ can be modi- 1 quantum correlated with the input probes and are di- fied if Vθ,−[1,1] depends on Nθ. For a fixed total energy a rectly sent to the measurement. For the specific tasks of 3/2 scaling Iθ Nθ can be achieved. This was proposed discriminating bosonic channels, the input is assumed to in (Barnett∝et al., 2003) for measuring a beam displace- be constrained in energy, so that we fix the mean num- ment. The quantum Cramér-Rao bound in eq.(29) can be ber of photonsn ¯ per input probe, or more strongly, the reached by homodyne detection with the local oscillator mean total number of photons which are globally irradi- in this detection mode. ated through the box (Weedbrook et al., 2012). By using compact expressions of the quantum Fisher Quantum channel discrimination is an open problem information for multi-mode Gaussian states, (Safránekˇ in general. However, when we fix the input state, it is and Fuentes, 2016) developed a practical method to find translated into an easier problem to solve, i.e., the quan- optimal Gaussian probe states for the estimation of pa- tum discrimination of the output states. In the binary rameters encoded by Gaussian unitary channels. Appli- case, this conditional problem has been fully solved by cations of the method to the estimation of relevant pa- the so-called Helstrom bound which provides the mini- rameters in single-mode and two-mode unitary channels, mum mean error probabilityp ¯ in the discrimination of such as phase, single-mode squeezing, two-mode squeez- any two states ρ0 and ρ1. Assuming equiprobable states ing, and transmissivity of a beam splitter, confirmed that (π = 1/2), this bound is simply given by their trace dis- 14 tance D, i.e., we have (Helstrom, 1976) Another application of quantum channel discrimination is quantum reading (Pirandola, 2011). Here the ba- 1 p¯ = [1 D(ρ0, ρ1)] . (30) sic aim is to discriminate between two different channels 2 − which are used to encode an information bit in a cell of In the case of multi-copy discrimination, in which we a classical memory. In an optical setting, this means to probe the box N times and we aim to distinguish the two discriminate between two different reflectivities, gener- N N outputs ρ0⊗ and ρ1⊗ , the mean error probabilityp ¯(N) ally assuming the presence of decoherence effects, such may be not so easy to compute and, therefore, we resort as background stray photons. The maximum amount of to suitable bounds. Using the quantum fidelity F (ρ0, ρ1) bits per cell that can be read is called “quantum reading from (11), and setting capacity” (Pirandola et al., 2011). This model has also

s 1 s been studied in the presence of thermal and correlated de- Q(ρ0, ρ1):= inf Tr(ρ0ρ1− ), (31) 0 s 1 coherence, as that arising from optical diffraction (Lupo ≤ ≤ et al., 2013). In all cases, the classical benchmark as- we may then write (Audenaert et al., 2007; Banchi et al., sociated with coherent states can be largely beaten by 2015; Fuchs and de Graaf, 1999) non-classical states, as long as the mean number of photons hitting the memory cells is suitably low. 1 1 F (ρ , ρ )N QN (ρ , ρ ) − − 0 1 p¯(N) 0 1 , (32) Depending on the regime, we may choose a different p 2 ≤ ≤ 2 type of non-classical states. In the presence of thermal N where Q (ρ0, ρ1)/2 is the quantum Chernoff bound decoherence induced by background photon scattering, (QCB) (Audenaert et al., 2007). In particular, the QCB two-mode squeezed vacuum states between signal modes is asymptotically tight for large N. Furthermore, it can (reading the cells) and idler modes (kept for detection) be easily computed for arbitrary multi-mode Gaussian are nearly-optimal. However, in the absence of decoher- states (Pirandola and Lloyd, 2008). ence, the sequential readout of an ideal memory (where Since the conditional output states can be optimally one of the reflectivities is exactly 100%) is optimized by distinguished, the non-trivial part in quantum channel number states at the input (Nair, 2011). (Roga et al., discrimination is the optimization of the mean error prob- 2015) showed that, in specific regimes, the quantum ad- abilityp ¯ over the input states. For this reason, it is an vantage can be related with a particular type of quantum extremely rich problem and depending on the types of correlations, the discord of response, which is defined as quantum channels, quantum correlations may play an im- the trace, or Hellinger, or Bures minimum distance from portant role or not. We now discuss some specific cases the set of unitarily perturbed states (Roga et al., 2014). in more detail. (Roga et al., 2015) also identified particular regimes in Quantum channel discrimination has various practical which strongly discordant states are able to outperform applications. One which is very well known is quantum pure entangled transmitters. illumination (Lloyd, 2008; Tan et al., 2008) which forms Let us consider the specific case of unitary channel dis- the basis for a “quantum radar” (Barzanjeh et al., 2015). crimination. Suppose that the task is to decide whether Despite the fact that entanglement is used at the input a unitary Uθ was applied or not to a probing subsystem between the signal (sent to probe a potential target) and A of a joint system (A, B). In other words, the aim is the idler (kept at the radar state for joint detection), en- to discriminate between the two possible output states I I tanglement is completely absent at the output between ρθ =(Uθ )ρ(Uθ )† (when the unitary Uθ has acted ⊗ ⊗ reflected and idler photons. Nevertheless the scheme as- on A) or ρ (equal to the input, when the identity has sures a superior performance with respect to the use of acted on A instead). In the limit of an asymptotically coherent states; in particular, an increase by a factor 4 large number N 1 of copies of ρ, the minimal proba- ≫ of the exponent (ln P (e))/M of the asymptotic error bility of error in distinguishing between ρθ and ρ, using probability P (e)− (where M is the number of transmis- an optimal discrimination strategy scales approximately N sions) (Tan et al., 2008). For this reason, the quantum as the QCB Q(ρ, ρθ) /2. illumination advantage has been studied in relation with It is clear that the quantity 1 Q(ρ, ρθ) plays a similar − the consumption of other discord-type quantum correla- role in the present discrimination context as the quantum tions beyond entanglement (Bradshaw et al., 2016; Weed- Fisher information in the parameter estimation scheme. brook et al., 2016). More precisely, the enhanced per- One can therefore introduce an analogous figure of merit formance of quantum illumination (with respect to sig- quantifying the worst case ability to discriminate, guar- nal probing not assisted by an idler) corresponds to the anteed by the state ρ. The discriminating strength of the amount of discord which is expended to resolve the tar- bipartite state ρ with respect to the probing system A is get (i.e., to encode the information about its presence or then defined as (Farace et al., 2014) absence). Quantum illumination was demonstrated ex- (A) D (ρ) = min[1 Q(ρ, ρθ)] , (33) perimentally in (Lopaeva et al., 2013; Zhang et al., 2015, H − 2013). where the minimization is performed once more over all 15 generators H within a given non-degenerate spectrum. et al., 2016) by defining the local average Wigner-Yanase As proven in (Farace et al., 2014), the discriminating skew information, which corresponds to the average ver- strength is another measure of discord-type correlations sion of the discriminating strength in case the probing in the input state ρ, which vanishes if and only if ρ is a subsystem A is a qubit (Farace et al., 2014). classical-quantum state as in eq. (20). The discriminating Unlike the minimum, the average skew information is strength is also computable in closed form for all finite- found not to be a measure of discord anymore. In par- A dimensional states such that subsystem A is a qubit. In ticular, it vanishes only on states of the form I τ B, dA the latter case, the discriminating strength turns out to that is, tensor product states between a maximally mixed⊗ be proportional to the local quantum uncertainty (Giro- state on A, and an arbitrary state on B (Farace et al., lami et al., 2013), a further measure of discord-type cor- 2016). This entails that, to ensure a reliable discrimina- relations defined as in eq. (22), but with the quantum tion of local unitaries on average, the input states need Fisher information replaced by the Wigner-Yanase skew to have either one of these two (typically competing) in- information (Girolami et al., 2013). The discriminating gredients: nonzero local purity of the probing subsystem, strength has also been extended to continuous-variable or nonzero correlations (of any nature) with the ancilla. systems, and evaluated for special families of two-mode The interplay between the average performance and the Gaussian states restricting the minimization in eq. (33) to minimum one, which instead relies on discord, as well Gaussianity-preserving generators (i.e., quadratic Hamil- as a study of the role of entanglement, are detailed in tonians) (Rigovacca et al., 2015). (Farace et al., 2016). A similar study has been recently Finally, we notice that the presence and use of quan- performed in continuous variable systems, in which the tum correlations beyond entanglement has also been in- average quantum Fisher information for estimating the vestigated in other tasks related to metrology and illumi- amount of squeezing applied to an input single-mode nation, such as ghost imaging with (unentangled) ther- probe, without previous knowledge on the phase of the mal source beams (Ragy and Adesso, 2012). Adopting a applied squeezing, was investigated with and without the coarse-grained two-mode description of the beams, quan- use of a correlated ancilla (Rigovacca et al., 2017). tum discord was found to be relevant for the implementation of ghost imaging in the regime of low illumination, while more generally total correlations in the thermal III. IDENTICAL PARTICLES source beams were shown to determine the quality of the imaging, as quantified by the signal-to-noise ratio. Measuring devices and sensors operating with many- body systems are among the most promising instances for which quantum-enhanced measurements can be actually H. Average precision in black-box settings experimented; indeed, their large numbers of elementary constituents play the role of resources according to which The results reviewed so far in this Section highlight a the accuracy of parameter estimation can be scaled. Typ- clear resource role for quantum discord, specifically mea- ical instances in which the quantum-enhanced measure- sured by operational quantifiers such as the interferomet- ment paradigm has been studied are in fact interferome- ric power and the discriminating strength, in black-box ters based on ultracold atoms confined in optical lattices metrology settings, elucidating in particular how quan- (Cronin et al., 2009; Gerry and Knight, 2005; Giorgini tum correlations beyond entanglement manifest them- et al., 2008; Haroche and Raimond, 2006; Inguscio et al., selves as coherence in all local bases for the probing sub- 2006; Köhl and Esslinger, 2006; Leggett, 2001, 2006; system. Discordant states, i.e., all states but those of Pethick and Smith, 2004; Pitaevskii and Stringari, 2003; eq. (20), are not only disturbed by all possible local mea- Yukalov, 2009) where a precise control on the state prepa- surements on A, but are also modified by — hence sensi- ration and on the dynamics can nowadays be obtained. tive to — all nontrivial unitary evolutions on subsystem These systems are made of spatially confined bosons or A. This is exactly the ingredient needed for the estima- fermions, i.e. of constituents behaving as identical partition and discrimination tasks described above. cles, a fact that has not been properly taken into account In practice, however, one might want to assess the gen- in most of the literature. eral purpose performance of probe states, rather than In systems of distinguishable particles, the no- their worst case scenario only. One can then introduce tion of separability and entanglement is well- alternative figures of merit quantifying how suitable a established (Horodecki et al., 2009): it is strictly state is, on average, for estimation or discrimination of associated with the natural tensor product structure of unitary transformations, when the average is performed the multi-particle Hilbert space and expresses the fact over all generators of a fixed spectral class. This can that one is able to identify each one of the constituent be done by replacing the minimum with an average ac- subsystems with their corresponding single-particle cording to the Haar measure, in Eqs. (22) and (33), re- Hilbert spaces. On the contrary, in order to describe spectively. Such a study has been carried out in (Farace identical particles one must extract from the tensor 16 product structure of the whole Hilbert space either the tified as those corresponding to permanents (the bosonic symmetric (bosonic) or the anti-symmetric (fermionic) analogue of Slater determinants); in other words, a pure sector (Feynman, 1994; Sakurai, 1994). This fact bosonic state is non-entangled not only when, as in the demands a more general approach to the notions of first approach mentioned above, it is a product state, but non-locality and entanglement based not on the particle also when all bosons are prepared in pairwise orthogonal aspect proper for first quantization, rather on the mode single particle states. The particle-based entanglement description typical of second quantization (Argentieri was then studied in (Eckert et al., 2002) both for bosons et al., 2011; Barnum et al., 2004, 2005; Benatti and and fermions, and then generalized within a more math- Floreanini, 2014; Benatti et al., 2010a, 2011, 2012a,b, ematical and abstract setting in (Grabowski et al., 2011, 2014a,b, 2017; Marzolino, 2013; Narnhofer, 2004; 2012). Summers and Werner, 1985, 1987a,b; Zanardi et al., From the point of view of particle description, a dif- 2004). ferent perspective was provided in (Ghirardi et al., 2002, The notion of entanglement in many-body systems has 2004), based on the fact that non-entangled pure states already been addressed and discussed in the literature: should possess a complete set of local properties, iden- for instance, see (Amico et al., 2008; Balachandran et al., tifiable by local measurements. For a more recent non 2013; Banuls et al., 2006; Bloch et al., 2008; Calabrese standard approach, still based on the particle descrip- et al., 2012; Dowling et al., 2006; Eckert et al., 2002; tion, see (Lo Franco and Compagno, 2016). Ghirardi et al., 2002; Grabowski et al., 2011; Hines In all these approaches, only bipartite systems consist- et al., 2003; Kraus et al., 2009; Lewenstein et al., 2007; ing of two identical particles are discussed; however, in Li et al., 2001; Micheli et al., 2003; Modi et al., 2012; the fermionic case, simple necessary and sufficient crite- Paskauskas and You, 2001; Schliemann et al., 2001; ria of many particle-entanglement, based on the single- Schuch et al., 2004; Song et al., 2012; Tichy et al., 2013; particle reduced density matrix, have been elaborated Wiseman and Vaccaro, 2003; Y.Shi, 2004) and references in (Plastino et al., 2009). therein. Nevertheless, only limited results actually apply A change of perspective occurred in (Barnum et al., to the case of identical particles and their applications 2004, 2005; Vedral, 2003; Zanardi et al., 2004) where to quantum-enhanced measurements. From the existing the focus moved from particles to orthonormal modes; literature on possible metrological uses of identical parti- within this approach, states that are not mode-entangled cles, there emerges as a controversial issue the distinction are characterized by correlations that can be explained between particle and mode entanglement. Before il- in terms of joint classical occupation probabilities of lustrating the general approach developed in (Benatti the modes. It then follows that entanglement and et al., 2010a, 2012a, 2014a) within which this matter can non-locality depend on the mode description which has be settled, we shortly overview the main aspects of the been chosen. In all cases, however, identical-particle problem. Readers who feel that the discussion whether states represented by fermionic Slater determinants or the states in question are to be considered as entangled bosonic permanents can be neither particle nor mode- or not is rather academic may be reassured by the very entangled, in direct conflict with claims that these states pragmatical result that independently of this discussion, are particle-entangled and hence metrologically useful systems of indistinguishable bosons offer a metrological (Demkowicz-Dobrzański et al., 2014). In an attempt advantage over distinguishable particles, in the sense to resolve the conflict, in (Killoran et al., 2014) it is that for certain measurements one would have to shown that such a pseudo, or “fluffy-bunny” entangle- massively entangle the latter for obtaining the same sen- ment as called in (Beenakker, 2006; Wiseman et al., sitivity as one obtains “for free” from the symmetrized 2003), which is due to bosonic state symmetrization, can states of the former. This we show explicitly in Sec.III.B. be turned into the entanglement of distinguishable, and thus metrologically accessible, modes; however, the op- Entanglement based on the particle description proper erations needed for such a transformation are non-local for first quantization has been discussed for pure states in the mode picture and thus ultimately responsible for in (Li et al., 2001; Paskauskas and You, 2001; Schliemann the achieved sub shot-noise accuracies. et al., 2001). In the fermionic case, Slater determinants The variety of approaches regarding entanglement in are identified as the only non-entangled fermionic pure identical particle systems and their use in metrological states, for, due to the Pauli exclusion principle, they are applications, in particular whether entanglement is nec- the least correlated many-particle states. In the case of essary or not to achieve sub-shot-noise accuracies, can bosons, two inequivalent notions of particle entanglement be looked at from the unifying point of view provided by have been put forward: in (Paskauskas and You, 2001), a the algebraic approach to quantum many-body systems pure bosonic state is declared non-entangled if and only proper to second quantization (Bratteli and Robinson, if all particles are prepared in the same single particle 1987; Strocchi, 1985). Within this scheme, we first ad- state, thus leading to a bosonic product state. In (Li dress the relations between mode and particle entangle- et al., 2001), bosonic non-entangled pure states are iden- ment. Then, we focus on specific quantum metrological 17 issues and show that neither entangled states, nor prelim- the same single particle Hilbert space that is used to inary state preparation as spin-squeezing are necessary in describe each one of a system of identicalH particles. In order to achieve sub shot-noise accuracies using systems practice, one fixes an orthonormal basis ψi in and of identical particles. populates the i-th mode by acting on the{| so-calledi} H vac-

