arXiv:1405.7703v3 [quant-ph] 16 Aug 2018 CONTENTS interferometry optical in limits Quantum I.Mc-ene nefrmty8 interferometry Mach-Zehnder III. V siainTheory Estimation IV. I unu iisi elsi nefrmty25 interferometry realistic in limits Quantum VI. I unu ttso light of states Quantum II. .Qatmlmt ndchrnefe nefrmty19 interferometry decoherence-free in limits Quantum V. .Introduction I. .Pril description Particle D. .Chrn qezdvcu nefrmty10 interferometry squeezed-vacuum + Coherent D. .Rl fetnlmn 24 entanglement of Role D. .Paesnigucrany9 uncertainty Phase-sensing A. .Casclprmtretmto 14 estimation parameter Classical A. .Md description Mode A. .Dchrnemdl 26 models Decoherence A. .QatmFse nomto prah20 approach Information Fisher Quantum A. .Dfiiepoo ubrsae 7 states number photon Definite C. .Fc tt nefrmty10 interferometry state Fock C. .Idfiiepoo-ubrsae 22 states photon-number Indefinite C. .Gusa states Gaussian B. .Chrn-tt nefrmty10 interferometry Coherent-state B. .Qatmprmtretmto 17 estimation parameter Quantum B. .Bysa approach Bayesian B. .Md spril nageet8 entanglement particle vs Mode E. .Dfiiepoo-ubrsaeitreoer 11 interferometry state photon-number Definite E. .Mlips rtcl 25 protocols Multi-pass E. .Ohritreoees12 interferometers Other F. .Tomd qezdsae 6 5 6 states Gaussian two-mode General states 4. squeezed Two-mode states 3. squeezed Single-mode 2. states Coherent 1. .Rl fterfrneba 22 23 states strategies Gaussian number photon 3. indefinite beam Optimal reference the 2. of Role 1. .Bysa prah19 15 17 14 approach 16 Bayesian approach Information 2. Fisher Quantum 1. estimation coefficient Transmission Example: 3. approach Bayesian approach 2. Information Fisher 1. aedtcosaecoet h pia ones. optimal the partic to in close stress are detectors and implemen wave being derived We are bounds which of bounds. interferometry the quantum-enhanced the tools of derive necessary attainability to the cal required all theory estimation introduce quantum We diffusion visibility. phase ometric including: fund processes derive interferometr decoherence relevant optical and in of interferometry precision developments quantum-enhanced optical the able on summarize we focus review, particular perfo this the In significantly hinders protocols. decoherence enhancin of grav presence in of the example applications notable with find experiments, light interferometric of states Non-classical Warszawa, PL-00-681 69, Poland Hoża ul. Warsaw, of University Physics, of Kołodyński Faculty J. and Jarzyna, M. Demkowicz-Dobrzański, R. 13 23 21 7 4 1 4 5 4 aigit con h most the account into taking y ossadipretinterfer- imperfect and losses , ttoa aedtcos Still, wave-detectors. itational mneo quantum-enhanced of rmance h efrac foptical of performance the g mna onso achiev- on bounds amental lrta h ehiusof techniques the that ular e nmdr gravitational modern in ted unu pisa elas well as optics quantum i ehiust h otsetclreape involv- examples spectacular most the The to interferomet- ba- techniques spectroscopic from ric screen. via ranges measurements and the length stunning on sic is interfere applications split of to path-length is number optical made beam tunable light and with single paths differences of a number when a e.g. into overlap- waves, more) light (or a two ping between from phases relative resulting in fringes) change lig (intensity observing variations about fundamental all intensity the is optical interferometry At of classical field level, the 2003). to (Hariharan, birth interferometry gave controlled and well effects involving of interference techniques num- process measurement a the of of into development ber prompted insight waves Conceptual light interfering corpuscular light. Newtonian of classical the of theory abandoning acceptance and final optics the interfe wave behind Light were interference. effects light ence of science the basically INTRODUCTION I. Conclusions VII. unu erlg with metrology quantum ihu uheagrto n a a htotc is optics that say may one exaggeration much Without , lodsustepracti- the discuss also References .Patclshmsstrtn h ons34 bounds the saturating schemes Practical D. .Bysa approach Bayesian C. .Bud nteQIapoc 28 approach QFI the in Bounds B. .Chrn qezdvcu taey35 34 strategy vacuum squeezed states + number Coherent photon indefinite 2. for Bounds 1. .Paediffusion Phase 2. losses Photonic 1. .Ipretvsblt 29 28 diffusion Phase 3. losses Photonic 2. visibility Imperfect 1. visibility Imperfect 3. losses Photonic 2. diffusion Phase 1. ht 37 36 34 33 33 32 30 27 27 r- 2 ing stellar interferometry and gravitational-wave detec- 1984; Yurke et al., 1986). A number of papers followed, tors (Pitkin et al., 2011). studying in more detail the phase-measurement probabil- Coherent properties of light as well as the degree of ity distribution and proposing various strategies leading overlap between the interfering beams determine the vis- to 1/ N precision scaling (Braunstein, 1992; Dowling, h i ibility of the observed intensity fringes and are crucial 1998; Holland and Burnett, 1993; Sanders and Milburn, for the quality of any interferometric measurement. Still, 1995). Similar observations have been made in the con- when asking for fundamental limitations on precision of text of precision spectroscopy, where spin-squeezed states estimating e.g. a phase difference between the arms of a have been shown to offer a 1/N scaling of atomic tran- Mach-Zehnder interferometer, there is no particular an- sition frequency estimation precision, where N denotes swer within a purely classical theory, where both the light the number of atoms employed (Wineland et al., 1994, itself as well as the detection process are treated classi- 1992). Sec. III provides a detailed framework for deriv- cally. In classical theory intensity of light can in principle ing the above results. However, already at this point a be measured with arbitrary precision and as such allows basic intuition should be conveyed that one cannot go to detect in principle arbitrary small phase shifts in an beyond the shot noise limit whenever an interferometric interferometric experiment. experiment may be regarded in a spirit that each pho- This, however, is no longer true when semi-classical ton interferes only with itself. In fact, the only possibil- theory is considered in which the light is still treated ity of surpassing this bound is to use light sources that classically but the detection process is quantized, so that exhibit correlations in between the constituent photons, instead of a continuous intensity parameter the number e.g. squeezed light, so that the interference process may of energy quanta (photons) absorbed is being measured. benefit from the properties of inter-photonic entangle- The absorption process within the semi-classical theory ment. has a stochastic character and the number of photons These early results provided a great physical insight detected obeys Poissonian or super-Poissonian statistics into the possibilities of quantum enhanced interferometry (Fox, 2006). If light intensity fluctuations can be ne- and the class of states that might be of practical interest glected, the number of photons detected, N, follows the for this purpose. The papers lacked generality, however, Poissonian statistics with photon number standard devia- by considering specific measurement-estimation strate- tion ∆N = N , where N denotes the mean number gies, in which the error in the estimated phase was related of photons detected.h i Thish impliesi that the determina- via a simple error-propagation formula to the variance of p tion of the relative phase difference ϕ between the arms some experimentally accessible observable, e.g. photon of the interferometer, based on the number of photons number difference at the two output ports of the inter- detected at the output ports, will be affected by the rel- ferometer, or by studying the width of the peaks in the ative uncertainty ∆ϕ ∆N/ N = 1/ N referred shape of the phase-measurement probability distribution. ∝ h i h i to as the shot noise. The shot noise plays a fundamen- Given a particular state of light fed into the interferom- p tal role in the semi-classical theory and in many cases eter, it is a priori not clear what is the best measurement it is indeed the factor limiting achievable interferometric and estimation strategy yielding the optimal estimation sensitivities. In modern gravitational wave detectors, in precision. Luckily, the tools designed to answer these particular, the shot noise is the dominant noise term in kinds of questions had already been present in the lit- the noise spectral density for frequencies above few hun- erature under the name of quantum estimation theory dred Hz (LIGO Collaboration, 2011, 2013; Pitkin et al., (Helstrom, 1976; Holevo, 1982). The Quantum Fisher 2011). Information (QFI) as well as the cost of Bayesian infer- Yet, the shot noise should not be regarded as a fun- ence provide a systematic way to quantify the ultimate damental bound whenever non-classical states of light limits on performance of phase-estimation strategies for are considered—we review the essential aspects of non- a given quantum state, which are already optimized over classical light relevant from the point of view of inter- all theoretically admissible quantum measurements and ferometry in Sec. II. The sub-Poissonian statistics char- estimators. The concept of the QFI and the Bayesian acteristic for the so-called squeezed states of light may approach to quantum estimation are reviewed in Sec. IV. offer a precision enhancement in interferometric scenar- As a side remark, we should note, that by treating the ios by reducing the photon number fluctuations at the phase as an evolution parameter to be estimated and sep- output ports. First proposals demonstrated that sending arating explicitly the measurement operators from the coherent light together with the squeezed vacuum state estimator function, quantum estimation theory circum- into the two separate input ports of a Mach-Zehnder in- vents some of the mathematical difficulties that arise if terferometer offers estimation precision beating the shot one insists on the standard approach to quantum mea- noise and attaining the 1/ N 2/3 scaling of the phase surements and attempts to define the quantum phase op- estimation precision (Caves,h 1981),i while consideration erator representing the phase observable being measured of more general two-mode squeezed states showed that (Barnett and Pegg, 1992; Lynch, 1995; Noh et al., 1992; even 1/ N scaling is possible (Bondurant and Shapiro, Summy and Pegg, 1990). h i 3

The growth of popularity of the QFI in the field of that the NOON states suffer from the 2π/N ambiguity quantum metrology was triggered by the seminal paper of in retrieving the estimated phase, and hence are designed Braunstein and Caves (1994) advocating the use of QFI only to work in the local estimation approach when phase as a natural measure of distance in the space of quantum fluctuations can be considered small. Derivations of the states. The QFI allows to pin down the optimal probe Heisenberg bounds for phase interferometry using both states that are the most sensitive to small variations of the QFI and Bayesian approaches are reviewed in Sec. V. the estimated parameter by establishing the fundamental We also discuss the problem of deriving the bounds for bound on the corresponding parameter sensitivity valid states with indefinite photon number in which case re- for arbitrary measurements and estimators. Following placing N in the derived bounds with the mean number these lines of reasoning the 1/N bound, referred to as of photons N is not always legitimate, so that in some the Heisenberg limit, on the phase estimation precision cases the “naive”h i Heisenberg bound 1/ N may in prin- using N-photon states has been claimed fundamental and ciple be beaten (Anisimov et al., 2010;h Giovannettii and the NOON states were formally proven to saturate it Maccone, 2012; Hofmann, 2009). (Bollinger et al., 1996). Due to close mathematical analo- Further progress in theoretical quantum metrology gies between optical and atomic interferometry (Bollinger stemmed from the need to incorporate realistic deco- et al., 1996; Lee et al., 2002) similar bounds hold for the herence processes in the analysis of the optimal esti- problem of atomic transition-frequency estimation and mation strategies. While deteriorating effects of noise more generally for any arbitrary unitary parameter esti- on precision in quantum-enhanced metrological proto- mation problem, i.e. the one in which an N-particle state cols have been realized by many authors working in the N evolves under a unitary U ⊗ , Uϕ = exp( iHϕˆ ), with Hˆ field (Caves, 1981; Datta et al., 2011; Gilbert et al., ϕ − being a general single-particle evolution generator and ϕ 2008; Huelga et al., 1997; Huver et al., 2008; Rubin and the parameter to be estimated (Giovannetti et al., 2004, Kaushik, 2007; Sarovar and Milburn, 2006; Shaji and 2006). Caves, 2007; Xiao et al., 1987), it has long remained an A complementary framework allowing to determine the open question to what extent decoherence effects may fundamental bounds in interferometry is the Bayesian ap- be circumvented by employing either more sophisticated proach, in which one assumes the estimated parameter to states of light or more advanced measurements strategies be a random variable itself and explicitly defines its prior including e.g. adaptive techniques. distribution to account for the initial knowledge about ϕ With respect to the most relevant decoherence pro- before performing the estimation. In the case of interfer- cess in optical applications, i.e. the photonic losses, ometry the typical choice is the flat prior p(ϕ)=1/2π strong numerical evidence based on the QFI (Demkowicz- which reflects the initial ignorance of the phase. The Dobrzanski et al., 2009; Dorner et al., 2009) indicated search for the optimal estimation strategies within the that in the asymptotic limit of large number of pho- Bayesian approach is possible thanks to the general the- tons the precision of the optimal quantum protocols orem on the optimality of the covariant measurements approaches const/√N, and hence the gain over classi- in estimation problems satisfying certain group symme- cal strategies is bound to a constant factor. This fact try (Holevo, 1982). In the case of interferometry, a flat has been first rigorously proven within the Bayesian ap- prior guarantees the phase-shift, U(1), symmetry and proach (Kołodyński and Demkowicz-Dobrzański, 2010) as a result the optimal measurement operators can be and then independently using the QFI (Knysh et al., given explicitly and they coincide with the eigenstates of 2011). Both approaches yielded the same fundamental the Pegg-Barnett “phase operator” (Barnett and Pegg, bound on precision in the lossy optical interferometry: 1992). This makes it possible to optimize the strategy ∆ϕ (1 η)/(ηN), where η is the overall power trans- over the input states and for simple cost functions allows mission≥ of an− interferometric setup. This bound is also p to find the optimal probe states (Berry and Wiseman, valid after replacing N with N when dealing with states 2000; Bužek et al., 1999; Luis and Perina, 1996). In par- of indefinite photon number,h andi moreover can be easily ticular, for the 4 sin2(δϕ/2) cost function which approx- saturated using the most popular scheme involving a co- imates the variance for small phase deviations δϕ, the herent and a squeezed-vacuum state impinged onto two corresponding minimal estimation uncertainty has been input ports of the Mach-Zehnder interferometer (Caves, found to read ∆ϕ π/N for large N providing again a 1981). This fact also implies that the presently imple- proof of the possibility≈ of achieving the Heisenberg scal- mented quantum enhanced schemes in gravitational wave ing, yet with an additional π coefficient. It should be detection, based on interfering the squeezed vacuum with noted that the optimal states in the above approach that coherent light, operate close to the fundamental bound have been found independently in (Berry and Wiseman, (Demkowicz-Dobrzański et al., 2013), i.e. they make al- 2000; Luis and Perina, 1996; Summy and Pegg, 1990) most optimal use of non-classical features of light for en- have completely different structure to the NOON states hanced sensing given light power and loss levels present which are optimal when QFI is considered as the figure in the setup. Based on the mathematical analysis of the of merit. This is not that surprising taking into account geometry of quantum channels (Fujiwara and Imai, 2008; 4

Matsumoto, 2010) general frameworks have been devel- then written as: oped allowing to find fundamental bounds on quantum precision enhancement for general decoherence models ρ = ρn,n′ n n′ , Tr(ρ)=1, ρ 0 (1) n n′ | ih | ≥ (Demkowicz-Dobrzański et al., 2012; Escher et al., 2011). X, These tools allow to investigate optimality of estimation n n strategies for basically any decoherence model and typ- with = n1,...,nM and = n1 nM rep- resenting a{ Fock state} with exactly| i |n iphotons ⊗ · · · ⊗ occupying | i ically provide the maximum allowable constant factor i improvements forbidding better than 1/√N asymptotic the i-th mode. States ni may be further expressed in terms of the respective| creationi and annihilation opera- scaling of precision. Detailed presentation of the above tors aˆ†, aˆ obeying [ˆa , aˆ†]= δ : mentioned results is given in Sec. VI. i i i j ij

Other approaches to derivation of fundamental metro- n logical bounds have been advocated recently. Making aˆi† ni = 0 , aˆ ni = √ni ni 1 , aˆi† ni = √ni +1 ni+1 , use of the calculus of variations it was shown in (Knysh | i √n!| i | i | − i | i | i et al., 2014) how to obtain exact formulas for the achiev- (2) where 0 is the vacuum state with no photons at all. able asymptotic precision for some decoherence models, | i while in (Alipour, 2014; Alipour et al., 2014) a variants In the context of optical interferometry, modes are typ- of QFI have been considered in order to obtain easier to ically taken to be distinguishable by their spatial sepa- calculate, yet weaker, bounds on precision. While de- ration, corresponding to different arms of an interferom- tailed discussion of these approaches is beyond the scope eter, whereas the various optical devices such as mir- of the present review, in Sec. VI.B.3 we make use of the rors, beam-splitters or phase-delay elements transform result from (Knysh et al., 2014) to benchmark the preci- the state on its way through the interferometer. Eventu- sion bounds derived in the case of phase diffusion noise ally, photon numbers are detected in the output modes model. allowing to infer the value of the relative phase difference The paper concludes with Sec. VII with a summary between the arms of the interferometer. and an outlook on challenging problems in the theory of In many applications, the above standard state repre- quantum enhanced metrology. sentation may not be convenient and phase-space descrip- tion is used instead—in particular, the Wigner function representation (Schleich, 2001; Wigner, 1932). Adopting II. QUANTUM STATES OF LIGHT the convention in which the quadrature operators read xˆi =a ˆi† +ˆai and pˆi = i(ˆai† aˆi), the Wigner function may be regarded as a quasi-probability− distribution on The advent of the laser, light-squeezing and single- the quadrature phase space: photon light sources triggered developments in interfer- ometry that could benefit from the non-classical features 1 M M i[p′(xˆ x) x′(pˆ p)] W (x, p)= d x′ d p′ Tr ρ e − − − , of light (Buzek and Knight, 1995; Chekhova, 2011; Torres (2π2)M ˆ et al., 2011). In this section, we focus on the quantum-  (3) light description of relevance to quantum optical inter- where x = x1,...,xM , xˆ = xˆ1,..., xˆM and similarly ferometry. We discuss the mode description of light and for p and pˆ{. As a consequence,} { the Wigner} function is the most commonly used states in quantum optics—the real, integrates to 1 over the whole phase space and its Gaussian states. In the end, we consider states of defi- marginals yield the correct probability densities of each nite photon-number and study their particle-description, of the phase space variables. Yet, since it may take nega- in particular, investigating their relevant entanglement tive values it cannot be regarded as a proper probability properties. distribution. Most importantly, it may be reconstructed from experimental data either by tomographic methods (Schleich, 2001) or by direct probing of the phase space A. Mode description (Banaszek et al., 1999), and hence is an extremely use- ful representation both for theoretical and experimental Classically, electromagnetic field can be divided into purposes. orthogonal modes distinguished by their characteristic spatial, temporal and polarization properties. This fea- ture survives in the quantum description of light, where B. Gaussian states formally we may associate a separate quantum subsys- tem with each of these modes. Each subsystem is de- From the practical point of view, the most interesting scribed by its own Hilbert space and, because photons class of states are the Gaussian states (D’Ariano et al., are bosons, can be occupied by an arbitrary number of 1995; Olivares and Paris, 2007; Pinel et al., 2012, 2013). particles (Mandel and Wolf, 1995; Walls and Milburn, The great advantage of using them is that they are rela- 1995). The most general M-mode state of light may be tively easy to produce in the laboratory with the help of 5 standard laser-sources and non-linear optical elements, where α = α eiθ and α , θ are respectively the ampli- which allow to introduce non-classical features such as tude and the| phase| of a| coherent| state. Equivalently, we squeezing or entanglement. Gaussian states have found may write numerous application in various fields of quantum infor- mation processing (Adesso, 2006) and are also extensively α = Dˆ(α) 0 , (7) | i | i employed in quantum metrological protocols. ˆ αaˆ† α∗aˆ Gaussian states of M modes are fully characterized where D(α)=e − is the so-called displacement op- by their first and second quadrature moments and are erator or write the coherent state explicitly as a super- most conveniently represented using the Wigner function position of consecutive Fock states: which is then just a multidimensional Gaussian distribu- n tion α 2/2 ∞ α α =e−| | n . (8) | i √ | i n=0 n! 1 1 (z ˆz )T σ−1(z zˆ ) X W (z)= e− 2 −h i −h i , (4) (2π)M √det σ From the formula above, it is clear that coherent states do not have a definite photon number and if a photon num- where for a more compact notation we have introduced: ber n is measured its distribution follows the Poissonian 2n the phase space variable z = x ,p ,...,x ,p , α 2 α 2 1 1 M M statistics P (n)=e−| | | | with average n = α and the vector containing mean quadrature{ values ˆz =} n! h i | | h i standard deviation ∆n = α . Thus, the relative uncer- xˆ1 , pˆ1 ,..., xˆM , pˆM , zˆi = Tr(ˆziρ) = tainty ∆n/ n in the measured| | photon number scales like {h 2Mi h i h i h i} h i h i d z W (z)zi, and the 2M dimensional covariance ma- 1/ n and hence in the classical limit of large n the ´trix σ: beamh poweri may be determined up to arbitrary precision.h i p 1 Moreover, the evolution of the coherent state amplitude σi,j = zˆizˆj +ˆzj zˆi zˆi zˆj . (5) is identical to the evolution of a classical-wave ampli- 2h i − h ih i tude. In particular, an optical phase delay ϕ transforms Gaussian states remain Gaussian under arbitrary evo- the state α into α eiϕ . These facts justify a common | i | i lution involving Hamiltonians at most quadratic in the jargon of calling coherent states the classical states of quadrature operators, what includes all passive devices light, even though for relatively small amplitudes differ- such as beam-splitters and phase-shifters as well as ent coherent states may be hard to distinguish due to 2 α β 2 single- and multi-mode squeezing operations. Below we their non-orthogonality α β = e| − | . More gener- |h | i| focus on a few classes of Gaussian states highly relevant ally, we call ρcl a classical state of light if and only if it to interferometry. can be written as a mixture of coherent states:

