Quantum metrology with nonclassical states of atomic ensembles

Quantum metrology with nonclassical states of atomic ensembles

Luca Pezzè QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy    Augusto Smerzi QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy    Markus K. Oberthaler Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany    Roman Schmied    Philipp Treutlein Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
July 21, 2019
Abstract

Quantum technologies exploit entanglement to revolutionize computing, measurements, and communications. This has stimulated the research in different areas of physics to engineer and manipulate fragile many-particle entangled states. Progress has been particularly rapid for atoms. Thanks to the large and tunable nonlinearities and the well developed techniques for trapping, controlling and counting, many groundbreaking experiments have demonstrated the generation of entangled states of trapped ions, cold and ultracold gases of neutral atoms. Moreover, atoms can couple strongly to external forces and light fields, which makes them ideal for ultra-precise sensing and time keeping. All these factors call for generating non-classical atomic states designed for phase estimation in atomic clocks and atom interferometers, exploiting many-body entanglement to increase the sensitivity of precision measurements. The goal of this article is to review and illustrate the theory and the experiments with atomic ensembles that have demonstrated many-particle entanglement and quantum-enhanced metrology.

pacs:
03.67.Bg, 03.67.Mn, 03.75.Dg, 03.75.Gg, 06.20.Dk, 42.50.Dv
Contents

I Introduction

Figure 1: Building blocks of phase estimation. Phase estimation follows four steps: ) preparation of the probe state ; ) encoding of a phase shift that depends on the parameter of interest; ) readout ( indicates a generic measurement result); ) deriving a phase estimate from the measurement results. The uncertainty of the phase estimation depends crucially on all of these operations.
Figure 2: Summary of experimental achievements. Gain of phase sensitivity over the standard quantum limit achieved experimentally with trapped ions (black symbols), Bose-Einstein condensates (red) and cold thermal ensembles (blue). The gain is shown on logarithmic [left, dB, ] and linear [right, ] scale. The solid thick line is the Heisenberg limit . Stars refer to directly measured phase sensitivity gains, performing a full phase estimation experiment. Circles are expected gains based on a characterization of the quantum state, e.g., calculated as , where is the spin-squeezing parameter, or as , where is the quantum Fisher information, see Sec. II. Filled (open) circles indicate results obtained without (with) subtraction of technical and/or imaging noise. Every symbol is accompanied by a number (in chronological order) corresponding to the reference reported in the side table. Here is the total number of particles or, in presence of fluctuations, the mean total.

The most precise measurement of a physical parameter (such as the strength of a field, a force, or time) is very often obtained by mapping it to a phase that can be determined using interferometric techniques. Phase estimation is thus a unifying framework for precision measurements. It follows the general scheme outlined in Fig. 1: a probe state of particles is prepared, acquires a phase shift , and is finally detected. This conceptually simple scheme is common to all interferometric sensors: from gravitational wave detectors to atomic clocks, gyroscopes, gravimeters, just to name a few. The goal is to estimate with the smallest possible uncertainty given finite resources such as time and number of particles. The noise that determines can be of a technical (classical) or fundamental (quantum) nature Helstrom (1976); Braunstein and Caves (1994); Holevo (1982). Current two-mode atomic sensors are limited by the so-called standard quantum limit, , inherent in probes using a finite number of uncorrelated Giovannetti et al. (2006) or classically-correlated Pezzè and Smerzi (2009) particles. Yet, the standard quantum limit is not fundamental Caves (1981); Yurke et al. (1986); Bondurant and Shapiro (1984). Quantum metrology studies how to exploit quantum resources, such as squeezing and entanglement, to overcome this classical bound Giovannetti et al. (2011, 2004); Pezzè and Smerzi (2014); Tóth and Apellaniz (2014). Research on quantum metrology with atomic ensembles also sheds new light on fundamental questions about many-particle entanglement Amico et al. (2008); Gühne and Tóth (2009) and related concepts, such as Einstein-Podolsky-Rosen correlations Reid et al. (2009) and Bell nonlocality Brunner et al. (2014). Since the systems of interest for quantum metrology often contain thousands or even millions of particles, it is generally not possible to address, detect, and manipulate all particles individually. Moreover, the finite number of measurements limits the possibility to fully reconstruct the generated quantum states. These limitations call for conceptually new approaches to the characterization of entanglement that rely on a finite number of coarse-grained measurements. In fact, many schemes for quantum metrology require only collective manipulations and measurements on the entire atomic ensemble. Still, the results of such measurements allow one to draw many interesting conclusions about the underlying quantum correlations between the particles.