uum vector with (powers of) the creation operator ai†, a† 0 = ψi , while the adjoint operators ai annihilate the A. Particle and mode entanglement i | i | i vacuum, ai 0 = 0. For bosons, [ai , aj†] = aiaj† aj†ai = δ and one| cani find arbitrarily many particles in− a given In quantum mechanics, indistinguishable particles can- ij mode, while for fermions a , a† = a a† + a† a = δ not be identified by specific labels, whence their states { i j} i j j i ij must be either completely symmetrized (bosons) or anti- and each mode can be occupied by one fermion, at most. symmetrized (fermions). This makes second quantization Typical modes are given by the eigenvectors of a given a most suited approach to deal with them, while the par- single particle Hamiltonian, orthogonal polarization di- ticle representation proper for first quantization results rections, the left and right position in a double-well po- to be too restrictive for a consistent treatment of entan- tential, or the atomic positions in an optical lattice. In gled identical particles. the case of free photons, typical modes are plane waves As a simple example of the consequences of indis- labeled by wave vector and polarization, arising from the tinguishability, consider two qubits, each of them de- quantization of classical electrodynamics in terms of inde- scribed by the Hilbert space = C2, where we select pendent harmonic oscillators. Within this picture, saying H that there are n photons in a certain energy mode can the orthonormal basis 0 , 1 of eigenvectors of σz: {| i | i} also be interpreted as a quantum oscillator being pro- σz 0 = 0 , σz 1 = 1 . If the two qubits describe distinguishable| i | i particles,| i −| thei tensor product structure of moted to its n-th excited state. the common Hilbert space C4 = C2 C2 exhibits the fact In dealing with distinguishable particles, quantum that one knows which is qubit 1 and⊗ which one qubit 2. entanglement (Horodecki et al., 2009) is basically ap- Instead, if the two qubits are indistinguishable, the corre- proached by referring to the tensor product structure of sponding Hilbert space loses its tensor product structure the total Hilbert space which embodies the fact that par- and becomes, in the bosonic case, the three-dimensional ticles can be identified. From the previous discussion, it subspace C3 spanned by the orthogonal two-particle sym- follows that, in the case of identical particles, one ought (2) to consider the entanglement of modes rather than the metric vectors 00 , 11 and ψ+ := ( 01 + 10 )/√2, whereas, in the| Fermionici | i case,| iti reduces| i to| thei anti- entanglement of particles. The relations between the two (2) approaches are studied in detail in (Benatti et al., 2014b); symmetric two-particle vector ψ := ( 01 10 )/√2. here, we briefly compare them in the case of two two- An important consequence of| − thisi fact| isi−| that generici mode indistinguishable bosons associated with orthogo- mixed states of indistinguishable particles must be rep- nal single-particle pure states ψ , as, for instance, resented by density matrices that arise from convex lin- 1,2 orthogonal polarization states,| describedi∈H by creation and ear combinations of projectors onto symmetrized or anti- annihilation operators a ,a†, i =, 2. The pure state symmetrized pure states. Indeed, while operators must i i be symmetrized in order to comply with particle in- 1, 1 = a†a† 0 (34) distinguishability, by symmetrizing density matrices of | i 1 2| i distinguishable particles, say sending ρ = ρ ρ into 1 2 belongs the two-particle sector of the symmetric Fock ρ ρ + ρ ρ /2 one cannot in general obtain⊗ an 1 2 2 1 space and represents a pure state with one boson in each appropriate⊗ bosonic⊗ or fermionic state, unlike sometimes mode. In the particle picture of first quantization, the stated in the literature (see e.g. (Demkowicz-Dobrzański same state corresponds to the symmetrized vector et al., 2014)). For instance, the two qubit symmetric mixed state ρ ρ cannot be a fermionic density ma- 1 ⊗ (2) (2) ψ = ψ ψ + ψ ψ , (35) trix since the only fermionic state is ψ ψ . On + √ 1 2 2 1 | − ih − | | i 2 | i⊗| i | i⊗| i the other hand, in the bosonic case, ρ ρ can be a ⊗ bona-fide bosonic state only if ρ is pure. Indeed, as which describes a balanced superposition of two states: ρ ρ ψ(2) = det(ρ) ψ(2) , if the determinant of the den- one with the first particle in the state ψ and the second | 1i sity⊗ matrix,| − i det(ρ) |=− 0,i then ρ ρ has support also in particle in the state ψ , the other with the two states | 2i the anti-symmetric6 component of⊗ the Hilbert space. A exchanged. The net result is that, given the state (35), careful discussion of the relationship between permuta- we can only say that one particle is surely in the state tionally invariant density matrices and symmetric/anti- ψ and that the other one is surely in the state ψ , but | 1i | 2i symmetric states can be found in (Damanet et al., 2016). we cannot attribute a specific state to a specific particle. Instead of on the standard particle picture, second As an example of the misunderstandings that can re- quantization is based on the so-called mode picture. In sult from sticking to the particle picture when dealing general, a mode is any of the normalized vectors ψ of with identical particles, state (35) is often referred to as | i 18 entangled: actually, it is only formally so in the parti- Similarly, the projector corresponding to the two qubits cle picture, while, as we shall show below, it is separable possessing two different properties ψ1,2 ψ1,2 must be symm | ih | in the mode picture. Indeed, (35) would clearly display P12 = ψ1 ψ1 ψ2 ψ2 + ψ2 ψ2 ψ1 ψ1 . If particle entanglement as the state cannot be written as ψ ψ = 0,| thenih P|⊗|symmih= | | .ih It then|⊗| followsih that,| h 1| 2i 12 Eψ1 Eψ2 a tensor product of single particle states. Instead, in despite the formal entanglement of ψ+ , the factorization (34), the same state is expressed as the action on the of mean-values as in (36) still holds;| indeed,i vacuum state of two independent creation operators and thus should correspond to a mode-separable state under ψ ψ = ψ ψ = ψ P sym ψ = 1 . h +|Eψ1 | +i h +|Eψ2 | +i h +| 12 | +i whichever meaningful definition of mode-entanglement (39) one should adopt. We shall later give solid ground to Therefore, from the particle point of view, namely of at- this latter statement, but firstly we show that, while for tributable properties, the formally entangled state ψ | +i distinguishable particles the state (35) is the prototype is indeed separable. of an entangled pure state, it is nevertheless separable in the sense specified below. 2. Mode entanglement

1. Particle entanglement Basing on the attribution of properties to identical particles, the previous discussion is developed in ﬁrst quan- For distinguishable particles, bipartite observables are tization terms, namely using symmetrized states and ob- termed local if they are tensor products O12 = O1 O2 servables. From the point of view of second quantization, I ⊗ of single particle observables O1 pertaining to particle separability and entanglement are instead to be related to 1 and I O pertaining to particle⊗ 2, where I denotes the ⊗ 2 the algebraic structure of Bose and Fermi systems rather identity operator, namely if they address each particle than to the possibility of attributing individual proper- independently. Consequently, in such a context, locality ties: this is the point of view usually adopted in the anal- is associated to the addressability of single particles and ysis of many-body systems, for which the primary object referred to as particle-locality in the following. Then, a of investigation are the algebras of operators rather than bipartite pure state ψ12 is separable if and only if the their representations on particular Hilbert spaces (Brat- expectation values |O i = ψ O O ψ of all h 12i12 h 12| 1 ⊗ 2| 12i teli and Robinson, 1987; Emch, 1972; Haag, 1992; Stroc- local observables factorize: chi, 1985, 2008a,b, 2012; Thirring, 2002).

O12 12 = O1 I 12 I O2 12 . (36) In order to appropriately formulate the notion of en- h i h ⊗ i h ⊗ i tanglement and non-locality in systems made of identical Indeed, if ψ = ψ ψ the above equality clearly 12 1 2 particles, the leading intuition is that there is no a pri- holds. On| thei other| hand,i⊗| ifi the equality holds, by using (1) (2) ori given tensor product structure, reminiscent of particle the Schmidt decomposition of ψ12 = i λi φi φi identification, either in the Hilbert space or in the alge- (1) (1)| i (2)| i⊗|(2) i and choosing O1 = φ φ , O2 = φ φ one bra of observables. Therefore, questions about entangle- | i ih i | P | j ih j | finds that only one Schmidt coefficient can be different ment and separability are meaningful only with reference from zero. Instead, for distinguishable particles, the state to specific classes of observables. Within this broader ψ+ in (35) violates the above equality for O1 = ψ1 ψ1 context, entanglement becomes a caption for non-local | i | ih | and O2 = ψ2 ψ2 and is thus entangled. quantum correlations between observables exhibited by | ih | If the two particles are identical, they cannot be ad- certain quantum states (the original discussion can be dressed individually; one has thus to refer to observables found in (Summers and Werner, 1985, 1987a,b); further that specify properties attributable to single particles developments can be found in (Clifton and Halvorson, without specifying to which one of them. It therefore 2001; Halvorson and Clifton, 2000; Keyl et al., 2006, follows that, in the case of two identical particles, appro- 2003; Moriya, 2006; Verch and Werner, 2005). priate single particle observables cannot be of the form Polynomials in creation and annihilation operators can O I or I O, but of the symmetrized form be used to generate bosonic and fermionic algebras ⊗ ⊗ A O I + I O . (37) containing all physically relevant many-body observables. ⊗ ⊗ Quite in general, physical states are given by positive and Consider the single-particle property described by the normalized linear functionals ω : C associating to one-dimensional projector ψ ψ ; for two indistinguish- any operator A its mean valueAω →(A), such that pos- | ih | able particles the property ”one particle is in the state itive observables∈AA 0 are mapped to positive numbers ψ ”, must then be represented by the symmetric projec- ω(A) 0 and ω(I≥) = 1. Typical instances of states | i tion (Ghirardi et al., 2002, 2004) are the≥ standard expectations obtained by tracing with respect to a given density matrix ρ, ω(A) = Tr(ρ A); no- = ψ ψ I ψ ψ + I ψ ψ ψ ψ Eψ | ih |⊗ −| ih | −| ih | ⊗| ih | tice, however, that in presence of infinitely many degrees + ψ ψ ψ ψ . (38) of freedom, not all physically meaningful states, like, for | ih |⊗| ih | 19 instance, the thermal ones, can be represented by density Henceforth, we shall focus on many-body systems, matrices (Narnhofer, 2004). whose elementary constituents can be found in M differ- Within such an algebraic approach, one can study the ent states or modes described by a discrete set of anni- entanglement between observables of with respect to hilation and creation operators ai,ai† i I ; this is a very a state ω by considering algebraic bipartitionsA (Benatti general framework, useful for the{ description} ∈ of physi- et al., 2010a, 2014a), namely any couple of subalgebras cal systems in quantum optics, in atomic and condensed ( 1, 2) of , having only the identity in common, 1 matter physics. A bipartition of the M-mode algebra A =AI, suchA that the linear span of products of theirA ∩ associated with the system can then be easily obtainedA A2 operators generate the entire algebra and by considering two disjoint sets a , a† i = 1, 2, . . . , m A { i i | } and a , a† j = m + 1, m + 2,..., M , M being possibly 1. [A ,A ] = 0 for all A and A in the j j 1 2 1 1 2 2 infinite.{ The| two sets form subalgebras} that indeed bosonic case; ∈ A ∈ A 1,2 constitute an algebraic bipartition of ;A in practice, it is e e e MA 2. [A1 ,A2] = 0 for all even elements A1 1 and determined by the integer 0

(1) (2) The differences between mode-entanglement and stan- ω(A1A2)= λk ωk (A1) ωk (A2) , (40) i dard entanglement are best appreciated in the case of X N bosons that can occupy M different modes; this is a (1) (2) in terms of other states ωi , ωi with λk > 0, i λk = 1; very general situation encountered, for instance, in ul- 2 otherwise, ω is called ( 1, 2)-entangled . tracold gases consisting of bosonic atoms confined in a A A P In the standard case of bipartite entanglement for pairs multiple site optical lattices. These systems turn out of distinguishable particles (Horodecki et al., 2009), en- to be a unique laboratory for studying quantum effects tangled states are all density matrices ρ which cannot be in many-body physics, e.g. in quantum phase transi- written in the form tion and matter interference phenomena, and also for applications in quantum information (e.g. see (Cronin ρ = λ ρ(1) ρ(2) , λ 0 , λ = 1 , (41) k k ⊗ k k ≥ k et al., 2009; Gerry and Knight, 2005; Giorgini et al., 2008; Xk Xk Haroche and Raimond, 2006; Inguscio et al., 2006; Köhl (1) (2) and Esslinger, 2006; Leggett, 2006; Pethick and Smith, with ρk and ρk density matrices of the two parties. That the algebraic definition reduces to the standard one 2004; Pitaevskii and Stringari, 2003; Yukalov, 2009), and becomes apparent in the case of two qubits by choosing references therein). The algebra is in this case generated by creation and the algebraic bipartition 1 = 2 = 2, where 2 is A A A M M † M the algebra of 2 2 complex matrices and the expectation annihilation operators ai , ai, i = 1, 2,..., , obeying × the commutation relations, [a , a†] = δ . The reference value ω(A1A2) = Tr(ρ A1 A2). i j ij ⊗ state ω is given by the expectations with respect to the

2 This generalized definition of separability can be easily extended to the case of more than two partitions; for instance, in the case 3 In the algebraic descripiton, Hilbert spaces are a byproduct of ··· of an n-partition, Eq.(40) would extend to ω(A1A2 An) = the algebraic structure and of the expectation functional (state) (1) (2) (n) Pk λk ωk (A1) ωk (A2) ··· ωk (An). defined on it (Bratteli and Robinson, 1987). 20 vacuum state 0 , ai 0 = 0 for all 1 i M, so that 1 2 is such that (compare with (42)) the natural states| i whose| i entanglement≤ properties≤ need be P P k,N k k,N k studied are vectors in the Fock Hilbert space spanned h − |P1P2| − i H 1 k k N k N k by the many-body Fock states = 0 a (a†) 0 0 a − (a†) − 0 k!(N k)!h | 1 P1 1 | i h | 2 P2 2 | i − n1,n2,...,nM = = k k N k N k , (46) | i h |P1| i h − |P2| − i 1 n1 n2 nM = (a1†) (a2†) (aM† ) 0 , (43) where k and N k are single-mode Fock states. Con- √n1! n2! nM! ··· | i | i | − i ··· sequently, mixed spearable states must be diagonal with or density matrices acting on it. respect to the Fock basis (45), i.e. density matrices of Given a bipartition ( 1, 2) defined by two disjoint the form (Benatti et al., 2010a): groups of creation andA annihilationA operators, any ele- N N ment A1 1, commutes with any element A2 2, ρ = p k,N k k,N k , p 0 , p = 1 . i.e. [ , ∈] A = 0. ∈ A k | − ih − | k ≥ k A1 A2 k=0 k=0 In this framework, a necessary and sufficient condition X X (47) for pure states ψ to be separable with respect to a given Unlike , most observables of physical interest are | i P1P2 bipartition ( 1, 2) is that they be generated by acting non-local with respect to the bipartition ( , ), i.e. A A A1 A2 on the vacuum state with ( 1, 2)-local operators (Be- they are not of the form = A A , with A and A A O 1 2 1 ∈ A1 natti et al., 2012a), A2 2. Prominent among them are those used in phase estimation∈A protocols based on ultra-cold atoms trapped ψ = (a†,...,a† ) (a† ,...,aM† ) 0 , (44) in double-well interferometers. Consider the generators | i P 1 m ·Q m+1 | i of the rotations satisfying the su(2) algebraic relations where , are polynomials in the creation operators [Ji,Jj] = iεijk Jk, i,j,k = x,y,z. Among their possible relativeP toQ the first m modes, the last M m modes, − representations in terms of two-mode creation and anni- respectively. Pure states that can not be cast in the above hilation operators, let us focus upon the following one form are thus ( , )-entangled. A1 A2 1 When the state of the bosonic many-body system is not Jx = a1†a2 + a1a2† , (48) pure, it can be described by a density matrix ρ, in general 2 1 not diagonal with respect to the Fock basis (43); since Jy = a1†a2 a1a2† , (49) density matrices form a convex set whose extremal points 2i − 1 are projectors onto pure states, one deduces that generic J = a†a a†a . (50) z 2 1 1 − 2 2 mixed states ρ can be ( 1, 2)-separable if and only if they are convex combinationsA A of ( , )-separable one- Notice that, although the operators in (48), as well as the A1 A2 dimensional projections. exponentials eiθJx and eiθJy , are non-local with respect An interesting application of these general considera- to the bipartition ( 1, 2), θ [0, 2π], the exponential A A ∈ tions is given by a system of N bosons that can be found of Jz turns out to be local: in just two modes, M = 2. In the Bose-Hubbard ap- iθJ iθa†a /2 iθa†a /2 proximation, N ultracold bosonic atoms confined in a e z = e 1 1 e− 2 2 , (51) double-well potential can be effectively described in this † † iθa a1/2 iθa a2/2 with e 1 1 and e− 2 2. By a linear way. The two creation operators a1† and a2† generate out ∈ A ∈ A of the vacuum bosons in the two wells, so that the Fock transformation, one can always pass to new annihilation basis (43) can be conveniently relabeled in terms of the operators integer k counting the number of bosons in the first well: a1 + a2 a1 a2 b1 = , b2 = − , (52) k N k √2 √2 (a†) (a†) k,N k = 1 2 − 0 , 0 k N . (45) | − i k!(N k)! | i ≤ ≤ and corresponding creation operators b1†,2, and rewrite − the three operators in (48) as p Furthermore, a1,a1†, respectively a2,a2†, generate two 1 commuting subalgebras and that, together, in Jx = b†b1 b†b2 , (53) 1 2 2 1 − 2 turn generate the whole algebraA A; it is the simplest bi- A 1 partition of the system one can obtain by means of the Jy = b1b† b†b2 , (54) 2i 2 − 1 operators a ,a† and a ,a†. 1 1 2 2 1 J = b b† + b†b . (55) Then, the states k,N k are separable: this agrees z 2 1 2 1 2 with the fact that they| are− createdi by the local operators k N k (a†) (a†) − . Indeed, for any polynomial operator To bi,bi†, i = 1, 2 one associates the bipartition ( 1, 2) 1 2 P1 ∈ B B and , the expectation value of the product of consisting of the subalgebras generated by b ,b†, A1 P2 ∈ A2 A 1 1 21 respectively b2,b2†: with respect to it, the exponential of information with respect to a parameter θ imprinted J becomes: onto a ( , )-separable state via a ( , )-local unitary x A B A B † † operator of the form U(θ) = exp(iθ(A(a,a†)+ B(b,b†)), iθJx iθb1b1/2 iθb2b2/2 e = e e− , (56) where A (respectively B) are hermitian functions of a,a†

