# Quantum metrology from a quantum information science perspective

## Abstract

We summarise important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger-Horne-Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér-Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

###### type:

Review ArticleThis article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ’50 years of Bell’s theorem’.

###### Contents:

Quantum metrology from a quantum information science perspectiveQuantum metrology from a quantum information science perspective

## 1 Introduction

Metrology plays a central role in science and engineering. In short, it is concerned with the highest achievable precision in various parameter estimation tasks, and with finding measurement schemes that reach that precision. Originally, metrology was focusing on measurements using classical or semiclassical systems, such as mechanical systems described by classical physics or optical systems modelled by classical wave optics. In the last decades, it has become possible to observe the dynamics of many-body quantum systems. If such systems are used for metrology, the quantum nature of the problem plays an essential role in the metrological setup. Examples of the case above are phase measurements with trapped ions [1], inteferometry with photons [2, 3, 4, 5] and magnetometry with cold atomic ensembles [6, 7, 8, 9, 10, 11].

In this paper, we review various aspects of quantum metrology with the intention to give a comprehensive picture to scientists with a quantum information science background. We will present simple examples that, while can be used to explain the fundamental principles, have also been realised experimentally. The basics of quantum metrology [12, 13, 14, 15, 16, 17, 18, 19, 20, 21] can be perhaps best understood in the fundamental task of magnetometry with a fully polarised atomic ensemble. It is easy to deduce the precision limits of the parameter estimation, as well as the methods that can improve the precision. We will also consider phase estimation with other highly entangled states such as for example Greenberger-Horne-Zeilinger (GHZ) states [22].

After discussing concrete examples, we present a general framework for computing the precision of the parameter estimation in the quantum case, based on the Cramér-Rao bound and the quantum Fisher information. In the many-particle case, most of the metrology experiments have been done in systems with simple Hamiltonians that do not contain interaction terms. Such Hamiltonians cannot create entanglement between the particles. For cold atoms, a typical situation is that the input state is rotated with some angle and this angle must be estimated. We will show that quantum states with particles exhibiting quantum correlations, or more precisely, quantum entanglement [23, 24], provide a higher precision than an ensemble of uncorrelated particles. The most important question is how the achievable precision scales with the number of particles. Very general derivations lead to, at best,

(1) |

for nonentangled particles. Equation (1) is called the shot-noise scaling, the term originating from the shot-noise in electronic circuits, which is due to the discrete nature of the electric charge. is a parameter of a very general unitary evolution that we would like to estimate. On the other hand, quantum entanglement makes it possible to reach

(2) |

which is called the Heisenberg-scaling. Note that if the Hamiltonian of the dynamics has interaction terms then even a better scaling is possible (see, e.g., references [25, 26, 27, 28, 29, 30, 31]).

All the above calculations have been carried out for an idealised situation. When a uncorrelated noise is present in the system, it turns out that for large enough particle numbers the scaling becomes a shot-noise scaling. The possible survival of a better scaling under correlated noise, under particular circumstances, or depending on some interpretation of the metrological task, is at the centre of attention currently. All these are strongly connected to the question of whether strong multipartite entanglement can survive in a noisy environment.

Our paper is organised as follows. In section 2, we will discuss examples of metrology with large particle ensembles and show simple methods to obtain upper bounds on the achievable precision. In section 3, we define multipartite entanglement and discuss how entanglement is needed for spin squeezing, which is a typical method to improve the precision of some metrological applications in cold gases. We also discuss some generalised spin squeezing entanglement criteria. In section 4, we introduce the Cramér-Rao bound and the quantum Fisher information, and other fundamental notions of quantum metrology. In section 5, we show that multipartite entanglement is a prerequisite for maximal metrological precision in many very general metrological tasks. We also discuss how to define macroscopic superpositions, how the entanglement properties of the quantum state are related to the speed of the quantum mechanical processes and to the quantum Zeno effect. We will also very briefly discuss the meaning of inter-particle entanglement in many-particle systems. In section 6, we review some of the very exciting recent findings showing that uncorrelated noise can change the scaling of the precision with the particle number under very general assumptions.

## 2 Examples for simple metrological tasks with many-particle ensembles

In this section, we present some simple examples of quantum metrology, involving an ensemble of spin- particles in an external magnetic field. We demonstrate how simple ideas of quantum metrology can help to determine the precision of some basic tasks in parameter estimation. We consider completely polarised ensembles, as well as GHZ states, symmetric Dicke states [32, 33, 10, 34] and singlet states [35, 36].