i.1 Entanglement useful for quantum metrology

In the context of phase estimation, the idea that quantum correlations are necessary to overcome the standard quantum limit emerged already in pioneering works Kitagawa and Ueda (1993); Yurke et al. (1986); Wineland et al. (1992). A major step forward was to recognize that can be decreased below by using spin-squeezed states Wineland et al. (1992, 1994), and that such states are entangled Sørensen et al. (2001); Sørensen and Mølmer (2001). However, spin-squeezed states are a relatively small class of states useful for quantum-enhanced metrology. Other prominent examples are NOON and Greenberger-Horne-Zeilinger (GHZ) states, which are not spin squeezed but can nevertheless provide phase sensitivities beating the standard quantum limit Bollinger et al. (1996); Lee et al. (2002). In recent years, it has been clarified that only a special class of quantum correlations can be exploited to estimate an interferometric phase with sensitivity overcoming . This class of entangled states is fully identified by the quantum Fisher information, . The quantum Fisher information is related to the maximum phase sensitivity achievable for a given probe state and interferometric transformation—the so-called quantum Cramér-Rao bound, Helstrom (1967); Braunstein and Caves (1994). The condition Pezzè and Smerzi (2009) is sufficient for entanglement and necessary and sufficient for the entanglement useful for quantum metrology: it identifies the class of states characterized by , i.e., those that can be used to overcome the standard quantum limit in any two-mode interferometer where the phase shift is generated by a local Hamiltonian. Spin-squeezed, GHZ and NOON states fulfill the condition . Phase uncertainties down to can be obtained with metrologically useful -particle entangled states Hyllus et al. (2012a); Tóth (2012). In the absence of noise, the ultimate limit is , the so-called Heisenberg limit Giovannetti et al. (2006); Yurke et al. (1986); Holland and Burnett (1993), which can be reached with metrologically useful genuine -particle entangled states ().

i.2 Generation of metrologically useful entanglement in atomic ensembles

A variety of techniques have been used to generate entangled states useful for quantum metrology with atomic ensembles. The crucial ingredient is interaction between the particles, for instance atom-atom collisions in Bose-Einstein condensates, atom-light interactions in cold thermal ensembles (including experiments performed with warm vapors in glass cells), or combined electrostatic and ion-light interaction in ion chains. Figure 2 summarizes the experimental achievements (gain of phase sensitivity relative to the standard quantum limit) as a function of the number of particles. Stars in Fig. 2 show the measured phase-sensitivity gain obtained after a full interferometer sequence using entangled states as input to the atom interferometer. Filled circles report witnesses of metrologically useful entanglement (i.e., spin squeezing and Fisher information) measured on experimentally generated states, representing potential improvement in sensitivity. Open circles are inferred squeezing, being obtained after subtraction of detection noise. The Heisenberg limit has been reached with up to 10 trapped ions. Attaining this ultimate bound with a much larger number of particles is beyond current technology as it requires the creation and protection of large amounts of entanglement. Nevertheless, metrological gains up to 100 have been reported with large atomic ensembles Hosten et al. (2016a). A glance at Fig. 2 reveals how quantum metrology with atomic ensembles is a very active area of research in physics. Moreover, the reported results prove that the field is now mature enough to take the step from proofs-of-principle to technological applications.

i.3 Outline

This Review presents modern developments of phase-estimation techniques in atomic systems aided by quantum-mechanical entanglement, as well as fundamental studies of the associated entangled states. In Sec. II, we give a theoretical overview of quantum-enhanced metrology with special emphasis on the role of entanglement. We first discuss the concepts of spin squeezing and Fisher information considering spin-1/2 particles. We then illustrate different atomic systems where quantum-enhanced phase estimation—or, at least, the creation of useful entanglement for quantum metrology—has been demonstrated. Sections III and IV review the generation of entangled states in Bose-Einstein condensates. Section V describes the generation of entangled states of many atoms through the common coupling to an external light field. Section VI describes metrology with ensembles of trapped ions. Finally, Sec. VII gives an overview of the experimentally realized entanglement-enhanced interferometers and the realistic perspective to increase the sensitivity of state-of-the-art atomic clocks and magnetometers. This section also discusses the impact of noise in the different interferometric protocols.