† † (respectively b,b†), strictly vanishes. Hence, in order to iθb b1/2 iθb b2/2 with e 1 1 and e− 2 2. This explicitly be able to estimate θ at all (and even more so to beat ∈ B ∈ B shows that an operator, local with respect to a given the standard quantum limit in terms of the total number bipartition, can become non-local if a different algebraic of bosons), the separability of the input state or the bipartition is chosen. locality of the unitary operator that imprints θ on the These considerations are relevant for metrological ap- state need to be broken. Fortunately, mode non-locality plications (see for instance (Boixo et al., 2009; Bollinger is easily achieved e.g. in quantum optics with a simple et al., 1996; Bouyer and Kasevich, 1997; Caves, 1981; beam-splitter, without any particle interactions (see Dorner et al., 2009; Dowling, 1998, 2008; Dunningham below). et al., 2002; Giovannetti et al., 2004, 2011; Higgins et al., 2007; Holland et al., 2002; Holland and Burnett, In the case of the double-well system introduced above, et al. 1993; Kacprowicz , 2010; Kitagawa and Ueda, 1993; the operator algebras defined by a1†, a1 and a2†, a2, are Korbicz et al., 2005; Pezzèand Smerzi, 2009; Sanders the natural ones. With respect to them, a balanced Fock and Milburn, 1995; Sørensen et al., 2001; Tóth et al., N N N N number state as in (45) of the form ρN/2 = 2 , 2 2 , 2 2009; Uys and Meystre, 2007; Wang and Sanders, 2003; is separable. By choosing the vector n in| the planeih or-| Wineland et al., 1994; Yurke, 1986; Yurke et al., 1986)). thogonal to the z direction one computes Iθ ρN/2,Jn = In fact, ultracold atoms trapped in a double-well optical N 2 + N, thus approaching the Heisenberg limit. potential realize a very accurate interferometric device: 2 state preparation and beam splitting can be precisely As a concrete example (Benatti and Braun, 2013), achieved by tuning the interatomic interaction and by consider the action of a beam-splitter described by the acting on the height of the potential barrier. The com- unitary UBS(α) = exp(αa1†a2 α∗a1a2†) involving two bination of standard Mach-Zehnder type interferometric modes. If the complex transparency− parameter α = iθ/2, operations, i.e. state preparation, beam splitting, phase with θ real, then UBS(α) = exp(iθJx). In the case of N shift and subsequent beam recombination, can be effec- distinguishable qubits, a state as the balanced Fock num- N N tively described as a suitable rotation of the initial state ber state 2 , 2 has half qubits in the state 0 and half | i i | i ρin by a unitary transformation (Sanders and Milburn, in the state 1 such that σ i =( ) i , i = 0, 1. Then, | i z| i − | i 1995; Yurke et al., 1986): N σ(j) J = i with i x,y,z have zero mean values, i 2 ∈ { } iθ Jn j=1 ρin ρθ = Uθ ρin Uθ† , Uθ = e . (57) 7→ X 2 while the purity of ρN/2 yields Iθ[ρN/2, Jx] = 4 Jx = N, The phase change is induced precisely by the operators since h i in (48) through the combination: N 2 N 1 (j) (k) 2 2 2 Jx = + σx σx . (59) Jn nx Jx + ny Jy + nz Jz , nx + ny + nz = 1 . (58) 4 4 ≡ j=k=1 6 X In practice, the state transformation ρ ρ inside the in 7→ θ Instead, in the case of indistinguishable bosonic qubits, interferometer can be effectively modeled as a pseudo- the mean values of J for all i x,y,z in (48) vanish, n i ∈ { } spin rotation along the unit vector = (nx,ny,nz), N 2 while using (48), I [ρ , J ] = 4 J 2 = + N. whose choice depends on the specific realization of the θ N x h x i 2 interferometric apparatus and of the adopted measure- Unlike in the case of distinguishable particles, the ment procedure. quantum Fisher information can thus attain a value As discussed in Section I, in the case of distinguishable greater than N even with initial states like ρN/2 that are particles, for any separable state ρsep the quantum Fisher separable with respect to the given bipartition. As men- information is bounded by Iθ[ρin, Jn] cN where c is a tioned before, the rotation operated by the beam-splitter constant independent of N that can be≤ taken as the max- is not around the z axis and is thus non-local with re- imum quantum Fisher information of a single system in spect to the chosen bipartition. From the point of view all components of the mixed separable state (see eq.(8)) of mode-entanglement, it is thus not the entanglement of (Fujiwara and Hashizumé, 2002; Giovannetti et al., 2006; the states fed into the beam-splitter that helps overcom- Pezzèand Smerzi, 2009). Then, entangled initial states ing the shot-noise-limit in the transparency parameter of distinguishable particles are needed in order to obtain θ estimation accuracy; rather, the non-local character of sub-shot-noise accuracies. A corresponding statement the rotations operated by the apparatus on initially sepa- was proven in (Benatti and Braun, 2013) for a system rable states allows σ(θest) to be smaller than 1/√N, with of N indistinguishable bosons: The quantum Fisher the possibility of eventually reaching the Heisenberg 1/N 22 limit (Benatti and Braun, 2013; Benatti et al., 2011). limit. However, under certain conditions the first beam- Notice that, if one does not take into account the identity splitter can generate enough mode-entanglement from a of particles, the beam-splitter action in (56) is particle- mode-separable state fed into the two input ports of the local according to the discussion at the beginning of sec- interferometer to beat the standard quantum limit. For tion A.1; indeed, more general settings, the question of what scaling of the quantum Fisher information can be achieved with N (j) the number of indistinguishable bosons is still open. iθJx iθσ /2 e = e x . (60) Non-locality is partially attributed to operations like j=1 O beamsplitting instead of entirely to states, even in cases Thus, a prior massive entanglement of the input state when ”fluffy-bunny” entanglement is turned into useful entanglement as discussed in (Killoran et al., 2014). 0 0 1 1 (61) | i⊗···| i⊗| i⊗···| i For massive bosons one might think that the ten- with k spins up, σz 0 = 0 , and N k spins down, sor product of Fock states is a very natural state: As σ 1 = 1 , is needed.| i | i − the boson-number is conserved at the energies consid- z| i −| i Instead, if particle identity is considered, then the op- ered, one cannot make coherent superpositions of differ- erator in (60), is not particle-local since it cannot be ent numbers of atoms. However, in typical experiments, written as a product of symmetrized single particle oper- one needs to average over many runs, and the real diffi- ators (see (37) for the case N = 2). Furthermore, in the culty consists in controlling the atom number with single- formalism of first quantization, a Fock number state as atom precision from run to run (Demkowicz-Dobrzański in (45) is represented in the symmetrized form (Benatti et al., 2014). One has therefore effectively a mixed state and Braun, 2013) with a distribution of different atom numbers. So far the best experimentally demonstrated approximations to 1 Fock states with massive bosons are number-squeezed 0 0 1 1 , (62) | π(1)i⊗···| π(k)i⊗| π(k+1)i⊗···| π(N)i states with a few dB of squeezing (Esteve et al., 2008; π N X Gross et al., 2010b; Riedel et al., 2010b). One may hope where the sum is over all possible permutations π of the that novel measurement techniques such as the quantum N indices and = N!k!(N k)!. Despite its formally gas microscope (Bakr et al., 2009) may enable precise entangled structure,N such a state− is the symmetrization knowledge of boson numbers in the future and thus the p of a tensor product state with the first k particles in the preparation of Fock states with a large number of atoms state 0 and the second N k particles in the state 1 . at least in a post-selected fashion. Generalizing| i the argument briefly− sketched in section A.1| i Recently, the authors of (Oszmaniec et al., 2016) used in the case of two identical particles, individual properties concentration of measure techniques to investigate the can then be attributed to each of its constituents. There- usefulness of randomly sampled probe states for unitary fore, the state in (62) carries no particle-entanglement, quantum metrology. They show that random pure states particle non-locality being instead provided by the par- drawn from the Hilbert space of distinguishable particles ticle non-local operator in (60). typically do not lead to super-classical scaling of preci- Of course, returning to the second quantization sion. However, random states from the symmetric sub- formalism, by changing bipartition from ( , ) to space, i.e. bosonic states, typically achieve the Heisenberg A1 A2 ( 1, 2) via equations (52), the action of the beam- limit, even for very mixed isospectral states. Moreover, splitter,B B as outlined in (56), is local. In this bipartition, the quantum-enhancement is typically robust against the the non-locality necessary for enhancing the sensitiv- loss of particles, in contrast to e.g. GHZ-states. It re- ity completely resides in the state. Therefore, the mains to be seen how entangled these states are in the mode-description, leaves the freedom to locate the sense of mode-entanglement, but independently of the resources necessary to accuracy-enhancing either in the outcome of such an assessment, these results are in line entanglement of the state or in the non-locality of the with the finding that the naturally symmetrized pure operations performed on it. states of bosons are a useful resource for quantum metrology. The fact that certain bosonic states can lead to the In (Benatti and Braun, 2013), also the paradigmatic Heisenberg limit while mode entanglement does not play case of phase estimation in a Mach-Zehnder interfer- any significant role has also been recently emphasized in ometer was considered with similar results: in the (Friis et al., 2015; Safránekˇ and Fuentes, 2016). mode-bipartition corresponding to the two modes after There are also metrological advantages achievable with the first beam-splitter, the phase shift operation in bosons that are beyond the context of “standard quan- one arm is a local operator. Hence, at that stage the tum limit” versus “Heisenberg limit” scaling: In (Duiv- state must be mode-entangled to allow estimating the envoorden et al., 2017) it was shown that by using a grid- phase shift with an accuracy better than the shot-noise state (Gottesman et al., 2001) of a single bosonic mode, 23 one can determine both amplitude and phase of a Fourier- like the ones appearing in the decomposition (40) that component of a small driving field that adds at most π/2 defines separable states. Specifically, as a consequence photons, or equivalently, both quadrature components of the result in (Araki and Moriya, 2003), any product of the displacement operator of the state. Slightly bi- (1) (2) ωk (A1) ωk (A2) vanishes whenever A1 and A2 both be- ased estimators were found whose sum of mean-square long to the odd components of their respective subalge- deviations from the true values scales as 1/√n with the bras. Then, given a mode bipartition ( 1, 2) of the average number of photons n in the probe-state. A “com- fermionic algebra , i.e. a decompositionA ofA in the pass state” was proposed in (Zurek, 2001) that achieves subalgebra generatedA by the first m modesA and the A1 similar sensitivity for small displacements up to order subalgebra 2, generated by the remaining M m ones, it 1/√n. These results should be contrasted with the lower follows thatA the decomposition (40) is meaningful− only for bound for any single-mode Gaussian state as a probe local operators A1A2 for which [A1,A2] = 0, so that the state that is of order one, independent of n, and regard- definition of separability it encodes is completely equiv- less the amount of squeezing. With two-mode Gaussian alent to the one adopted for bosonic systems. states one can beat this constant lower bound, but then As a further consequence of the result in (Araki and the two modes must be necessarily entangled (Genoni Moriya, 2003), one derives that if a state ω is non vanish- et al., 2013). It is possible that the result in (Duiven- ing on a local operator AoAo, with the two components voorden et al., 2017) may still be improved upon with 1 2 Ao o, Ao o both belonging to the odd part of other states, as in (Duivenvoorden et al., 2017) also a 1 1 2 2 the∈ two A subalgebras,∈ A then ω is entangled with respect to lower bound for all single-mode probe states was found the bipartition ( , ). Indeed, if ω(AoAo) = 0, then that scales as 1/(2n+1). This is the same lower bound as 1 2 1 2 ω cannot be writtenA A as in (40), and therefore cannot6 be for arbitrary (in particular: entangled) two-mode Gaus- separable. sian states (Genoni et al., 2013). Using these results, as for the bosonic case, one shows that (Benatti et al., 2014a), given a bipartition of the C. Mode entanglement and metrology: fermions fermionic algebra determined by the integer m, a pure state ψ results separableA if and only if it can be written As in the case of bosonic systems, we shall consider in the| formi (44). Examples of pure separable states of N generic fermionic many-body systems made of N elemen- fermions are the Fock states in (63); indeed, they can be tary constituents that can occupy M different states or recast in the form M modes, N < . The creation ai† and annihilation ai operators for mode i obey now the anticommutation relations a , a† = δ and generate the fermion algebra k1,...,km; pm+1,...,pM (64) { i j} ij | i , i.e. the norm closure of all polynomials in these opera- k1 km pm+1 pM = (â1†) (âm† ) (âm† +1) (âM† ) 0 , tors.A As already specified before, a bipartition ( , ) ··· × ··· | i 1 2 h i h i of this algebra is the splitting of the collectionA ofA creation and annihilation operators into two disjoint sets. where the and appearing in (44) are now monomials The Hilbert space of the system is again generated in the creationP operatorsQ of the two partitions. out of the vacuum stateH 0 by the action of the creation Concerning the metrological use of fermionic systems, operators; it is spanned by| i the many-body Fock states, the situation may appear more problematic than with n1 n2 nM bosons, for each mode can accommodate at most one n1,n2,...,nM =(a1†) (a2†) (aM† ) 0 , (63) | i ··· | i fermion; therefore, the scaling with N of the sensitivity where the integers n1,n2,...,nM are the occupation in the estimation of physical parameters may worsen. In- numbers of the different modes, with i ni = N; they deed, while a two-mode bosonic apparatus, as a double- can now take only the two values 0 or 1. well interferometer, filled with N particles is sufficient to As already clear from the definitionP of fermionic alge- reach sub shot-noise sensitivities, with fermions, a multi- braic bipartitions, because of the anti-commutation re- mode interferometer is needed in order to reach compa- lations, in dealing with fermions, one must distinguish rable sensitivities (for the use of multi-mode interferome- between even and odd operators. While the even com- ters see, for instance (Cooper et al., 2009; D’Ariano et al., ponent e of consists of elements Ae such that 1998; D’Ariano and Paris, 1997; Söderholm et al., 2003; ϑ(Ae) =A Ae,A the odd component o of∈ Aconsists of Vourdas and Dunningham, 2005), and (Cooper et al., those elements Ao such that ϑ(AAo)= AAo. Even el- 2012) in the fermionic case). As an example, consider a ements of commute∈A with all other elements,− while odd system of N fermions in M modes, with M even, and elements commuteA only with even ones. let us fix the balanced bipartition (M/2, M/2), in which The anticommuting character of the fermion algebra each of the two parts contain m = M/2 modes, taking puts stringent constraints on the form of the fermion for simplicity N m. As generator of the unitary trans- Astates that can be represented as product of other states, formation ρ ρ≤inside the measuring apparatus let us → θ 24 take the following operator: g , e , respectively. If these have energies ~ω/2, then the| i Hamiltonian| i giving rise to the Ramsey oscillation± is m 1 N = ωk ak† am+k + am† +kak , (65) J 2 H = ~ωS ~ω s(i) (68) k=1 z ≡ z X i=1 p X where ωk is a given spectral function, e.g. ωk k , 1 ≃ where sz ( e e g g ) is a pseudo spin-1/2 op- with p positive. The apparatus implementing the above ≡ 2 | ih |−| ih | erator describing the transition and the superscript (i) state transformation is clearly non-local with respect to iθ indicates the i-th atom. This Hamiltonian is mani- the chosen bipartition: e J can not be written as the festly linear in Sz, and can be trivially decomposed into product A1A2 of two components made of operators A1 (i) (i) micro-Hamiltonians H = h ~ωsz that de- and A2 belonging only to the first, second partition, re- i i scribe the uncoupled precession of≡ each atom in the en- spectively. It represents a generalized, multimode beam P P splitter, and the whole measuring device behaves as a semble. For a single Ramsey sequence of duration T , multimode interferometer. so that the unknown parameter is θ = ωT , the Stan- dard Quantum Limit and Heisenberg-limit sensitivities Let us feed the interferometer with a pure initial state, 1 are as described in Section I, Var(θest)SQL = N − and ρ = ψ ψ , 2 | ih | Var(θest)HL = N − . The assumption of uncoupled parti- ψ = 1,..., 1, 0,..., 0 ; 0,..., 0 = a†a† a† 0 , cles is often physically reasonable, for example when de- | i | i 1 2 ··· N | i scribing photons in a linear interferometer or low-density N m N m − atomic gases for which collisional interactions can be (66) neglected. In other systems, including Bose-Einstein where the| fermions{z } | occupy{z } the| {z first} N modes of the first condensates (Gross et al., 2010a; Riedel et al., 2010a), partition; ψ is a Fock state and therefore it is separable, laser interferometers at high power (Aasi et al., 2013; as already| discussed.i The quantum Fisher information The LIGO Scientific Collaboration, 2011), high-density can be easily computed (Benatti et al., 2014a) atomic magnetometers (Dang et al., 2010; Kominis et al., N 2003; Shah et al., 2010; Vasilakis et al., 2011) and high- I ρ, = ω2 . (67) density atomic clocks (Deutsch et al., 2010), the assump- θ J k k=1 tion of uncoupled particles is unwarranted. This moti- X vates the study of nonlinear Hamiltonians. Unless ωk is k-independent, Iθ ρ, is larger than N and The unusual features of nonlinear Hamiltonians are J therefore the interferometric apparatus can beat the shot- well illustrated in the following example (Boixo et al., noise limit in θ-estimation, even starting with a separable 2008b). First, define the collective operator S0 p 2p+1 ≡ state. Actually, for ωk k , one gets: Iθ ρ, N , s(i) 1(i), where 1 indicates the identity oper- reaching sub-Heisenberg≃ sensitivities with aJ linear≃ device. i 0 ≡ i ator. S0 is clearly the total number of particles. Now Note that this result and the ability to go beyond the considerP theP nonlinear Hamiltonian Heisenberg limit is not a “geometrical” phenomenon at- N N tributable to a phase accumulation even on empty modes ~ ~ (i) (j) H = ΩS0Sz = Ω s0 sz . (69) (D’Ariano and Paris, 1997); rather, it is a genuine quan- i=1 j=1 tum effect, that scales as a function of the number of X X fermions, the resource available in the measure. This is linear in the unknown Ω, but of second order Again, as in the bosonic case, it is not the entanglement in the collective variables S. At the microscopic level, of the initial state that helps overcoming the shot-noise- the Hamiltonian describes a pair-wise interaction, with ~ (i) (j) limit in the phase estimation; rather, it is the non-local energy Ωs0 sz , between each pair of particles (i,j). character of the rotations operated by the apparatus on For a system with a fixed number N of particles, the an initially separable state that allows one beating the consequence for the dynamics of the system is very sim- shot-noise limit. ple: the operator S0 can be replaced by its eigenvalue N, leading to an effective Hamiltonian