2M ρcl = d α P (α) α α (9) ˆ | ih | 1. Coherent states with P (α) 0, which is equivalent to the statement ≥ Coherent states are the Gaussian states with identity that ρcl admits a non-negative Glauber P -representation covariance matrix σ = 11, so that the uncertainties are (Glauber, 1963; Walls and Milburn, 1995). Classical equal for all quadratures saturating the Heisenberg un- states are often used as a benchmark to test the degree of 2 2 certainty relations ∆ xi∆ pi = 1 and there are no cor- possible quantum enhancement which may be obtained relations between the modes. Mean values of quadra- by using more general states outside this class. tures may be arbitrary and correspond to the coherent state complex amplitude α = ( xˆ + i pˆ )/2. These are h i h i the states produced by any phase-stabilized laser, what 2. Single-mode squeezed states makes them almost a fundamental tool in the theoret- ical description of many quantum optical experiments. Heisenberg uncertainty principle imposes that Moreover, coherent sates have properties that resemble ∆x∆p 1 for all possible quantum states. Single- features of classical light, and thus enable to establish a mode states≥ that saturate this inequality are called bridge between the quantum and classical descriptions of the single-mode squeezed states (Walls and Milburn, light. 1995). As mentioned above, coherent states fall into In the standard representation, an M-mode coherent such a category serving as a special example for which state α = α1 αM is a tensor product of single- ∆x = ∆p = 1. Yet, as for general squeezed states | i | i⊗· · ·⊗| i mode coherent states, whereas a single-mode coherent ∆x = ∆p, the noise in one of the quadratures can be state is an eigenstate of the respective annihilation oper- made6 smaller than in the other. Formally, a single-mode ator: squeezed state may always be expressed as

aˆ α = α α , (6) α, r = Dˆ(α)Sˆ(r) 0 , (10) | i | i | i | i 6

(a) (b) 3. Two-mode squeezed states

The simplest non-classical two-mode Gaussian state is the so-called two-mode squeezed vacuum state or the twin-beam state (Walls and Milburn, 1995). Mathemat- ically, such a state is generated from the vacuum by a FIG. 1 Phase-space diagrams denoting uncertainties in dif- two-mode squeezing operation, so that ferent quadratures for momentum squeezed states (a) and for ξ = Sˆ (ξ) 0, 0 , (13) position squeezed states (b). Dashed circles denote corre- | i2 2 | i sponding uncertainties for coherent states ∆x =∆p. iθ where Sˆ2(ξ) = exp(ξ∗aˆˆb ξaˆ†ˆb†) and ξ = ξ e , whereas in the Fock basis it reads− | | where Sˆ(r) = exp( 1 r aˆ2 1 raˆ 2) is the squeezing opera- 2 ∗ 2 † 1 ∞ tor, r = r eiθ is a complex− number and r and θ are the ξ = ( 1)neiθ tanhn ξ n,n . (14) | | | | | i2 cosh ξ − | i squeezing factor and the squeezing angle respectively. In n=0 X fact, any pure Gaussian one-mode state may be written A notable feature of the twin-beam state, which may be in the above form. For θ =0, uncertainties in the quadra- clearly seen from Eq. (14), is that it is not a product of tures x and p read ∆x = e r and ∆p = er—reduction − squeezed states in modes a and b, but rather it is cor- of noise in one quadrature is accompanied by an added related in between them being a superposition of terms noise in the other one. This may be conveniently visual- with the same number of photons in both modes. Its first ized in the phase-space picture by error disks represent- moments of all quadratures are zero, zˆ =0, whereas in ing uncertainty in quadratures in different directions, see the case of ξ = ξ its covariance matrixh hasi a particularly Fig. 1. In such a representation squeezed states corre- simple form: | | spond to ellipses while coherent states are represented by circles. Importantly, the fact that the uncertainty of cosh(2ξ) 0 sinh(2ξ) 0 one of the quadratures can be less relatively to the other 0 cosh(2ξ) 0 sinh(2ξ) σ = . makes it possible to design an interferometric scheme  sinh(2ξ) 0 cosh(2ξ)− 0  where the measured photon number fluctuations are be-  0 sinh(2ξ) 0 cosh(2ξ)   −  low that of a coherent state and allows for a sub-shot  (15) noise phase estimation precision, see Sec. III. As squeezed Such a covariance matrix clearly indicates the presence states in general cannot be described as mixtures of co- of correlations between the modes and since the state herent states, they are non-classical and their features (13) is pure this implies immediately the presence of the cannot be fully described by classical electrodynamics. mode-entanglement. In fact, in the limit of large squeez- Nevertheless, they can be relatively easily prepared using ing coefficient ξ such twin-beam state becomes the non-linear optical elements in the process of parametric original famous| | Einstein-Podolsky-Rosen → ∞ state (Adesso, down conversion (Bachor and Ralph, 2004). 2006; Banaszek and Wódkiewicz, 1998) that violates as- The special type of squeezed states which is most rel- sumptions of any realistic local hidden variable theory. evant from the metrological perspective is the class of Twin-beam states may be generated in a laboratory by the squeezed vacuum states that possess vanishing mean various non-linear processes such as four- and three-wave values of their quadratures, i.e. zˆ =0: mixing (Bachor and Ralph, 2004; Reid and Drummond, h i 1988). Alternatively, they may be produced by mixing r = Sˆ(r) 0 . (11) two single-mode squeezed vacuum states with opposite | i | i squeezing angles on a fifty-fifty beam-splitter. In the Fock basis a squeezed vacuum state reads

n 4. General two-mode Gaussian states 1 ∞ Hn(0) tanh r 2 i n θ r = e 2 n , (12) | i √ √ 2 | i cosh r n=0 n! X   General two-mode Gaussian state is rather difficult to write in the Fock basis, so it is best characterized by its where H (0) denotes values of n-th Hermite polynomial n 4 4 real symmetric covariance matrix at x = 0. As for odd n Hn(x) is antisymmetric and × thus H (0) = 0, it follows that squeezed vacuum states n [σ11][σ12] are superpositions of Fock states with only even photon σ = , (16) [σ21][σ22] numbers. The average number of photons in a squeezed   2 vacuum state is given by n = sinh r, what means that, where σij represent blocks with 2 2 matrices de- despite their name, a squeezedh i vacuum states contain scribing correlations between the i-th× and the j-th photons, possibly a lot of them. mode, and the vector of the first moments ˆz = h i 7

xˆ1 , pˆ1 , xˆ2 , pˆ2 . This in total gives up to four- D. Particle description {hteeni realh i parametersh i h i} describing the state: ten covari- ances, two displacement amplitudes and two phases of When dealing with states of definite photon number, displacement. General Gaussian states are in principle instead of thinking about modes as quantum subsystems feasible within current technological state of art, as any that possess some number of excitations (photons), we pure Gaussian state can theoretically be generated from may equivalently consider the “first quantization” formal- the vacuum by utilizing only a combination of one-, two- ism and regard photons themselves as elementary sub- mode squeezing and displacement operations with help systems. Fundamentally, photons are indistinguishable of beam-splitters and one-mode rotations (Adesso, 2006). particles and since they are bosons their wave function Furthermore, mixed Gaussian states are obtained as a re- should always be permutation-symmetric. Still, it is com- sult of tracing out some of the system degrees of freedom, mon in the literature to use a description in which pho- which is effectively the case in the presence of light losses, tons are regarded as distinguishable particles and adopt or by adding a Gaussian noise to the state. a notation such that m = m m m (18) | i | 1i1 ⊗ | 2i2 ⊗ · · · ⊗ | N iN denotes a product state of N photons, where the i-th pho- C. Definite photon number states ton occupies the mode mi. We explicitly add subscripts to the kets above labeling each constituent photon, in order to distinguish this notation from the mode descrip- Gaussian states are important from the practical point tion of Eq. (1), where kets denoted various modes and not of view due to the relative ease with which their may be the distinct particles. The description (18) is legitimate prepared. From a conceptual point of view, however, provided there are some degrees of freedom that ascribe a when asking fundamental questions on limits to quan- meaning to the statement “the i-th photon”. For example, tum enhancement in interferometry, states with a defi- in the case when photons are prepared in non-overlapping nite photon number prove to be a better choice. The time-bins, the time-bin degree of freedom plays the role main reason is that photons are typically regarded as a of the label indicating a particular photon, whereas the resource in interferometry and when benchmarking dif- spatial characteristics determine the state of a given pho- ferent interferometric schemes it is natural to restrict the ton. Nevertheless, if we assume the overall wave function class of states with the same number of photons, i.e. the describing also the temporal degrees of freedom of the same resources consumed. A general M-mode state con- complete state to be fully symmetric, the notion of the sisting of N photons is given by: “i-th photon” becomes meaningless. A general pure state of N “distinguishable” photons

ρ = ρn n′ n n′ , (17) has the form: N , | ih | n = n′ =N | | X| | ψN = cm m (19) | i m | i X where n = ni, so that the summation is restricted 2 i where cm =1. If indeed there is no additional de- only to| terms| with exactly N photons in all the modes. m | | P gree of freedom that makes the notion of “the i-th photon” Apart from the vacuum state 0 no Gaussian state falls meaningful,P the above state should posses the symmetry into this category. | i property such that cm = cΠ(m), where Π is an arbitrary States with an exact photon number are extensively permutation of the N indices. used in other fields of quantum information processing, Consider for example a Fock state n = n ... n | i | 1i | M i including quantum communication and quantum com- of N = n1 + + nM indistinguishable photons in M puting (Kok et al., 2007; Pan et al., 2012). Most of the modes. In the··· particle description the state has the form: quantum computation and communication schemes are designed with such states in mind, as they provide the n ! ...n ! n = 1 M most intuitive and clear picture of the role the quantum | i r N! × features play in these tasks. For large N, however, states Π( 1,..., 1, 2 ..., 2,...,M,...,M ) , (20) with a definite photon numbers are notoriously hard to | { } i Π prepare and states with N of the order of 10 can only be X n1 n2 nM produced with the present technology pushed to its limits where the sum is| performed{z } | {z over} all non-trivial| {z } permuta- (Hofheinz et al., 2008; Sayrin et al., 2011; Torres et al., tions Π of the indices inside the curly brackets (Shankar, 2011). When considering states with definite N, it is also 1994). Since all quantum states may be written in the possible to easily switch between the mode- and particle- Fock basis representation, by the above construction one description of the states of light, which is a feature that can always translate any quantum state to the particle we discuss in the following section. description. 8

E. Mode vs particle entanglement which is both mode- and particle-entangled. Mode en- tanglement emerges because the beam-splitter is a joint One of the most important features which makes the operation over two modes that introduces correlations quantum theory different from the classical one is the between them. On the other hand, it is a local opera- notion of entanglement (Horodecki et al., 2009). This tion with respect to the particles, i.e. it can be written as phenomenon plays also an important role in quantum U U in the particle representation, and does not cou- metrology and is often claimed to be the crucial resource ple⊗ photons with each other. Thus, using a beam-splitter for the enhancement of the measurement precision (Gio- one may change mode entanglement but not the content vannetti et al., 2006; Pezzé and Smerzi, 2009). Conflict- of particle entanglement. ing statements can be found in the literature, however, As a second example, consider two modes of light, a as some of the authors claim that entanglement is not and b, each of them in coherent state with the same am- indispensable to get a quantum precision enhancement plitude α, α a α b. This state clearly has no mode en- (Benatti and Braun, 2013). This confusion stems simply tanglement.| i Since| i this state does not have a definite from the fact that entanglement is a relative concept de- photon number, in order to ask questions about the par- pendent on the way we divide the relevant Hilbert space ticle entanglement we first need to consider its projection into particular subsystems. In order to clarify these is- on one of the N-photon subspaces—one can think of a sues, it is necessary to explicitly study relation between non-demolition total photon-number measurement yield- mode- and particle-entanglement, i.e. entanglement with ing result N. After normalizing the projected state we respect to different tensor product structures used in the obtain: two descriptions. N (N) 1 N Firstly, let us go through basic definitions and notions ψN = [ α a α b] = n a N n b, | i | i | i √ N n | i | − i of entanglement. The state ρAB of two parties A and 2 n=0 s  B is called separable if and only if one can write it as a X (24) mixture of product of states of individual subsystems: which in the particle representation reads: N (AB) (A) (B) 1 ρ = pi ρ ρ , pi 0. (21) i ⊗ i ≥ ψN = ( a i + b i) (25) i | i √ N | i | i 2 i=1 X O Entangled states are defined as all states that are not and is clearly a separable state. The fact that products separable. A crucial feature of entanglement is that it of coherent states contain no particle entanglement is in depends on the division into subsystems. For example, agreement with our definition of classical state given in consider three qubits A, B and C and their joint quantum Eq. (9) being a mixture of products of coherent states. 1 state ρ(ABC) = 1 i j i j 0 0 . This state is A classical state according to this definition will contain i,j=0 2 | ih |⊗| ih |⊗| ih | separable with respect to the AB C cut but is entangled neither mode nor particle entanglement. P | with respect to the A BC cut. As a last example, consider the case of particular inter- | As a first example, consider a two-mode Fock state est for quantum interferometry, i.e. a coherent state of 1 a 1 b which represents one photon in mode a and one mode a and a squeezed vacuum state of mode b: α r . | i | i a b photon in mode b. This state written in the mode formal- Again this state has no mode entanglement. On the| i other| i ism of Eq. (1) is clearly separable. On the other hand, hand it is particle entangled. To see this, consider e.g. the photons are indistinguishable bosons and if we would like two-photon sector, which up to irrelevant normalization to write their state in the particle formalism of Eq. (18) factor reads: we have to symmetrize over all possible permutations of [ α r ](N=2) α2 2 0 + tanh r 0 2 = particles, thus obtaining the state | i| i ∝ | ia| ib | ia| ib 2 1 = α a 1 a 2 + tanh r b 1 b 2 (26) 1 a 1 b = ( a 1 b 2 + b 1 a 2). (22) | i | i | i | i | i | i √2 | i | i | i | i and contains particle entanglement provided both α and In this representation the state is clearly entangled. We r are non-zero. We argue and give detailed arguments may thus say that the state contains particle entangle- in Sec. V that it is indeed the particle entanglement and not ment but not the mode entanglement. the mode entanglement that is relevant in quantum- et al. If, however, we perform the Hong-Ou-Mandel exper- enhanced interferometry. See also Killoran (2014) for more insight into the relation between mode and par- iment and send the 1 a 1 b state through a balanced beam-splitter which transforms| i | i mode annihilation oper- ticle entanglement. ators as aˆ (ˆa + ˆb)/√2, ˆb (ˆa ˆb)/√2, the resulting → → − state reads: III. MACH-ZEHNDER INTERFEROMETRY 1 1 1 a 1 b ( 2 a 0 b 0 a 2 b)= ( a 1 a 2 b 1 b 2), | i | i → √2 | i | i −| i | i √2 | i | i −| i | i We begin the discussion of quantum-enhancement ef- (23) fects in optical interferometry by discussing the paradig- 9

α n of the MZ interferometer in terms of the algebra of the | i ϕa a a a¢ angular momentum operators (Yurke et al., 1986). Let us define the operators:

b ˆ 1 ˆ ˆ ˆ i ˆ ˆ ˆ 1 ˆ ˆ r b¢ n Jx = (ˆa†b+b†aˆ), Jy = (b†aˆ aˆ†b), Jz = (ˆa†aˆ b†b), | i ϕb b 2 2 − 2 − (28) FIG. 2 The Mach-Zehnder interferometer, with two input which fulfill the angular momentum commutation rela- ′ ′ light modes a, b and two output modes a ,b . In a standard tions [Jˆi, Jˆj ] = iǫijkJˆk while the corresponding square of configuration a coherent state of light |αi is sent into mode a. the total angular momentum reads: In order to obtain quantum enhacement, one needs to make use of the b input port also, sending e.g. the squeezed vacuum ˆ ˆ ˆ2 N N ˆ ˆ ˆ state |ri. J = +1 , N =ˆa†aˆ + b†b, (29) 2 2 !

where Nˆ is the total photon number operator. The action matic model of the Mach-Zehnder (MZ) interferometer. of linear optical elements appearing in the MZ interfer- We analyze the most popular interferometric schemes in- ometer can now be described as rotations in the abstract volving the use of coherent and squeezed states of light spin space: aˆ′ = UaUˆ †, ˆb′ = UˆbU †, U = exp( iαJˆ s), accompanied by a basic measurement-estimation proce- − · where Jˆ = Jˆ , Jˆ , Jˆ and α, s are the angle and the dure, in which the phase is estimated based on the value { x y z} of the photon-number difference between the two out- axis of the rotation respectively. In particular, the bal- put ports of the interferometer. Such a protocol provides anced beam splitter is a rotation around the x axis by an angle π/2: U = exp( i π Jˆ ), while the phase delay is a us with a benchmark that we may use in the following − 2 x ϕ rotation around the z axis: U = exp( iϕJˆ ). Instead sections when discussing the optimality of the interfer- − z ometry schemes both with respect to the states of light of analysing the transformation of the annihilation oper- used as well as the measurements and the estimation pro- ators, it is more convenient to look at the corresponding cedures employed. transformation of the Ji operators themselves:

Jˆ′ 1 0 0 cos ϕ sin ϕ 0 x − A. Phase-sensing uncertainty Jˆ = 0 0 1 sin ϕ cos ϕ 0  y′     × Jˆ 0 1 0 0 01 z′ − In the standard MZ configuration, depicted in Fig. 2,      Jˆ Jˆ a coherent state of light is split on a balanced beam- 10 0 x cos ϕ 0 sin ϕ x 0 0 1 Jˆy = 0 1 0 Jˆy , splitter, the two beams acquire phases ϕa, ϕb respec- × −      01 0 Jˆ sin ϕ 0 cos ϕ Jˆ tively, interfere on the second beam-splitter and finally z − z       the photon numbers na, nb are measured at the output (30) ports. Let a,ˆ ˆb and aˆ , ˆb be the annihilation operators cor- ′ ′ which makes it clear that the sequence of π/2, ϕ and responding to the two input and the two output modes π/2 rotations around axes x, z and x respectively, re- respectively. The combined action of the beam-splitters sults− in an effective ϕ rotation around the y axis. and the phase delays results in the effective transforma- Using the above formalism, let us now derive a sim- tion of the annihilation operators: ple formula for uncertainty of phase-sensing based on the

iϕ measurement of the photon-number difference at the out- aˆ′ 1 1 i e a 0 1 i aˆ = = put. Note that nˆa nˆb = 2Jˆz, so the photon-number ˆ iϕb − ˆ b′ 2 i 1 0 e i 1 b − ˆ        −    difference measurement is equivalent to the Jz measure- cos(ϕ/2) sin(ϕ/2) aˆ ment. Utilizing Eq. (30) in the Heisenberg picture, the =ei(ϕa+ϕb)/2 − , (27) sin(ϕ/2) cos(ϕ/2) ˆb average Jz evaluated on the interferometer output state     may be related to the average of Jz′ of the input state where ϕ = ϕb ϕa is the relative phase delay and for ψ in as convenience we− assume that the beams acquire a π/2 | i − Jˆ = cos ϕ Jˆ in sin ϕ Jˆ in. (31) or π/2 phase when transmitted through the first or the h zi h zi − h xi second beam-splitter respectively. The common phase In order to assess the precision of ϕ-estimation, we also factor ei(ϕa+ϕb)/2 is irrelevant for further discussion in calculate the variance of the Jˆ operator of the output this section and will be omitted. z state of the interferometer: In order to get a better insight into the quantum- enhancement effects in the operation of the MZ interfer- 2 2 2 2 2 ∆ J = cos ϕ ∆ J in + sin ϕ ∆ J in+ ometer, it is useful to make use of the so-called Jordan- z z| x| 2 sin ϕ cos ϕ cov(J ,J ) in, (32) Schwinger map (Schwinger, 1965) and analyse the action − x z | 10

1 ˆ ˆ ˆ ˆ ˆ ˆ where cov(Jx,Jz) = 2 JxJz + JzJx Jx Jz is the estimation precision. This, however, is scarcely of any covariance of the two observables.h Thei − precisionh ih i of es- use in practice, since one may always perform rough in- timating ϕ can now be quantified via a simple error- terferometric measurements and bring the setup close to propagation formula: the optimal operating points before performing more pre- cise measurements there. Moreover, the ϕ-dependence in ∆J ∆ϕ = z . (33) Eq. (35) can be easily removed by taking into account not d Jˆ z only the photon-number difference observable Jˆz but also hdϕ i the total photon number measured. By using their ratio

as an effective observable in the r.h.s. of Eq. (33), or in other words by considering the “visibility observable”, the B. Coherent-state interferometry formula (35) is replaced by 36. Let us now analyze the precision given by Eq. (33) for the standard optical interferometry with the input state ψ in = α 0 , representing a coherent state and no D. Coherent + squeezed-vacuum interferometry light at| i all being| i| senti into the input modes a and b of the interferometer in Fig. 2. The relevant quantities required We thus need to use more general input states in order for calculating the precision given in Eq. (33) read: to surpass the shot noise limit. Firstly, let us note that sending the light solely to one of the input ports will not 1 2 2 2 1 2 provide us with the desired benefit. As in the end only Jˆ in = α , Jˆ in =0, ∆ J in = ∆ J in = α , h zi 2| | h xi z| x| 4| | a photon-number measurement is assumed, which is not cov(Jx,Jz) in =0 (34) sensitive to any relative phase differences between various | Fock terms of the output state, any scenario involving a yielding the precision: single-beam input may always be translated to the the situation in which an incoherent mixture of Fock states 1 α 0 2 α 1 1 is sent onto the input port. Since the variance is a convex ∆ϕ| i| i = | | = = , 1 α 2 sin ϕ α sin ϕ N sin ϕ function with respect to state density matrices, and thus 2 | | | | | | h i| (35)| increasing under mixing, such strategies are of no use for p where the average photon number N = Nˆ = α 2. our purposes. h i | | Let us now consider a scheme were apart from the The above formula represents 1/ N shotD noiseE scal- coherent light we additionally send a squeezed-vacuum ing of precision characteristic forh thei classical inter- p state into the other input port (Caves, 1981): ψ in = ferometry. The shot noise is a consequence of the α r , see Fig. 2. This kind of strategy| isi be- ∆J effectively representing the Poissonian fluctuations z |ingi⊗ implemented | i in current most advanced interferom- of the photon-number difference measurements at the eters designed to detect gravitational waves like LIGO or output ports. Yet, although such fluctuations are ϕ- GEO600 (LIGO Collaboration, 2011, 2013; Pitkin et al., independent, the average photon-number difference J z 2011). Assuming for simplicity that r is real, the relevant changes with ϕ with speed proportional to sin ϕ hap-i quantities required to calculate the estimation precision pearing in Eq. (35), so that the optimal operating| points| read: are at ϕ = π/2, 3π/2.