Ii Fundamentals

In this Review we consider systems and operations involving particles and assume that all of their degrees of freedom are restricted to only two modes (single-particle states), which we identify as and . These can be two hyperfine states of an atom, as in a Ramsey interferometer Ramsey (1963), two energy levels of a trapping potential, or two spatially-separated arms, as in a Mach-Zehnder interferometer Zehnder (1891); Mach (1892), see Fig. 3. The interferometer operations are collective, acting on all particles in an identical way. The idealized formal description of these interferometer models is mathematically equivalent Wineland et al. (1994); Lee et al. (2002): as discussed in Sec. II.1, it corresponds to the rotation of a collective spin on the generalized Bloch sphere of maximum radius .

Figure 3: Two-mode interferometers. In a Mach-Zehnder interferometer (a) two spatial modes and are combined on a balanced beam splitter, followed by a relative phase shift between the two arms, and finally recombined on a second balanced beam splitter. In a Ramsey interferometer (b) a resonant Rabi rotation creates a balanced superposition between two internal states and , followed by a relative phase shift given by the energy difference between these states multiplied by the interrogation time, . Finally a second resonant Rabi rotation recombines the two modes. (c) Equivalent representation of Mach-Zehnder and Ramsey interferometer operations as rotations of the collective spin on the Bloch sphere. The initial state here, , is pointing toward the north pole. The full sequence is equivalent to the rotation of an angle around the axis.

ii.1 Collective spin systems

Single spin.

By identifying mode with spin-up and mode with spin-down, a (two-mode) atom can be described as an effective spin-1/2 particle: a qubit Nielsen and Chuang (2000); Peres (1995). Any pure state of a single qubit can be written as , with and the polar and azimuthal angle, respectively, in the Bloch sphere. Pure states satisfy , where is the Pauli vector and is the mean spin direction. Mixed qubit states can be expressed as , and have an additional degree of freedom given by the length of the spin vector , such that the effective state vector lies inside the Bloch sphere.

Many spins.

To describe an ensemble of distinguishable qubits, we can introduce the collective spin vector , where

(1)

and is the Pauli vector of the th particle. In particular, is half the difference in the populations of the two modes. The operators (1) satisfy the angular-momentum commutation relations

(2)

and have a linear degenerate spectrum spanning the -dimensional Hilbert space. The well-known set of states forms a basis, where and , and as well as Zare (1988).

Many spins in a symmetrized state.

The Hilbert space spanned by many-qubit states symmetric under particle exchange is that of total spin , which is the maximum allowed spin length for particles.It has dimension , linearly increasing with the number of qubits. Symmetric qubit states are naturally obtained for indistinguishable bosons and are described by the elegant formalism developed by Schwinger in the 1950s Biederharn and Louck (1981). Angular momentum operators are expressed in terms of bosonic creation, and , and annihilation, and , operators for the two modes and :

(3)

They satisfy the commutation relations (2) and commute with the total number of particles . The common eigenstates of and are called Dicke states Dicke (1954) or two-mode Fock states,

(4)

where is the vacuum. They correspond to the symmetrized combinations of particles in mode and particles in mode , where . The eigenstates along an arbitrary spin direction can be obtained by a proper rotation of : and , for instance. Finally, it is useful to introduce raising and lowering operators, ( and ), transforming the Dicke states as .

Collective rotations.

Any unitary transformation of a single qubit is a rotation on the Bloch sphere, where and are the rotation axis and rotation angle, respectively. With qubits, each locally rotated about the same axis and angle , the transformation is , where is the generation of the collective rotation. This is the idealized model of most of the interferometric transformations discussed in this Review. In the collective-spin language, a balanced beam splitter is described by , and a relative phase shift by . Combining the three transformations, , the whole interferometer sequence (Mach-Zehnder or Ramsey), is equivalent to a collective rotation around the -axis Yurke et al. (1986), see Fig. 3(c).