HN = ~NΩSz. (70) IV. MORE GENERAL HAMILTONIANS Estimation of the product NΩT = NΘ now gives the A. Non-linear Hamiltonians same uncertainties that we saw earlier in the estima- 1 tion of θ. That is, Var(NΘest)SQL = N − so that 3 4 Most discussions of quantum-enhanced measurements Var(Θest)SQL = N − . Similarly, Var(Θest)HL = N − . consider, implicitly or explicitly, evolution under a More generally, a nonlinear Hamiltonian containing k- Hamiltonian that is linear in a collective variable of the order products of collective variables will contain N k mi- system. For illustration, consider Ramsey spectroscopy croscopic interaction terms that contribute to the Hamil- on a collection of N atoms with ground and excited states tonian and thus to the rate of change of an observable 25 such as Sz under time evolution. In contrast, the variance B. Proposed experimental realizations of such a macroscopic observable, e.g. Var(Sz), scales as N 1 (Standard Quantum Limit) or N 0 (Heisenberg-limit). A number of physical systems have been proposed for 2k 1 This allows signal-to-noise ratios scaling as N − (Stan- nonlinear quantum-enhanced measurements: Propaga- dard Quantum Limit) and as N 2k (Heisenberg-limit) tion through nonlinear optical materials (Beltránand (Boixo et al., 2007). These noise terms depend only on Luis, 2005; Luis, 2004, 2007), collisional interactions in the size of the system and nature of the initial state but Bose-Einstein condensates (Boixo et al., 2009), Duff- not on the Hamiltonian (Boixo et al., 2007). ing nonlinearity in nanomechanical resonators (Woolley et al., 2008), and nonlinear Faraday rotation probing of That nonlinearities lead to improved scaling of the sen- an atomic ensemble (Napolitano and Mitchell, 2010). sitivity appears to have been independently discovered by A. Luis and J. Beltrán(Beltránand Luis, 2005; Luis, 2004, 2007) and by S. Boixo and co-workers (Boixo et al., 1. Nonlinear optics 2008a,b, 2007; Datta and Shaji, 2012). A related pro- N The first proposals concerned nonlinear optics (Beltrán posal using interactions to give a scaling σ(θest) 2− is described in (Roy and Braunstein, 2008). ∝ and Luis, 2005; Luis, 2004, 2007), in which a nonlinear optical susceptibility is directly responsible for a phase- shift θ N k, where k is the order of the nonlinear contri- Prior to the appearance of these results, the term ∝ “Heisenberg limit,” which was introduced into the lit- bution to the refractive index. In the simplest example, erature by (Holland and Burnett, 1993) in the context (Beltránand Luis, 2005) showed that an input coherent state α , experiencing a Kerr-type nonlinearity described of interferometric phase estimation with the definition | i 2 by the unitary exp[iΘ(a†a) ], and detected in quadrature σ(φest) = 1/N, had been used, often indiscriminately, to 1 X = (a + a†), gives an outcome distribution describe 1) the sensitivity σ(φest) = 1/N, 2) the scaling √2 σ(φest) 1/N, 3) the best possible sensitivity with N iΘ(a†a)2 2 particles,∝ and 4) the best possible scaling with N parti- P (X = x Θ) = x e α . (71) | |h | | i| cles (Giovannetti et al., 2004, 2006, 2011). Clearly these multiple definitions are not all compatible in a scenario If we consider the case of small Θ, imaginary α = ¯ ¯ with a nonlinear Hamiltonian. Taking as a definition “er- i N , and the estimator Θest = X/ ∂X/∂θ , where ¯ h i M | | ror . . . bounded by the inverse of the physical resources,” X Xi is the mean of the observed quadratures, p≡ i=1 and implicitly considering scaling, (Zwierz et al., 2010) we find the standard deviation P (see also (Zwierz et al., 2011) ) showed that an appropri- σ(X) σ(X) ate definition for “physical resource” is the query com- σ(Θest)= = (72) √M d X /dΘ √M [X, (a a)2] plexity of the system viewed as a quantum network. | h i | |h † i| 1 = 1/2 3/2 . (73) For the simplest optical nonlinearity, θ N 2 and 4M N − ∝ quadrature detection, it has been shown that quadrature Here we have used σ(X) = 1/√2 for the quantum me- squeezed states are near-optimal (Maldonado-Mundo chanical uncertainty of X in the initial state, and which and Luis, 2009). Considering the same nonlinearity and up to corrections of order Θ2 holds also for the evolved limiting to classical inputs, i.e. coherent states and mix- state when Θ is small. For large N, this strategy sat- tures thereof, it is argued in (Rivas and Luis, 2010) that urates the quantum Cramér-Rao bound; the quantum non-linear strategies can out-perform linear ones by con- Fisher information is straightforwardly calculated to be 2 3 centrating the available particles in a small number of IΘ = 4N + 24N + 16N . high-intensity probes. (Tilma et al., 2010) analyzed a variety of entangled coherent states for nonlinear interferometry of varying orders, and found that in most cases 2. Ultra-cold atoms entanglement degraded the sensitivity for high-order nonlinearities. (Berrada, 2013) considered the use of two- Coherent interaction-based processes are well devel- mode squeezed states as inputs to a non-linear interfer- oped in Bose-Einstein condensates and have been used ometer, including the effects of loss, and showed a robust extensively for squeezing generation. For example, a con- advantage for such states. fined two-species Bose-Einstein condensate experiences collisional energy shifts described by an effective Hamil- As already mentioned in Section I.C, the above results tonian concern local measures implying some prior knowledge of H a n (n 1) + 2a n n + a n (n 1)(74) the parameter being estimated. The situation for global eff ∝ 11 1 1 − 12 1 2 22 2 2 − =(a 2a + a )S2 + 2(a a )S S measures without prior information is considered in (Hall 11 − 12 22 z 11 − 22 z 0 and Wiseman, 2012). +terms in Sz,S0 (75) 26 where S 1 (n n ) and S 1 (n + n ) are pseudo- fective Hamiltonian for the interaction of polarized light, z ≡ 2 1 − 2 0 ≡ 2 1 2 spin operators, n1 and n2 are the number of atoms of described by the Stokes operators S, with the collective species 1 and 2, respectively, and aij are the collisional orientation and alignment spin variables J of an atomic scattering lengths. In 87Rb, and with 1 F = 1, m = ensemble: | i≡| 1 , 2 F = 2, m = 1 , the scattering lengths (near (2) (4) 3 − i | i≡| i Heff = H + H + O(S ) (77) zero magnetic field) have the ratio a11 : a12 : a22 = 1.03 : eff eff (2) (1) (2) 1 : 0.97. A proven method to generate spin squeezing in Heff = α SzJz + α (SxJx + SyJy) (78) this system is to increase a12 using a Feshbach resonance (4) (0) 2 (0) 2 (1) 2 Heff = βJ Sz J0 + βN Sz NA + β S0SzJz to give the single-axis twisting Hamiltonian Heff Sz ∝ (2) (Muessel et al., 2014), plus terms proportional to Sz and +β S0(SxJx + SyJy). (79) S , which induce a global rotation and a global phase 0 where the α and β coefficients are linear and non-linear shift, respectively. polarizabilities that depend on the detuning of the It was observed in (Boixo et al., 2008a) that the zero- probe photons from the atomic resonance. By proper field scattering lengths naturally give a 2a +a 0, 11 12 22 choice of detuning and initial atomic polarization J, the making small the coefficient of S2 and leaving− the S≈S z 0 z term β(1)S S J can be made dominant, making the term as the dominant nonlinear contribution. Detailed 0 z z Hamiltonian formally equivalent to that of Eq. (69). analyses of the Bose-Einstein condensate physics beyond Note that β(1)J , proportional to the atomic polarization the simplified single-mode treatment here are given in z J , plays the role of the unknown interaction energy (Boixo et al., 2009, 2008b; Tacla and Caves, 2013). The z ~Ω. The photons are thus made to interact, mediated strategy gives N 3/2 scaling for estimation of the relative − by and proportional to the atomic polarization J . scattering length a a . (Mahmud et al., 2014) de- z 11 22 For a different detuning, the term α(1)S J becomes scribe a strategy of dynamical− decoupling to suppress the z z dominant, allowing a linear measurement of the same second-order terms in the Hamiltonian and thus make quantity J with the same atomic system. dominant three-body interactions, giving a sensitivity z scaling of N 5/2 for measurements of three-body colli- − The experimental realization using a cold, optically- sion strengths. trapped 87Rb atomic ensemble is described in (Napoli- tano et al., 2011). The experiment observed the predicted scaling of Var(J ) N 3 over a range of photon numbers 3. Nano-mechanical oscillators z − from N = 5 105 ∝to 5 107. For larger photon numbers the scaling worsened,× i.e.× Var(J (N)) had a logarithmic (Woolley et al., 2008) propose a nonlinear interferome- z derivative > 3. Due to this limited range of the N 3 ter using two modes of a nano-mechanical oscillator, with − scaling, and the− difference in pre-factors β(1) versus α(1), amplitudes x and x , experiencing the nonlinear Hamil- a b the nonlinear estimation never surpassed the sensitivity tonian Var(Jz) of the linear measurement for the same number 1 1 of photons. Nonetheless, due to a shorter measurement H = H(a) + H(b) + χ mω2x4 + χ mω2x4 + C(t) eff SHO SHO 4 a a 4 b b time τ, the nonlinear measurement did surpass the lin- (76) ear measurement in spectral noise density Var(Jz)τ, a where HSHO indicates the simple harmonic oscillator common figure of merit for time- or frequency-resolved Hamiltonian, χ is the Duffing nonlinearity coefficient, measurements. ω is the low-amplitude resonance frequency, and C is an externally-controlled coupling between modes a and b that produces a beam-splitter interaction. With an inter- C. Observations and commentary ferometric sequence resembling a Mach-Zehnder interferometer, the Duffing nonlinearity can be estimated with Several differences between linear and nonlinear strate- 3/2 uncertainty scaling as N − , where N is the number of gies, perhaps surprising, deserve comment. excitations. First, it should be obvious that there is no conflict with the Heisenberg uncertainty principle. θ and Θ are parameters, not observables, and as such are not sub- 4. Nonlinear Faraday rotation ject to operator-based uncertainty relations, neither the Heisenberg uncertainty principle nor generalizations such Whereas Luis and co-workers considered phenomeno- as the Robertson uncertainty relation (Robertson, 1929). logical models of optical nonlinearities, (Napolitano and Moreover, the advantageous scaling in δΘ is the result of Mitchell, 2010) describes an ab-initio calculation of the a rapidly-growing signal, rather than a rapidly decreasing optical nonlinearity produced on a particular atomic statistical noise. A nonlinear Hamiltonian immediately transition, using degenerate perturbation theory and a leads to a strong change in the scaling of the signal: even collective quantum variable description. This gives an ef- the simplest k = 2 nonlinearity gives signal growing as 27

ω N 2 and thus Standard Quantum Limit uncertainty spin-squeezing was taken as the figure of merit, and ∝ 3/2 (δΘ)SQL N − , which scales faster with N than does the fully-optimized nonlinear measurement gave more ∝ 1 (δθ) N − . squeezing than the fully-optimized linear measurement. HL ∝ Second, the estimated phases θ and Θ necessarily re- This shows that for some quantities of practical inter- flect different physical quantities. ~ω describes a single- est, a nonlinear measurement can out-perform the best particle energy such as that due to an external field, possible linear measurement. Similar conclusions have whereas ~Ω describes a pair-wise interaction energy. As been drawn for the case of number-optimized saturable such, the uncertainties δθ and δΘ are not directly com- spectroscopy (Mitchell, 2017). parable. Any comparison of the efficacy of the measurements must introduce another element, a connection between a third physical quantity, θ, and Θ. This we have seen in Section IV.B.4, where the unknown Jz appears in both the linear and nonlinear Hamiltonians. In what E. Signal amplification with nonlinear Hamiltonians follows, we describe an optimized linear/nonlinear com- 2 parison. The single-axis twisting Hamiltonian Htwist = χSz , in addition to producing spin squeezed states, has been proposed as a nonlinear amplifier to facilitate state readout D. Nonlinear measurement under number-optimized in atom interferometry (Davis et al., 2016). Starting from conditions an x-polarized coherent spin state x , and defining the unitary U exp[ iH τ/~], the| Wigneri distribution ≡ − twist A more extensive study of nonlinear spin measure- of the squeezed state U x is thin in the z direction, and | i ments, using the same system as (Napolitano et al., 2011) is thus sensitive to rotations (φ) about the y-axis, so Ry is reported in (Sewell et al., 2014). This work compared that states of the form (φ)U x have large quantum Ry | i two estimation strategies, one linear and one non-linear, Fisher information with respect to φ. Exploitation of this for measuring the collective variable Jy, which describes a in-principle sensitivity is challenging, however, because component of the spin alignment tensor used in a style of it requires low-noise readout, detecting Sz at the single- optical magnetometry known as alignment-to-orientation atom level if the Heisenberg limit is to be approached. conversion (Budker et al., 2000; Pustelny et al., 2008; In contrast, a sequence that applies Htwist, waits for ro- Sewell et al., 2012). The linear estimation used the term tation about the y-axis and then applies Htwist for an (2) − α S J , which appears in Eq. (78) and produces a ro- equal time generates the state U † (φ)U x . Because U † y y Ry | i tation from linearly polarized light toward elliptically- is unitary, the quantum Fisher information is unchanged, polarized light. The nonlinear estimation in contrast but the perturbation implied by (φ) now manifests it- Ry used Eq. (78) in second order: in the first step, due to the self at the scale of the original coherent spin state, which (2) α SxJx term and the input Sx optical polarization, an is to say it is amplified from the Heisenberg-limit scale up initial Jy atomic polarization is rotated toward Jz by an to the Standard Quantum Limit scale, greatly facilitat- angle φ Sx , and thus N, where N is the number of ing detection. Implementations include a cold-atom cav- ∝h i ∝ (1) photons. In the second step, the term α SzJz produces ity QED system (Hosten et al., 2016) and Bose-Einstein a Faraday rotation, i.e. from Sx toward Sy, by an angle condensates (Linnemann et al., 2016). While this strat- proportional to the Jz polarization produced in the first egy clearly uses entanglement, it is nonetheless striking 2 step. The resulting Sy polarization is Sy JyN , while that un-doing the entanglement-generation step provides 1/2 ∝ the statistical noise is σ(Sy) N , giving sensitivity an important benefit. 3/2 ∝ scaling σ(J ) N − . Importantly, the two estimation y ∝ strategies use the same Sx-polarized input, and thus have identical statistical noise and cause identical damage in the form of spontaneous scattering, which adds noise to the atomic polarization. F. Other modifications of the Hamiltonian The experimentally-observed nonlinear sensitivity was N compared against the calculated ideal sensitivity of the The assumption of a Hamiltonian H = k=1 hk con- linear measurement. Owing to its faster scaling, and sidered for the derivation of eq.(1), where Λ and λ are the P more favorable pre-factors than in (Napolitano et al., largest and smallest eigenvalues of hk, respectively, not 2011), the nonlinear measurement’s sensitivity surpassed only implies distinguishable subsystems, it is also restric- that of the ideal linear measurement at about 2 107 tive in two other important regards: a.) The existence × photons. A comparison was also made when each mea- of such bounds on the spectrum of hk may not be war- surement was independently optimized for number N and ranted, and b.) interactions between the subsystems are detuning, which affects both the pre-factors α and the excluded. In this section we explore the consequences of scattering. The ability to produce measurement-induced lifting these restrictions. 28

1. Lifting spectral limitations the energy eigenstates. Hence, exactly the same minimal uncertainty of θest can be obtained by superposing the A large portion of the work on quantum-enhanced ground state of a single mode with a Fock state of measurements stems from quantum optics, where given maximum allowed energy as with an arbitrarily the basic dynamical objects are modes of the entangled multi-mode state containing components of e.m. field, corresponding to simple harmonic oscil- up to the same maximum energy. lators, hk = ~ωk(nk + 1/2). A phase shift in mode k For a specific example, consider phase estimation in a can be implemented by U = exp(inkθ). Clearly, for Mach-Zehnder interferometer. It has N = 2 modes, and the relevant Hamiltonian hk = nk acting as generator a phase shift just in one of them, i.e. the relevant Hamil- of the phase shift, Λ = and λ = 0. Hence, eq.(1) tonian is H = θn . Adding energy conservation of the ∞ 1 would imply a minimal uncertainty Var(θest) = 0. Of two modes at the beam-splitters (i.e. the fact that the course, one may argue that in reality one can never use accessible states are two-mode Fock states of the form states of infinite energy, such that there is effectively n n2,n2 , where n2 with 0 n2 n is the number a maximum energy. However, it need not be that the of| − photonsi in the second mode),≤ one≤ immediately finds maximum energy sustainable by the system must be that the optimal two mode state is ( n, 0 + 0,n )/√2, distributed over N modes. Indeed, what is typically i.e. the highly entangled N00N state (Boto| i et| al.,i 2000). counted in quantum optics in terms of resources is not However, we can achieve exactly the same variance the number of modes N, but the total number of photons of G and hence sensitivity with the single-mode state n, directly linked to the total energy. It turns out, that ( n + 0 )/√2 ρ , i.e. a product state where we keep the | i | i ⊗ 2 the total number of modes (or subsystems, in general) is second mode in any state ρ2. In both cases the maximum completely irrelevant for achieving optimal sensitivity, energy of the first mode is n~ω (assuming ω1 = ω2 = ω), even if the parameter is coded in several modes or and the average energy in the interferometer n~ω/2 subsystems, e.g. with a general unitary transformation (neglecting the vacuum energy ~ω/2). Hence, also N of the form U = exp(iθ k=1 hk), if one can stock the from the perspective of maximum energy deposit in the same amount of energy in a single system as in the total optical components, there is no advantage in using two system. Note that thisP is often the case in quantum entangled modes. If the Mach-Zehnder interferometer optics, where different modes can be spatially confined or is realized abstractly via Ramsey-pulses on N two-level parametrically influenced by the same optical elements systems (states 0 , 1 ) for the beam-splitters, and such as mirrors, beam-splitters, and phase shifters whose | i | i N (i) a phase shift exp(iθJz), Jz = i=1 σz /2, the state material properties ultimately determine the maximum that maximizes Var(Jz) is the (maximally entangled) amount of energy that can be used. GHZ state ( 0 ... 0 + 1 ... 1 )/√P2. But exactly the same uncertainty| cani be| obtainedi with a single spin-j To see the liberating effect of unbound spectra, recall (j = N/2) in the state ( j,j + j, j )/√2 (in the usual that for any initial pure state ψ propagated by a Hamil- j, m notation, where j|is thei total| − angulari momentum | i tonian of the form H = θG with a hermitian generator |and im its z-component). Clearly, allowing as large a G for a time T the quantum Fisher information is given spectrum for a single system as for the combined systems by (Braunstein and Caves, 1994; Braunstein et al., 1996) makes entanglement entirely unnecessary here.