2 2 1 2 2 Nˆ = α + sinh r, J in = ( α sinh r), J in =0 h i | | h zi 2 | | − h xi C. Fock state interferometry 2 1 2 1 2 ∆ J in = α + sinh 2r , cov(J ,J ) in =0, z 4 2 x z We can attempt to reduce the estimation uncertainty | | | | 2 1 2
2 2 using more general states of light at the input. For ex- ∆ J in = α cosh2r Re(α ) sinh 2r + sinh r . x 4 ample, we can replace the coherent state with an N- | | | −
(37) photon Fock state, so that ψ in = N 0 . This is an | 2i | i| i eigenstate of Jˆz and hence ∆ Jz in = 0, and only the 2 | Hence, the usage of squeezed-vacuum as a second in- ∆ Jx in contributes to the Jˆz variance at the output: 2 2 | 1 2 1 put allows to reduce the variance ∆ Jx in thanks to ∆ J = N sin ϕ. Since Jˆ = N cos ϕ, the corre- 2 | z 4 h zi 2 the Re(α ) sinh 2r term above, which is then maximized sponding estimation uncertainty reads: by choosing the phase of the coherent state such that α = Re(α). This corresponds to the situation, when the N 0 1 ∆ϕ| i| i = , (36) coherent state is displaced in phase space in the direc- √N tion in which the squeezed vacuum possesses its lowest being again shot-noise limited. The sole benefit of us- variance. With such an optimal choice of phase, substi- ing the Fock state is the lack of ϕ-dependence of the tuting the above formulas into Eq. (33), we obtain the 11

final expression for the phase-estimation precision: an improved precision compared with a coherent-state– based strategy. Therefore, we need to consider states 2 2 1 2 2 2r 2 cot ϕ ( α + sinh 2r)+ α e− + sinh r with photons being simultaneously sent into both input α r 2 ∆ϕ| i| i = | | | | . ports. A general N-photon two-mode input state can be q 2 2 α sinh r written down using the angular momentum notation as | | − (38)

The optimal operation point are again clearly ϕ = j π/2, 3π/2, since at them the first term under the square ψ in = c j, m , (40) | i m| i root, which is non-negative, vanishes. For a fair compar- m= j X− ison with other strategies we should fix the total average number of photons N , which is regarded as a resource, where j =N/2 and j, m = j +m j m in the standard h i mode-occupation notation.| i | In particular,i| − i j, j = N 0 and optimize the split of energy between the coherent | i | i| i and the squeezed vacuum beams in order to minimize corresponds to a state with all the photons being sent ∆ϕ. This optimization can only be done numerically, into the upper input port. One can easily check that but the solution can be well approximated analytically angular momentum operators introduced in Eq. (28) act in a standard way on the j, m states. For concreteness in the regime of N 1. In this regime the squeezed | i vacuum should carryh i approximately ≫ N /2 of photons, assume N is even and consider the state (Yurke et al., h i 1986): so the squeezing factor obeys sinh2 r 1 e2r N /2 p 4 while the majority of photons belongs≈ to the≈ coherenth i p 1 beam. The resulting precision reads ψ = ( j, 0 + j, 1 )= | iin √2 | i | i 1 N N N N N 1 N /(2 N )+ N /2 N 1 = + +1 1 , (41) α r h i≫ h i≫ 1 2 2 2 2 ∆ϕ| i| i h i h i h i √2 − ≈ q N N /2 ≈ N 3/4 p p h i
h i− h i (39) for which: p and proves that indeed this strategy offers better than 1 1 2 1 Jˆ in = , Jˆ in = j(j + 1), ∆ J in = , shot noise scaling of precision. h zi 2 h xi 2 z| 2 While the above example shows that indeed quantum 2 1 1p ∆ J in = j(j + 1) , cov(J ,J ) in =0. states of light may lead to an improved sensitivity, the x| 2 − 4 x z | issue of optimality of the proposed scheme has not been (42) addressed. In fact, keeping the measurement-estimation scheme unchanged, it is possible to further reduce the Plugging the above expressions into Eq. (33), we get: estimation uncertainty by sending a more general two- mode Gaussian states of light with squeezing present in cos2 ϕ + sin2 ϕ[j(j + 1) 1] both input ports and reach the 1/ N scaling of pre- ∆ϕ = − . (43) ∝ h i q sin ϕ + cos ϕ j(j + 1) cision (Olivares and Paris, 2007; Yurke et al., 1986). We | | skip the details here, as the optimization of the general The optimal operation point correspondsp to sin ϕ = 0, Gaussian two-mode input state minimizing the estima- 2 where we benefit from large Jˆx in and low ∆ Jz in mak- tion uncertainty of the scheme considered is cumbersome. ing the state very sensitive toh rotationsi around the| y axis. More importantly, using the tools of estimation theory The resulting precision reads: introduced in Sec. IV, we will later show in Sec. V that even with α r class of input states it is possible to 1 N 1 2 | i ⊗ | i ∆ϕ = ≫ , (44) reach the 1/ N scaling of precision, but this requires j(j + 1) ≈ N a significantly∝ modifiedh i measurement-estimation scheme (Pezzé and Smerzi, 2008) and different energy partition indicating the possibilityp of achieving the Heisenberg between the two input modes. scaling of precision. We should note here, that a sim- pler state ψ in = j, 0 = N/2 N/2 called the twin- Fock state |wherei N/| 2 iphotons| arei| simultaneouslyi sent E. Definite photon-number state interferometry into each of the input ports, is also capable of providing the Heisenberg scaling of precision (Holland and Burnett, Even though definite photon-number states are techni- 1993), but requires a different measurement-estimation cally difficult to prepare, they are conceptually appealing scheme which goes beyond the analysis of the average and we make use of them to demonstrate explicitly the photon-number difference at the output, see Sec. V.C.3. possibility of achieving the 1/N Heisenberg scaling of es- One should bear in mind that the expressions for at- timation precision in Mach-Zehnder interferometry. We tainable precision presented in this section base on a sim- have already shown that sending an N-photon state into ple error propagation formula calculated at a particular a single input port of the interferometer does not lead to operating point. Therefore, in order to approach any of 12

resulting input-output relation reads: ϕ aˆ 1 ′ = ˆb′ (2 T )i sin θ T cos θ × ϕ θ   − − 2i√1 T sin θ T aˆ − (45) × T 2i√1 T sin θ ˆb  −   FIG. 3 Other popular two-input/two-output mode interfer- and up to an irrelevant global phase may be rewritten as: ometers: (a) Michelson interferometer with one-way phase aˆ′ cos ϕ/2 i sin ϕ/2 aˆ delays ϕa, ϕb in the respective arms, (b) Fabry-Pérrot inter- = , ˆb′ i sin ϕ/2 cos ϕ/2 ˆb ferometer with one-way phase delay θ and power transmission       (46) T of the mirrors. ϕ = 2 arcsin T . √T 2+4(1 T ) sin2 θ  −  In terms of the angular momentum operators the above the precisions claimed, one needs to first lock the interfer- transformation is simply a ϕ-rotation around the x axis. ometer to operate close to an optimal point, which also Thus, up to a change of the rotation axis, the action requires some of the resources to be consumed. Rigor- of the Fabry-Pérrot interferometer with phase delay θ ous quantification of the total resources needed to attain is equivalent to the one of the MZ interferometer with a given estimation precision starting with a completely phase delay ϕ. Knowing the formulas for the estimation unknown phase may be difficult in general. We return to precision of ϕ in the MZ interferometer, we can easily this issue in Sec. V, where we are able to resolve this prob- calculate the corresponding estimation precision of θ via lem by approaching it with the language of the Bayesian the error propagation formula obtaining: inference. ∆ϕ T 2 + 4(1 T ) sin2 θ ∆θ = = ∆ϕ − . (47) ∂ϕ 4T √1 T cos θ | ∂θ | − As a consequence, the above expression allows to trans- F. Other interferometers late all the results derived for the MZ interferometer to the Fabry-Pérrot case. Even though we have focused on the Mach-Zehnder in- Ramsey interferometry is a popular technique for per- terferometer setup, analogous results could be obtained forming precise spectroscopic measurements of atoms. It for other optical interferometric configurations, such as is widely used in atomic clock setups, where it allows to the Michelson interferometer, the Fabry-Pérrot inter- lock the frequency of an external source of radiation, ω, ferometer, as well as the atomic interferometry setups to a selected atomic transition frequency, ω0, between the utilized in: atomic clocks operation (Diddams et al., single-atom excited and ground states, e and g (Did- | i | i 2004), spectroscopy (Leibfried et al., 2004), magnetome- dams et al., 2004). In a typical Ramsey interferometric try (Budker and Romalis, 2007) or the BEC interferom- experiment N atoms are initially prepared in the ground etry (Cronin et al., 2009). We briefly show below that state and are subsequently subjected to a π/2 Rabi pulse, despite physical differences the mathematical framework which transforms each of them into an equally weighted is common to all these cases and as such the results pre- superposition of ground and excited states. Afterwards, sented in this review, even though derived with the sim- they evolve freely for time t, before finally being sub- ple optical interferometry in mind, have a much broader jected to a second π/2 pulse, which in the ideal case of scope of applicability. ω0 = ω would put all the atoms in the exited state. In The Michelson interferometer is depicted in Fig. 3a. case of any frequency mismatch, the probability for an atom to be measured in an excited state is cos2(ϕ/2), Provided the output modes a′, b′ can be separated from where ϕ = (ω ω)t. Hence, by measuring the number the input ones a, b via an optical circulator, the Michel- 0 − son interferometer is formally equivalent to the Mach- of atoms in the excited state, one can estimate ϕ and Zehnder interferometer. The input output relations are consequently knowing t the frequency difference ω0 ω. Treating two-level atoms as spin-1/2 particles with their− identical as in Eq. (27) with ϕ = 2(ϕb ϕa), as the light acquires the relative phase twice—traveling− both to and two levels corresponding to up and down projections of from the end mirrors. the spin z component, we may introduce the total spin operators Jˆ = 1 N σˆ(k), i = x,y,z, where σˆ(k) are Consider now the Fabry-Pérrot interferometer de- i 2 k=0 i i standard Pauli sigma matrices acting on the k-th parti- picted in Fig. 3b. We assume for simplicity that both cle. Evolution ofP a general input state can be written mirrors have the same power transmission coefficient T as: and the phase θ is acquired while the light travels from iJˆxπ/2 iJˆzϕ iJˆxπ/2 one mirror to the other inside the interferometer. The ψ =e− e e− ψ in. (48) | ϕi | i 13

states. In particular, starting with atoms in a ground state the two-axis spin-squeezed states may be obtained via:

χ (Jˆ2 Jˆ2 ) N ψ =e− 2 +− − g ⊗ , (50) | χi | i

where Jˆ = Jˆx iJˆy. The above formula resembles the definition± of an± optical squeezed state given in Eq. (10), where aˆ, aˆ† operators are replaced by Jˆ , Jˆ+. With an appropriate choice of squeezing strength−χ as a function of N it is possible to achieve the Heisenberg scaling of precision ∆ω 1/N (Ma et al., 2011; Wineland et al., FIG. 4 Formal equivalence of Ramsey and MZ interferometry. ∝ The two atomic levels which are used in the Ramsey interfer- 1994). ometry play analogous roles as the two arms of the interfer- Analogous schemes may also be implemented in BEC ometer, while the π/2 pulses equivalent from a mathematical (Cronin et al., 2009; Gross et al., 2010). In particu- point of view to the action of the beam-splitters. Quantum- lar, BEC opens a way of realizing a specially appealing enhancement in Ramsey interferometry may be obtained by matter-wave interferometry, in which, similarly to the op- preparing the atoms in a spin-squeezed input state reducing tical interferometers, the matter-wave is split into two the variances of the relevant total angular momentum oper- spatial modes, evolves and finally interferes resulting in ators, in a similar fashion as using squeezed states of light leads to an improved sensitivity in the MZ interferometry. a spatial fringe pattern that may be used to estimate the relative phase acquired by the atoms (Shin et al., 2004). Such a scheme may potentially find applications in pre- which is completely analogous to the MZ transforma- cise measurements of the gravitational field (Anderson tion (30) with π/2 pulses playing the role of the beam- and Kasevich, 1998). Yet, in this case, the detection splitters, see Fig. 4. The total spin z operator can be involves measurements of positions of the atoms form- written as Jˆz = 2(ˆng nˆe), where nˆg, nˆe denote the ing the interference fringes, what makes the estimation − ground and excited state atom number operators respec- procedure more involved than in the simple MZ scheme tively. Therefore, measurement of Jˆz is equivalent to (Chwedeńczuk et al., 2012, 2011), but eventually the pre- the measurement of the difference of excited and non- cisions for the optimal estimation schemes should coin- excited atoms analogously to the optical case where it cide with the ones obtained for the MZ interferometry. corresponded to the measurement of photon-number dif- Finally, we should also mention that atomic ensembles ference at the two output ports of the interferometer. interacting with light are excellent candidates for ultra- Fluctuation of the number of atoms measured limits the precise magnetometers (Budker and Romalis, 2007). Col- estimation precision and in case of uncorrelated atoms lective magnetic moment of atoms rotates in the pres- is referred to as the projection noise, which may be re- ence of magnetic field to be measured, what again can be garded as an analog of the optical shot noise. An im- seen as an analogue of the MZ transformation on Jˆi in portant difference from the optical case, though, is that Eq. (30). The angle of the atomic magnetic-moment rota- when dealing with atoms we are restricted to consider tion is determined by sending polarized light which due to states of definite particle-number. Thus, there is no exact the Faraday effect is rotated proportionally to the atomic analogue of coherent or squeezed states that we consider magnetic-moment component in the direction of the light in the photonic case. We can therefore regard atomic propagation. In standard scenarios, the ultimate pre- Ramsey interferometry as a special case of the MZ in- cision will be affected by both the atomic projection terferometry with inputs restricted to states of definite noise, due to characteristic uncertainties of the collec- particle-number, discussed in Sec. III.E, and further re- tive magnetic-moment operator for uncorrelated atoms, late the precision of estimating the frequency difference and the light shot noise. The quantum enhancement of to the precision of phase estimation via precision may again be achieved by squeezing the atomic ∆ϕ states (Sewell et al., 2012; Wasilewski et al., 2010) as ∆ω = . (49) well as by using the non-classical states of light (Horrom t et al., 2012), what in both cases allows to go beyond the Beating the projection noise requires the input state of projection and shot-noise limits. the atoms to share some particle entanglement. From an experimental point of view the most promising class of states are the so-called one-axis or two-axis spin-squeezed IV. ESTIMATION THEORY states, which may be realized in BEC and atomic sys- tems interacting with light (Kitagawa and Ueda, 1993; In this section we review the basics of both classical Ma et al., 2011). In fact, these states may be regarded as and quantum estimation theory. We present Fisher In- a definite particle-number analogues of optical squeezed formation and Bayesian approaches to determining the 14

2 optimal estimation strategies and discuss tools particu- ∆ ϕN for all ϕ. Looking for the optimal estimator may larly useful for analysis of optical interferometric setups. be difficult and it may even be the case that there is no single estimator that minimizes the MSE for all ϕ. Still, one may always construct the so-called Cramer- A. Classical parameter estimation Rao Bound (CRB) that lower-bounds the MSE of any unbiased estimator ϕ˜N (see e.g. (Kay, 1993) for a re- The essential question that has been addressed by view): statisticians long before the invention of quantum me- 2 1 chanics is how to most efficiently extract information ∆ ϕ˜N , (53) from a given data set, which is determined by some non- ≥ N F [pϕ] deterministic process (Kay, 1993; Lehmann and Casella, where F is the Fisher Information (FI), and can be ex- 1998). pressed using one of the formulas below: In a typical scenario we are given an N-point data set 2 x = x1, x2,...,xN which is a realization of N indepen- 1 ∂pϕ(x) { } N F [pϕ]= dx = dent identically distributed random variables, X , each ˆ pϕ(x) ∂ϕ distributed according to a common Probability Density   ∂ 2 ∂2 Function (PDF), pϕ(X), that depends on an unknown = ln p = ln p , (54) ∂ϕ ϕ − ∂ϕ2 ϕ parameter ϕ we wish to determine. Our goal is to con- *  +   struct an estimator ϕ˜ (x) that should be interpreted as N The basic intuition is that the bigger the FI is the a function which outputs the most accurate estimate of higher estimation precision may be expected. The FI the parameter ϕ based on a given data set. Importantly, is non-negative and additive for uncorrelated events, so as the estimator ϕ˜N is build on a sample of random data, (1,2) (1) (2) (1,2) it is a random variable itself and the smaller are its fluc- that F pϕ = F pϕ + F pϕ , for pϕ (x1, x2)= tuations around the true value ϕ the better it is. (1) h (2) i h i h i N pϕ (x1) pϕ (x2) and in particular: F pϕ = N F [pϕ], Typically, two approaches to the problem of the choice which can be easily verified using the last expression in of the optimal estimator are undertaken. In the so-called definition (54). The FI is straightforward  to calculate frequentist or classical approach, ϕ is assumed to be a and once an estimator is found that saturates the CRB deterministic variable with an unknown value that, if it is guaranteed to be optimal. In general estimators sat- known, could in principle be stated to any precision. In urating the CRB are called efficient. The sufficient and this case, one of the basic tools in studying optimal esti- necessary condition for efficiency is the following condi- mation strategies is the Fisher information, and hence tion on the PDF and the estimator (Kay, 1993): we will refer to this approach as the Fisher informa- ∂ tion approach. In contrast, when following the Bayesian ln pϕ(x)= N F [pϕ] (˜ϕk(x) ϕ) . (55) paradigm, the estimated parameter is a random variable ∂ϕ − itself that introduces some intrinsic error that accounts An estimator ϕ˜ satisfying the above equality exist only for the lack of knowledge about ϕ we possess prior to for a special class of PDFs belonging to the so called performing the estimation. We describe both approaches exponential family of PDFs, for which: in detail below. a′(ϕ) ln pϕ(x)= a(ϕ)+ b(x)+ c(ϕ)d(x), = ϕ, (56) c′(ϕ) − 1. Fisher Information approach where a(ϕ), c(ϕ) and b(x), d(x) are arbitrary functions and primes denote differentiation over ϕ. In general, In this approach pϕ(X) is regarded as a family of PDFs however, the saturability condition cannot be met. parametrized by ϕ—the parameter to be estimated based Note that in general FI is a function of ϕ, so that on the registered data x. The performance of a given depending on the true value of the parameter, the CRB estimator ϕ˜N (x) is quantified by the Mean Square Error puts weaker or stronger constraints on the minimal MSE. (MSE) deviation from the true value ϕ: Actually, one is not always interested in the optimal esti- mation strategy that is valid globally—for any potential ∆2ϕ˜ = (˜ϕ (x) ϕ)2 = dNx p (x)(˜ϕ (x) ϕ)2 . N N − ˆ ϕ N − value of ϕ—but may want to design a protocol that works D E (51) optimally for ϕ confined to some small parameter range. A desired property for an estimator is that it is unbiased: In this case one can take a local approach and analyze the CRB at a given point ϕ = ϕ0. Formally derivation N of the CRB at a given point requires only a weaker local ϕ˜N = d x pϕ(x)ϕ ˜N (x)= ϕ , (52) h i ˆ unbiasedness condition: so that on average it yields the true parameter value. ∂ ϕ˜N = 1 (57) The optimal unbiased estimator is the one that minimizes ∂ϕ h i ϕ=ϕ0