ii.2 Phase estimation

Broadly speaking, an interferometer is any apparatus that transforms a probe state depending on the value of an unknown phase shift , see Fig. 1. The parameter cannot be measured directly and its estimation proceeds from the results of measurements performed on identical copies of the output state . There are good and bad choices for a measurement observable. Good ones (that we will quantify and discuss in more details below) are those characterized by a statistical distribution of measurement results that is maximally sensitive to changes of . We indicate as the probability of a result111 In a simple scenario, is the eigenvalue of an observable . In a more general situation, the measurement is described by a positive-operator-values measure (POVM). A POVM is a set of Hermitian operators parametrized by Nielsen and Chuang (2000) that are positive, , to guarantee non-negative probabilities , and satisfy , to ensure normalization . given that the parameter has the value . The probability of observing the sequence of independent measurements is . An estimator is a generic function associating each set of measurement outcomes with an estimate of . Interference fringes of a Ramsey interferometer are a familiar example of such an estimation (they belong to a more general estimation technique known as the method of moments, discussed in II.2.6). Since the estimator is a function of random outcomes, it is itself a random variable. It is thus characterized by a -dependent statistical mean value and variance

(5)

the sum extending over all possible sequences of measurement results. Different estimators can yield very different results when applied to the same measured data. In the following, we will be interested in locally-unbiased estimators, i.e., those for which and , so that the statistical average yields the true parameter value.

ii.2.1 Cramér-Rao bound and Fisher information

How precise can a statistical estimation be? Are there any fundamental limits? A first answer came in the 1940s with the works of Cramér (1946), Rao (1945), and Fréchet (1943), who independently found a lower bound to the variance (5) of any arbitrary estimator. The Cramér-Rao bound is one of the most important results in parameter-estimation theory. For an unbiased estimator and independent measurements, the Cramér-Rao bound reads

(6)

where

(7)

is the Fisher information Fisher (1922, 1925), the sum extending over all possible values of . The factor in Eq. (6) is the statistical improvement when performing independent measurements on identical copies of the probe state. The Cramér-Rao bound assumes mild differentiability properties of the likelihood function and thus holds under very general conditions,222The Cramér-Rao theorem follows from , that implies , and the Cauchy-Schwarz inequality . The equality is obtained if and only if with independent on . Equation (6) is recovered for unbiased estimators, i.e., , using the additivity of the Fisher information, . see for instance Kay (1993). No general unbiased estimator is known for small . In the central limit, , at least one efficient and unbiased estimator exists in general: the maximum of the likelihood, see Sec. II.2.5.

ii.2.2 Lower bound to the Fisher information: average moments

A lower bound to the Fisher information can be obtained from the rate of change with of specific moments of the probability distribution Pezzè and Smerzi (2009):

(8)

where , and . The Fisher information is larger because it depends on the full probability distribution rather than some moments.

A bound to the Fisher information can be also obtained when estimating the phase shift from the one-body density, e.g., from the intensity of a spatial interference pattern Chwedeńczuk et al. (2012). One find

(9)

where is the Fisher information corresponding to the one-body density , and the coefficient further depends on the two-body density .

ii.2.3 Upper bound to the Fisher information: the quantum Fisher information

An upper bound to the Fisher information is obtained by maximizing Eq. (7) over all possible generalized measurements in quantum mechanics Braunstein and Caves (1994), , called the quantum Fisher information (see footnote 1 for the notion of generalized measurements and their connection to conditional probabilities). We have , and the corresponding bound on the phase sensitivity for unbiased estimators and independent measurements is

(10)

called the quantum Cramér-Rao bound Helstrom (1967). The quantum Fisher information and the quantum Cramér-Rao bound are fully determined by the interferometer output state . Hence they allow to calculate the optimal phase sensitivity of any given probe state and interferometer transformation Helstrom (1976); Holevo (1982), for recent reviews see Paris (2009); Giovannetti et al. (2011); Pezzè and Smerzi (2014). In general, the quantum Fisher information can be expressed as the variance of a -dependent Hermitian operator called the symmetric logarithmic derivative and defined as the solution of Helstrom (1967). A general expression of the quantum Fisher information can be found in terms of the spectral decomposition of the output state where both the eigenvalues and the associated eigenvectors depend on Braunstein and Caves (1994):

(11)

showing that depends solely on and its first derivative . We can decompose this equation as

(12)

The first term quantifies the information about encoded in and corresponds to the Fisher information obtained when projecting over the eigenstates of . The second term accounts for change of eigenstates with (we indicate ). For pure states, , the first term in Eq. (12) vanishes, while the second term simplifies dramatically to .