First, let us explain the characteristics of the physical system we use for our discussion. In a large particle ensemble, typically only collective quantities can be measured. For spin- particles, such collective quantities are the angular momentum components defined as

(3) |

for where are the components of the angular momentum of the particle. More concretely, we can measure the expectation values and the variance of the angular momentum component where is a unit vector describing the component.

The typical Hamiltonians involve also collective observables, such as the Hamiltonian describing the action of a magnetic field pointing in the -direction

(4) |

where is the gyromagnetic ratio, is the strength of the magnetic field, is the direction of the field, and is the angular momentum component parallel with the field. Hamiltonians of the type (4) do not contain interaction terms, thus starting from a product state we arrive also at a product state. Interferometry with dynamics determined by (4) is discussed in the context of SU(2) interferometers [37], as the are the generators of the SU(2) group. We will mostly study this type of interferometry in multiparticle systems, as it gives a good opportunity to relate the entanglement of many-particle states to their metrological performance.

The Hamiltonian (4), with the choice of generates the dynamics

(5) |

where we defined the angle that depends on the evolution time

(6) |

A basic task in quantum metrology is to estimate the small parameter by measuring the expectation value of a Hermitian operator, which we will denote by in the following. If the evolution time is a constant then estimating is equivalent to estimating the magnetic field The precision of the estimation can be characterised with the error-propagation formula as

(7) |

where is the expectation value of the operator and its variance is given as

(8) |

Thus, the precision of the estimate depends on how sensitive is to the change of and also on how large the variance of is. Based on the formula (7), one can see that the larger the slope the higher the precision. On the other hand, the larger the variance the lower the precision.

The formula (7) above can be calculated for any given Thus, this formalism can be used to characterise small fluctuations around a given For simplicity we will calculate the precision typically for (To be more precise, if both the numerator and the denominator in (7) are zero, then we will take the limit instead.) This approach is connected to the estimation theory based on the quantum Fisher information discussed in this review and could be called a local approach. Figure 1 helps to interpret the quantities appearing in (7). We note that the global alternative is the Bayesian estimation theory. There, the parameter to be estimated is a random variable with a certain probability density For a recent review discussing this approach in detail, see reference [19]. For an application, see reference [38].

Finally, note that often, instead of one calculates which is large for a high precision. It scales as for the shot-noise scaling, and as for the Heisenberg scaling. [Compare with (1) and (2).]

### 2.1 Ramsey-interferometry with spin squeezed states

Let us start with a basic scheme for magnetometry using an almost completely polarized state. The total spin of the ensemble, originally pointing into the -direction, is rotated by a magnetic field pointing to the -direction, as can be seen in figure 2(a). Hence, the unitary giving the dynamics of the system is

(9) |

The stronger the field, the faster the rotation. The rotation angle can be obtained by measuring a spin component perpendicular to the initial spin. The estimation of rotation angle in such an experiment is a particular case of Ramsey-interferometry (see, e.g., references [13, 19]) and has been realised for magnetometry in cold atoms [39]. This idea has been used in experiments with cold gases to get even a spatial information on the magnetic field and its gradient [6, 7, 8, 9, 11].

So far, it looks as if the mean spin behaves like a clock arm and its position tells us the value of the magnetic field. At this point one has to remember that we have an ensemble of particles governed by quantum mechanics, and the uncertainty of the spin component perpendicular to the mean spin is not zero. For the completely polarised state, the squared uncertainty is

(10) |

Hence, the angle of rotation can be estimated only with a finite precision as can be seen in figure 2(a). Intuitively speaking, we can detect a rotation only if the uncertainty ellipses of the spin in the original position and in the position after the rotation do not overlap with each other too much. Based on these ideas and (10), with elementary geometric considerations, we arrive at

(11) |

Thus, we obtained the shot-noise scaling (1), even with very simple, qualitative arguments.

A more rigorous argument is based on the formula (7), where we measure the operator

(12) |

The expectation value and the variance of this operator, as a function of are

(13) | |||||

Hence, using =0, we obtain for the precision

(14) |

which equals demonstrating a shot-noise scaling for the totally polarised states.