15 at a given parameter value ϕ0. FI at ϕ0 is a local quantity which, in contrast to Eq. (51), is also averaged over all that depends only on p (X) and ∂ pϕ(X) , asex- the values of the parameter with the Bayesian prior p(ϕ). ϕ ϕ=ϕ0 ∂ϕ | ϕ=ϕ0 Making use of the Bayes theorem we can rewrite the plicitly stated in Eq. (54). As a result, the FI is sensitive above expression in the form to changes of the PDF of the first order in δϕ = ϕ ϕ . − 0 Looking for the optimal locally unbiased estimator at a 2 2 ∆ ϕ˜N = dx p(x) dϕ p(ϕ x)(˜ϕN (x) ϕ) . given point ϕ0 makes sense provided one has a substan- ˆ ˆ | −   tial prior knowledge that the true value of ϕ is close to (61) ϕ0. This may be the case if the data is obtained from From the above formula it is clear that the optimal esti- a well controlled physical system subjected to small ex- mator is the one that minimizes terms in square bracket ternal fluctuations or if some part of the data had been for each x. Hence, we can explicitly derive the form of used for preliminary estimation narrowing the range of the Minimum Mean Squared Error (MMSE) estimator compatible ϕ to a small region around ϕ0. In this case, even if the condition for saturability of the CRB cannot ∂ 2 dϕ p(ϕ x)(˜ϕN (x) ϕ) =0 = be met, it still may be possible to find a locally unbiased ∂ϕ˜Nˆ | − ⇒ estimator which will saturate the CRB at least at a given MMSE ϕ˜N (x)= dϕ p(ϕ x) ϕ= ϕ p(ϕ x) (62) point ϕ0. The explicit form of the estimator can easily ˆ | h i | be derived from Eq. (55) by substituting ϕ = ϕ : 0 which simply corresponds to the average value of the pa- rameter computed with respect to the posterior PDF, ϕ0 1 ∂ ln pϕ(x) ϕ˜N (x)= ϕ0 + . (58) p(ϕ x). The posterior PDF represents the knowledge we NF [pϕ] ∂ϕ ϕ=ϕ0 | possess about the parameter after inferring the informa-

Fortunately, difficulties in saturating the CRB are only tion about it from the sampled data x. The correspond- present in the finite-N regime. In the asymptotic limit of ing MMSE reads: infinitely many repetitions of an experiment, or equiva- 2 2 ∆ ϕ˜ = dx p(x) dϕ p(ϕ x) ϕ ϕ x lently, for an infinitely large sample, N , a particular N ˆ ˆ | − h ip(ϕ ) →∞  |  estimator called the Maximum Likelihood (ML) estima-   tor saturates the CRB (Kay, 1993; Lehmann and Casella, = dx p(x) ∆2ϕ , (63) ˆ p(ϕ x) 1998). The ML estimator formally defined as |

so that it may be interpreted as the variance of the pa- ML ϕ˜N (x) = argmax pϕ(x) (59) rameter ϕ computed again with respect to the posterior ϕ PDF p(ϕ x) that is averaged over all the possible out- | is a function that for a given instance of outcomes, x, comes. outputs the value of parameter for which this data sam- The optimal estimation strategy within the Bayesian ple is the most probable. For finite N the ML estimator approach depends explicitly on the prior PDF assumed. is in general biased, but becomes unbiased asymptoti- If either the prior PDF will be very sensitive to varia- ML tions of ϕ or the physical model will predict the data cally limN ϕN (x) = ϕ and saturates the CR bound →∞2h ML x i to be weakly affected by any parameter changes, so that limN N∆ ϕ˜ ( )= F [pϕ]. 2 →∞ p(x ϕ)p(ϕ) p(ϕ), the minimal ∆ ϕ˜N will be predomi- nantly| determined≈ by the prior distribution p(ϕ) and the 2. Bayesian approach sampled data will have limited effect on the estimation process. Therefore, it is really important in the Bayesian In this approach, the parameter to be estimated, ϕ, approach to choose an appropriate prior PDF that, on is assumed to be a random variable that is distributed one hand, should adequately represent our knowledge according to a prior PDF, p(ϕ), representing the knowl- about the parameter before the estimation, but, on the edge about ϕ one possesses before performing the estima- other, its choice should not dominate the information ob- tion, while p(x ϕ) denotes the conditional probability of tained from the data. obtaining result| x for parameter value ϕ. Notice, a sub- In principle, nothing prevents us to consider more gen- tle change in notation from pϕ(x) in the FI approach to eral cost functions, C(˜ϕ, ϕ), that in some situations may p(x ϕ) in the Bayesian approach reflecting the change in be more suitable than the squared error. The correspond- the| role of ϕ which is a parameter in the FI approach and ing optimal estimator will be the one that minimizes the a random variable in the Bayesian approach. If we stick average cost function to the MSE as a cost function, we say that the estimator C = dϕ dx p(ϕ) p(x ϕ) C (˜ϕ (x), ϕ) . (64) ϕ˜N (x) is optimal if it minimizes the average MSE h i ¨ | N In the context of optical interferometry the estimated ∆2ϕ˜ = dϕ dx p(x ϕ)p(ϕ)(˜ϕ (x) ϕ)2 , (60) N ¨ | N − quantity of interest ϕ is the phase which is a circular

16 parameter, i.e. may be identified with a point on a circle equivalent to a coin-tossing experiment, where we assume or more formally as an element of the circle group U(1) an unfair coin, which “heads” and “tails” occurring with and in particular ϕ ϕ +2nπ. Following (Holevo, 1982) probabilities p and q=1 p respectively. the cost function should≡ respect the parameter topology Probability that n out− of N photons get transmitted and the squared error is clearly not the proper choice. is governed by the binomial distribution We require the cost function to be symmetric, C(˜ϕ, ϕ)= N N n N n C(ϕ, ϕ˜), group invariant, i.e. φ U(1) : C(˜ϕ+φ, ϕ+φ)= p (n)= p (1 p) − . (69) ∀ ∈ p n − C(˜ϕ, ϕ), and periodic, C(ϕ +2nπ, ϕ˜) = C(ϕ, ϕ˜). This   restricts the class of cost functions to: The FI equals F [pN ]= N/[p(1 p)] and hence the CRB p − ∞ imposes a lower bound on the achievable estimation vari- C(˜ϕ, ϕ)= C(δϕ)= c cos[n δϕ] with δϕ =ϕ ˜ ϕ . n − ance: n=0 X (65) p(1 p) ∆2˜p − . (70) Furthermore, we require C(δϕ) must rise monotonically N ≥ N from C(0)=0 at δϕ =0 to some C(π)= Cmax at δϕ = π, so that C′(δϕ) 0, so that the coefficients cn must fulfill Luckily the binomial probability distribution belongs to following constraints:≥ the exponential family of PDFs specified in Eq. (56), and by inspecting saturability condition Eq. (55) it may be ∞ ∞ c =0, ( 1)nc = C , easily checked, that the simple estimator p˜N (n) = n/N n − n max n=0 n=0 saturates the CRB. It is also worth mentioning that the X X optimal estimator also coincides with the ML estimator, ∞ ∞ n2c 0, n2( 1)nc 0, (66) hence in this case the ML estimator is optimal also for n ≤ − n ≥ n=1 n=1 finite N and not only in the asymptotic regime. X X In the context of optical interferometry, we will deal which may be satisfied by imposing n>0 : cn 0 and ∀ ≤ with an analogous situation, where photons are sent into taking cn to decay at least quadratically with n. Lastly, for the sake of compatibility we would like the cost func- one input port of an interferometer and p, q correspond tion to approach the standard variance for small δϕ to the probabilities of detecting a photon in one of the so that C(δϕ) = δϕ2 + O δϕ4 , which is equivalent to two output ports. For a Mach-Zehnder interferometer, 2 see Sec. III, the probabilities depend on the relative phase n∞=1 n cn = 2. In all the Bayesian− estimation
problems considered in delay difference between the arms of an interferometer ϕ: 2 2 Pthis work, we will consider the simplest cost function that p = sin (ϕ/2), q = cos (ϕ/2), which is the actual param- eter of interest. Probability distribution as a function of satisfies all above-mentioned conditions with c0 = c1 =2, : c =0 which reads explicitly: − ϕ then reads: ∀n>1 n N 2 ϕ˜ ϕ N 2n 2(N n) C(˜ϕ, ϕ) = 4 sin − . (67) pϕ (n)= sin(ϕ/2) cos(ϕ/2) − (71) 2 n     Following the same argumentation as described in and the corresponding FI and the CRB take the form Eq. (62) when minimizing the averaged MSE, one may N 2 1 prove that for the above chosen cost function the aver- F pϕ = N, ∆ ϕ˜N . (72) C ≥ N age cost C is minimized if an estimator, ϕ˜N (x), can h i   be found that for any possible data sample x collected Interestingly, FI does not depend on the actual value satisfies the condition ϕ, what suggests that the achievable estimation preci- sion may be independent of the actual parameter value. dϕ p(ϕ x) sin ϕ˜C (x) ϕ =0. (68) ˆ | N − However, for such a parametrization, the CRB satura-
bility condition Eq. (55) does not hold and there is no unbiased estimator saturating the bound. Nevertheless, 3. Example: Transmission coefficient estimation using Eq. (58), we may still write a locally unbiased esti- mator saturating CRB at ϕ : ϕ˜ (n)= ϕ tan(ϕ /2)+ 0 N 0 − 0 In order to illustrate the introduced concepts let us 2n/(N sin ϕ0), which is possible provided sin ϕ0 =0. consider a simple model of parameter estimation, where Since the CRB given in (72) can only be saturated6 a single photon impinges on the beamsplitter with power locally it is worth looking at the MLE which we know transmission and reflectivity equal respectively p and will perform optimally in the asymptotic regime N . q=1 p. The experiment is repeated N times and based Solving Eq. (59) we obtain → ∞ on the− data obtained—number of photons transmitted and the number of photons reflected—the goal is to es- N n ϕ˜N (n)= argmax ln pϕ (n)= 2 arctan (73) timate the transmission coefficient p. This problem is ϕ ± N n r − 17 and in general we notice that there are two equivalent las simplify in the limit N . The optimal es- maxima. This ambiguity is simply the result of invariance timator approaches the ML→ estimator ∞ ϕ˜C (n) 2 N ≈ × of the PDF pϕ(n) with respect to the change ϕ ϕ arctan n/(N n) while the average cost function ap- → − and might have been expected. Hence in practice we proaches C −1/N indicating saturation of the CRB p would need some additional information, possibly com- and confirmingh i ≈ again that in the regime of many experi- ing from prior knowledge or other observations, in order mental repetitions the two discussed approaches coincide. to distinguish between this two cases and be entitled to claim that the MLE saturates the CRB asymptotically for all ϕ. B. Quantum parameter estimation We now analyze the same estimation problem employ- ing the Bayesian approach. Let us first consider the case In a quantum estimation scenario the parameter ϕ where p is the parameter to be estimated and the rel- is encoded in a quantum state ρϕ which is subject to N N evant conditional probability p (n p) = pp (n) is given a quantum measurement Mx yielding measurement re- by Eq. (69). Choosing a flat prior distribution| p(p)=1 sult x with probability pϕ(x) = Tr ρϕMx . The esti- and the mean square as a cost function, we find using mation strategy is complete once an{ estimator} function Eq. (62) and Eq. (63) that the MMSE estimator and the ϕ˜(x) is given ascribing estimated parameters to partic- corresponding minimal averaged MSE equal: ular measurement results. The quantum measurement may be a standard projective von-Neumann measure- n +1 2 1 p˜N (n)= , ∆ p˜N = , (74) ment, MxMx′ = Mxδx x′ , or a generalized measurement N +2 6 (N + 2) , where the measurement operators form a Positive Oper- which may be compared with the previously discussed ator Valued Measure (POVM) with the only constraints FI approach where the optimal estimator was p˜(n) = being Mx 0, dx Mx = 11 (Bengtsson and Zyczkowski, ≥ n/N and the resulting variance when averaged over all 2006; Nielsen and´ Chuang, 2000). Establishing the opti- 2 1 p(p 1) − mal estimation strategy corresponds then not only to the p would yield ∆ p˜N = 0 dp N = 1/(6N). Hence, in the limit Nh ithese´ results converge to the ones most accurate inference of the parameter value from the obtained previously→ ∞ in the FI approach. This is a typical data, but also to a non-trivial optimization over the class situation that in the case of large amount of data the of all POVMs to find the measurement scheme maximiz- two approaches yield equivalent results (van der Vaart, ing the precision. However, as soon as we decide on a par- 2000). ticular measurement scheme, we obtain a model that de- We now switch to ϕ parametrization and consider termines probabilities describing the sampled data pϕ(x) N N and their quantum mechanical origin becomes irrelevant. p (n ϕ)= pϕ (n) as in Eq. (71). Assuming flat prior dis- tribution| p(ϕ)=1/(2π) and the previously introduced Hence, the estimation problem becomes then fully classi- natural cost function in the case of circular parameter cal and all the techniques developed in Sec. IV.A apply. C(˜ϕ, ϕ) = 4 sin2[(ϕ ˜ ϕ)/2], due to ϕ estimation ambi- guity we realize that− the condition± for the optimal esti- mator given in Eq. (68) is satisfied for a trivial estimator 1. Quantum Fisher Information approach ϕ˜C (n)=0 which does not take into account the mea- surement results at all. This can be understood once The problem of determining the optimal measurement we realize that the ambiguity in the sign of estimated scheme for a particular estimation scenario is non-trivial. phase ϕ and the possibility of estimating the phase with Fortunately analogously as in the classical estimation it is the wrong sign is worse than not taking into account relatively easy to obtain useful lower bounds on the min- the measured data at all. In order to obtain a more in- imal MSE. The Quantum Cramér-Rao Bound (QCRB) teresting result we need to consider a subset of possible (Braunstein and Caves, 1994; Helstrom, 1976; Holevo, values of ϕ over which the reconstruction is not ambigu- 1982) is a generalization of the classical CRBs (53), which ous. If we choose ϕ [0, π), and the corresponding prior lower bounds the variance of estimation for all possible p(ϕ)=1/π, Eq. (68)∈ yields the optimal form of the es- locally unbiased estimators and the most general POVM timator and the corresponding minimal cost calculated measurements 1: according to Eq. (64) are as follows: 2 1 2 ∆ ϕN with FQ[ρϕ]= Tr ρϕ L[ρϕ] C 2 ≥ N FQ[ρϕ] ϕ˜N (n) = arctan (75) f(N,n) n (77)o  N C =2 1 1 4+ f(N,n)2 , (76) h i − π(N+1) n=0 ! X p where f(N,n) = (N 2n)(n 1/2)!(N n 1/2)!/n!(N 1 For clarity of notation, in what follows we drop the tilde symbol n)!. Despite its complicated− − form− the− above formu-− when writing estimator variance. 18 where FQ is the Quantum Fisher Information (QFI) est QFI of its purifications Ψϕ while the Hermitian operator L[ρ ] is the so called Sym- | i ϕ 2 metric Logarithmic Derivative (SLD), which can be un- FQ[ρϕ] = min FQ[ Ψϕ ] = 4 min Ψ˙ ϕ Ψ˙ ϕ Ψ˙ ϕ Ψϕ . Ψϕ | i Ψϕ | − | ambiguously defined for any state ρϕ via the relation ρ˙ϕ =   1 D E D (80)E (ρϕL[ρϕ]+L[ρϕ]ρϕ). Crucially, QFI is solely deter- 2 Even though minimization over all purification may still mined by the dependence of ρϕ on the estimated param- eter, and hence allows to analyze parameter sensitivity be challenging, the above formulation may easily be em- of given probe states without considering any particular ployed in deriving upper bounds on QFI by considering measurements nor estimators. Explicitly, the SLD when some class of purifications. Since upper bounds on QFI translate to lower bounds on estimation uncertainty this written in the eigenbasis of ρϕ = i λi(ϕ) ei(ϕ) ei(ϕ) reads: | ih | approach proved useful in deriving bounds in quantum P metrology in presence of decoherence (Escher et al., 2012, 2 e (ϕ) ρ˙ e (ϕ) 2011). L[ρ ]= h i | ϕ | j i e (ϕ) e (ϕ) , (78) ϕ λ (ϕ)+ λ (ϕ) | i ih j | Independently, in (Fujiwara and Imai, 2008) another i,j i j X purification-based QFI definition has been constructed: where the sum is taken over the terms with non-vanishing FQ[ρϕ] = 4 min Ψ˙ ϕ Ψ˙ ϕ . (81) denominator. Analogously to the FI (54), the QFI is an Ψϕ | additive quantity when calculated on product states and D E N Despite apparent difference, Eqs. (80) and (81) are equiv- in particular FQ ρϕ⊗ = N FQ[ρϕ]. Thus the N term in the denominator of Eq. (77) may be equivalently inter- alent and one can prove that any purification minimiz-   preted as the number of independent repetitions of an ing one of them is likewise optimal for the other caus- experiment with a state ρ to form the data sample x ing the second term of Eq. (80) to vanish. Although for ϕ any suboptimal Ψ Eq. (80) must provide a strictly of size N, or a single shot experiment with a multi-party | ϕi N tighter bound on QFI than Eq. (81), the latter defini- state ρ⊗ . ϕ tion, owing to its elegant form allows for a direct and ef- Crucially, as proven in (Braunstein and Caves, 1994; ficient procedure for derivation of the precision bounds in Nagaoka, 2005), there always exist a measurement quantum metrology (Demkowicz-Dobrzański et al., 2012; strategy—a projection measurement in the eigenbasis of Kołodyński and Demkowicz-Dobrzański, 2013). Deriva- the SLD—for which the FI calculated for the resulting tions of the precision bounds using the above two tech- probability distribution equals the QFI, and consequently niques in the context of optical interferometry are dis- the bounds (53) and (77) coincide. Hence the issue of sat- cussed in Sec. VI urability of the QCRB amounts to the problem of satura- For completeness, we list below some other important bility of the corresponding classical CRB. As discussed properties of the QFI. QFI does not increase under a in detail in Sec. IV.A.1, the bound is therefore globally parameter independent quantum channel saturable for a special class of probability distributions belonging to the so called exponential family, and if this FQ(ρϕ) FQ[Λ(ρϕ)], (82) is not the case the saturability is achievable either in ≥ the asymptotic limit of many independent experiments where Λ is an arbitrary completely positive (CP) map. N or in the local approach when one estimates QFI appears in the lowest order expansion of the measure → ∞ of fidelity between two quantum states small fluctuation of the parameter in the vicinity of a F known value ϕ0. 2 For pure states, ρ = ψ ψ , the QFI in Eq. (77) (ρ ,ρ )= Tr √ρ ρ √ρ , (83) ϕ | ϕih ϕ| F 1 2 1 2 1 simplifies to  q  1 2 4 (ρϕ,ρϕ+dϕ)=1 FQ(ρϕ)dϕ + O(dϕ ). (84) 2 d ψϕ F − 4 FQ[ ψϕ ]=4 ψ˙ϕ ψ˙ϕ ψ˙ϕ ψϕ , ψ˙ϕ = | i. | i | − | | i dϕ QFI is convex D E D E  (79)

Yet, for general mixed states, calculation of the QFI in- (i) (i) FQ piρϕ piFQ(ρϕ ), pi =1, pi 0, volves diagonalization of the quantum state ρϕ, in or- ≤ ≥ i ! i i der to calculate the SLD, and becomes tedious for probe X X X (85) states living in highly dimensional Hilbert spaces. which reflects the fact that mixing quantum states can Interestingly, the QFI may be alternatively calcu- only reduce achievable estimation sensitivity. In a com- lated by considering purifications Ψ of a given fam- | ϕi monly encountered case, specifically in the context of op- ily of mixed states on an extended Hilbert space ρϕ = tical interferometry, when the estimated parameter is en- TrE Ψϕ Ψϕ , where by E we denote an ancillary space coded on the state by a unitary needed{| forih the|} purification. It has been proven by Escher iHϕˆ et al. (2011) that the QFI of any ρϕ is equal to the small- ρϕ =UϕρUϕ† ,Uϕ =e− , (86) 19 where Hˆ is the generating “Hamiltonian”, the general for- and leaves us with the problem of minimization C over h i mula for QFI reads: a general POVM Mϕ˜ with standard constraints Mϕ˜ 0, dϕ˜ ≥ ˆ 2 2 2π Mϕ˜ = 11. Note that for clarity we use the normalized 2 ei H ej (λi λj ) dϕ FQ(ρϕ)= |h | | i| − (87) ´measure . λ + λ 2π i,j i j X Provided the problem enjoys the ϕ shift symmetry, so that p(ϕ + ϕ0) = p(ϕ), C(˜ϕ + ϕ0, ϕ + ϕ0) = C(˜ϕ, ϕ), where ei , λi form the eigendecomposition of ρ. Note that in| thisi case the QFI does not depend on the actual it may be shown that one does not loose optimality by value of ϕ. For the pure state estimation case, ρ = ψ ψ , restricting the class of POVM measurements to the QFI is proportional to the variance of Hˆ : | ih | Mϕ˜ = Uϕ˜ Ξ Uϕ†˜, (92) 2 2 FQ( ψϕ )=4∆ H = 4( ψ Hˆ ψ ψ Hˆ ψ ) (88) | i h | | i − h | | i which is a special case of the so-called covariant mea- and the QCRB (77) takes a particular appealing form surements (Bartlett et al., 2007; Chiribella et al., 2005; resembling the form of the energy-time uncertainty rela- Holevo, 1982). If we take flat prior distribution p(ϕ)=1, tion: and the natural cost function C(˜ϕ, ϕ) = 4 sin2[(ϕ ˜ ϕ)/2] − 1 introduced in Sec. IV.A.2, symmetry conditions are ful- ∆2ϕ∆2H . (89) ≥ 4N filled and substituting (92) into (91) we get a simple ex- We conclude the discussion of QFI properties by men- pression: tioning a very recent and elegant result proving that for C = Tr ρϕ Ξ , (93) unitary parameter encodings, QFI is proportional to the h i h iC convex roof of the variance of Hˆ (Tóth and Petz, 2013; dϕ  2 ϕ where ρϕ C=4 2π UϕρUϕ† sin 2 is the final quantum Yu, 2013): state averagedh i with´ the cost function. Looking for the
2 (i) optimal Bayesian strategy now amounts to minimization FQ(ρϕ) = 4 min pi∆ H , (90) (i) pi, ψ of the above quantity over Ξ with the POVM constraints { | i} i dϕ˜ X requiring that Ξ 0, 2π Uϕ˜ΞUϕ†˜ = 11. As shown in where the minimum is performed over all decompositions Sec. V and Sec. VI≥ this´ minimization is indeed possible of ρ = p ψ(i) ψ(i) , and ∆2H(i) denotes the variance i i| ih | for optical interferometric estimation models and allows of Hˆ calculated on ψ(i) . to find the optimal Bayesian strategy and the correspond- P | i ing minimal average cost.