For unitary transformations generated by some Hermitian operator , we have , and Eq. (12) becomes333We use the notation to indicate the quantum Fisher information for a generic transformation of the probe state, and for unitary transformations. Braunstein and Caves (1994); Braunstein et al. (1996)

(13)

For pure states , Eq. (13) reduces to . For mixed states, . It is worth recalling here that , where and are the maximum and minimum eigenvalues of with eigenvectors and , respectively. This bound is saturated by the states , with arbitrary real , which are optimal input states for noiseless quantum metrology. In presence of noise, the search for optimal quantum states is less straightforward, as discussed in Sec. VII.1.

Convexity and additivity.

The quantum Fisher information is convex in the state:

(14)

with . This expresses the fact that mixing quantum states cannot increase the achievable estimation sensitivity. The inequality (14) can be proved using the fact that the Fisher information is convex in the state Cohen (1968); Pezzè and Smerzi (2014).

The quantum Fisher information of independent subsystems is additive:

(15)

In particular, for an -fold tensor product of the system, , we obtain an -fold increase of the quantum Fisher information: . A demonstration of Eq. (15) can be found in Pezzè and Smerzi (2014).

Optimal measurements.

The equality can always be achieved by optimizing over all possible measurements. A possible optimal choice of measurement for both pure and mixed states is given by the set of projectors onto the eigenstates of Braunstein and Caves (1994). This set of observables is necessary and sufficient for the saturation of the quantum Fisher information whenever is invertible, and only sufficient otherwise. In particular, for pure states and unitary transformations, the quantum Cramér-Rao bound can be saturated, in the limit , by a dichotomic measurement given by the projection onto the probe state itself, , and onto the orthogonal subspace, Pezzè and Smerzi (2014). It should be noted that the symmetric logarithmic derivative, and thus also the optimal measurement, generally depends on , even for unitary transformations. Nevertheless, without any prior knowledge of , the quantum Cramér-Rao bound can be saturated in the asymptotic limit of large using adaptive schemes Hayashi (2005); Fujiwara (2006).

Optimal rotation direction.

Given a probe state, and considering a unitary transformation generated by , it is possible to optimize the rotation direction in order to maximize the quantum Fisher information Hyllus et al. (2010). This optimum is given by the maximum eigenvalue of the matrix

(16)

with , and the optimal direction by the corresponding eigenvector. For pure states, .

Figure 4: Schematic representation of different phase estimation protocols. (a) Maximum likelihood estimation. is the absolute maximum (dashed line) of the likelihood function (solid line) corresponding to the observed sequence of results . As schematically shown in the inset, the phase sensitivity is identified as the root-mean-square fluctuation (shaded region) of the statistical distribution of , obtained by repeating the measurements several times (at a fixed value of ). (b) Method of moments. The blue line is as a function of the parameter . Via this functional monotonic behavior, it is possible to associate the phase estimate with the measured mean value . The phase uncertainty follows from the to statistical uncertainty of (grey region). For , has a simple expression, Eq. (21), obtained by error propagation, locally approximating by the tangent to the curve (thin black line). (c) Bayesian estimation. Posterior probability distribution (solid line). The phase estimate can be chosen as the weighted averaged phase. The integral gives the probability that the true phase value falls into the confidence interval (grey region) around .

ii.2.4 Phase sensitivity and statistical distance

Parameter estimation is naturally related to the problem of distinguishing neighboring quantum states along a path in the parameter space Wootters (1981); Braunstein and Caves (1994). Heuristically, the phase sensitivity of an interferometer can be understood as the smallest phase shift for which the output state of the interferometer can be distinguished from the input . We introduce a statistical distance between probability distributions,

(17)

called the Hellinger distance, where is the statistical fidelity, or overlap, between probability distributions, also known as Bhattacharyya coefficient Bhattacharyya (1943). is non-negative, , and its Taylor expansion reads

(18)

This equation reveals that the Fisher information is the square of a statistical speed, . It measures the rate at which a probability distribution varies when tuning the phase parameter . Equation (18) has been used to extract the Fisher information experimentally Strobel et al. (2014), see Sec. III.3. As Eq. (17) depends on the specific measurement, it is possible to associate different statistical distances to the same quantum states. This justifies the introduction of a distance between quantum states by maximizing over all possible generalized measurements (i.e., over all POVM sets, see footnote 1), Fuchs and Caves (1995), called the Bures distance Bures (1969). Hübner (1992) showed that

(19)

where is the transition probability Uhlmann (1976) or the quantum fidelity between states Jozsa (1994), see Bengtsson and Zyczkowski (2006); Spehner (2014) for reviews. Uhlmann’s theorem Uhlmann (1976) states that , where the maximization runs over all purifications of and of Nielsen and Chuang (2000). In particular, for pure states. A Taylor expansion of Eq. (19) for small gives