I = 4Var(G)T 2 4( G2 G 2)T 2 . (80) θ ≡ h i−h i These considerations teach us that the relevant Let G = i ei i i be the spectral decomposition of quantity to be maximized is the quantum uncertainty | i|hL| G, and ψ = ci i , where we assume that 1 of the generator G. This can be understood in terms of | Pi i=1 | i | i ( L ) are the states of lowest (largest) energy avail- generalized Heisenberg uncertainty relations, in which | i P L L able. Then Var(G) = p e2 ( p e )2 with the generator G plays the quantity complementary to i=1 i i − i=1 i i 2 L θ, as was found early on (Braunstein et al., 1996). In pi = ci and i=1 pi = 1. The Popoviciu inequality | | P P 2 (Popoviciu, 1935) states Var(G) (eL e1) /4. It is a multi-component system maximizing Var(G) may P ≤ − saturated for p1 = pL = 1/2, pi = 0 else. The state be achieved with highly entangled states, but if the ψ = ( 1 + eiϕ L )/√2 with an arbitrary phase ϕ spectral range of a single system admits the same | i | i | i saturates the inequality and thus maximizes Iθ. If eL Var(G), there is no need for entanglement. If unbound or e1 is degenerate, only the total probability for the spectra are permitted, one can in fact do much better degenerate energy levels is fixed to 1/2, and arbitrary than the Heisenberg-limit: In (Berry et al., 2015) the √3 n n ∞ linear combinations in the degenerate subspace are single-mode state 2 n=0 2− 2 was pointed out allowed. But the value of Var(G) remains unchanged that has diverging Var(n) with, at| thei same time finite under such redistributions, and we may still choose just n¯. It therefore allows,P at least in principle and in an two non-vanishing probabilities p1 = pL = 1/2. The ideal setting, arbitrarily precise phase measurements derivation did not make use of a multi-mode structure of while using finite energy. 29

(Haroche, 2013; Wineland, 2013) for historical accounts In (Braun, 2011, 2012) a state of the form ( 0 + of the development of these fields and many more refer- 2n )/√2 was found to be optimal for mass measurements| i ences, as well as the literature citing (Montina and Arec- |withi a nano-mechanical oscillator given a maximum al- chi, 1998) where superpositions of coherent states in a lowed number of excitation quanta 2n and times much Mach-Zehnder interferometer were studied with respect larger than the oscillation period (for shorter times there to the limitations arising from imperfect photodetectors). are contributions also from the dependence on frequency The use of superpositions of coherent states for metrol- of the energy eigenfunctions). The same state of a single ogy was examined in (Gilchrist et al., 2004; Ralph, mode of the e.m. field is optimal for measuring the speed 2002) and it was found that the Heisenberg limit can of light (Braun et al., 2017). In both cases the quantum be reached. In (Montina and Arecchi, 1998) . superposi- uncertainties scale as 1/n (quantum Fisher information tions of coherent states in atom interferometers may even proportional to n2), and obviously no entanglement is exhibit quantum-enhanced sensitivity to parameters that needed. Of course, a state of the form ( 0 + 2n )/√2 have have no classical analog. For example, in (Riedel, (called “half a N00N” state in (Braun,| 2011))i | isi still 2015) it was shown that monitoring the decoherence rate highly non-classical (see also (De Pasquale et al., 2015)). of an superposition of atomic coherent states may un- In fact, already a single Fock state n is highly non- cover clues about so-far undetected particles that couple classical, as is witnessed by its highly| oscillatoryi Wigner softly (i.e. via weak momentum transfer, but not weakly) function (Schleich, 2001) with substantial negative parts. to the atoms. This is reminiscent of previous ideas of The superposition ( 0 + 2n )/√2 leads in addition to 2n using decoherence as a sensitive probe (Braun and Mar- lobes in azimuthal direction| i | i that explain the sensitivity tin, 2011). The decoherence rate can be detected with 1 of phase measurements n− . Alternatively, one can use sensitivity that is limited only by the spatial size of the superpositions of coherent∝ states (Bimbard et al., 2010; superposition, and the situation is quite similar to the Braginsky et al., 1995; Lund et al., 2004; Neergaard- estimation of boson loss discussed in II.E. Nielsen et al., 2006; Suzuki et al., 2006; Wakui et al., 2007; Yukawa et al., 2013), i.e. ”Schrödinger-cat” type states of the form ( α + α )/√2. They have been cre- 2. Decoherence-enhanced measurements ated in quantum optics| i with|− i values of α = 0.79 in (Our- joumtsev et al., 2006). In (Lund et al., 2004) a “breeding Decoherence is arguably the most fundamental is- method” based on weak squeezing, beam mixing with sue that plagues quantum enhancements of all kinds, an auxiliary coherent field, and photon detecting with and quantum enhanced measurements are no exception. threshold detectors was proposed to achieve values up to However, decoherence has interesting physical properties α 2.5, but the success probability has been found to which imply that it can also be useful for precision mea- be≤ too low for a realistic iterated protocol. An alterna- surements. This goes beyond the benefits of decoherence tive based on homodyning was proposed in (Etesse et al., and open system dynamics found as early as the late 2014; Laghaout et al., 2013) and implemented in (Etesse 1990s and the early 2000s, when it was realized that en- et al., 2015), leading to α 1.63. The current record in tanglement can be created through decay processes or the optical domain for “large”≃ α appears to be α √3, more generally through coupling to common environ- achieved from two-mode squeezed vacuum and n-photon≃ ments (Benatti et al., 2009, 2010b, 2003, 2008; Braun, detection on one of the modes (Huang et al., 2015). In 2002, 2005; Plenio et al., 1999), and, paradoxically, that the microwave regime, superpositions of coherent states decoherence of quantum computations can be reduced by with α = √7 have been generated, as well as superposi- rather strong dissipation that confines the computation tion of coherent states with smaller phase differences with to a decoherence-free subspace through a Zeno-type effect up to 111 photons (Vlastakis et al., 2013). In (Monroe due to the rapid decay of states outside the decoherence- et al., 1996) superpositions of coherent states of the vi- free subspace (Beige et al., 2000a,b). Recently such ideas brational motion of a 9Be+ ion in a one-dimensional trap have found renewed interest, and meanwhile techniques with α 3 were reported. Almost arbitrary superposi- have been proposed to create steady state entanglement tions with≃ a small number of photons can be generated in driven open quantum systems, such as cold Rydberg by using couplings of a mode with two-level systems that gases in the Rydberg-blockade regime (Lee et al., 2015). can be tuned in and out of resonance, and a plethora of It remains to be seen whether such stabilized entangled methods for generating superpositions of coherent states states are useful for precision measurements. were proposed, but reviewing the entire literature of non- Here, on the contrary, we focus on the dynamics cre- classical states in general and even all the proposals for ated by decoherence processes themselves. Decoherence generating superpositions of coherent states is beyond the arises from an interaction with an environment described scope of the present review (see e.g. (Deléglise et al., 2008; by a non-trivial interaction Hamiltonian Hint = i SiBi Gottesman et al., 2001; Hofheinz et al., 2009) and the with a similar structure as the non-linear Hamiltonians P Nobel lectures of Serge Haroche and David J. Wineland considered above, where in Hint, however, one distin- 30 guishes operators Si pertaining to the system and others cavity limit in which superradiance arises. Thus, at least (Bi) pertaining to the environment. The environment is in principle, Heisenberg-limited scaling can arise here typically considered as heat bath with a large number of without the need for an entangled state, and in spite degrees of freedom that may not be entirely accessible of the inherently decoherent nature of superradiance. (Benatti and Floreanini, 2005; Breuer and Petruccione, However, the prefactor matters also here: Superradiance 2002; Weiss, 1999). In addition, system and heat bath leads to a rapid decay of all states that are not dark, have their own Hamiltonian, such that the total Hamil- such that the available signal and with it the prefactor tonian reads H = HS + HB + Hint. From simple model of the 1/N scaling law deteriorate rapidly with time. systems it is known that decoherence tends to become extremely fast for quantum superpositions of states that Given the delocalized nature of the cavity mode in this differ macroscopically in terms of the eigenvalues of one example, it is possible in principle to make the number of the Si. For example it was shown that superposing of atoms that interact with the mode arbitrarily large, two Gaussian wave packets of a free particle in one di- in contrast to the non-linear schemes above, for which menson that is coupled through its position to a heat the interactions have to decrease if the total energy is to bath of harmonic oscillators leads to decoherence times remain an extensive quantity. But if the volume is kept that scale as powers of ~ that depend on how the wave fixed the atoms will start to interact so that the simple packets are localized in phase space: The shortest de- independent atom model of superradiance breaks down. coherence time, scaling as ~/ q q results from wave For even larger N one has to increase the size of the cav- | 1 − 2| packets distinguished only by positions q1,q2, the longest ity in order to accommodate all atoms. When the largest one ~1/2/ p p from wave packets that differ only possible density is reached, the volume will have to grow ∼ | 1 − 2| in their mean momenta p ,p , and an intermediate one N, which implies coupling constants of the atoms to p 1 2 scaling as ~2/3/ (q q )(p p ) 1/3. For systems of finite ∝the cavity that decay as 1/√N and leads back to Stan- | 1− 2 1− 2 | Hilbert-space dimensions such as angular momenta, one dard Quantum Limit scaling. In addition, the number of can often identify an effective ~ that scales like the inverse atoms has to be macroscopic in order to compete with the Hilbert-space dimension which suggests that monitoring best classical sensitivities reached with interferometers the decoherence process can lead to highly sensitive mea- such as LIGO: Assuming that the prefactor in the 1/N- surements, possibly surpassing the 1/√N scaling of the scaling is of order one, one needs 1021 atoms for a min- 21 ∼ Standard Quantum Limit. imal uncertainty of 10− . A cubic optical lattice with That this intuition is correct was shown in (Braun one atom every µm, the lattice would have to span 10m and Martin, 2011), where a method war proposed for in x,y,z-direction in order to accommodate that many measuring the length of a cavity by monitoring the deco- atoms. When using a diamond with regularly arranged herence process of N atoms inside the cavity. The atoms NV-centers every 10nm (such dense samples have been are initially prepared in a highly excited dark state, fabricated (Acosta et al., 2009)), one would still require in which destructive interference prevents them from a cubic diamond of edge length 10cm. These examples transferring their energy to a mode of the cavity with show that competing with the best classical techniques is which they are resonant. For example, if one has two very challenging even if one can achieve Heisenberg-limit (1) (2) atoms coupling via an interaction (g1σ +g2σ )† +h.c. scaling, as in classical protocols it is relatively easy to to a mode of the cavity with annihilation− operator− a, scale up the number of photons, compensating thus for a a state (1/g1) 10 (1/g2) 01 of the atoms (where less favorable scaling with N. 0 and ∝1 are the| i ground − and| i excited states of the atoms)| i is| ai dark state, also known as decoherence-free state: the amplitudes of photon transfer from the two 3. Coherent averaging atoms to the cavity cancel. However, the couplings gi depend on the position of the atoms relative to the There is nothing inherently quantum about the cavity due to the envelope of the e.m. field. If the 1/√N scaling of the Standard Quantum Limit. Rather, cavity changes its length L with the atoms at fixed this behavior is a simple consequence of the central positions, the gi change, such that the original state limit theorem applied to N independently acquired becomes slightly bright. This manifests itself through measurement results that are averaged as part of a the transfer of atomic excitations to the mode of the classical noise-reduction procedure. The idea of coherent cavity, from where photons can escape and be detected averaging is to replace the averaging by a coherent outside. In (Braun and Martin, 2011) it was shown that procedure, in which the N probes all interact with a through this procedure the minimal uncertainty with central quantum system (a “quantum bus”). In the end which L can be estimated (according to the quantum one measures the quantum bus or the entire system. Cramér-Rao bound) scales as 1/N even when using an initial product state of N/2 pairs of atoms. This A simple example shows how this can lead to scaling applies both for a perfect cavity, and in the bad Heisenberg-limit sensitivity without needing any 31 entanglement: Consider N spins-1/2 interacting the standard scenario in decoherence, it is assumed here with a central spin-1/2 with the Ising-interaction that at least part of the “environment” is accessible. The N (0) (i) above example shows that the “environment” can be as Hint = i=1 giσz σz , where the index zero indicates the central spin. The⊗ interaction commutes with the free simple as a single spin-1/2. One can also extend the P ~ N (i) model to include additional decoherence processes. In Hamiltonian of all spins Hs = i=0 ωiσz , and we can solve the time-evolution exactly, starting from the initial (Braun and Martin, 2011) a phase-flip channel with rate 1 P N Γ on all spins was considered. It was found that phase product state ψ(0) = ( 0 0 + 1 0) i=1 0 i. | i √2 | i | i ⊗ | i flips on any of the N probes has no effect, whereas phase At time t, the state has evolved to ψ(t) = 1 i(ω0/2 Ng¯)t i(ω0/2 Ng¯)t | N i flip of the central spin introduces a prefactor that decays (e − 0 0 + e− − 1 0) 0 i, √2 | i | i ⊗i=1 | i exponentially as exp( 2Γt) with time in σ(0)(t) . Since up to an unimportant global phase factor. In par- x the prefactor is independent− of N the power-lawh i scaling ticular, the state remains a product state at all (0) of the sensitivity with N is unchanged, but it is clear times. If we measure σx of the central spin, we find (0) that the prefactor matters and leads to a sensitivity that σx (t) = cos((ω0 2Ng¯)t), i.e. the oscillation frequency deteriorates exponentially with time. increasesh i for large N− proportional to N. As the quantum fluctuations of the central spin are independent of N, The fact that the parameter estimated in the above this implies a standard deviation in the estimation of the example is the interaction strength between the bus and average couplingg ¯ that scales as 1/N, which can be con- the N probes prevents a comparison with other schemes firmed by calculating the quantum Fisher information. that do not use interactions. In (Fra¨ısse and Braun, Clearly, this is not an effect of entanglement, but simply 2015) a more comprehensive study of two different spin- of a phase accumulation. In this respect the approach systems was undertaken where also parameters describ- is reminiscent to the sequential phase accumulation ing the probes themselves and the bus were examined. protocols, in which the precision of a phase shift ϕ Regimes of strong and weak interaction were analyzed, measurement is enhanced by sending the light several and different initial states considered. The two models, et al. times through the same phase shifter (Higgins , called ZZZZ and ZZXX models, are respectively given by 2007). However, there the losses increase exponentially the Hamiltonians with the number of passes, and the sequential nature of the interaction leads to a bandwidth penalty that N N ω0 (0) ω1 (i) g (0) (i) is absent for the simultaneous interaction described by H1 = σ + σ + σ σ , 2 z 2 z 2 z z i=1 i=1 Hint. In (Birchall et al., 2017) the exponential loss X X of photons with the number of passes was taken into ω ω N g N H = 0 σ(0) + 1 σ(i) + σ(0)σ(i) , (81) account, and quantum Fisher information per scattered 2 2 z 2 z 2 x x i=1 i=1 photon optimized. It was found that when both probe X X and reference beam are subject to photon loss, the where ~ = 1. The ZZZZ model is an exactly solvable de- reduction of σ(ϕest) by a non-classical state compared to phasing model, the ZZXX can be analyzed numerically an optimal classical multi-pass strategy is only at most and with perturbation theory. The analysis is simplified 19.5%, and an optimal number of passes independent ∼ when starting with a product state that is symmetric un- of the initial photon number was found, resulting in a der exchange of the probes, in which case the probes can loss of about 80% of all input photons. For a single-mode be considered as a single spin with total spin quantum lossy phase the possible sensitivity gain is even smaller. number j = N/2. It was found that for ω1, Heisenberg- Multi-pass microscopy was proposed in (Juffmann limit scaling can be achieved in the ZZXX model when et al. , 2016a) and it was experimentally demonstrated measuring the entire system, but not when only mea- that at a constant number of photon sample inter- suring the quantum bus, and not for the ZZZZ model. actions retardance and transmission measurements Heisenberg-limit scaling for the uncertainty of ω0 is not with a sensitivity beyond the single-pass shot-noise possible in either model. For the interaction strength g, limit could be achieved. Similar ideas are being de- Heisenberg-limit scaling is found for a large set of initial et al. veloped for electron-microscopy (Juffmann , 2016b). states and range of interaction strengths when measuring the entire system, but only for a small set of initial The fact that an interaction with N probes and a states in the vicinity of the state considered in the simple central quantum bus can lead to Heisenberg-limit scaling example above, when measuring only the quantum bus. of the sensitivity was first found in (Braun and Martin, Interestingly, in the ZZZZ model Heisenberg-limit scal- 2011), where a more general model was studied. It will ing of the sensitivity for measuring g also arises with the be noted that the total Hamiltonian has exactly the probes in a thermal state at any finite temperature, as same structure as for a decoherence model, with the N long as the quantum bus can be brought into an initially probes playing the role of the original system, and the pure state. This is reminiscent of the “power of one- bus the role of an environment. However, in contrast to qubit” (Knill and Laflamme, 1998): With a set of qubits 32 of which only a single one is initially in a pure state, tion was investigated in (Braun and Popescu, 2014) in a quantum enhancement is already possible in quantum a purely classical model of harmonic oscillators, in which computation, providing evidence for an important role of N “probe” oscillators interact with a central oscillator. quantum discord (Datta et al., 2005; Lanyon et al., 2008) It was found that indeed for weak interaction strengths (see Sec. II.C). As detailed in Sec.II.C, the DQC1-scheme a regime of Heisenberg-limit scaling of the sensitivity ex- can provide better-than-Standard Quantum Limit sensi- ists, even though the scaling crosses over to Standard tivity as soon as the ancillas have a finite purity. The Quantum Limit scaling for sufficiently large N. Nev- control qubit plays the role of the quantum bus, and the ertheless, it was proposed that this could be useful for dipole-dipole interaction between the Rydberg atoms im- measuring very weak interactions, and notably improve plements the XX-interaction considered in (Fra¨ısse and measurements of the gravitational constant. Braun, 2015). An important limitation of such schemes was proven in (Boixo et al., 2007; Fra¨ısse and Braun, 2016), where 4. Quantum feedback schemes it was shown very generally that with a Hamiltonian extension to an ancilla system the sensitivity of a phase In the context of having probes interact with ad- shift measurement cannot be improved beyond the best ditional (ancilla) systems, quantum feedback schemes sensitivity achievable with the original system itself. should be mentioned. This is a whole field by itself (see Coherent averaging is nevertheless interesting, as it e.g. (D’Alessandro, 2007; Serafini, 2012; Wiseman and allows one to achieve without injecting entanglement Milburn, 2009) for reviews). Quantum feedback general- better-than-Standard Quantum Limit sensitivity for izes classical feedback loops to the quantum world: one which the non-interacting phase shift measurement tries to stabilize, or more generally dynamically control, would need a highly entangled state. In (Fra¨ısse and a desired state of or an operation on a quantum system Braun, 2017) it was shown that for general parameter by obtaining information about its actual state or opera- dependent Hamiltonians H(θ) the largest sensitivity is tion, and feeding back corrective actions into the controls achieved if the eigenvectors of (d/dθ)H(θ) to the largest of the system that bring it back to that desired state or and smallest eigenvalue are also eigenvectors of H(θ). operation if any deviation occurs. As a consequence, the If these eigenvalues are non-degenerate, the condition field can be broadly classified according to two different is also necessary. For a phase shift-Hamiltonian the categories: Firstly, the object to be controlled maybe a condition is obviously satisfied. This insight opens the quantum state or an entire operation. And secondly, the way to Hamiltonian engineering techniques by adding type of information fed back can be classical or quan- parameter-independent parts to the Hamiltonian that tum. With classical information is meant information remove or overwhelm parts that spoil the commuta- that is obtained from a measurement, and which is then tivity of (d/dθ)H(θ) and H(θ) in the subspace of the typically processed on a classical computer and used to largest and smallest eigenvalues of (d/dθ)H(θ). These re-adjust the classical control knobs of the experiment. techniques were called “Hamiltonian subtraction” and Such schemes are called “measurement-based feedback”. “signal flooding”, respectively, and proposed to improve In contrast to this, “coherent feedback” schemes directly magnetometry with NV-centers. use quantum systems that are then manipulated unitarily and fed back to the system. Another opportunity for Hamiltonian engineering Measurement-based feedback schemes in metrology arises if the eigenvalues of H(θ) do not depend on are also known as “adaptive measurements” (see also the parameter. It was shown in (Pang and Brun, Sec.V.D for adaptive measurements in the context of 2014, 2016) that in such a case the quantum Fisher phase transitions). An adaptive scheme was proposed information becomes periodic in time. This is partic- as early as 1988 by Nagaoka for mending the problem ularly pernicious for quantum-enhanced measurement that the optimal POVM obtained in standard quantum schemes that allow long measurement times, as under parameter estimation depends on the a priori unknown the condition of quantum coherence the quantum parameter (Nagaoka, 1988). One starts with a random Fisher information typically increases quadratically with estimate, uses its value to determine the corresponding time if the eigenvalues of H(θ) depend on θ. Adding optimal POVM, measures the POVM, updates the es- another parameter-independent Hamiltonian might lead timate, uses that new value to determine a new optimal to parameter-dependent eigenvalues and hence unlock POVM, and so on. The scheme was shown to be strongly unbound increase of the quantum Fisher information consistent (meaning unbiased for the number of iterations with time. going to infinity), and asymptotically efficient (i.e. saturates the quantum Cramér-Rao bound in that limit) by The existence of Heisenberg-limit scaling of sensitiv- Fujiwara (Fujiwara, 2006). It was experimentally imple- ities for product states, suggests that “coherent aver- mented in (Okamoto et al., 2012) as an adaptive quantum aging” might even be possible classically. This ques- state estimation scheme for measuring polarization of a 33 light beam, but is in principle a general purpose estima- are obtained not only from data measured up to the tion scheme applicable to any quantum statistical model time when one wants to estimate the phase, but also on using identical copies of an unknown quantum state. later data. This implies, of course, that these smoothed In the context of quantum optics, an adaptive homo- estimates can only be calculated after a sufficient delay dyne scheme was proposed by Wiseman for measuring or at the end of the experiment, whereas feedback the phase of an optical mode, in which the reference itself at time t can only use data from times t′ t (or ≤ phase of the local oscillator Φ(t) is adjusted in real-time even t′ < t when considering finite propagation times). to Φ(t) π/2+ ϕ(t), where ϕ(t) is the latest estimate of Theory predicts a reduction of the mean square error by the phase≃ carried by a continuous-wave, phase-squeezed a factor 2√2 compared to the Standard Quantum Limit light signal (Wiseman, 1995). This reference phase (achievable by non-adaptive filtering, i.e. without feed- corresponds to the highest sensitivity in a homodyne back and using only previous data at any time), and an scheme. Keeping the local oscillator phase through experimental reduction of about 2.24 0.14 was achieved. ± feedback close to this optimal operating point can beat non-adpative heterodyning in single shot phase decod- Quantum error correction for quantum-enhanced mea- ing, as experimentally demonstrated in (Armen et al., surements, recently introduced in (Dür et al., 2014; 2002). In (Pope et al., 2004) it was shown that adaptive Kessler et al., 2014), can also be seen in the context of measurements have a finite factor advantage even in quantum feedback schemes (Ahn et al., 2002). Quan- the limit of arbitrarily weak coherent states. Phase tum error correction is one of the most important ingre- estimation using feedback was also studied in (Berry dients of quantum computing (Gottesman, 1996; Shor, and Wiseman, 2000, 2002, 2006; Higgins et al., 2007). In 1995; Steane, 1996). The general idea, both for quan- (Berry and Wiseman, 2002) it was investigated how well tum computing and quantum-enhanced measurements, a stochastically varying, white noise-correlated phase is that one would like to apply recovery operations R can be estimated. The theoretical analysis showed that to a state that after encoding the desired information the variance of the phase estimation could be reduced through an operation has been corrupted by a noise M by a factor √2 by a simple adaptive scheme compared process , such that (ρ) ρ. In (Kessler E R◦E◦M ∝ M to a non-adaptive heterodyne scheme, resulting in a et al., 2014) it was shown that this can be achieved for 1/2 value of n− /√2, where n is the number of photons the sensing of a single qubit subject to dephasing noise per coherence time. With a squeezed beam and a more if it is coupled to a pure ancilla bit. Syndrome measure- 2/3 accurate feedback, the scaling can be improved to n− . ments (i.e. measurements of collective observables which The latter result was also found for a narrow-band do not destroy the relevant phase information, a concept squeezed beam (Berry and Wiseman, 2006, 2013). In developed in quantum error correction) of both qubits at (Yonezawa et al., 2012) 15 4% reduction of mean a rate faster than the dephasing rate allow one to de- square error of the phase below± the coherent-state limit tect whether a phase flip has occurred and to correct it, was reported with this scheme in optical-phase tracking, extending in this way the coherence time available for i.e. in a case without any a priori information about Ramsey interferometry to much longer times and thus the value of the signal phase. The broad support of the better maximum sensitivities. When using N qubits in signal phase implies that there is an optimal amount of parallel, an ancilla is not necessary. The method then squeezing, and the sensitivity enhancement is directly operates directly on an initially entangled state, such as given by the squeezing. The scheme can therefore be the GHZ-state, and measures error-syndromes on pairs seen as a generalization of Caves’ idea of reducing the of spins. Thus, the idea is here not so much to avoid en- uncertainty with which a (fixed) phase shift in one arm tanglement but rather to stabilize through rapid multi- of an interferometer can be measured (Caves, 1981). spin error-syndrome measurements the correct imprint- Instead of having a fixed phase reference by the beam in ing of the information on the quantum state against un- the other arm of the interferometer, the feedback allows wanted decoherence processes. For phase estimation on one to continuously adjust the phase of the reference in N qubits evolving in parallel under individual and iden- the homodyning scheme to the optimal operating point. tical Pauli rank-one noisy channels, fast control schemes (Clark et al., 2016) proposed a feedback scheme based based on quantum error correction allow one to restore on measured temporal correlations (g(2) correlation the Heisenberg-limit by completely eliminating the noise, function) for estimating the phase of a coherent state at the cost of slowing down the unitary evolution by a inside a cavity and find that the uncertainty scales better constant factor, unless the noise is dephasing noise that 0.65 than the Standard Quantum Limit, namely as n− , couples to the same Pauli-operator as the Hamiltonian where n is the mean photon number of the coherent generating the phase shift (Sekatski et al., 2017). state. In (Wheatley et al., 2010) an “adaptive quantum More generally, one can prove that sequential metro- smoothing method” was used experimentally for esti- logical schemes involving an initial probe entangled with mating a stochastically fluctuating phase on a coherent an ancilla, with the probe undergoing N passes of a beam. “Smoothing” refers to the fact that estimates transformation encoding the parameter of interest, inter- 34 spersed by arbitrary feedback control operations acting A. Thermodynamical states and thermal phase transitions on probe and ancilla, and followed by a joint measurement on the two particles at the output, can outper- Thermodynamic states at equilibrium are derived by form any parallel metrological scheme relying on an ini- the maximization of the information-theoretic Shannon’s tial N-particle entangled state (Demkowicz-Dobrzański entropy (Jaynes, 1957a,b), equivalent to the maximiza- and Maccone, 2014; Huang et al., 2016; Sekatski et al., tion of the number of microscopic configurations com- 2017; Yuan, 2016; Yuan and Fung, 2015). In particular, patible with physical constraints. Given the probability in (Yuan, 2016) it was shown that a sequential feedback distribution p of a set of configurations j , the con- { j}j { }j scheme allows one to realize a joint quantum-enhanced straints are the normalization j pj = 1 and the aver- measurement of all the three components of a magnetic (k) (k) (k) ages of certain quantities F P= j pjfj , fj being field on a single-qubit probe. As remarked in Sec. II.A, h i the values of the quantity F (k) corresponding to the j-th the use of sequential schemes assisted by suitable control P configuration. The solution of the maximization is the reduces the input demand from multipartite to bipartite well-known Boltzmann-Gibbs distribution entanglement, resulting in a notable technological advantage. (k) Pk θkfj e− P θ f (k) p = , Z = e− k k j , (82) j Z j X where Z is the partition function, and θk is the Lagrange multiplier corresponding to the quantity F (k) fixed on average. V. THERMODYNAMICAL AND NON-EQUILIBRIUM This formalism is equally adequate for both classical STEADY STATES and quantum thermodynamic systems. In the quantum case all the quantities F (k) are commuting operators, the configurations are labeled by the set of eigenvalues of In this section, we discuss precision parameter estima- these operators and possibly additional quantum num- tions when probes are thermal states or non-equilibrium bers in the case of degeneracy, and the thermal state is steady states of dissipative dynamics. These states have the density matrix ρ diagonal in the common eigenbasis (k) the advantage to be stationary, and describe mesoscopic of the F with eigenvalues pj: systems. From measurements on these probes, intensive parameters, like temperature, chemical potential, or cou- ρ = p j j , F (k) j = f (k) j . (83) plings of Hamiltonians or of dissipators, are infered with j| ih | | i j | i j a sensitivity given by the inverse of the quantum Fisher X Information. Thermal probes are crucial for both funda- Lagrange multipliers are the thermodynamic parame- mental issues and technological applications (Benedict, ters to be estimated. Since the density matrix in (83) 1984; Childs, 2001; Giazotto et al., 2006). Estimations depends on them only through its eigenvalues pj, the with dissipative dynamics (Alipour et al., 2014; Bellomo ′ quantum Fisher matrix I = [Iθk,θk′ ]k,k coincides with et al., 2009, 2010a,b; Zhang and Sarovar, 2015) are also the classical Fisher matrix of the probability distribution instances of process tomography (Baldwin et al., 2014; pj j. A straightforward computation shows Bendersky and Paz, 2013; Merkel et al., 2013; Mohseni { } et al., 2008) with partial prior knowledge. The identifi- 2 ∂ ln Z ′ cation of the quantum Fisher Information with the Bures (k) (k ) Iθk,θk′ = = Cov F ,F . (84) ∂θk∂θk′ metric clarifies the role of criticality as a resource for estimation sensitivity. With extensive, i.e. linear in the See also (Jiang, 2014) for the computation of the quan- system size, interactions and away from critical behav- tum Fisher Information with density matrices in expo- iors, the Bures distance between the probe state and its nential form. The diagonal element Iθk,θk is the largest infinitesimal perturbation is at most extensive. Criti- inverse sensitivity for a single estimation of the param- cal behaviors, e.g. separation between different states of eter θk, while the Fisher matrix I bounds the inverse matter or long-range correlations, are thus characterised covariance matrix of the multiparameter estimation, see by superextensive Bures metric and the quantum Fisher eq.(17). The Cramér-Rao bound hence reads Information. We thus focus on superextensivity of the (k) (k′) 1 quantum Fisher Information as a signature of enhance- Cov(θk,est,θk′,est) Cov F ,F , (85) ments in precision measurements on thermodynamical ≥ M and non-equilibrium steady states. We lay emphasis on which is the uncertainty relation for conjugate variables highly sensitive probes that do not need to be entangled, in statistical mechanics, see e.g. (Davis and Gutiérrez, and in certain cases not even quantum. 2012; Gilmore, 1985). 35