2. Bayesian approach V. QUANTUM LIMITS IN DECOHERENCE-FREE As the quantum mechanical estimation scenario, with a INTERFEROMETRY particular measurement scheme chosen, represents noth- ing but a probabilistic model with outcome probabilities pϕ(x), we may also apply the Bayesian techniques de- |αi ϕ na scribed in Sec. IV.A.2. The quantum element of the prob- lem, however, i.e. minimization of the average cost func- ϕ˜(na, n b) tion over the choice of measurements is in general highly r n non-trivial. Fortunately, provided the problem possesses | i b a particular kind of symmetry it may be solved using the x concept of covariant measurements (Holevo, 1982). |ψiin Mx ϕ˜( ) In the context of optical interferometry, it is sufficient to consider the unitary parameter encoding case as de- FIG. 5 Instead of a particular Mach-Zehnder interferometric fined in (86), where the estimated parameter ϕ will cor- strategy a general quantum interferometric scheme involves a general input probe state |ψiin that is subject to a unitary respond to the phase difference in an interferometer, see phase delay operation Uϕ followed by a general quantum mea- Fig. 5 in Sec. V. Let us denote a general POVM mea- surement (POVM) Mx. Finally, an estimator ϕ˜(x) is used to surement as Mx and the corresponding estimator as ϕ˜. obtain the estimated value of the phase delay. Since we need to minimize the average cost over both the measurements and the estimators it is convenient In order to analyze the ultimate precision bounds in to combine these two elements into one by labeling the interferometry one needs to employ the quantum esti- POVM elements with the estimated values themselves: mation theory introduced in Sec. IV.B. In this section M = dxMxδ[˜ϕ ϕ˜(x)]. The expression for the aver- ϕ˜ − we review the most important results of decoherence-free age cost´ function, (64), takes the form: interferometry leaving the analysis of the impact of de- dϕ dϕ˜ coherence for Sec. VI. This will provide us with precision C = p(ϕ)Tr(U ρU † M ) C (˜ϕ, ϕ) . (91) h i ¨ 2π 2π ϕ ϕ ϕ˜ benchmarks to which we will be able to compare realistic 20 estimation schemes and analyze the reasons for the de- sponding QFI read: parture of practically achievable precisions from idealized scenarios. N ψ in = c n N n , (95) | i n| i| − i Formally, interferometry may be regarded as a chan- n=0 X nel estimation problem where a known state ψ in is sent N N 2 ˆ | i iϕJz 2 2 2 through a quantum channel Uϕ = e− , with an un- FQ =4 cn n cn n . (96) 1  | | − | |  known parameter ϕ, where Jˆ = (a†a b†b) is the z n=0 n=0 ! z 2 − X X component of angular momentum operator introduced   in Sec. III.A. Quantum measurement Mˆ x is performed Let us first consider the situation in which ψ in is a at the output state and the value of ϕ is estimated based state resulting from sending N photons on the| balancedi on the obtained outcome x through an estimator ϕ˜(x), beam-splitter, as discussed in Sec. III.C. This time, how- see Fig. 5. Pursuing either QFI or Bayesian approach ever, we do not insist on a particular measurement nor it is possible to derive bounds on achievable precision estimation scheme, but just want to calculate the cor- that are valid irrespectively of how sophisticated is the responding QCRB on the sensitivity. Written in the measurement-estimation strategy employed and how ex- photon-number basis the state takes the form: otic the input states of light are. We start by consid- ering definite photon-number states using both QFI and N 1 N ψ in = n N n , (97) Bayesian approaches and then move on to discuss issues | i 2N n | i| − i n=0 s that arise when discussing fundamental bounds taking X   into account states of light with indefinite photon num- for which FQ = N and this results in the shot noise bound ber. on precision:

N 0 1 ∆ϕ| i| i . (98) ≥ √N A. Quantum Fisher Information approach Recall, that this bound is saturated with the sim- ple MZ interferometric scheme discussed in Eq. (36), As discussed in detail in Sec. IV, the QFI approach is which is a proof that for the considered probe state particularly well suited to analyze problems where one this measurement-estimation scheme is optimal. From a wants to estimate small deviations of ϕ around a known particle-description point of view, see Sec. II.D, the above value ϕ0, as in this local regime the QCRB, (77), is considered state is a pure product state with no entan- known to be saturable. This is, for example, the case glement between the photons. More generally, the shot of gravitational-wave interferometry in which one sets noise limit, sometimes referred to as the standard quan- the interferometer at the dark fringe and wants to es- tum limit, is valid for all N-photon separable states (Gio- timate small changes in the interference pattern induced vannetti et al., 2011; Pezzé and Smerzi, 2009), and going by the passing wave (LIGO Collaboration, 2011, 2013; beyond this limit requires making use of inter-photon en- Pitkin et al., 2011). tanglement, see Sec. V.D. Let us now investigate general input states, which pos- Since the state of light at the output, ψ = | ϕi sibly may be entangled. Consider the state of the form iϕJˆz e− ψ in, is pure, the QFI may be calculated using NOON = 1 ( N 0 + 0 N ) which is commonly re- | i | i √2 | i| i | i| i the simple formula given in Eq. (79). Realizing that ferred to as the “NOON” state (Bollinger et al., 1996; Lee ψ˙ϕ = iJˆz ψϕ we get the QFI and the corresponding 2 | i − | i et al., 2002). QFI for such state is given by FQ = N and QCRB on the estimation precision: consequently

1 ˆ2 ˆ 2 2 1 ∆ϕ , (99) FQ =4 ψϕ Jz ψϕ ψϕ Jz ψϕ = 4∆ Jz, ∆ϕ . ≥ N h | | i − |h | | i| ≥ 2∆Jz   (94) which is referred to as the Heisenberg limit. In fact, Note that the form of the QCRB above may be regarded NOON state gives the best possible precision as it has the as an analogue of the Heisenberg uncertainty relation for biggest variance of Jˆz among the states with a given pho- phase and angular momentum. ton number N (Bollinger et al., 1996; Giovannetti et al., Clearly, according to the above bound, the optimal 2006). Still, the practical usefulness of the NOON states probe states for interferometry are the ones that max- is doubtful. The difficulty in their preparation increases imize ∆Jz. We fix the total number of photons and dramatically with increasing N, and with present tech- look for N-photon states maximizing ∆Jz. A General nology the experiments are limited to relatively small N, N-photon input state and the explicit form of the corre- e.g. N = 4 (Nagata et al., 2007) or N = 5 (Afek et al., 21

2010). Moreover, even if prepared, their extreme suscep- which coincides with the standard shot-noise limit de- tibility to decoherence with increasing N, see Sec. VI, rived within the QFI approach. makes them hardly useful in any realistic scenario unless Note a subtle difference between the above solution N is restricted to small values. Taking into account that and the solution of the optimal Bayesian transmission experimentally accessible squeezed states of light offer coefficient estimation problem discussed in Sec. IV.A.3 a comparable performance in the decoherence-free sce- with the ϕ parametrization employed. The formulas for nario, see Sec. V.C, and basically optimal asymptotic probabilities in Sec. IV.A.3, can be regarded as arising performance in the presence of decoherence, see Sec. VI, from measuring each of uncorrelated photons leaving the there is not much in favor of the N00N states apart from interferometer independently, while in the present con- their conceptual appeal. siderations we have allowed for arbitrary quantum mea- surements, which are in general collective. Importantly, we account for the adaptive protocols in which a mea- B. Bayesian approach surement on a subsequent photon depends on the re- sults obtained previously (Kołodyński and Demkowicz- Let us now look for the fundamental precision bounds Dobrzański, 2010)—practically these are usually addi- in the Bayesian approach (Berry and Wiseman, 2000; tional controlled phase shifts allowing to keep the setup Hradil et al., 1996). We assume the flat prior distribu- at the optimal operation point (Higgins et al., 2007). tion p(ϕ)=1/2π reflecting our complete initial ignorance This approach is therefore more general and in partic- on the true phase value, and the natural cost function ular, does not suffer from a ϕ ambiguity that forced C(˜ϕ, ϕ) = 4 sin2[(ϕ ˜ ϕ)/2], see Sec. IV.A.2. Thanks to ± − us to restrict the estimated region to [0, π) in order to the phase shift symmetry of the problem, see Sec. IV.B.2, obtain nontrivial results given in Eq. (75). we can restrict the class of measurements to covari- Let us now look for the optimal input states. From ant measurements Mϕ˜ = Uϕ˜ΞUϕ†˜, where Ξ is the seed Eq. (102) it is clear that we may restrict ourselves to measurement operator, to be optimized below. Using real cn. Denoting by c the vector containing the state 2π dϕ 2 iϕ(n m) Eq. (93), and noting that 0 2π 4 sin (ϕ/2)e − = coefficients c , we rewrite the formula for C in a more n h i 2δnm (δn,m 1 +δn,m+1), the´ averaged cost for a general appealing form − − N N-photon input state ψ = n=0 cn n,N n reads: | i | − i T C =2 c Ac, An,n 1 = An 1,n =1, (104) N P h i − − − C =2 2Re cn∗ cn 1Ξn,n 1 . (100) h i − − − from which it is clear that minimizing the cost function is n=1 ! X equivalent to finding the eigenvector with maximal eigen- Because Ξ is Hermitian, the completeness condition value of the matrix A, which has all its entries zero ex- dϕ ˆ ˆ 2π UϕΞUϕ† = 11 implies that Ξnn = 1, while due to cept for its first off-diagonals. This can be done analyti- the´ positive semi-definiteness condition Ξ 0, Ξ ≥ | nm| ≤ cally (Berry and Wiseman, 2000; Luis and Perina, 1996; √ΞnnΞmm = 1. Therefore, The real part in the sub- N Summy and Pegg, 1990) and the optimal state, which we tracted term in Eq. (100) can at most be n=1 cn cn 1 , will refer to as the sine state, together with the resulting i(ξn ξm| ) || − | which will be the case for Ξn,m = e − , where P cost read ξn = arg(cn). This is a legitimate positive semi-definite N operator, as it can be written as Ξ = eN eN , with 2 n +1 N iξn | ih | ψ in = sin π n N n , (105) eN = n=0 e n,N n (Chiribella et al., 2005; | i 2+ N N +2 | i| − i | i | − i n=0 r   Holevo, 1982). Thus, for a given input state the opti- X 2 P π N π mal Bayesian measurement-estimation strategy yields C =2 1 cos →∞ . (106) h i − N +2 ≈ N 2 N    C =2 1 cncn 1 . (101) Again, in the large N limit we may identify the aver- h i − | − | n=1 ! X age cost with the average MSE, so that the asymptotic For the uncorrelated input state (97), the average cost precision reads: reads: N π N 1 ∆ϕ →∞ . (107) 1 − N N N 1 ≈ N C =2 1 →∞ . h i − 2N n n +1 ≈ N n=0 s ! Analogously, as in the QFI approach we arrive at the X    (102) 1/N Heisenberg scaling of precision, but with an addi- Since in the limit of small estimation uncertainty the con- tional constant factor π, reflecting the fact that Bayesian sidered cost function approaches the MSE, we may con- approach is more demanding as it requires the strategy clude that in the limit of large N: to work well under complete prior ignorance of the value N 1 of the estimated phase. Note also that the structure of ∆ϕ →∞ , (103) ≈ √N the optimal states is completely different from the NOON 22 states. In fact, the NOON states are useless in absence phase delay - it is defined with respect to reference beam of any prior knowledge on the phase, since they are in- and there is no such thing like absolute phase delay. variant under 2π/N phase shift, and hence cannot unam- More formally, let ψ ar = α a β r, be the original co- biguously resolve phases differing by this amount. herent state used for| sensingi | thei | phase,i accompanied by a coherent reference beam β . The corresponding out- | iϕ put state reads ψϕ ar = αe− a β r. Now, the phase ϕ C. Indefinite photon-number states plays the role of| thei relative| phasei | shifti between the two modes, with a clear physical interpretation. The com- We now consider a more general class of states with bined phase shift in the two modes, i.e. an operation indefinite photon numbers and look for optimal probe a r iθ(ˆna+ˆnr ) Uθ Uθ = e− has no physical significance as it is states treating the average photon number N as a fixed h i not detectable without an . . . additional reference beam. resource. A state with an indefinite number of photons Hence, before calculating the QFI or any other quantity may posses in general coherences between sectors with determining fundamental precision bounds, one should different total numbers of particles. These coherences first average the state ψ ar over the combined phase shift may in principle improve estimation precision. However, and treat the resulting| densityi matrix as the input probe a photon number measurement performed at the out- state put ports projects the state on one of the sectors and 2π ∞ necessarily all coherences between different total photon dθ a r a r ρ = U U ψ ψ β β U †U † = p ρ . number sectors are destroyed. In order to benefit from ˆ 2π θ θ | iarh | ⊗ | ih | θ θ N N 0 N=0 these coherences, one needs to make use of a more general X (109) scheme such as e.g. homodyne detection, where an ad- This operation destroys all the coherences between sec- ditional phase reference beam is needed, typically called tors with different total photon number, and the resulting the local oscillator, which is being mixed with the signal state is a mixture of states ρN with different total photon light at the detection stage. Usually, the local oscilla- numbers N, appearing with probabilities pN . tor is assumed to be strong, classical field with a well In the absence of a reference beam, i.e. when β =0, the defined phase. In other words, it provides one with ref- above averaging kills all the coherences between terms erence frame with respect to which phase of the signal with different photon numbers in the mode a: beams can be measured (Bartlett et al., 2007; Mølmer, 1997). Thus, it is crucial to explicitly state whether the dθ iθ iθ ρ = αe− αe− 0 0 = reference beam is included in the overall energy budget or ˆ 2π | ih | ⊗ | ih | is it treated as a free resource. Otherwise one my arrive 2n α 2 ∞ α at conflicting statements on the achievable fundamental =e−| | | | n n 0 0 (110) n! | ih | ⊗ | ih | bounds (Jarzyna and Demkowicz-Dobrzański, 2012). n=0 X which results in a state insensitive to phase delays and 1. Role of the reference beam gives F = 0, restoring our physical intuition. On the other hand, F = 4 α 2 obtained previously is recovered in the limit β 2 | | , meaning that reference beam is As an illustrative example, consider an artificial one | | → ∞ mode scheme with input in coherent state ψ = α classical—consists of many more photons than the signal which passes through the phase delay ϕ. Strictly| i speak-| i beams. ing, this is not an interferometer and one may wonder The above averaging prescription, is valid in general how one can possibly get any information on the phase also when label a refers to more than one mode. Consid- by measuring the output state. Still ψ evolves into a ering the standard Mach-Zehnder interferometer fed with | i iϕ a state ψ ab with an indefinite photon numbers and no formally different state ψϕ = Uϕ α = αe− , where | i iϕnˆ | i | i | i additional reference beam available one again needs to Uϕ =e− , and since the corresponding QFI is non-zero perform the averaging over a common phase shift. If this 2 2 2 FQ =4 α nˆ α α nˆ α =4 α , (108) is not done, one may obtain conflicting results on e.g. h | | i − |h | | i| | | QFI for seemingly equivalent phase shift operations such
iˆnaϕ i(ˆna nˆb)ϕ/2 it is in principle possible to draw some information on as Uϕ = e− or Uϕ′ = e− − . The reason is the phase by measuring ψϕ . Clearly, the measure- that, without the common phase averaging, one implic- ment required cannot be a| directi photon-number mea- itly assumes the existence of a strong external classical surement, and an additional phase reference beam needs phase reference to with respect to which the phase shifts to be mixed with the state before sending the light to the are defined. In particular, Uϕ assumes that the second detectors. In a fair approach one should include the ref- mode is perfectly locked with the external reference beam erence beam into the setup and assume that whole state and is not affected by the phase shift, whereas Uϕ′ as- of signal+reference beam is averaged over a global un- sumes that there are exactly opposite phase shifts in the defined phase. This formalizes the notion of the relative two modes with respect to the reference. Different choices 23 of „phase-shift distribution” may lead to a factor of 2 or we get FQ = N N . Therefore, while keeping N fixed even factor of 4 discrepancies in the reported QFIs in ap- we may increaseh Ni arbitrarily and in principleh i reach parently equivalent optical phase estimation schemes— FQ = , suggesting the possibility of arbitrary good see (Jarzyna and Demkowicz-Dobrzański, 2012) for fur- sensing∞ precisions (Rivas and Luis, 2012; Zhang et al., ther discussion and compare with some of the results 2013). Note in particular, that a naive generalization that were obtained without the averaging performed (Joo of the Heisenberg limit to ∆ϕ 1 , does not hold, ≥ N et al., 2011; Spagnolo et al., 2012). and the strategies beating this boundh i are referred to as One can also understand why ignoring the need for sub-Heisenberg strategies (Anisimov et al., 2010). A uni- a reference beam may result in underestimating the re- versally valid bound may be written as ∆ϕ 1/ Nˆ 2 quired energy resources. Consider a singe mode state ≥ h i N (Hofmann, 2009), but the question remains, whetherq the with an indefinite photon number ψ = n=0 cn n iϕnˆ | i | i bound is saturable, and in particular does quantum me- evolving under e− and note that, from the phase sens- ing point of view, this situation is formallyP equivalent chanics indeed allows for practically useful estimation to a two-mode state with a definite photon number N: protocols leading to the sub-Heisenberg scaling of pre- N cision. Closer investigations of that problem proves such ψ = c n N n , evolving under e iϕnˆa . Still, ab n=0 n − et al. |thei average photon| i| number− i consumed in the one mode hypothesis to be false (Berry , 2012; Giovannetti and P N 2 Maccone, 2012; Tsang, 2012; Zwierz et al., 2010). In prin- case is N = n=0 cn n and is in general smaller than N. h i | | ciple we may achieve the sub-Heisenberg precision in the P local estimation regime but in order for the local strat- egy to be valid, we should know the value of estimated 2. Optimal indefinite photon number strategies parameter with prior precision of the same order as the one we want to obtain, what makes the utility of the pro- Looking for the optimal states with fixed average pho- cedure questionable. Actually, if no such assumption on ton number is in general more difficult than in the definite the priori knowledge is made, the Heisenberg scaling in the form ∆ϕ = const/ N is the best possible scaling of photon-number case. Still, if we agree with the above- h i advocated approach to average all the input states over a precision. common phase-shift transformation as in Eq. (109), then This claim can also be confirmed within the Bayesian the resulting state is a probabilistic mixture of definite approach with flat prior phase distribution. For large photon number states. Intuitively, it is then clear that, N the minimal Bayesian cost behaves like C(ρN ) = 2 2 h i instead of sending the considered averaged state, it is π /N , see Eq. (105). Since this function is convex, tak- more advantageous to have information which particular ing convex combinations of the cost for two different total component ρ of the mixture is being sent. This would photon numbers N1, N2, such that p1N1 + p2N2 = N N 2 2 h i allow to adjust the measurement-estimation procedure to will yield the cost higher than π / N , and the corre- sponding uncertainty ∆ϕ π/ N h, indicatingi the uni- a given component and improve the overall performance. ≥ h i This intuition is reflected by the properties of both the versal validity of the Heisenberg scaling of precision. QFI and the Bayesian cost, which are respectively convex and concave quantities (Helstrom, 1976): 3. Gaussian states