(20)

The quantum Fisher information is thus the square of a quantum statistical speed, , maximized over all possible generalized measurements. The quantum Fisher information has also been related to the dynamical susceptibility Hauke et al. (2016), while lower bounds have been derived by Apellaniz et al. (2017) and Frérot and Roscilde (2016).

ii.2.5 The maximum likelihood estimator

The maximum likelihood estimator is the phase value that maximizes the likelihood of the observed measurement sequence , see Fig. 4(a): . The key role played by in parameter estimation is due to its asymptotic properties for independent measurements. For sufficiently large , the distribution of the maximum likelihood estimator tends to a Gaussian centered at the true value and of variance equal to the inverse Fisher information Lehmann and Casella (2003): . Therefore, the maximum likelihood estimator is asymptotically unbiased and its variance saturates the Cramér-Rao bound: . In the central limit, any estimator is as good as—or worse than—the maximum likelihood estimate.

ii.2.6 Method of moments

The method of moments exploits the variation of collective properties of the probability distribution—such as the mean value and variance —with the phase shift . Let us take the average of measurements results . The estimator is the value for which is equal to , see Fig. 4(b). Applying this method requires to be a monotonous function of the parameter , at least in a local region of parameter values determined from prior knowledge. The sensitivity of this estimator can be calculated by error propagation,444A Taylor expansion of around the true value gives . We obtain Eq. (21) by identifying (valid for )and . giving

(21)

As expected on general grounds and proved by Eq. (8), the method of moments is not optimal in general, , with no guarantee of saturation even in the central limit. The equality is obtained when the probability distribution is Gaussian, , and , such that the changes of the complete probability distribution are fully captured by the shift of its mean value Pezzè and Smerzi (2014). Nevertheless, due to its simplicity, Eq. (21) is largely used in the literature to calculate the phase sensitivity of an interferometer for various input states and measurement observables Wineland et al. (1994); Dowling (1998); Yurke et al. (1986). For instance, in the case of unitary rotations generated by (as in Ramsey and Mach-Zehnder interferometers) and taking as measurement observable, Eq. (21) in the limit can be rewritten as

(22)

This equation is useful to introduce the concept of metrological spin-squeezing, see Sec. II.3.5. We recall that Eqs. (21) and (22) are valid for a sufficiently large number of measurements.

Finally, there are many examples in the literature where a small is obtained for phase values where , while the ratio remains finite Yurke et al. (1986); Kim et al. (1998). These “sweet spots” are very sensitive to technical noise: an infinitesimal amount of noise may prevent to vanish, while leaving unchanged , such that diverges Lücke et al. (2011).

ii.2.7 Bayesian estimation

The cornerstone of Bayesian inference is Bayes’ theorem. Le us consider two random variables and . Their joint probability density can be expressed as in terms of the conditional probability and the marginal probability distribution . Bayes’ theorem

(23)

follows from the symmetry of the joint probability .

In the Bayesian subjective interpretation of probabilities, and are both considered as random variables with as the posterior probability distribution given the measurement results . is the prior probability distribution that quantifies our (subjective) ignorance of the true value of the interferometric phase, i.e., before any measurements were done. One often has no prior knowledge on the phase (maximum ignorance), which is expressed by a flat prior distribution . Bayes’ theorem allows to update our knowledge about the interferometric phase by including measurement results, since can be calculated directly (see the introduction of Sec. II.2) and is determined by the normalization . Bayesian probabilities express our (lack of) knowledge of the interferometric phase as a probability distribution . This is radically different from the standard frequentist view where the probability is defined as the infinite-sample limit of the outcome frequency of an observed event. Having the posterior distribution , we can consider any phase as the estimate. In practice, it is convenient to choose the weighted averaged , or the phase corresponding to the maximum of the probability , since the corresponding mean square fluctuations saturate the Cramér-Rao bound (see below). We can further calculate the probability that the chosen estimate falls into a certain interval by integrating . To take into account the periodicity of the probability, quantities like can be calculated. Remarkably, Bayesian estimation is asymptotically consistent: as the number of measurements increases, the posterior probability distribution assigns more weight in the vicinity of the true value. The Laplace-Bernstein-von Mises theorem Lehmann and Casella (2003); Pezzè and Smerzi (2014); Gill (2008) demonstrates that, under quite general conditions, , to leading order in , for . In this limit, the posterior probability becomes normally distributed, centered at the true value of the parameter, and with a variance inversely proportional to the Fisher information. See Van Trees and Bell (2007) for a review of bounds in Bayesian phase estimation.