The computation of the Fisher matrix (84), together with the Cramér-Rao bound (85), tells us that the best ∂µ N 2 sensitivity of Lagrange multipliers θk k is inversely pro- Iβ,β =Var(µN H)= h i + kBT CV (87) portional to squared thermal fluctuations,{ } and thus sus- − ∂β ceptibilities, see e.g. (Reichl, 1998). For connections N 2 I =β2 Var(N)= h i κ . (88) among metric of thermal states, Fisher information, and µ,µ βV T susceptibilities see (Brody and Rivier, 1995; Brody and Ritz, 2003; Crooks, 2007; Davis and Gutiérrez, 2012; Another parameter that can be estimated within this Diósi et al., 1984; Dolan, 1998; Janke et al., 2003, 2002; framework is the magnetic field. For certain classical Janyszek, 1986b, 1990; Janyszek and Mrugala, 1989; magnetic or spin systems, the interaction with a magnetic field B is B M, with M being the total magne- Mrugala, 1984; Nulton and Salamon, 1985; Prokopenko · et al., 2011; Ruppeiner, 1979, 1981, 1991, 1995; Salamon tization. This interaction term can represent a contri- et al., 1984; Weinhold, 1974) for classical systems and bution to the Hamiltonian as well as additional “fixed- B M (Paunkovićand Vieira, 2008; Janyszek, 1986a; Janyszek on-average quantities” with Lagrange multipliers β . The magnetic field is also linked to magnetic susceptibil- and Mrugala, 1990; Marzolino and Braun, 2013, 2015; ∂ M Quan and Cucchietti, 2009; You et al., 2007; Zanardi ity χ = ∂Bh i (where we consider for simplicity only a single component of B and M, M M , Bz): et al., 2007a, 2008) for quantum systems. ≡ z ≡ (k) 2 Due to the pairwise commutativity of the F , the IB,B = β Var(M)= βχ, (89) estimations of the parameters θk k are reduced to parameter estimations with the{ classical} probability dis- This picture of the magnetic field as a Lagrange multi- tribution p . Thus, the maximum likelihood esti- plier is valid also for coupling constants whenever the { j}j mator is asymptotically unbiased and optimal, in the Hamiltonian is H = j λjHj, where βλj is the La- sense of achieving the Cramér-Rao bound, in the limit of grange multiplier of Hj. For general quantum systems, P infinitely many measurements (Helstrom, 1976; Holevo, the non-commutativity of magnetization or other Hamil- 1982). This estimator consists in measuring each F (k) tonian contributions Hj with the rest of the Hamiltonian (k) gives rise to quantum phase transitions that occur also with outcomes f j=1,...,M , and in maximizing the av- { j } at zero temperature without thermal fluctuations. The erage logarithmic likelihood ℓ = 1 ln p with respect M j j above considerations also apply to the so-called general- to the parameters θk k. { } Q ized Gibbs ensembles, i.e. with arbitrary fixed-on-average (k) Among the most common statistical ensembles for quantities F , for which estimations of parameters θk equilibrium systems, there are the canonical ensemble are under experimental (Langen et al., 2015) and theo- and the grandcanonical ensemble. The canonical ensem- retical (Foini et al., 2017) study. ble describes systems that only exchange energy with Thermal susceptibilities are typically extensive except their surrounding: the only quantity F1 = H fixed on in the presence of phase transitions. Thus, their con- average is the Hamiltonian, the Lagrange multiplier is nection with Fisher information suggests that thermal θ = β = 1 where k is the Boltzmann constant and states at critical points (Baxter, 1982; Brody and Rivier, 1 kB T B T is the absolute temperature, and the Fisher informa- 1995; Brody and Ritz, 2003; Diósi et al., 1984; Dolan, ∂ H 1998; Janke et al., 2003, 2002; Janyszek, 1990; Janyszek tion is proportional to the heat capacity CV = ∂Th i , V and Mrugala, 1989; Prokopenko et al., 2011; Reichl, 1998; Ruppeiner, 1991, 1995) with divergent susceptibilities are