F p ρ p F (ρ ), (111) From a practical point of view, rather than looking for Q N N ≤ N Q N N ! N the optimal indefinite photon number states for interfer- X X ometry it is more important to analyze experimentally C p ρ p C(ρ ) . (112) accessible Gaussian states. The paradigmatic example of N N ≥ N h N i * N !+ N a Gaussian state applied in quantum enhanced interfer- X X ometry is the two mode state α r —coherent state in This, however, implies that knowing the solution for the mode a and squeezed vacuum in| modei| i b. We have already optimal definite photon number probe states, by adjust- discussed this example in Sec. III.D, and calculated the ing the probabilities pN with which different optimal ρN precision for a simple measurement-estimation scheme. are being sent, we may determine the optimal strategies For such states, sent through a fifty-fifty beam splitter, with the average photon-number fixed. quantum Fisher information can been calculated explic- Taking the QFI approach for a moment, we recall that itly (Jarzyna and Demkowicz-Dobrzański, 2012; Ono and the optimal N-photon state, the NOON state, yields Hofmann, 2010; Pezzé and Smerzi, 2008): 2 FQ(ρN ) = N . Let us consider a strategy where a vac- 2 2r 2 uum state and the NOON state are sent with probabili- FQ = α e + sinh r. (113) ties 1 p and p respectively, with the constraint on the | | average− photon number pN = N . The corresponding For the extreme cases α 2 = 0, sinh2 r = N and QFI reads F = (1 p) 0+pN 2.h Substitutingi p = N /N α 2 = N , sinh2 r = 0|this| formula gives F h =i N , Q − · h i | | h i Q h i 24 implying the shot noise scaling. Most importantly, op- 2012). Hence, assuming the path-symmetry the optimal timization of Eq. (113) over α and r with constraint Gaussian state is given by r r with sinh2 r = N /2— α 2 + sinh2 r = N gives asymptotically the Heisen- two squeezed vacuums send| i| intoi the input portsh ofi the |berg| limit ∆ϕ h1/iN , making this strategy as good interferometer—and its corresponding QFI leads to the as the NOON -one∼ forh largei number of photons. More- QCRB over, this bound on precision can be saturated by esti- 1 1 mation strategies based on photon-number (Pezzé and ∆ϕ (117) et al. ≥ N ( N + 2) ≈ N Smerzi, 2008; Seshadreesan , 2011) or homodyne h i h i h i (D’Ariano et al., 1995) measurements. This also proves The state also achievesp the Heisenberg limit for a large that a simple measurement-estimation strategy discussed number of photons in the setup but it does not require in Sec. III.D which yielded 1/ N 3/4 scaling of precision any external phase reference. It is also worth noting is not optimal. Unlike the simpleh i interferometric scheme that this state, while being mode-separable is particle- where it was optimal to dedicate approximately N entangled and is feasible to prepare with current tech- photons to the squeezed beam, from the QFI pointh ofi p nology for moderate squeezing strengths. However, the view it is optimal to equally divide the number of pho- enhancement over optimal squeezed-coherent strategy is tons between the coherent and the squeezed beam. rather small and vanish for large number of photons. More generally, finding the fundamental limit on pre- Precisions obtained in squeezed-coherent and squeezed- cision achievable with general Gaussian states, requires squeezed scenarios are depicted in Fig. 6. optimization of the QFI or the Bayesian average cost function over general two-mode Gaussian input states, specified by the covariance matrix and the vector of first 1 ∆ϕ ≥ moments, see Sec. II.B. For the decoherence-free case this hNi was done in Pinel et al. (2012, 2013). Crucial observation is that for pure states, the overlap between two M-mode Gaussian states ψϕ and ψϕ +dϕ is given by (up to the second order in d| ϕ)i | i 1 ∆ϕ ≤ hNi dϕ2 dW (z) 2 ψ ψ 2 =1 2(4π)2 d2M z |h ϕ| ϕ+dϕi| − 4 ˆ dϕ   ! (114) hNi where W (x) is the Wigner function (5) of state ψϕ . 2 1 2 | i FIG. 6 Limits on precision obtained within QFI approach Thus, because ψϕ ψϕ+dϕ =1 4 FQdϕ we may write that |h | i| − when using two optimally squeezed states in both modes |ri|ri (black, solid), coherent and squeezed-vacuum states |αi|ri 1 (black, dashed). For comparison precision achievable with dW (z) 2 − 2 ∆ϕ 2(4π)2 d2M z . (115) simple coherent and squeezed vacuum MZ interferometric ≥ ˆ dϕ scheme discussed in Sec. III.D is also depicted (gray, dashed).   ! In terms of the covariance matrix σ and the first moments z the formula takes an explicit form: One can also study Gaussian states within the h i Bayesian framework. Optimal seed operator can be eas- 1 T 2 − 2 ily generalized from the definite photon number case to d z 1 d z 1 dσ 1 ∆ϕ h i σ− h i + Tr σ− . Ξ= ∞ e e . Conceptually, the whole treatment ≥ dϕ dϕ 4 dϕ N=0 | N ih N |   !! is the same as in the definite photon number case. How- (116) ever,P the expressions and calculations are very involved Formal optimization of the above equation was done by and will not be presented here. Pinel et al. (2012), however, the result was a one-mode squeezed-vacuum state, which in order to carry phase in- formation needs to be assisted by a reference beam. Un- D. Role of entanglement fortunately, performing a common phase-averaging pro- cedure described in Sec. V.C.1 in order to calculate the The issue of entanglement is crucial in quantum in- precision in the absence of additional phase reference de- terferometry as it is known that only entangled states stroys the Gaussian structure of the state and makes the can beat the shot noise scaling (Pezzé and Smerzi, 2009). optimization intractable. Luckily, for path symmetric This statement is sometimes questioned, pointing out states, i.e. the states invariant under the exchange of the example of the squeezed+coherent light strategy, interferometer arms, the phase averaging procedure does where the interferometer is fed with seemingly unentan- not affect the QFI (Jarzyna and Demkowicz-Dobrzański, gled α r input state. The reason of confusion is the | i| i 25 conflict of notions of mode and particle entanglement. As discussed in detail in Sec. II.E, the two notions are not compatible, and there are states which are particle entangled, while having no mode entanglement and vice versa. In the context of interferometry it is the particle entanglement that is the source of quantum enhancement of precision. In order to avoid criticism based on the ground of fundamental indistinguishability of particles FIG. 7 A multi-pass interferometric protocol. A standard and therefore a questionable physical content of the dis- phase delay element is replaced by a setup which makes the tinguishable particle-based entanglement picture on the beam to pass through the phase delay multiple number of fundamental level (Benatti et al., 2010), we should stress times. that when considering models involving indistinguishable particles one should regard this statement as a formal (but still a meaningful and useful) criterion where the the light bounce back and forth through the phase de- particles are treated as formally distinguishable as de- lay element many times so that the phase delay signal scribed in Sec. II.D. is enhanced as shown in Fig. 7. This method is used in To see this, let us consider first a separable input state GEO600 experiment (LIGO Collaboration, 2011), where of N photons of the form ρ = ρ1 ρN , where ρi the light bounces twice in each of the sensing arms, mak- denotes the state of the i-th photon.⊗···⊗ Since the phase ing the detector as sensitive as the one with arms twice shift evolution affects each of the photons independently as long. Up to some approximation, one can also view N N the Fabry-Perrot cavities placed on top of the Mach- ρϕ = Uϕ⊗ ρUϕ⊗ † and the QFI is additive on product states we may write: Zehnder design as devices forcing each of the photon to pass multiple-times through the arms of the interferome- N ter and acquire a multiple of the phase delay (Berry et al., F (ρ)= F (ρ ) NF (ρ ) (118) Q Q i ≤ Q max 2009; Demkowicz-Dobrzański et al., 2013). i=1 X Consider a single photon in the state (after the first 1 where ρ denotes state from the set ρ for beam splitter) ψ in = ( 01 + 10 ). After passing max { i}i=1,...,N | i √2 | i | i which QFI takes the largest value. But for a one photon through the phase shift N times it evolves into ψϕ = 1 iNϕ | i state, the maximum value of QFI is equal to 1, so ( 01 +e− 10 ). The phase is acquired N times √2 | i | i 1 faster compared with a single pass case, mimicking the F (ρ) N, ∆ϕ . (119) behavior of a single pass experiment with a NOON state. ≤ ≥ √ N Hence, the precision may in principle be improved by a (i) (i) factor of N. Treating the number of single photon passes For general separable states ρ = i piρ1 ρN it is sufficient to use the convexity of QFI, together⊗···⊗ with as a resource, it has been demonstrated experimentally Eq. (119) to obtain the same conclusion.P Above results (Higgins et al., 2007) that in the absence of noise such imply that QFI, or precision, can be interpreted as a a device can indeed achieve the Heisenberg scaling with- particle-entanglement witness, i.e. all states that give pre- out resorting to entanglement and efficiency of various cision scaling better than the shot noise must be particle- multi-pass protocols has been analyzed in detail in (Berry entangled (Hyllus et al., 2012; Tóth, 2012). et al., 2009). This is not to say, that all quantum strate- The seemingly unentangled state α r when pro- gies are formally equivalent to single-photon multi-pass jected on the definite total photon number| i| sector,i indeed strategies. As will be discussed in the next section, the contains particle entanglement as was demonstrated in NOON states are extremely susceptible to decoherence, Sec. II.E. This fact should be viewed as the fundamental in particular loss, and this property is shared by the source of its ability for performing quantum-enhanced multi-pass strategies. Other quantum strategies prove sensing. It is also worth stressing, that unlike mode more advantageous in this case, and they do not have entanglement, particle entanglement is invariant under a simple multi-pass equivalent (Demkowicz-Dobrzański, passive optical transformation like beam splitters, delay 2010; Kaftal and Demkowicz-Dobrzanski, 2014). lines and mirrors, which makes it a sensible quantity to be treated as a resource for quantum enhanced interfer- ometry. VI. QUANTUM LIMITS IN REALISTIC INTERFEROMETRY

E. Multi-pass protocols In this section we revisit the ultimate limits on preci- sion derived in Sec. V taking into account realistic noise A common method, used in e.g. gravitational wave effects. We study three decoherence processes that are detectors, to improve interferometric precision is to let typically taken into account when discussing imperfec- 26 tions in interferometric setups. We consider the effects (i) (ii) (iii) measurement of phase diffusion, photonic losses and the impact of im- a ηa perfect visibility, see Fig. 8. In order to establish the ul- timate limits on the estimation performance, we first an- alyze the above three decoherence models using the QFI perspective and secondly compare the bounds obtained b ηb with the ones derived within the Bayesian approach. For the most part of this section, we will consider input |ψiin states with definite number of photons, N, so that

N ρin = ψ in ψ , ψ in = c n N n . (120) | i h | | i n| i| − i FIG. 8 Schematic description of the decoherence processes n=0 X discussed that affect the performance of an optical interferom- Similarly as in Sec. V, this will again be sufficient to eter: (i) phase diffusion representing stochastic fluctuations draw conclusions also on the performance of indefinite of the estimated phase delay, (ii) losses in the respective a/b photon number states, which will be discussed in detail arms represented by fictitious beam splitters with 0≤ηa/b ≤1 transmission coefficients, (iii) imperfect visibility indicated by in Sec. VI.D.1. a mode mismatch of the beams interfering at the output beam In what follows it will sometimes prove useful to switch splitter from mode to particle description, see Sec. II.D, and treat photons formally as distinguishable particles but prepared in a symmetrized state. This approach is il- (i) (ii), (iii) lustrated in Fig. 9 where each photon is represented by U Λ a different horizontal line, and travels through a phase ϕ (i) iϕσˆ(i)/2 (i) encoding transformation Uϕ = e− z , where σˆz is Uϕ Λ |ψiin a z Pauli matrix acting on the i-th qubit. The com- Λ ρϕ bined phase encoding operation is a simple tensor prod- (i) N iϕ σˆ iϕJˆz uct U ⊗ =e− i z =e− , recovering the familiar ϕ P formula but with Jˆz being now interpreted as the z com- Uϕ Λ ponent of the total angular momentum which is the sum of individual angular momenta. The photons are then subject to decoherence that acts in either correlated or uncorrelated manner. In the case of phase diffusion (i) FIG. 9 General metrological scheme in case of photons being the decoherence has a collective character since each of treated as formally distinguishable particles. Each photon travels through a phase encoding transformation Uϕ. Apart the photons experiences the same fluctuation of the phase from that all photons are subject to either correlated (i) being sensed, while in the case of loss (ii) and imperfect (phase diffusion), or uncorrelated (ii), (iii) (loss, imperfect visibility (iii) the decoherence map has a tensor structure visibility) decoherence process. N Λ⊗ reflecting the fact that it affects each photon inde- pendently. In the latter case of independent decoherence models the overall state evolution is uncorrelated and where K are called the Kraus operators. may be written as: i N Effects of decoherence inside an interferometer are ρ =Λ⊗ (ρin), Λ ( )=Λ(U U † ). (121) ϕ ϕ ϕ · ϕ · ϕ taken into account by replacing the unitary transforma- tion Uϕ describing the action of the ideal interferometer, see Sec. V, with its noisy variant Λϕ: A. Decoherence models

In general, decoherence is a consequence of the uncon- ρϕ =Λϕ(ρin)= KiUϕρinUϕ† Ki† . (123) trolled interactions of a quantum system with the en- i ! vironment. Provided the system is initially decoupled X from the environment, the general evolution of a quan- The formal structure of the above formula corresponds to tum system interacting with the environment mathemat- a situation in which decoherence happens after the uni- ically corresponds to a completely positive trace preserv- tary phase encoding. This of course might not be true in ing map Λ. Every Λ can be written using the Kraus general. Still, for all the models considered in this review representation (Nielsen and Chuang, 2000): the decoherence part and the unitary part commute and therefore the order in which they are written is a matter ρout = Λ(ρin)= KiρinKi†, Ki†Ki = 11, (122) i i of convenience. X X 27

1. Phase diffusion 2009):

la lb Phase diffusion, also termed as the collective dephasing (1 ηa) † (1 ηb) ˆ†ˆ aˆ aˆ la b bˆlb Kla,lb = − ηa aˆ − ηb b (126) or the phase noise represents the effect of fluctuation of s la! s lb! the estimated phase delay ϕ. Such effect may be caused by any process that stochastically varies the effective op- where the values of index la/b corresponds to the number tical lengths traveled by the photons, such as thermal de- of photons lost in mode a/b respectively. For a general N- formations or the micro-motions of the optical elements. photon input state of the form (120), the density matrix We model the optical interferometry in the presence of representing the output state of the lossy interferometer phase diffusion process by the following map: reads

ρ = Λ(U ρinU † )= ∞ ϕ ϕ ϕ ρ =Λ (ρ )= dφp (φ) U ρ U † (124) N N N ′ ϕ ϕ in ˆ ϕ φ in φ − = pla,lb ξla,lb (ϕ) ξla,lb (ϕ) , (127) −∞ ′ | ih | N =0 la=0 M (l =NX−N′−l ) where the phase delay is a random variable φ distributed b a according to probability distribution pϕ(φ). Note that N 2 (la,lb) where pl ,l == cn bn is the binomially dis- the above form is actually the Kraus representation of a b n=0 | | tributed probability of losing la and lb photons in arms the map Λϕ with Kraus operators Kφ = pϕ(φ)Uφ. In a and b respectively,P with case p (φ) is a Gaussian distribution with variance Γ and ϕ p the mean equal to the estimated parameter ϕ, pϕ(φ)= (la,lb) n n la la N n N n lb lb 2 bn = ηa − (1 ηa) − ηb − − (1 ηb) , 1 −(φ−ϕ) l − l − e 2Γ , the output state reads explicitly (Genoni  a  b  √2πΓ (128) et al., 2011): while the corresponding conditional pure states read:

N N lb inϕ Γ 2 − ⋆ (n m) i(n m)ϕ cn e− (la,lb) ρ = c c e− 2 − e− − n,N n m,N m , ξ (ϕ) = b n l ,N n l . ϕ n m | − ih − | la,lb n a b n,m=0 | i √pla,lb | − − − i X n=la q (125) X (129) where cn are parameters of the input state given as in The direct sum in Eq. (127) indicates that the output (120). The above equation indicates that due to the states of different total number of surviving photons, N ′, phase diffusion the off-diagonal elements of ρϕ are ex- belong to orthogonal subspaces, which in principle could ponentially suppressed at a rate increasing in the anti- be distinguished by a non-demolition, photon-number diagonal directions. counting measurement. In the particle-approach when photons are considered as formally distinguishable particles, the loss process acts 2. Photonic losses on each of the photons independently, see Fig. 9, so that the overall decoherence process has a tensor product N In the lossy interferometer scenario, the fictitious structure Λ⊗ , with Λ being a single particle loss trans- beam-splitters introduced in the interferometer arms formation. At the input stage, each photon occupies a with respective power transmission coefficient η ac- two-dimensional Hilbert space spanned by vectors a , b a/b | i | i count for the probability of photons to leak out. Such a representing the photon traveling in the mode a/b respec- loss model is relatively general, as due to the commutativ- tively. In order to describe loss, however, and formally ity of the noise with the phase accumulation (Demkowicz- keep the number of particles constant, one needs to intro- Dobrzanski et al., 2009), it accounts for the photonic duce a third photonic state at the output, vac , repre- | i losses happening at any stage of the phase sensing pro- senting the state of the photon being lost. Then formally, cess. Moreover, losses at the detection as well as the Λ maps states from the input two-dimensional Hilbert preparation stages can be moved inside the interferom- space to the output three-dimensional Hilbert space, and eter provided they are equal in both arms. This makes can be fully specified by means of the Kraus representa- 3 the model applicable in typical experimental realization tion, Λ(ρ) = i=1 KiρKi†, where K1, K2, K3 are given of quantum enhanced interferometry (Kacprowicz et al., respectively by the following matrices: P 2010; Spagnolo et al., 2012; Vitelli et al., 2010), and √ηa 0 0 0 0 0 most notably, when analyzing bounds on quantum en- 0 √ηb , 0 0 , 0 0 . (130) hancement in gravitational-wave detectors (Demkowicz-  0 0  √1 η 0   0 √1 η  Dobrzański et al., 2013). − a − b       Loss decoherence map Λ may be formally described Intuitively, the above Kraus operators account for no using the following set of Kraus operators (Dorner et al., photon loss, photon loss in mode a and photon loss in 28 mode b respectively. When applied to symmetrized in- Note that similarly to the loss model we have modeled put states, this loss model yields output states the noise with the use of fictitious beam-splitters to vi- sualize the effects of decoherence. Now, as we know that N N N N ρϕ =Λ⊗ (Uϕ⊗ ρinUϕ†⊗ )=Λϕ⊗ (ρin), (131) a beam-splitter acts on the photons contained in its two- mode input state in an uncorrelated manner, the effective N equivalent to the ones given in Eq. (127), where Uϕ is a map on the full N-photon input state is Λ⊗ . In case of iϕσˆz/2 single photon phase shift operation Uϕ =e− . atomic systems, this would be a typical local dephasing model describing uncorrelated loss of coherence between the two relevant atomic levels (Huelga et al., 1997). Still, 3. Imperfect visibility there is a substantial difference from the loss models as the dephased photons are assumed to remain within the In real-life optical interferometric experiments, it is al- spatially confined beams of the interferometer arms. ways the case that the light beams employed do not con- We can relate the two models by a simple observation, tribute completely to the interference pattern. Due to namely that if the photons lost in the loss model with spatiotemporal or polarization mode-mismatch, caused ηa = ηb = η were incoherently injected back into the for example by imperfect wave-packet preparation or arms of the interferometer, we would recover the local misalignment in the optical elements, the visibility of dephasing model with the corresponding parameter η. the interference pattern is diminished (Leonhardt, 1997; It should therefore come as no surprise, when we derive Loudon, 2000). This effect may be formally described as bounds on precisions for the two models in Sec. VI.B.1 an effective loss of coherence between the two arms a and and Sec. VI.B.2, that for the same η the local dephas- b of an interferometer. ing (imperfect visibility) model implies more stringent Consider a single photon in a superposition state of bounds on achievable precision than the loss model. In- being in modes a and b respectively: ψ = α a + β b . | i | i | i tuitively, it is better to get rid of the photons that lost If other degrees of freedom such as e.g. polarization, their coherence and do not carry information about the temporal profile etc. were identical for the two states phase, rather than to inject them back into the setup. a , b , we could formally write (α a + β b ) 0 , where X The structure of the output state ρϕ is more complex |0i | denotesi the common state of| additionali | i | degreesi of | iX than in the case of phase diffusion and loss models. This freedom. Loss of coherence may be formally described as is because the local dephasing noise not only transforms the transformation of the state ψ 0 into | i| iX the input state into a mixed state, but due to tracing out some degrees of freedom, the output state