ii.3 Entanglement and phase sensitivity

In this section we show how entanglement can offer a precision enhancement in quantum metrology. We start with the formal definition of multiparticle entanglement and then clarify, via the Fisher information introduced in the previous section, the notion of useful entanglement for quantum metrology.

ii.3.1 Multiparticle entanglement

Let us consider a system of particles (labeled as ), each particle realizing a qubit. A pure quantum state is separable in the particles if it can be written as a product

(24)

where is the state of the th qubit. A mixed state is separable if it can be written as a mixture of product states Werner (1989),

(25)

with and . States that are not separable are called entangled Horodecki et al. (2009); Gühne and Tóth (2009). In the case of particles, any quantum state is either separable or entangled. For , we need further classifications Dür et al. (2000). Multiparticle entanglement is quantified by the number of particles in the largest non-separable subset. In analogy with Eq. (24), a pure state of particles is -separable (also indicated as -producible in the literature) if it can be written as

(26)

where is the state of particles and . A mixed state is -separable if it can be written as a mixture of -separable pure states Gühne et al. (2005)

(27)

A state that is -separable but not -separable is called -particle entangled: it contains at least one state of particles that does not factorize. Using another terminology Sørensen and Mølmer (2001), it has an entanglement depth larger than . In maximally entangled states () each particle is entangled with all the others. Finally, note that -separable states form a convex set containing the set of -separable states with Gühne and Tóth (2009).

ii.3.2 Sensitivity bound for separable states: the standard quantum limit

The quantum Fisher information of any separable state of qubits is upper-bounded Pezzè and Smerzi (2009):

(28)

This inequality follows from the convexity and additivity of the quantum Fisher information and uses Pezzè and Smerzi (2014). As a consequence of Eqs. (10) and (28), the maximum phase sensitivity achievable with separable states is Giovannetti et al. (2006)

(29)

generally indicated as the shot-noise or standard quantum limit. This bound is independent of the specific measurement and estimator, and refers to unitary collective transformations that are local in the particles. In Eq. (29) and play the same role: repeating the phase estimation times with one particle has the same sensitivity bound as repeating the phase estimation one time with particles in a separable state.

Figure 5: Rotation of different quantum states. Wigner distribution (normalized to its maximum value – left column), see Sec. II.4, and spin probability along the direction (red histograms – right column) of different quantum states : (a) a coherent spin state pointing along the positive -axis, , (Sec. II.3.3); (b) a spin-squeezed state with (Sec. II.3.5); (c) a twin-Fock state (Sec. II.3.6) and (d) a NOON state (Sec. II.3.7). The thin blue histogram is obtained after a rotation of the state by an angle in (a), in (b), in (c), and in (d). Here .

ii.3.3 Coherent spin states

The notion of coherent spin states was introduced by Arecchi et al. (1972); Radcliffe (1971) as a generalization of the field coherent states first discussed by Glauber (1963), see Zhang et al. (1990) for a review. Coherent spin states are constructed as the product of qubits (spins-1/2) in pure states all pointing along the same mean-spin direction :

(30)

Equation (30) is the eigenstate of with the maximum eigenvalue of . The coherent spin state is a product state and no quantum entanglement is present between the particles. can also be written as a binomial sum of Dicke states with Arecchi et al. (1972). When measuring the spin component of along any direction orthogonal to , each individual atom is projected with equal probability into the up and down eigenstates along this axis, with eigenvalues , respectively: we thus have , and Itano et al. (1993); Yurke et al. (1986).

Coherent spin states are optimal separable states for metrology. They saturate the equality sign in Eq. (28) and thus reach the standard quantum limit. Let us consider the rotation of around a direction perpendicular to the mean spin direction (here , and are mutually orthogonal). This rotation displaces the coherent spin state on the surface of the Bloch sphere, see Fig. 5(a). The initial and final states become distinguishable after rotating by an angle heuristically giving the phase sensitivity of the state. This rotation angle can be obtained from a geometric reasoning Yurke et al. (1986): we have , giving for . More rigorously, the squared Bures distance, Eq. (19), between and the rotated is

(31)

that is