2 probes for enhanced measurements. Thermal suscepti- Iβ,β = Var(H)= kBT CV . (86) bility divergences occur also in classical systems, proving precision measurements without entanglement. The grandcanonical ensemble describes systems that exchange energy and particles with the surrounding: the 1. Role of quantum statistics quantities fixed on average are the Hamiltonian F1 = H and the particle number F2 = N, with Lagrange mul- We now discuss the estimation of Lagrange multipliers tipliers being the inverse temperature θ = β = 1 1 kB T of quantum gases in the grandcanonical ensemble and the and θ2 = βµ where µ is the chemical potential. The role of quantum statistics therein (Marzolino and Braun, − Fisher information of temperature and chemical poten- 2013, 2015). Consider ideal gases in a homogeneous or tial are linked to thermal fluctuations, i.e. heat capac- harmonic trap. Iβ,β is always extensive in the average ∂ H particle number N . The corresponding relative error ity CV = ∂Th i and isothermal compressibility κT = V found in (Marzolinoh i and Braun, 2013, 2015) for temper- 1 ∂V V ∂P respectively, where V is the volume and P is − T ature estimation is still one order of magnitude smaller the pressure (Marzolino and Braun, 2013, 2015): than the standard deviations obtained experimentally via 36 density measurements of Bosons (Leanhardt et al., 2003) Krueger, 1968; Mullin, 1997; Mullin and Sakhel, 2012; and Fermions (Müller et al., 2010; Sanner et al., 2010). Rehr, 1970; Zobay and Garraway, 2004), in connection Estimations of chemical potentials are more sensitive with liquid helium in thin films (Douglass et al., 1964; to quantum statistics than estimations of temperature, Goble and Trainor, 1965, 1966, 1967; Khorana and Dou- because the chemical potential is the conjugate Lagrange glass, 1965; Mills, 1964; Osborne, 1949), magnetic flux of multiplier of the particle number which in turn reveals superconducting rings (Sonin, 1969), and gravito-optical clear signatures of quantum statistics, such as bunching traps (Wallis, 1996). Experimental realizations of ef- and antibunching. Effects of quantum statistics are more fective low dimensional Bose-Einstein condensates with evident in quantum degenerate gases, i.e. at low temper- trapped atoms have been reported in (van Amerongen, atures. 2008; van Amerongen et al., 2008; Armijo et al., 2011; In Fermion gases, Iµ,µ is extensive but diverges at Bouchoule et al., 2011; Esteve et al., 2006; Görlitz et al., zero temperature. A change in the chemical potential 2001; Greiner et al., 2001). Iµ,µ in these Bose-Einstein corresponds to the addition or the subtraction of par- condensate phases is superextensive and interpolates be- ticles, thus achieving a state orthogonal to the previ- tween the scaling above the critical temperature and ous one. This sudden state change makes the chem- I = β2 N 2 for a standard Bose-Einstein conden- µ,µ O h i ical potential estimation very sensitive. A generaliza- sate only consisting of the ground state. The advantage tion of the Cramér-Rao bound, called Hammerseley- of Bose-Einstein condensate probes for precision estima- Chapman-Robbins-Kshirsagar bound that is suitable for tions is that Iµ,µ is superextensive in the entire Bose- non-differentiable statistical models (Tsuda and Mat- Einstein condensate phase and not only at critical points sumoto, 2005) must be applied. This may lead to as for susceptibilities of other thermal phase transition. superextensive Iµ,µ, depending on the degree of rota- In a mean field model with interactions treated pertur- tional symmetry breaking or confinement anisotropy and batively, if the ideal system exhibits a superlinear scaling dimension (see appendix B in (Marzolino and Braun, of the Fisher information, the interaction strength λ has 2013)). to go to zero for N for the perturbation theory to → ∞ Bose gases undergo a phase transition to a Bose- remain valid. In this limit, the superlinear scaling disap- Einstein condensate in three dimensions for homogeneous pears for any non-zero value of λ, but at finite N there confinement, and in three or two dimensions in a har- are values of λ which do not destroy the sub-shot-noise. monic trap. Approaching from above the critical tem- Moreover, superextensive quantum Fisher Information perature, or zero temperature with large density when in one dimension at fixed volume Lx and small contact interaction coupling c results there is no phase transition, Iµ,µ is superextensive: for Lx homogeneous and harmonic trap respectively 2 2 3 2 4 4 4π N β λ N 2 β λ N λ2 ρ2 T T − T h i Iµ,µ h 2 i + β N 2h 4 i 1 e ≃ 2πLx h i− 8π Lx − 4 2 3 β N in three dimensions 3 6 7 4π 3β λ N 2 N O h i 2 T − λ ρ2 h i 2 N + c h i 1 e T Iµ,µ .  β h i in two dimensions , (90) 3 7 log N 16π Lx − O h i  β2 N 2 in one dimension 3 4 6 4π N O h i β λT N λ2 ρ2 h i 2+ e− T h i , (92)  β2 N in three dimensions − 4π2L4  x ! O 2h i Iµ,µ .  β N log N in two dimensions . (91) O h i 2 h i where λ is the thermal wavelength, in agreement with  2 N T  β logh iN in one dimension 87 O h i experimental measurements on atom chips using Rb et al.  atoms (Armijo , 2011). Superextensive grandcanon- Below the critical temperature, the Bose-Einstein con- ical fluctuations of particle number, and thus superexten- densate phase depends on the anisotropy of the external sive Iµ,µ, have been observed in a photon Bose-Einstein potential. If the gas is much less confined along certain condensate, see e.g. (Schmitt et al., 2014), which can be directions, the Bose-Einstein condensate is an effective realized at room temperature (Klaers, 2014). low dimensional gas with excitations restricted to directions along the less confined dimensions. A hierarchy of condensations to subsequent lower-dimensional gases 2. Interferometric thermometry is possible. These Bose-Einstein condensates have been studied both at finite size and in the thermodynamic We now discuss a protocol for precision thermometry limit focusing on mathematical structures and general proposed in (Stace, 2010), using a Mach-Zehnder interfer- properties (Beau and Zagrebnov, 2010; van den Berg, ometer coupled to an ideal gas consisting of N two-level 1983; van den Berg and Lewis, 1982; van den Berg et al., atoms in the canonical ensemble. The gas Hamiltonian N 1986a,b; Casimir, 1968; van Druten and Ketterle, 1982; is H0 = j=1 ǫ ǫ j ǫ , where ǫ j is the j-th particle ex- Girardeau, 1960, 1965; Ketterle and van Druten, 1982; cited state with| single-particlei h | | energyi ǫ, while the single- P 37 particle lowest energy is zero. We skip the label j at the etry with a single probe was realized experimentally in bra-vector in the projector for brevity. an NMR setup in (Raitz et al., 2015) and the role of The interferometer is injected with K two-level atoms quantum coherence emphasized. Experimental simula- that interact with the gas in one arm of the in- tions in quantum optical setups and investigations of the terferometer with the interaction Hamiltonian HI = role of quantum coherence were reported in (Mancino K N α j=1 l=1 ǫ j ǫ ǫ l ǫ , where the index j labels the et al., 2016; Tham et al., 2016). The theoretical study atoms in the interferometer| i h |⊗| i h | and l refers to the atoms in in (Jevtic et al., 2015) examined thermometry with two theP gas.P In order for the interaction not to sensitively per- qubits. Numerical evidence suggested that while initial turb the gas, the interferometer should be much smaller quantum coherences can improve the sensitivity, the op- than the gas, thus K N. Each atom in the inter- timal initial state is not maximally entangled. ferometers acquires a relative≪ phase φ = αmτ between the arms, where τ is the interaction time and m is the number of excited atoms in the gas whose expectation B. Thermodynamical states and quantum phase transitions m = N/(1 + eβǫ) depends on the temperature. h Thei inverse temperature can be estimated from the Quantum phase transitions are sudden changes of the interferometric phase measurement with the sensitivity ground state for varying Hamiltonian parameters. If the θ θ θ θ Hamiltonian H( ) = j 0 Ej( ) Ej( ) Ej( ) , with ≥ | ih | Ej Ej+1, has a unique pure ground state E0(θ) , the 2 ≤ P | i δφ 1+ eβǫ δφ Fisher matrix (Campos Venuti and Zanardi, 2007; Gu σ(β)= = , (93) d m ǫeβǫN · ατ and Lin, 2009; You et al., 2007; Zanardi et al., 2007c) is ατ hdβ i resulting from error propagation, where δφ is the best E0 ∂θ H Ej Ej ∂θ ′ H E0 I = 4Re h | k | ih | k | i (94) sensitivity of the phase estimation according to the quan- θk,θk′ 2 j>0 (Ej E0) tum Cramér-Rao bound. The scaling with the number K X − of probes comes from δφ. For distinguishable atoms, sep- which follows from the differentiation of the eigenvalue 1/2 arable states imply shot-noise δφ = 1/K , while sub- equation H Ej = Ej Ej or from the standard time- shot-noise can be achieved with separable states of identi- independent| perturbationi | i theory with respect to small cal atoms as discussed in section III, leading to δβ 1/K variations dθ. The quantum Fisher Information is also for ideal non-interacting bosons, or δβ 1/Kp+1∝/2 for expressed in terms of the imaginary time correlation func- ideal fermions with a dispersion relation∝ of the probe tion or dynamical response function (Campos Venuti and atoms as discussed after (67). Note also that the scaling Zanardi, 2007; Gu and Lin, 2009; You and He, 2015; You of the sensitivity (93) with respect to the particle number et al., 2007), and has been used to count avoided cross- N in the gas looks like a Heisenberg scaling irrespectively ings (Wimberger, 2016). of the probe state, and it is not possible to achieve this Equation (94) tells us that the Fisher matrix can di- scaling by direct measurement of the gas (i.e. without verge only for divergent Hamiltonian derivatives or for any probe) because the Fisher information is extensive gapless systems E1 E0 0 in the thermodynamic limit. except at critical points. What is conventionally consid- Thus, the divergence− or the→ superextensivity of the quan- ered is however the scaling with the number of probes, tum Fisher Information reveals a quantum phase transi- in this case K, which can be controlled. tion but the converse does not hold: see (Gu, 2010) for a Optimal thermometry with a single quantum probe review. Finite size scaling of the quantum Fisher Infor- (and hence by definition without entanglement) was dis- mation (Campos Venuti and Zanardi, 2007; Gu and Lin, cussed in (Correa et al., 2015). It was found that the 2009; Zanardi et al., 2008) can be derived using finite size quantum Cramér-Rao bound for T of a thermalizing scaling at criticality (Brankov et al., 2000; Continentino, probe reproduces the well-known relation of temperature 2001). fluctuations to specific heat C (T ), (T/Var(T ))2 Superextensive quantum Fisher Information was ob- V est ≤ CV (T ) (with Boltzmann constant kB = 1, see also served at low order symmetry breaking quantum phase (Jahnke et al., 2011)). The level structure of the probe transitions, topological quantum phase transitions, and was then optimized to obtain maximum heat capacity gapless phases, but this criterion may fail at high order and it was found that the probe should have only two symmetry breaking quantum phase transitions (Tzeng different energy levels, with the highest one maximally et al., 2008) and Berezinskii-Kosterlitz-Thouless (BKT) degenerate and a non-trivial dependence of the optimal quantum phase transitions (Chen et al., 2008; Sun et al., energy gap on temperature (see also (Reeb and Wolf, 2015). We now review quantum critical systems at zero 2015)). The sensitivity of the probe increases with the temperature exhibiting superextensive quantum Fisher number of levels, but the role of the quantumness of the Information of Hamiltonian parameters without entan- initial state of the probe on the sensitivity of thermome- glement. We adapt and unify the notation of existing lit- try is not fully understood yet. Interferometric thermom- erature, by writing down a general parameterized Hamil- 38 tonian whose different special cases are studied in the where L is a measure of the system volume. Translation- literature. ally invariant Hamiltonians (96) with periodic boundary

conditions and tunneling in the modes cj,cj† j of range r have { } 1. Quasi-free Fermion models

A =(J µ)δ Jθ(r j l ) Jθ( j l L + r), We start with non-interacting many-body Hamiltoni- j,l − j,l − −| − | − | − |− ans, i.e. Bj,l = Jγ sign(j l) θ(r j l ) θ( j l L + r) , − −| − | − | − |− (97) θ θ θ θ H( )= ωj( )aj†( )aj( ), (95) j where J > 0, sign(0) = 0, and θ( ) is the unit step func- X tion with θ(0) = 1. Here J corresponds· to a tunneling energy between different sites, and Jγ to an effective inter- where aj† (aj) creates (annihilates) a Fermion in the j-th action; µ multiplies the total particle number and hence eigenmode. The dependence of the eigenmodes a†,a on j j corresponds to a chemical potential. This Hamiltonian the parameters corresponds in second quantization to the can be analytically diagonalized (Cozzini et al., 2007). dependence of the Hamiltonian eigenvectors on θ, as re- For large L and a fully connected system, r = L with quired for the ground state to be sensitive to variations of 2 periodic boundary conditions, the Fisher matrix with re- θ. The eigenstates of (95), and thus thermal probes in- spect to (µ,γ) is cluding the ground state, are not entangled in the eigenmodes. Therefore, measurements on the ground state 2 2 provide parameter estimation without entanglement and Iµ,µ = γ S, Iγ,γ =(µ J) S, Iµ,γ = (µ J)γS, with enhanced precision at phase transitions, as shown − − − (98) by the following examples. with Under Bogoliubov transformations the Hamiltonian (95) can be mapped into many models studied in litera- L J 1 ture. Bogoliubov transformations preserve neither mode if (µ = J, γ = 0) 2 (µ J)γ 2 6 6 entanglement nor operation locality, as discussed in Sec- µ J + Jγ  2 − | − | | | tion III. The relativity of entanglement with respect to  L S  if (µ = J, γ = 0) ≃  2 4 the basis of modes provides complementary pictures of 3J γ 6 the Hamiltonian and of estimation protocols of its param-  L2J 2 4 if (µ = J, γ = 0) eters but does not change the physics: either the probe (µ J) 6  − state is entangled in one basis, or it is not in another and  (99) the enhanced estimation precision is achieved by non- The lines (µ = J, γ = 0) and (µ = J, γ = 0) reveal 6 6 local measurements as a consequence of long-range cor- second-order quantum phase transitions, as well as su- relations at criticality. This situation is reminiscent of perextensivity of the Fisher matrix in the volume L. At the case of interferometry with identical particles, dis- the critical line γ = 0, the ground state is unentangled cussed in Section III, for which rotations of modes re- not only in the eigenmodes but also in the original modes distributes quantum resources between initial entangle- c ,c† . { j j}j ment and non-local interferometers. Moreover, Bogoli- The fully connected translationally invariant Hamilto- bov transformations and corresponding rotated modes nian with open boundary conditions reads are experimentally addressed in several physical systems (Davis et al., 2006; Hu et al., 2014; Inguscio and Fallani, A =(µ J)δ + J, B = Jγ sign(l j), (100) 2013; Moritz et al., 2003; Robillard et al., 2008; Sattler, j,l − j,l j,l − 2011; Segovia et al., 1999; Vogels et al., 2002; Yan et al., with second-order quantum phase transitions at lines 2016), showing that quasi-particles represent legitimate µ = J and γ = 0, and superextensive Fisher matrix and experimentally relevant subsystems. as for periodic boundary conditions with different pref- Hamiltonians equivalent to (95) under Bogoliubov actors (Cozzini et al., 2007; Zanardi et al., 2007b). For transformations are quasi-free Fermion models (Cozzini instance, at (µ>J,γ = 0) et al., 2007), i.e. quadratic Hamiltonians in the creation cj† and annihilation cj operators of L Fermionic modes: L2J 2 I = 0, I = , I = 0. (101) µ,µ γ,γ 3(µ J)2 µ,γ − L 1 H = c†A c + c†B c† + h.c. , Another interesting case is the Hamiltonian with near- quasi-free j j,l l 2 j j,l l est neighbor tunneling in the modes c ,c† , periodic j,lX=1 j j j (96) boundary conditions, and J > 0: { } 39

with j and l labeling the sites in the x direction. Con- sider L = Fm, the m-th Fibonacci number, and φ = Fm−1 √5 1 Aj,l =(J µ) δj,l Jθ(1 j l ), − , the inverse of the golden ratio in − − −| − | Fm −−−−→m 2 B = Jγ sign(l j) θ(1 j l ), (102) →∞ j,l the thermodynamic limit. At the critical line tc = ta, − −| − | 4.9371 numerical computations result in It /t = L if Such Hamiltonian (Zanardi et al., 2008) is also equal to c a O m = 3L + 1, otherwise I = L2.0 . I has the tc/ta tb/ta same size dependence at the criticalO line 2t = t . A more b a L complicated quasi-free Fermion Hamiltonian describes a ⌊ 2 ⌋ H = ǫ σz +∆ σy , (103) superconductor with a magnetic impurity (Paunković quasi-spin − n n n n et al., 2008). A sudden increase of the Bures distance n=1 X between two reduced ground states of few modes around 2πn µ 2πn the impurity at very close exchange interaction with the with ǫn = J cos L 2 , ∆n = Jγ sin L , and σy,z are− Pauli matrices− on n orthogonal− ❈2 subspaces. impurity was numerically observed, but it is unclear n n Eq.{ (103)} provides an alternative representation of the whether the quantum Fisher Information with respect Hamiltonian in terms of non-interacting quasi-spins, and to exchange interaction is superextensive. its eigenstates are separable with respect to both the eigenmodes and the quasi-spins. The same quantum Fisher Information scaling holds in both representations 2. Hubbard models as theoretical independent models. The quantum Fisher Information is linear in L away Another class of Fermionic systems is described by L- from the critical points, but superextensive at criticality mode Hubbard Hamiltonians in the leading order for large L (Zanardi and Paunković, 2006; Zanardi et al., 2008): H = t c† c + h.c. HM − σ j,σ j+1,σ j=1,...,L 2 σX= , L ↑ ↓ I µ = 2J, γ = , (104) L J,J | | O J 2γ2 + U nj, nj, µ nj,σ, (107) 2 ↑ ↓ − Iγ,γ µ 2J, γ = 0 = L . (105) j=1 j,σ | |≤ O X X The origin of the superextensivity of IJ,J stems from the with nj,σ = cj,σ† cj,σ, tσ the hopping constants, U the fact that the symmetric logarithmic derivative, and thus interaction strength, and µ the external potential. When the optimal estimation of J, is close to a single particle t = t , the system undergoes a Berezinskii-Kosterlitz- operator in the Fermion representation away from the Thouless↑ ↓ (BKT) quantum phase transition at U = 0 and critical point, but is a genuine multi-particle operator 1 half filling n = nj,σ = 1. IU is extensive, but close to the critical point. In the canonical ensemble, at L j,σh i I /L diverges as 1/n for n 0 and U = 0, and as 1/U 4 µ J . 1 , the quantum Fisher Information has a di- U P → 2 − β around the BKT critical point only if the system size is vergence around zero temperature, I = Lβ . The J,J O Jγ much larger than the correlation length (Campos Venuti superextensivity of I , together with the divergence| | of J,J et al., 2008). At zero interaction (107) is a free Fermion the derivative of the geometric phase, is also a universal Hamiltonian, and thus the ground state is not entangled feature depending only on the slope for the closing of the in the eigenmodes. energy gap between one and zero Fermion occupations of 2 In the large U limit and at n = 3 , a quantum phase certain eigenmodes (Cheng et al., 2017). transition occurs with control parameter t /t . The Superextensive quantum Fisher Information is ob- ↓ ↑ quantum Fisher Information It↓/t↑ at critical points is su- served also for tight-binding electrons on the triangular perextensive Lα , where the exponent was numerically φ O lattice with magnetic flux 2 within each triangle, hop- computed α 5.3 (Gu et al., 2008). In the large U limit, ≃ ping constants ta (tb) at edge along the x (y) direction the eigenstates of the Hamiltonian are perturbations of and tc at the third edge (Gong and Tong, 2008). Assum- those of the interaction term, that are Fock states, thus ing zero momentum in the y direction (Ino and Kohmoto, with vanishingly small entanglement with respect to the 2006), the Hamiltonian can be transformed into (96), modes cj,σ,cj,σ† j,σ. with Bj,l = 0 and { }