Ψ = α √η a 0 X + 1 η a + X + | i | i| i − | i| i N   ρϕ =Λϕ⊗ (ρin)= + β √η b 0p + 1 η b , (132) | i| iX − | i|−iX 1 N N  p  = Ki1 KiN Uϕ⊗ ρinUϕ⊗ †Ki† Ki† ⊗···⊗ 1 ⊗···⊗ N where + X , X are states orthogonal to 0 X , corre- i ,...,i =0 | i |−i | i 1 XN sponding to photon traveling in e.g. orthogonal transver- (135) sal spacial modes as depicted in Fig. 8 (iii), in which case parameter η can be interpreted as transmission of ficti- is no longer supported on the bosonic space spanned by tious beam splitters that split the light into two orthog- the fully symmetric states n N n . This makes it im- onal modes. Assuming we do not control the additional possible to use the mode-description| i| − i in characterization degrees of freedom the effective state of the photon is ob- of the process. Even though it is possible to write down tained by tracing out the above state over X, obtaining the explicit form of the above state (Fröwis et al., 2014; the effective single-photon decoherence map: Jarzyna and Demkowicz-Dobrzanski, 2014) decomposing the state into SU(2) irreducible subspaces, we will not 2 α αβ∗η present it here for the sake of conciseness, especially that Λ( ψ ψ )= TrX ( Ψ Ψ )= | | 2 , (133) | ih | | ih | α∗βη β it will not be needed in derivation of the fundamental  | |  bounds. where the off-diagonal terms responsible for coherence, are attenuated by coefficient η, what corresponds to the standard dephasing map (Nielsen and Chuang, 2000). B. Bounds in the QFI approach Written using the Kraus representation, the above map reads: Once we have the formula for the output states 2 ρϕ, given a particular decoherence model, we may use 1+ η 1 η Λ(ρ)= K ρK†,K = 11,K = − σ . Eq. (77) to calculate QFI, FQ(ρϕ), which sets the limit on i i 1 2 2 2 z i=1 r r practically achievable precision of estimation of ϕ. In or- X (134) der to obtain the fundamental precision bound for a given 29 number N of photons used, we need to maximize the re- sion achievable with a state of N uncorrelated photons sulting FQ over input states ψ in, which will in general be as given in Eq. (97). The ratio between this quantities very different from the NOON| i states which maximize the bounds the amount of quantum-precision enhancement QFI in the decoherence-free case. This is due to the fact that can be achieved with the help of quantum correla- that the NOON states are extremely susceptible to deco- tions present in the input state of N photons. herence, as loss of e.g. a single photon makes the state completely useless for phase sensing. Unfortunately, for mixed states, the computation of the QFI requires in gen- 1. Imperfect visibility eral performing the eigenvalue decomposition of ρϕ and such a minimization ceases to be effective for large N. The fundamental QFI bound on precision in case of Therefore, while it is relatively easy to obtain numeri- imperfect visibility or equivalently the local dephasing cal bounds on precision and the form of optimal states model has been derived in (Demkowicz-Dobrzański et al., for moderate N (Demkowicz-Dobrzański, 2010; Dorner 2012; Escher et al., 2011; Knysh et al., 2014) and reads: et al., 2009; Huelga et al., 1997), going to the large N 1 η2 1 regime poses a huge numerical challenge, making deter- ∆ϕ − , (136) mination of the asymptotic bounds for N with ≥ s η2 √N brute force optimization methods infeasible. → ∞ where η is the dephasing parameter, see Sec. VI.A.3. Over the past few years, elegant methods have been For the optimal uncorrelated input state, ψin = [( a + proposed that allow to circumvent the above men- | i | i √ N 2 tioned difficulties and obtain explicit bounds on precision b )/ 2]⊗ we get ∆ϕ =1/ η N, and hence the quan- tum| i precision enhancement which is the ratio of the based on QFI for arbitrary N, and in particular grasp p the asymptotic precision scaling (Demkowicz-Dobrzański bound on precision achievable for the optimal strat- et al., 2012; Escher et al., 2011; Knysh et al., 2014). egy and the precision for the uncorrelated strategy is bounded by a constant factor of 1 η2. These methods include: the minimization over channel − purifications method (Escher et al., 2011) which is ap- p plicable in general but requires some educated guess to a. Classical simulation method The derivation of the for- obtain a useful bound, as well as classical and quantum mula (136) presented below makes use of the classical simulation methods (Demkowicz-Dobrzański et al., 2012) simulation method (Demkowicz-Dobrzański et al., 2012), which are applicable when the noise acts in an uncorre- which requires viewing the quantum channel represent- lated manner on the probes, but have an advantage of ing the action of the interferometer from a geometrical being explicit and convey some additional physical intu- perspective. The set of all physical quantum channels, Λ: ition on the bounds derived. Description of the newly ( ) ( ), that map between density matrices published method (Knysh et al., 2014) which is based in out LdefinedH on→ theL H input/ouput Hilbert spaces ( ) con- on continuous approximation of the probe states and the in/out stitutes a convex set (Bengtsson and Zyczkowski,H 2006). calculus of variations is beyond the scope of this review. This is to say that given any two quantum channels We will present the methods by applying them directly Λ , Λ , their convex combination Λ = pΛ + (1 p)Λ , to interferometry with each of the decoherence models in- 1 2 1 2 0 p 1 is also a legitimate quantum channel.− Physi- troduced above. We invert the order of presentation of cally≤ Λ≤corresponds to a quantum evolution that is equiv- the bounds for the decoherence models compared with alent to a random application of Λ , Λ transformations the order in Sec. VI.A, as this will allow us to discuss 1 2 with probabilities p, 1 p respectively. the methods in the order of increasing complexity. The As derived in Sec. VI.A.3,− within the imperfect visibil- simplest of the methods, the classical simulation, will be ity (local dephasing) decoherence model: ρ =Λ N (ρ ), applied to the imperfect visibility model, while the quan- ϕ ϕ⊗ in and hence the relevant quantum channel, has a simple tum simulation will be discussed in the context of loss. tensor structure. Consider a single-photon channel Λ , Finally, minimization over channel purification method ϕ which ϕ-dependence we may depict as a trajectory within will be described in the context of the phase diffusion the set of all single-photon quantum maps, see Fig. 10. model, to which classical and quantum simulation meth- The question of sensing the parameter ϕ can now be ods are not applicable due to noise correlations. We translated to the question of determining where on the should note that methods of (Escher et al., 2011; Knysh trajectory a given quantum channel Λ lies. et al., 2014) can also be successfully applied to uncorre- ϕ Consider a local classical simulation (CS) of a quantum lated noise models. Still, classical and quantum simula- channel trajectory Λ in the vicinity of a given point tion approaches are more intuitive and that is why we ϕ ϕ , ϕ = ϕ + δϕ (Demkowicz-Dobrzański et al., 2012; present derivations based on them even though the other 0 0 Matsumoto, 2010), techniques yield equivalent bounds. In order to appreciate the significance of the derived 2 Λϕ[̺]= pϕ(x) Πx[̺]+ O δϕ , (137) bounds, we will always compare them with the preci- x X
30

from XN , i.e. ϕ XN ϕ˜, can perform only better than the scheme where→ the→ information about ϕ is firstly en- coded into the quantum channel which acts on the probe state and afterwards decoded from the measurement re- sults performed on the output state. This way, we may always construct a classically scaling lower bound on the precision, or equivalently an upper bound on the QFI of ρϕ (135):

N FQ[ρ ] Fcl p = N Fcl[p ] , (139) ϕ ≤ ϕ ϕ FIG. 10 The space of all quantum channels, Λ, which which is determined by the classical FI (54) evaluated map between density matrices specified on two given Hilbert for the probability distribution pϕ(X). Importantly, spaces, Λ : L (Hin) → L (Hout), represented as a convex set. Demkowicz-Dobrzański et al. (2012) have shown that for The estimated parameter ϕ specifies a trajectory, Λ (black ϕ the estimation problems in which the parameter is unitar- curve), in such a space. From the point of view of the QFI, ily encoded, it is always optimal to choose a CS depicted any two channel trajectories, e.g. Λϕ and Λ˜ ϕ (gray curve), are equivalent at a given ϕ0 as long as they and their first deriva- in Fig. 10, which employs for each ϕ only two channels tives with respect to ϕ coincide there. Moreover, they can be Π lying at the points of intersection of the tangent to the ± optimally classically simulated at ϕ0 by mixing two channels trajectory with the boundary of the quantum maps set. lying on the intersection of the tangent to the trajectory and Such an optimal CS leads to the tightest upper bound the boundary of the set: Π±. specified in Eq. (139): FQ[ρϕ] N/(ε+ε ), where ε are the “distances” to the boundary≤ marked− in Fig. 10,± dΛϕ Π =Λϕ ε ϕ=ϕ . which represents the variation of the channel up to ± 0 ± ± dϕ | 0 Looking for ε parameters amounts to a search of the first order in δϕ as a classical mixture of some ϕ- ± the distances one can go along the tangent line to the independent channels Π where the ϕ dependence is x x trajectory of Λ so that the corresponding map is still present only in the mixing{ } probabilities p (x). Under ϕ ϕ a physical quantum channel, i.e. a completely posi- such a construction the random variable X distributed tive trace preserving map. This is easiest to do mak- according to p (x) specifies probabilistic choice of chan- ϕ ing use of the Choi-Jamiolkowski isomorphism (Choi, nels Π that reproduces the local action of Λ in the x ϕ 1975; Jamiołkowski, 1972) which states that with each vicinity of ϕ0. Crucially, the QFI is a local quantity— quantum channel, Λ: ( in) ( out), we can asso- see discussions in Sec. IV.B.1—and at a given point ϕ0 L H → L H ciate a positive operator ΩΛ ( out in), so that is a function only of the quantum state considered and ∈ L H dim⊗( Hin) Ω = (Λ )( I I ), where I = H i i is a its first derivative with respect to the estimated param- Λ ⊗ I | ih | | i i=1 | i| i maximally entangled state on in in, while is the eter. Consequently, when considering the parameter be- H ⊗P H I identity map on ( in). Since the complete positivity of ing encoded in a quantum channel, all the channel tra- L H Λ is equivalent to positivity of the ΩΛ operator, one needs jectories at a given point ϕ0 are equivalent from the dΩ to analyze Ω ε Λϕ and find maximum ε point of view of QFI if they lead to density matrices Λϕ0 dϕ ϕ=ϕ0 ± ± | ± that are identical up to the first order in δϕ. In other so that the above operator is still positive-semidefinite. words we can replace Λϕ with any Λ˜ ϕ and obtain the Taking the explicit form of the Λϕ for the case of optical interferometry with imperfect visibility, see same QFI at a given point ϕ0 provided Λϕ = Λ˜ ϕ and 0 0 2 dΛϕ dΛ˜ ϕ Sec. VI.A.3, one can show that ε = 1 η /η ± dϕ = dϕ , see Fig. 10. This means that when − ϕ=ϕ0 (Demkowicz-Dobrzański et al., 2012), which yields the constructing a local CS of the quantum channel Λ at p ϕ ultimate quantum limit on precision given by Eq. (136). ϕ0, we need only to satisfy x pϕ0 (x)Πx =Λϕ0 , as well as dpϕ(x) Π = dΛϕ . x dϕ ϕ=ϕ0 x dϕ Pϕ=ϕ0 | | 2. Photonic losses Crucially, as the maps Λϕ in Eq. (135) act indepen- P N dently on each photon, we can simulate the overall Λϕ⊗ with N independent random variables, XN , associated The expression for the QFI of the output state (127) with each channel. The estimation procedure can now in the asymptotic limit of large N has been first derived be described as by (Knysh et al., 2011). Yet, the general frameworks proposed by (Demkowicz-Dobrzański et al., 2012; Escher N N N et al. ϕ X Λϕ⊗ Λϕ⊗ [ ψin ] ϕ,˜ . (138) , 2011) for generic decoherence allowed to recon- → → → | i → struct this bound on precision with the following result: where N classical random variables are employed to gen- erate the desired quantum map Λ N . It is clear that a 1 1 η 1 η 1 ϕ⊗ ∆ϕ − a + − b ,. (140) strategy in which we could infer the parameter directly ≥ 2 η η √ r a r b  N 31

σϕ Eq. (82). Last equality follows from the additivity prop- ̺ ̺ϕ Φ ̺ ̺ϕ erty of the QFI, which, similarly to Eq. (139), constrains N FQ ρϕ to scale at most linearly for large N. Similarly to the case of CS, in order to get the tightest bound one   FIG. 11 The Quantum Simulation (QS) of a channel. The ac- should find QS that yields the smallest FQ[σϕ], which in tion of the channel Λϕ is simulated up to the first order in the principle is a non trivial task. vicinity of a given point ϕ0 using a ϕ-independent Φ channel Fortunately, Kołodyński and Demkowicz-Dobrzański and an auxiliary state σϕ that contains the full information about the estimated parameter ϕ. (2013) have demonstrated that the search for the opti- mal channel QS corresponds to the optimization over the Kraus representation of a given channel. Without loss of generality we may assume that σ = ϕ ϕ is a pure where ηa, ηb are transmission in the two arms of the inter- ϕ | ih | ϕ-dependent state while Φ[ ] = † is unitary. For a ferometer respectively, see VI.A.2. This bound simplifies · U·U to given QS we may write the corresponding Kraus repre- sentation of the channel by choosing a particular basis 1 η i E in the E space: K (ϕ) =E i ϕ E. In order for ∆ϕ − (141) | i i h |U| i ≥ ηN the QS to be valid, these Kraus operators should corre- r spond to a legitimate Kraus representation of the channel in the case of equal losses, and since the precision achiev- Λϕ[ ]= i Ki(ϕ) Ki(ϕ). Two Kraus representation of a able with uncorrelated states is given by 1/√ηN, the given· quantum channel· are equivalent if and only if they maximal quantum-enhancement factor is √1 η. In the are relatedP by a unitary matrix u: following, we derive the above bounds using the− quantum simulation approach of (Demkowicz-Dobrzański et al., K˜i(ϕ)= uij (ϕ)Kj (ϕ), (143) 2012; Kołodyński and Demkowicz-Dobrzański, 2013). j X which may in principle be also ϕ dependent. Since we require QS to be only locally valid in the vicinity of ϕ , a. Quantum Simulation method Unfortunately, in the 0 the above equation as well as its first derivative needs case of loss the simple CS method yields a trivial bound to be fulfilled only at ϕ . Because of that, the search ∆ϕ 0, since the tangent distances to the boundary of 0 for the optimal Kraus representation K˜ (or equivalently the set≥ of quantum channels are ε =0 in this case. It is i ± the optimal QS) may be restricted to the class of trans- possible, however, to derive a useful bound via the Quan- formations where u(ϕ) = ei(ϕ ϕ0)h with h being any tum Simulation (QS) method which is a natural gener- − Hermitian matrix that shifts the relevant derivative of alization of the CS method. The QS method has been K (ϕ) at ϕ , so that K˜ (ϕ ) = K (ϕ ) and K˜˙ (ϕ ) = described in detail and developed for general metrologi- i 0 i 0 i 0 i 0 K˙ (ϕ )+ i h K (ϕ ). As shown by (Kołodyński and cal schemes with uncorrelated noise by Kołodyński and i 0 j ij j 0 Demkowicz-Dobrzański, 2013), the problem of finding the Demkowicz-Dobrzański (2013) stemming from the works optimal QSP i.e. the minimal F [ ϕ ϕ ] which we term of Demkowicz-Dobrzański et al. (2012) and Matsumoto Q as the F , can be formally rewritten| ih as| (2010). QS

As shown in Fig. 11, local QS amounts to re-expressing FQS = min s s.t. h the action of Λϕ for ϕ = ϕ0+δϕ by a larger ϕ-independent ˙ ˙ s ˙ map Φ that also acts on the auxiliary ϕ-dependent input K˜ (ϕ )†K˜ (ϕ )= 11 , K˜ (ϕ )†K˜ (ϕ )= 0, i 0 i 0 4 2 i 0 i 0 σϕ, up to the first order in δϕ: i i X X (144) 2 Λϕ[̺]= TrEΦ[σϕ ̺]+ O(δϕ ). (142) ⊗ where the parameter s has the interpretation of Note that for σ = p (x) x x , and Φ = x x FQ[ ϕ ϕ ] for the particular QS at ϕ0 and the constraints ϕ x ϕ | ih | | ih |⊗ | ih | Πx, QS becomes equivalent to the CS of Eq. (137), so imposed in Eq. (144) are necessary and sufficient for the that CS is indeed a specificP instance of the more general QS required transformation and the state ϕ to exist. U | i QS. An analogous reasoning as in the case of CS leads to The above optimization problem may not always be the conclusion that we may upper-bound the QFI of the easy to solve. Still, its relaxed version: overall output state, here (127) for the case of losses, as ˙ ˙ ˙ min K˜i(ϕ0)†K˜i(ϕ0) , K˜i(ϕ0)†K˜i(ϕ0)= 0, h k k N N N i i FQ Λ⊗ [ ψ in ψ ] = FQ TrE Φ⊗ σ⊗ ψ in ψ X X ϕ | i h | ϕ ⊗ | i h | (145) N    FQ σϕ⊗ = N FQ[σϕ] ,  where is the operator norm, can always be cast in the ≤ form ofk·k an explicit semi-definite program, which can be N   since TrEΦ⊗ [ ] is just a parameter independent map, easily solved numerically (Demkowicz-Dobrzański et al., under which the· overall QFI may only decrease—see 2012). Numerical solution of the semi-definite program 32 provides a form of the optimal h which may then be taken 3. Phase diffusion as an ansatz for further analytical optimization. Plugging in the Kraus operators Ki(ϕ)= KiUϕ repre- Since the phase diffusion model, see Sec. VI.A.1, is senting the lossy interferometer, see Eq. (130), and fol- an example of a correlated noise model, it cannot be ap- lowing the above described procedure one obtains proached with the CS and QS methods. The study of the behavior of the QFI within the phase-diffusion model was 4 for the first time carried out by (Genoni et al., 2011) con- FQS = 2 (146) 1 ηa 1 ηb sidering indefinite-photon-number Gaussian input states − + − ηa ηb and studied numerically the achievable precision and the q q  structure of optimal input states. Yet, the fundamen- for the optimal h given by tal analytical bounds on precision have not been veri- et al. 1 ηa 4 ηb 4 fied until Escher (2012), where the phase noise has hopt = diag χ, χ , + χ , been approached using the minimization over purifica- −8 1 ηa ηa − −1 ηb ηa  −   −  (147) tions method of Escher et al. (2011) and most recently ηb ηa using the calculus of variations approach of Knysh et al. where χ=FQS − . This indeed reproduces the bounds ηaηb given in (140). (2014). In order to provide the reader with a simple intuition concerning the QS method, we shall present an elemen- a. Minimization over purification method The minimiza- tary construction of the QS for lossy interferometer in tion over purification method of Escher et al. (2011) the special case of η = η =1/2. In this case the bound a b is based on the observation, already mentioned in (140) yields ∆ϕ 1/√N which implies that using opti- Sec. IV.B.1, that QFI for a given mixed quantum state, mal entangled probe≥ state at the input under 50% losses here ρ (127), is not only upper bounded by the QFI of cannot beat the precision which can be obtained by un- ϕ any of its purifications, but there always exists an opti- correlated probes in ideal scenario of no losses. opt opt mal purification, Ψϕ , for which FQ[ρϕ]= FQ Ψϕ , Consider the action of the single photon lossy channel opt opt where ρϕ = TrE Ψϕ Ψϕ . As such statement does Λϕ on the pure input state ψ = α a + β b :   | i | i | i not rely on the form of the transformation ψ in ρ  | i → ϕ 1 1 but rather on the properties of the output state itself, Λ ( ψ ψ )= ψ ψ + vac vac (148) ϕ | ih | 2| ϕih ϕ| 2| ih | the framework of Escher et al. (2011) in principle does not put any constraints on the noise-model considered. with ψ = αeiϕ a + β b , which represents 1/2 proba- Note that, even if the optimal purification itself is diffi- | ϕi | i | i bility of photon sensing the phase undisturbed and the cult to find, any suboptimal purification yields a legiti- 1/2 probability of the photon being lost. Let the auxil- mate upper bound on the QFI and hence may provide a iary state for QS be ϕ = (eiϕ 0 + 1 )/√2. The joined non-trivial precision bound. | i | i | i input + auxiliary state reads: In order to get a physical intuition regarding the pu- rification method, consider a physical model of the phase 1 diffusion where light is being reflected from a mirror φ ψ = αeiϕ 0 a + β 1 b + | i| i √2 | i| i | i| i which position fluctuations are randomly changing the 1
effective optical length. Formally, the model amounts to + αeiϕ 0 b + β 1 a . (149) ˆ √2 | i| i | i| i coupling the phase delay generator Jz to the mirror posi- 1
tion quadrature xˆE = aˆE +ˆaE† (Escher et al., 2012). The map Φ realizing the QS consists now of two steps. √2 Assuming the mirror, serving as the environment E, to First the controlled NOT operation is performed with the reside in the ground state of a quantum oscillator 0 auxiliary system being the target qubit, this transform E before interaction with the light beam, the pure output| i the above state to: 1 0 (αeiϕ a +β b )+ 1 1 (αeiϕ b + √2 | i | i | i √2 | i | i state reads: β a ). The second step is the measurement of the aux- | i iϕJˆz i√2ΓJˆzxˆE iliary system. If the result 0 is measured (probability Ψ = e− e ψ in 0 E. (150) | i | ϕi | i | i 1/2), the system is left in the correct state ψϕ and the | i 2 x2 map leaves it unchanged, if the 1 is measured the state Thanks to the fact that x 0 = e− /√π, the reduced |h | i| of the photon is not the desired| one,i in which case the state map returns the vac state. This map is therefore a ∞ proper QS of the| desiredi lossy interferometer transfor- ρϕ = dx E x Ψϕ Ψϕ x E (151) mation for ηa = ηb = 1/2. Since the auxiliary state em- ˆ h | ih | i ployed in this construction was ϕ = (eiϕ 0 + 1 )/√2 for −∞ N | i | i | i which FQ( ϕ ϕ ⊗ ) = N this leads to the anticipated indeed coincides with the correct output phase diffused | ih | result ∆ϕ 1/√N. state (125). Therefore, this is a legitimate purification ≥ 33 of the interferometer output state in presence of phase C. Bayesian approach diffusion. iδϕHˆE Consider now another purification Ψ˜ ϕ = e Ψϕ Minimizing the average Bayesian cost, as given by | i Eq. (93), over input probe states ψ in is in general more generated by a local (ϕ=ϕ0 +δϕ) rotation E of the mirror demanding than minimization of the| i QFI due to the fact modes, i.e. a unitary operation on the system E. We look that it is not sufficient to work within the local regime for a transformation of the above form which hopefully and analyze only the action of a channel and its first erases as much information on the estimated phase as derivative at a given estimation point as in the QFI ap- ˜ possible, so that QFI for the purified state Ψϕ will be proach. For this reason we do not apply the Bayesian minimized leading to the best bound on the QFIE of ρϕ. approach to the imperfect visibility model as obtaining

Choosing HˆE =λpˆE we obtain the following upper bound the bounds requires a significant numerical and analyt- on the QFI ical effort (Jarzyna and Demkowicz-Dobrzanski, 2014), and constrain ourselves to loss and phase diffusion mod-

iϕλpˆE els. FQ[ρϕ] min FQ e Ψ(ϕ) = ≤ λ | i   2  4 ∆2J = min 2λ2 +4 1 √2Γλ ∆2J = z , z 2 1. Photonic losses λ − 1+4Γ∆ Jz     and thus a lower limit on the precision The optimal Bayesian performance of N-photon states has been studied by Kołodyński and Demkowicz- Dobrzański (2010). Assuming the natural cost function 1 1 ∆ϕ Γ+ = Γ+ , (152) (67) and the flat prior phase distribution, the average 4 ∆2J N 2 ≥ r z r cost (93) reads:

2 2 where we plugged in ∆ Jz = N /4 corresponding to C = Tr ρϕ C Ξ (154) the NOON state which maximized the variance for N- h i h i photon states. See also an alternative derivation of the dϕ 2 ϕ where ρϕ C=4 2π ρϕ sin 2 and ρϕ is given by (127). above result that has been proposed recently in (Ma- The optimalh i measurement´ seed operator Ξ can be found cieszczak, 2014). Crucially, the above result proves that analogously as in the decoherence-free
case. The block the phase diffusion constrains the error to approach a diagonal form of ρϕ, implies that without losing op- constant value √Γ as N , which does not vanish N → ∞ timality one can assume Ξ = N ′=0 eN ′ eN ′ with in the asymptotic limit, what contrasts the 1/√N be- N ′ | ih | e ′ = n,N n . Physically, the block-diagonal havior characteristic for uncorrelated noise models. Note N n=0 ′ L structure| i of Ξ|indicates− i that the optimal covariant mea- also that due to the correlated character of the noise, P the bound (152) predicts that it may be more benefi- surement requires a non-demolition photon number mea- cial to perform the estimation procedure on a group of surement to be performed before carrying out any phase k particles and then repeat the procedure independently measurements, so that the orthogonal subspaces, labeled ν times obtaining 1/√ν reduction in estimation error, by the number of surviving photons N ′, may be firstly rather than employing N = kν in a single experimental distinguished, and subsequently the measurement which shot (Knysh et al., 2014). is optimal in the lossless case is performed (Kołodyński and Demkowicz-Dobrzański, 2010). Plugging in the ex- Only very recently, the exact ultimate quantum limit plicit form of Ξ together with the explicit form of the for the N-photon input states has been derived in (Knysh output state ρ , we arrive at et al., 2014) ϕ n,N n − 2 T (la,lb) (la,lb) π C =2 c Ac, An,n 1 = An 1,n = bn bn 1 , h i − − − − ∆ϕ Γ+ 2 , (153) ≥ N laX,lb=0 q r (155) showing that the previous bound was not tight, with the where A is a symmetric (N +1) (N +1) matrix that is × (la,lb) second term following the HL-like asymptotic scaling of non-zero only on its first off-diagonals, bn are the bi- the noiseless decoherence-free Bayesian scenario stated nomial coefficients previously defined in Eq. (128), while in Eq. (107). In fact, as proven by (Knysh et al., 2014), c is a state vector containing coefficients cn of the N- the optimal states of the noiseless Bayesian scenario, i.e. photon input state (120). the sine states (105), attain the above correct quantum The minimal average cost (154) for the lossy interfer- limit. In Sec. VI.C.2, we show that within the Bayesian ometer then equals C min = 2 λmax, where λmax is approach, with the phase-diffusion effects incorporated, the maximal eigenvalueh i of the matrix− A and the cor- the sine states are always the optimal inputs. responding eigenvector cmax provides the optimal input 34

state coefficients. C min quantifies the maximal achiev- of (Berry and Wiseman, 2000) for the noiseless scenario, h i Γ π able precision and in the N limit may be interpreted which leads then to λmax =2e− 2 cos and hence as the average MSE (60) due→∞ to the convergence of the N+2   cost function (67) to the squared distance as ϕ˜ ϕ. → Γ π N The procedure described above allows only to obtain C =2 1 e− 2 cos →∞ h imin − N +2 ≈ numerical values of the achievable precision, and ceases to    2 be feasible for N . The main result of (Kołodyński Γ Γ π → ∞ 2 1 e− 2 + e− 2 . (159) and Demkowicz-Dobrzański, 2010) was to construct a − N 2 valid analytical lower bound on the minimal average cost   (154): The optimal input states are the same as in the decoherence-free case, i.e. they are the N-photon sine π states of Eq. (105). Note that in contrast to the pho- C 2 1 Amax cos , (156) h imin ≥ − N +2 tonic loss which is an example of an uncorrelated noise,    the minimal average cost (159) does not asymptotically where Amax =max1 n N An,n 1 is the largest element coincide with the QFI-based precision limit (153) unless ≤ ≤ { − } of the matrix A, contained within its off-diagonal entries Γ 1. (155). The bound yields exactly the same formula as the ≪ QFI bound (140), proving that in this case the Bayesian and QFI approaches are equivalent: D. Practical schemes saturating the bounds

1 1 ηa 1 ηb 1 ∆ϕ C C min − + − , Deriving the fundamental bounds on quantum en- ≈ h i≥ h i ≥ 2 ηa ηb √N q r r  hanced precision in presence of decoherence is interest- p (157) ing in itself from a theoretical a point of view. Still, where represents the fact that Bayesian cost approxi- ≈ a practical question remains whether the bounds de- mated the variance only in the limit of large N. The fact rived are saturable in practice. Note that NOON states that both approaches lead to the same ultimate bounds and the sine states that are optimal in case of QFI and on precision suggests that the optimal input states may Bayesian approaches in the decoherence-free case are be approximated for N up to an arbitrary good pre- → ∞ notoriously hard to prepare apart from regime of very cision with states manifesting only local finite-number of small N. For large photon numbers, the only practically particle correlations and may in particular be efficiently accessible states of light are squeezed Gaussian states simulated with the concept of matrix-product states and one of the most popular strategies in performing (Jarzyna and Demkowicz-Dobrzański, 2013; Jarzyna and quantum-enhanced interferometry amounts to mixing a Demkowicz-Dobrzanski, 2014). coherent beam with a squeezed vacuum state on the in- put beam splitter of the Mach-Zehnder interferometer, see Sec. III.D. We demonstrate below that in presence 2. Phase diffusion of uncorrelated decoherence, such as loss or imperfect visibility, this strategy is indeed optimal in the asymp- Similarly to the case of losses discussed in the previ- totic regime of large N and allows to saturate the fun- ous section, we study the estimation precision achieved damental bounds derived above. We will not discuss the within the Bayesian approach but in the presence of phase phase-diffusion noise, since the estimation uncertainty is diffusion. The analysis follows exactly in the same way, finite in the asymptotic limit, and the issue of saturat- so that likewise assuming no prior knowledge and the ing the asymptotic bound becomes trivial as practically natural cost function introduced in Eq. (93) the formula all states lead to the same asymptotic precision value, for the average cost reads while saturating the bound for finite N requires the use C = Tr ρ Ξ =2 cT Ac, (158) of experimentally inaccessible sine states. h i h ϕiC − where this time one may think of the effective state, as of the input state ρin which is firstly averaged over the 1. Bounds for indefinite photon number states Gaussian distribution dictated by the evolution (124) and then over the cost function in accordance with Eq. (93). Derivation of the bounds presented in this section both The optimal seed element of the covariant POVM is iden- in the QFI and Bayesian approaches assumed definite- tical as in the decoherence-free case Ξ = eN eN and photon number states at the input. We have already dis- the matrix A possesses again only non-zero| entriesih on| its cussed the issue of translating the bounds from a definite first off-diagonals, but this time all of them are equal to photon number input state case to a general indefinite- Γ e− 2 . As a result, the minimal average cost (158) may be photon number state case in Sec. V.C in the case of evaluated analytically following exactly the calculation decoherence-free metrology, where we have observed that 35 due to quadratic dependence of QFI on number of pho- splitters in the description of the state transformation— tons used, maximization of QFI over states with fixed in terms of Fig. 8 this corresponds to moving ψ in to | i averaged photon number N is ill defined and arbitrary the left and ρϕ to the right of the figure. In case of high QFI are in principleh achievable.i Controversies re- loss the decoherence map has the same form as given in ϕ i σy lated to this observation, discussed in Sec. V.C.2, are Sec. VI.A.2, but with Uϕ = e− 2 , while in the case fortunately not present in the noisy metrology scenario. of imperfect visibility the Kraus operators (134) will be For the decoherence models, analyzed in this paper, 1+η 1 η modified to K1 = 11, K2 = − σy, so that the lo- the QFI scales at most linearly with N. Following the 2 2 cal dephasing is definedq with respectq to the y rather than reasoning presented in Sec. V.C, consider a mixture of the z axis. For the two decoherence models, the resulting different photon number states p ρ . Since in the N N N Heisenberg picture transformation of the J observable presence of decoherence F (ρ ) cN, where c is a con- z Q N yields (Ma et al., 2011): stant coefficient that depends onP≤ the type and strength of the noise considered, thanks to the convexity of the Jˆz h i = cos ϕ Jˆ in sin ϕ Jˆ in, (162) QFI we can write: η h zi − h xi 2 ∆ Jz N 2 2 2 2 = f(η)h i + cos ϕ ∆ Jz in + sin ϕ ∆ Jx in + F p ρ p F (ρ ) p cN = c N . η2 4 | | Q N N ≤ N Q N ≤ N h i N ! N 2 sin ϕ cos ϕ cov(Jx,Jz) in. X X X (160) − | Hence the bounds on precision derived in Sec. VI.B.2 where f(η) = (1 η)/η for the loss model and f(η) = (1 η2)/η2 in the− case of local dephasing model. The (losses) and Sec. VI.B.1 (imperfect visibility) are valid − also under replacement of N by N . Still, one may come above expressions have a clear intuitive interpretation. h i The signal Jˆ is rescaled by a factor η compared with across claims of precisions going beyond the above men- h zi tioned bounds typically by a factor of two (Aspachs et al., the decoherence-free case, while the variance apart from 2009; Joo et al., 2011). This is only possible, however, the analogous rescaling is enlarged by an additional noise contribution f(η) N /4 due to lost or dephased photons. if classical reference beam required to perform e.g. the h i homodyne detection is not treated as a resource. As dis- In order to calculate the precision achievable with cussed in detail in Sec. V.C.1, we take the position that coherent+squeezed-vacuum strategy, we may use the al- such reference beams should be treated in the same way ready obtained quantities presented in Eq. (37). Af- as the light traveling through the interferometer and as ter substituting the input variances and averages into such also counted as a resource. Eq. (162) and optimally setting α = Re(α) as before, we arrive at a modified version of the formula (38) for the phase estimation precision:

2. Coherent + squeezed vacuum strategy α r ∆ϕ| i| i =

2 2 In section Sec. III.A, we have derived an error- 2 2 1 2 2 −2r 2 |α| +sinh r cot ϕ( α + 2 sinh 2r)+ α e +sinh r+f(η) 2 = r | | | | sin ϕ . propagation formula for the phase-estimation uncertainty α 2 sinh2 r | | − (33) for the standard Mach-Zehnder interferometry in ab- | | (163) sence of decoherence. For this purpose we have adopted the Heisenberg picture and expressed the precision in The optimal operation points are again ϕ = π/2, 3π/2. terms of expectation values, variances and covariances of Considering the asymptotic limit N = α 2 + sinh2 r the respective angular momentum observables calculated and assuming the coherent beamh i to| carry| the dom-→ for the input state. Here, we follow the same procedure inant∞ part of the energy α sinh2 r, the formula for but take additionally into account the effect of imperfect precision at the optimal operation| | ≫ point reads: visibility (local dephasing) and loss. For simplicity, in 2r 2r the case of loss we restrict ourselves to equal losses in α r N e− + f(η) N e− + f(η) ∆ϕ| i| i h i h i = . both arms. The Heiseberg picture transformation of an ≈ N N p h i p h i observable Oˆ corresponding to a general map Λϕ (123) (164) reads Clearly, even for relatively small squeezing strengthp r the 2r e− term becomes negligible, and hence we can effec- ˆ ˆ Uϕ† Ki†OKiUϕ =Λϕ∗ (O), (161) tively approach arbitrary close precision given by: i X α r f(η) ∆ϕ| i| i , (165) where Λ∗ is called the conjugated map. ≈ N For a more direct comparison with the decoherence- ph i free formulas of Sec. III.A, we explicitly include the action which recalling the definitionp of f(η) for the two de- of the Mach-Zehnder input and output balanced beam coherence models considered coincides exactly with the 36

account detection efficiency, optical instruments imper- fections and imperfect coupling was estimated at the level of 38% (LIGO Collaboration, 2011). In (Demkowicz- 1 ∆ϕ Dobrzański et al., 2013) it has been demonstrated that ≥ √N the sensing precision achieved in (LIGO Collaboration, 2011) using the 10dB squeezed vacuum (corresponding 2r to the squeezing factor e− 0.1), was strikingly close to the fundamental bound, and≈ only 8% further reduc- 1 ∆ϕ tion in estimation uncertainty would be possible if more ≤ N advanced input states of light were used.

VII. CONCLUSIONS FIG. 12 The phase estimation precision of an interferome- ter with equal losses in both arms (η = 0.9). The perfor- In this review we have showed how the tools of quan- mance of the optimal N-photon input states (120) is shown tum estimation theory can be applied in order to derive (solid black) that indeed saturate the asymptotic quantum fundamental bounds on achievable precision in quantum- limit (140) (dotted): p(1 − η) / (ηN). The NOON states (solid grey) achieve nearly optimal precision only for low N enhanced optical interferometric experiments. The main (≤10) and rapidly diverge becoming out-performed by classi- message to be conveyed is the fact that while the power of cal strategies. For comparison, the precision attained for an quantum enhancement is seriously reduced by the pres- indefinite photon number scheme is presented, i.e. a coher- ence of decoherence, and in general the Heisenberg scal- ent state and squeezed vacuum optimally mixed on a beam- ing cannot be reached, non-classical states of light of- splitter (Caves, 1981) (dashed), which in the presence of loss fer a noticeable improvement in interferometric precision also saturates the asymptotic quantum limit (140). and simple experimental schemes may approach arbi- trary close the fundamental quantum bounds. It is also worth noting that in the presence of uncorrelated deco- fundamental bounds (136), (141) derived before. This herence the Bayesian approaches coincide asymptotically proves that the fundamental bounds can be asymptoti- with the QFI approaches easing the tension between this cally saturated with a practical interferometric scheme. two often competing ways of statistical analysis. One should note that this contrasts the noiseless case and We would also like to mention an inspiring alterna- the suboptimal performance of simple estimation scheme tive approach to the derivation of limits on precision based on the photon-number difference measurements, of phase estimation, where the results are derived mak- see Eq. (39). ing use of information theoretic concepts such as rate- To summarize the results obtained in this section, in distortion theory (Nair, 2012) or entropic uncertainty Fig. 12, we present a plot of the maximal achievable pre- relations (Hall and Wiseman, 2012). Even though the cision for the lossy interferometer in the equal-losses sce- bounds derived in this way are weaker than the bounds ¯ nario with η =0.9, i.e. ∆ϕ =1/ FQ[ρϕ] as a function presented in this review and obtained via Bayesian or of N compared with the NOON state–based strategy as p QFI approaches, they carry a conceptual appeal encour- well as the asymptotic bound (140). On the one hand, aging to look for deeper connections between quantum the NOON states remain optimal for relatively small estimation and communication theories. N( 10), for which the effects of losses may be disre- ≤ Let us also point out, that while we have focused our garded. This fact supports the choice of NOON -like discussion on optical interferometry using the paradig- states in the quantum-enhanced experiments with small matic Mach-Zehnder model, the same methods can be number of particles (Krischek et al., 2011; Mitchell et al., applied to address the problems of fundamental precision 2004; Nagata et al., 2007; Okamoto et al., 2008; Resch bounds in atomic interferometry (Cronin et al., 2009), et al., 2007; Xiang et al., 2010). However, one should magnetometry (Budker and Romalis, 2007), frequency note that in the presence of even infinitesimal losses, the stabilization in atomic clocks (Diddams et al., 2004) as precision achieved by the NOON states quickly diverges well as the limits on resolution of quantum enhanced with N, because their corresponding output state QFI, lithographic protocols (Boto et al., 2000). All these se- NOON N 2 FQ =η N , decays exponentially for any η<1. tups can be cast into a common mathematical frame- Most importantly, it should be stressed that the co- work, see Sec. III.F, but the resulting bounds will depend herent+squeezed vacuum strategy discussed above has strongly on the nature of dominant decoherence effects been implemented in recent gravitational-wave interfer- and the relevant resource limitations such as: total ex- ometry experiments (LIGO Collaboration, 2011, 2013). perimental time, light power, number of atoms etc., as The main factor limiting the quantum enhancement of well as on the chosen figure of merit. In particular, it is precision in this experiments is loss, which taking into not excluded that in some atomic metrological scenarios 37 one may still obtain a better than 1/√N of precision if Arrad, G., Y. Vinkler, D. Aharonov, and A. Retzker (2014), decoherence is of a special form allowing for use of the Phys. Rev. Lett. 112, 150801. decoherence-free subspaces (Dorner, 2012; Jeske et al., Aspachs, M., J. Calsamiglia, R. Muñoz Tapia, and E. Bagan 79 2013) or when its impact may be significantly reduced by (2009), Phys. Rev. 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