1 3. Spin- systems 2πiφ(j ) 1/2 Aj,l = 2tb cos (2πφj) δl,j ta + tce− − 2 δl,j 1 − − − 2πiφ j+ 1 We now discuss systems of N spins-1/2, which provide t + t e ( 2 ) δ , (106) − a c l,j+1 alternative representations of quasi-free models. Con- 40 sider the following complete Hamiltonian, to be spe- Thus, the exponent of N in average Ih,h and Iγ,γ is cialised later on, slightly reduced but the superextensivity is broadened in the parameter range. Superextensivity of I at the N 1 h,h − 1+ γ x x 1 γ y y critical Ising point h = J becomes a broad peak, and Hspin = J σj σj+1 + − σj σj+1 | | − 2 2 the superextensive Iγ,γ at the γ = 0 critical line is split j=1 X into two symmetric broad peaks around γ = 0, where z z +∆σj σj+1 at γ = 0 a local minimum with extensive Iγ,γ arises. N 1 N Furthermore, the scaling of Ih,h of the XY model at the − z + d σ σ h gj σz, (108) critical point h = J is preserved when a periodic time- j ∧ j+1 − − j | | j=1 j=1 oscillating transverse magnetic field is considered, and X X the state is the Floquet time-evolution of the ground with anisotropies γ and ∆, Dzyaloshinskii-Moriya cou- state at t = 0 up to times that scale linearly with the pling d, magnetic field with uniform value h and gradient system size for small time-dependent driving (Lorenzo σ x y z g, and j = σj ,σj ,σj . et al., 2017). These robustness features are suitable for For the moment, we focus on the XY model with trans- practical implementations of precision measurements. verse field, i.e. ∆ = d = g = 0, with J > 0 and periodic Another interesting system is the XXZ model, i.e. γ = boundary conditions. This Hamiltonian can be trans- 1 0 and J < 0, whose low-energy spectrum for ∆ < 2 is formed into (96) with (102), µ = 2h, and L = N using equivalent to a quasi-free Boson Hamiltonian| known| as the Jordan-Wigner transformation, see e.g. (Giamarchi, Luttinger liquid, see e.g. (Giamarchi, 2003). Thus, su- 2003). Thus, the XY Hamiltonian provides an alterna- perextensive ground state quantum Fisher Information tive physical setting to implement precision metrology implies precision measurements without entanglement in without entanglement in the Fermionic eigenmodes. the Boson eigenmodes. Given the Luttinger liquid pa- J 2 π At zero temperature, Ih,h = 2 IJ,J , since the ground h rameter K = 2 arccos( 2∆ ) , superextensive quantum state depends on h and J only via the ratio h . The Fisher − √1+2d/J J Fisher Information I is observed with periodic boundary matrix is extensive with divergent prefactors around crit- d conditions at d = h = g = 0, ical regions (Cheng et al., 2017; Zanardi et al., 2007c), and is superextensive at criticality as in (104,105). In the Ising model, i.e. γ = 1, corrections to the quan- N ln N 1+√5 J 2 if ∆= 8 tum Fisher Information around the critical points h = J O 6−8K | | N 1+√5 1 (Chen et al., 2008; You and He, 2015; Zhou and Barjak- Id =  J 2 if 8 < ∆ < 2 , (109) O tareviˇc, 2008; Zhou et al., 2008) and at small temperature  N 2 1 J ln N if ∆= 2 (Invernizzi et al., 2008; Zanardi et al., 2007a) are also O  2 2 linear in N with divergent prefactors. Ih,h was also ex- while, with open boundary conditions, I = J KN d O (J 2+4d2)2 plicitly computed in (Damski, 2013; Damski and Rams, KN 4 π√1 4∆2 at h = g = 0 and I = , with u = − , 2014; You and He, 2015). A good estimator of the param- g O J 2u2 2 arccos(2∆) at d = h = g = 0 (Greschner et al., 2013). Here, the eter J is the value inferred by measurements of the total magnetization, at least at small system size (Invernizzi quantum Fisher Information is superextensive mainly et al., 2008). because the system is gapless, while the quantum phase The divergence of Ih,h/N was numerically observed transition only affects subleading in N, even though also in the XX model, i.e. γ = 0, approaching the crit- superextensive, orders of the quantum Fisher Informa- ical field h = J, with the reduced state of spin blocks tion. Id is also equal to the quantum Fisher Information | | (Sacramento et al., 2011). This implies precision estima- Iφ with respect to a twist phase φ of spin operators + + iφ σ σ− σ σ− e (Thesberg and Sørensen, 2011). tions of the magnetic field looking only at a part of the j j+1 → j j+1 system. Via the Jordan-Wigner transformation, the ∆ = γ = 0 Divergent Ih,h/N at critical field h = J was numer- case in (108) is equivalent to a quasi-free Fermion ically computed also when an alternating| | magnetic field Hamiltonian with matrix elements given by (102) with N ( )j+1δσz is added to the XY Hamiltonian (You γ = µ = 0 and with the Dzyaloshinskii-Moriya and j=1 j id N − the gradient field terms c†c + h.c. and and He, 2015). The corresponding quasi-free Fermion − 2 j=1 j j+1 P N j+1 N Hamiltonian has the additional term j=1( ) δcj†cj. j=1 gjcj†cj, respectively. Id Pand Ig have the same The XY and the Ising models are also interesting− be- scaling in a generalization of the latter model with cause the superextensivity of the FisherP matrix I = Pspin-1/2 Fermions (Greschner et al., 2013), providing [Iθ,θ′ ]θ,θ′=h,γ is robust against disorder, i.e. with Hamilto- a further example of precision measurements without nian parameters being Gaussian random variables (Gar- Fermion entanglement. nerone et al., 2009b). In the presence of disorder, quantum phase transitions are broadened to Griffiths phases Another model, mapped to a quasi-free Fermionic (Fisher, 1992, 1995; Griffiths, 1969; Sachdev, 1999). Hamiltonian and exhibiting superextensive quantum 41

Fisher Information, is the quantum compass chain with In the honeycomb lattice with K = 0, Jx + Jy + Jz = J periodic boundary conditions (Motamedifar et al., 2013) and Jx = Jy, the quantum Fisher Information is IJz = 1.2535 0.00005 J N N ± at the critical point Jz = 2 (Yang 2 N O α α z z y et al., 2008), and IJz = (N ln N) in the gapless phase HQCC = Jασ2j 1σ2j + Jzσ2jσ2j+1 h σj . J O − − Jz < (Gu and Lin, 2009). Superextensivity in the gap- j=1 α=x,y ! j=1 2 X X X less phase is due to the algebraic decay of the correlation (110) function of the z-edge, in contrast to exponential decay in Quantum phase transitions occur at critical ﬁelds h = 1,2 the gapped phase J > J leading to extensive quantum J (J J )/2 when these values are real. Numerical z 2 x y z Fisher Information. computations± show that I = N α is superextensive p h with O C. Non-equilibrium steady states

Jx Jy 1.80 0.02 for > 0, > 1 and h = h1 ± Jz Jz Jx Jy Consider now parameter estimation with probes in 1.98 0.02 for > 0, > 1 and h = h2 α  ± Jz Jz . non-equilibrium steady states of spin-1/2 chains with Jx Jy ≃ 1.94 0.02 for < 0, < 1 and h = h2  ± Jz Jz boundary noise (Marzolino and Prosen, 2016a; Prosen, 2.02 0.02 for Jx > 0, Jy < 1 and h = h Jz Jz 1 2015; Zunkovic and Prosen, 2010) described by the  ±  (111) Markovian master equation (Benatti and Floreanini,  2005; Breuer and Petruccione, 2002)

4. Topological quantum phase transitions ∂ρt = i[HXYZ, ρt] The quantum Fisher Information is superextensive also ∂t − in topological quantum phase transition which have non- 1 + λ ρ † † , ρ , local order parameters (Zeng et al., 2015). Mosaic mod- Lα,j tLα,j − 2 Lα,jLα,j t α=1,2 els defined on two-dimensional lattices with trivalent ver- =1X,N n o tices, i.e. each vertex is the border among three polygons, (113) and with three-body interaction were numerically studied. The N-particle Hamiltonian is with XYZ Hamiltonian, i.e. (108) with d = g = 0, and Lindblad operators 1(2),j = (1 µ)/2 σj±. Superextensive quantumL Fisher± Information of the H = J α σασα K σxσyσz, (112) p mosaic − j,l j l − j l k non-equilibrium steady states ρ is observed at non- α=x,y,z j,l,k ∞ (j,lX) S(α) X equilibrium phase transition and in phases with long- ∈ range correlations. In the XY model, ∆ = 0, superex- where S(α) is the set of edges in the α x,y,z ∈ { } tensive quantum Fisher Information was computed at direction. H can be mapped onto a free Majo- 6 mosaic the critical lines h = 0(Ih,h = (N )) and γ = 0 rana Fermion Hamiltonian, thus without entanglement with h < J 1 γ2 (I = (NO2)), at the critical | | | − | γ,γ O in Fermion eigenmodes. When the edge numbers of 2 ′ ′ 6 x,y points h = J 1 γ ([Iθ,θ ]θ,θ =h,γ = (N )), and in z | | | − | O 2 the three polygons are (4, 8, 8), Jj,l = J, Jj,l = Jz, the phase with long-range correlations h < J 1 γ and K = 0, the quantum Fisher Information is I = 3 | | | − | Jz ([Iθ,θ′ ]θ,θ′=h,γ = (N )) (Banchi et al., 2014). 1.07615 0.00005 O N ± at the critical point Jz = √2 J > 0 In the XXZ model γ = 0, the quantum Fisher In- O(Garnerone et al., 2009a). When the edge numbers are x,y,z formation I∆ is superextensive in the limit of small λ 1 arccos ∆ (3, 12, 12), Jj,l = J for edges within triangular ele- and for ∆ at irrational (Marzolino x,y,z J | | ≤ 2 π mentary subcells (Yao and Kivelson, 2007), J = J ′ arccos ∆ j,l and Prosen, 2014, 2016b, 2017). If π is rational for other links, and K = 0, the quantum Fisher Infor- λ2µ2 ˜ 2 ˜ 1.078 0.005 I∆ = J 2 (ξN + ξN) where ξ and ξ are constants in mation is IJ ′ = N ± at the critical point O N and for λ < 1 , thus the quantum Fisher Informa- J = √3 J > 0 (Garnerone et al., 2009a). H on the J √N ′ mosaic honeycomb lattice with J α = J has topological quan- tion is not superextensive. Nevertheless, after inserting j,l α the value of ∆ in arccos ∆ and reducing the fraction to tum phase transition at the boundaries Jx = Jy + Jz , π | | | | | | lowest term arccos ∆ = q with coprime integers q,p, one Jy = Jz + Jx , and Jz = Jx + Jy with superexten- π p | | | | | | | | | | | | realizes that the coefficient ξ is unbounded when the de- sive quantum Fisher Information, e.g. IJx = (N ln N) Jx O nominator p grows. Therefore, I∆, as a function of ∆, for Jy = Jz = 2 and K = 0 (Zhao and Zhou, 2009), and 1.08675 0.00005 J J Jz exhibits a fractal-like structure with a different size scal- I = N ± for J = , J = J = − Jz O z 2 x y 2 ing at irrational arccos ∆ . Moreover, the limit of arccos ∆ and K = 1 (Garnerone et al., 2009a). π π 15 Topological quantum phase transition are particularly approaching irrational numbers from rationals and the relevant because of superextensive quantum Fisher Infor- thermodynamic limit do not commute. If the thermody- λ2µ2 5 namic limit is first performed then I = 2 N ∼ mation in gapless phases and not only at critical points. ∆ O J 42

λ 1 with J < √ . If the thermodynamic limit is post- θ to be estimated, and assume the quantum Fisher Infor- N ξN arccos ∆ poned after the limit to irrational then the quan- mation Iθ θc = θ θ α close to a critical point θc with π − | − c| tum Fisher Information is still fitted by a superexten- α > 0 and a prefactor ξ. We use the notation Iθ θc − sive power law with exponent depending on the value to make clear that the quantum Fisher Information de- 2 2 λ µ 2.32788 0.0009 λ 1 pends on the difference θ θc, rather than on θ alone. of ∆, e.g. I∆ = 2 N ± with < O J J √N Sub-shot-noise sensitivity− is shown only at θ = θ . As- arccos ∆ 1 c and π being the golden ratio. When ∆ = 2 , sume also that θ is initially known within a fixed interval λ2µ2 4 λ 1 I∆ = J 2 N with J < N in both cases. θ [θmin,θmax] enclosing the value θc which can be con- O trolled∈ by other system parameters. First, set the critical In addition, the quantum Fisher Information of the (1) (1) reduced state of a single spin at position k scales su- point to θc = θmax, ensuring that θ<θc , i.e. one is perextensively also for arbitrary dissipation strength λ on a well-defined side of the phase transition. Find a (1) but only at the critical points ∆ = 1, and with a power first estimate θest for the parameter θ, with an uncer- law depending on the position| k| of the spin (Marzolino tainty that saturates the quantum Cramér-Rao bound. 2 and Prosen, 2017): e.g. at λ = 1, I∆ = (N ∼ ) for k = 1, Use therefore a number of measurements M that ensures N O4 N a small error compared to the original confidence inter- k = 2 , or k = N, and I∆ = (N ∼ ) for k = 4 or ⌊ 3N⌋ O ⌊ ⌋ 1/2 k = 4 ). This proves that the anisotropy at ∆ = 1 (1) 1 val, σ θ = (θmax θmin), where ⌊ ⌋ | | est MI (1) can be precisely estimated measuring single spin magneti- θ−θc ≪ − 1/2 sations along the z axis, or measuring the magnetisations once more σ θ(1) = Var θ(1) is the standard devia- z est est j σj for any non-centrosymmetric portion of the tion of the estimate. Then, update the critical parameter ∈P h i P z (2) (1) (1) (1) chain, or j f σj with even functions f( ) for any to θc = θ + σ θ . Since σ θ (θ θ ), P ∈P h i · est est est max min set , even centrosymmetric ones. (2) ≪ − P P the new critical point θc is now much closer to the true For either small λ or small µ the steady states ρ of the (1) ∞ value of θ than θc , assuming that the obtained estimate above models are perturbations of the completely mixed (1) state and thus not entangled, see e.g. (Bengtsson and θest (which is random and only on average agrees for an Zyczkowski,˙ 2006). In the XX model γ =∆=0, there is unbiased estimator with θ) is indeed within an interval (1) no nearest neighbor spin entanglement for a wide range of order σ(θest ) of the true θ. Hence, in the next round, of parameters (Znidariˇc,ˇ 2012). Non-equilibrium steady the quantum Fisher Information should be substantially state probes are favorable because the quantum Fisher larger. Perform then again sufficiently many times a Information is superextensive in a whole phase and not POVM that allows saturating the quantum Cramér-Rao bound for a new estimate θ(2), σ2 θ(2) = 1 only at exceptional parameters. Moreover, the distin- est est MI (2) θ−θc ≃ guishability of non-equilibrium steady states via Fisher (1) (σ(θ )α 2+α est 2 information, thus detectability of non-equilibrium criti- ξMN (MN)− . After k iterations, the sensitiv- ∝ (k) cality and metrological performances, are enhanced com- ity of the estimate θ saturating the quantum Cramér- pared to thermal equilibrium systems: e.g. the BKT Rao bound is quantum phase transition at ∆ = 1 in the ground state 2 k 1 k−1 αj 1−(α/2) of the XXZ model does not correspond to superexten- 2 (k) Pj=0 j α/2−1 σ θest = (MN)− 2 =(MN) , sive quantum Fisher Information (Chen et al., 2008; Sun MI (k) ∝ θ θc et al., 2015). − (114) achieving sub-shot-noise for α< 2 with the limiting scal- 2 ing σ2 θ (MN) α−2 for k . est ∝ → ∞ D. Adaptive measurements Within the above adaptive scheme, control of Hamil- tonian parameters, estimations at each step, and re- Since critical points without critical phases are isolated materialization or re-stabilization of states at each step values, they should be known in advance in order to set are assumed to be efficiently implementable. This adap- the system at criticality and benefit of superextensive tive measurement was proposed to estimate the critical quantum Fisher Information. A partial solution is an magnetic field h = J of the XX model at small non- adaptive approach (Mehboudi et al., 2016) that we gen- zero temperature when the quantum phase transition is eralize to phase transitions under reasonable conditions. smoothened to a phase crossover (Mehboudi et al., 2016), 2 4 The idea is to perform several estimates changing the achieving the limiting scaling σ θest (MN)− 3 . The thermal state or the non-equilibrium steady state, in par- optimal estimate is derived from measurements∝ of the z ticular the critical point, at each step according to previ- magnetization along the z direction j σj , since the ous estimates, in order to approach the phase transition magnetic field commutes with the spin interaction restor- ensuring enhanced sensitivity of the control parameter ing the classical picture of LagrangeP multipliers and estimation. quantities fixed on average. The same estimate and that x Consider any phase transition with control parameter derived from measurements of the variance of j σj are P 43 nearly optimal for the XY model. It is remarkable that, who estimated the ultimate achievable precision of when the adaptive measurement is applied to the model atomic clocks: increasing the energy of the used clock (97) approaching the critical lines in (99) from the non- states more and more for improving its precision leads critical region, the critical scalings are consistently recov- ultimately to the formation of a black hole and hence ered. renders reading off the clock impossible (Wigner, 1957). Similar limitations of this kind exist for measurements of lengths (Amelino-Camelia, 1999; Ng and Dam, 2000), VI. OUTLOOK and have been recently explored in more detail for the speed of light in vacuum (Braun et al., 2017). Most work on quantum-enhanced measurements has investigated the benefits of using quantum entanglement While current technology is still far from probing (see (Giovannetti et al., 2011; Paris, 2009; Pezzèand such extreme conditions, we nevertheless arrive to the Smerzi, 2014; Pezzè et al., 2016; Tóth and Apellaniz, conclusion that the importance of scaling is to give the 2014) for recent reviews). Indeed, under certain restric- functional form for an extrapolation of the sensitivity, tive assumptions (see Introduction), entanglement can a prediction of how well one could measure if one had be shown to be necessary if one wants to improve over a given large N. The validity of this extrapolation will classical sensitivity. However, going beyond these re- be limited by the range over which the scaling persists, strictive assumptions opens up a host of new possibilities an additional datum not described by the scaling nor of which we have explored a large number in this review. by the prefactor. Moreover, there is the possibility Given the difficulty of producing and maintaining of an optimum N, beyond which the sensitivity wors- entangled states of a large number of subsystems, some ens (Nichols et al., 2016). In such a case, the interesting of these may open up new roads to better sensitivities questions are “is the optimum Nopt achievable given than classically possible with a comparable number of available resources” and “what is the actual sensitivity resources in actual experiments. at this optimum?” At such an optimum the local scaling is flat, i.e. the smallest possible uncertainty of We conclude this review by challenging yet another an unbiased estimate of the parameter is independent common mind-set in the field (which we could not quite of N. Ironically, our discussion of advantageous scaling escape in this review either), namely the hunt for faster leads us to the conclusion that the best scaling may be scaling of the sensitivity, in particular the quest for a no scaling at all. scaling faster than 1/√N with the number of subsystems N, and the goal of reaching “Heisenberg-limited” scaling The relevance of the actually achievable smallest un- 1/N: It should be clear (see also Sec.IV.C and the lin- certainty rather than its scaling with the number of re- ear/nonlinear comparison in (Napolitano et al., 2011)) sources makes it particularly important that alternatives that scaling of the sensitivity is not per se a desideratum. to the use of massive entanglement be investigated, as Any given instrument or measurement is judged by its so far the number of subsystems that could be entangled sensitivity, not the scaling thereof. When the sensitivity experimentally has remained relatively small. We hope d is σ(θest) = αN , it is sometimes argued that the that the present review will stimulate further research in pre-factor α is irrelevant, because a more rapid scaling this direction. necessarily leads to better sensitivity for sufficiently large N. While mathematically impeccable, this argument assumes that the scaling persists to sufficiently large N ACKNOWLEDGMENTS where the possibly small prefactor can be compensated — an assumption that may not be valid in practice. DB thanks Julien Fra¨ısse for useful discussions, a Typically, at some point the model breaks down, and careful reading of the manuscript, and pointing out systematic errors arise that scale with a positive power ref. (Popoviciu, 1935). GA acknowledges fruitful dis- of N and at some point become comparable to the cussions with Kavan Modi. SP would like to thank S. stochastic error quantified by the quantum Cramér-Rao L. Braunstein, Cosmo Lupo and Leonardo Banchi for bound. And finally, some large-N catastrophe destroys comments on Section II. This work was supported in the instrument and its measurement capability. Such part by the Deutsche Forschungsgemeinschaft through concerns are of course very relevant for real-life experi- SFB TRR21 and Grant BR 5221/1-1, by the Euro- ments, for which material properties have to be taken pean Research Council (ERC) through the StG GQ- into account. But they may also determine fundamental COP (Grant Agreement No. 637352) and AQUMET bounds to achievable precision for various physical (Grant Agreement No. 280169) and the PoC ERID- quantities that are ultimately linked to the fabric of IAN (Grant Agreement No. 713682), by the Royal So- space-time at extremely small length- and time-scales. ciety through the International Exchanges Programme Such ideas were advanced early on notably by Wigner, (Grant No. IE150570), by the Foundational Questions 44

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