We can see that could be smaller if we decrease [40]. A comparison between figure 2(a) and (b) also demonstrates the fact that a smaller leads to a higher precision. The variances of the angular momentum components are bounded by the Heisenberg uncertainty relation [41]

(15) |

Thus, the price of decreasing is increasing

Let us now characterise even quantitatively the properties of the state that can reach an improved metrological precision. For fully polarised states, (15) is saturated such that

(16) |

Due to decreasing , our state fulfils

(17) |

where is the direction of the mean spin, and the bound in (17) is the square root of the bound in (15). Such states are called spin squeezed states [41, 42, 43, 44]. In practice this means that the mean angular momentum of the state is large, and in a direction orthogonal to the mean spin the uncertainty of the angular momentum is small. An alternative and slightly different definition of spin squeezing considers the usefulness of spin squeezed states for reducing spectroscopic noise in a setup different from the one discussed in this section [42]. Spin squeezing has been realised in many experiments with cold atomic ensembles. In some systems the particles do not interact with each other, and light is used for spin squeezing [13, 39, 45, 46, 47, 48], while in Bose-Einstein condensates the spin squeezing can be generated using the inter-particle interaction [49, 50, 51, 11].

Next, we can ask, what the best possible phase estimation precision is for the metrological task considered in this section. For that, we have to use the following inequality based on general principles of angular momentum theory

(18) |

Note that equation (18) is saturated only by symmetric multiqubit states. Together with the identity connecting the second moments, variances and expectation values

(19) |

equation (18) leads to a bound on the uncertainty in the squeezed orthogonal direction

(20) |

Introducing the maximal spin length

(21) |

we arrive at the inequality

(22) |

This leads to a simple bound on the precision

(23) |

which indicates that the precision is limited for almost completely polarised spin squeezed states with Here, the equality in (23) is based on (14), while the first inequality is due to (15), and the second one comes from (22). The bound in (23) is not optimal, as for the fully polarised state we would expect while (23) allows for a higher precision for

It is possible to obtain the best achievable precision numerically for our case, when is measured for a state that is almost completely polarised in the -direction in a field pointing into the -directon. For even states giving the smallest for a given can be obtained as a ground state of the Hamiltonian [43]

(24) |

where plays the role of a Lagrange multiplier. This also means that the ground states of give the best for a given when collective operators are measured for estimating Since the ground state of (24) is symmetric, it is possible to make the calculations in the symmetric subspace and hence model large systems. We plotted the precision as a function of the polarisation for different values of in figure 3, which demonstrates that scales as Hence, for the precision of phase-estimation the Heisenberg scaling (2) can be reached.

Paradoxically the maximum is reached in the limit of zero mean spin. An added noise can radically change this situation. If the mean spin is small, and its direction is the information that we use for metrology, then a very small added noise can change the direction of the spin, making the metrology for this case impractical. Thus, if we consider local noise acting on each particle independently, then the maximum will be reached at a finite spin length.

### 2.2 Metrology with a GHZ state

Next, we will show another example where the Heisenberg scaling for the precision of phase estimation can be reached. The scheme is based on a GHZ state defined as

(25) |

where we follow the usual convention defining the and states with the eigenstates of as and Such states have been created in photonic systems [52, 53, 54, 55, 56] and in cold trapped ions [1, 57, 58]. Let us consider the dynamics given by

(26) |

Under such dynamics, the GHZ state evolves as

(27) |

hence the difference of the phases of the two terms scales as Let us consider measuring the operator

(28) |

which is essentially the parity in the -basis. Note that this operator needs an individual access to the particles. For the dynamics of the expectation value and the variance we obtain

(29) |

Hence, based on (7), for small the precision is

(30) |

which means that we reached the Heisenberg scaling (2). In reference [1], the scheme described above has been realised experimentally with three ions and a precision above the shot-noise limit has been achieved.

Note, however, that the GHZ state is very sensitive to noise. Even if a single particle is lost, it becomes a separable state. Thus, it is a very important question, how well such a state can be created, and how noise is influencing the scaling of the precision with the particle number. This question will be discussed in section 6. Concerning spin squeezed states and GHZ states, it has been observed that under local noise, such as dephasing and particle loss, for large particle numbers, the GHZ state becomes useless while the spin squeezed states, discussed in the previous section, are optimal [59].

A related metrological scheme for two-mode systems is based on a Mach-Zender interferometer [60, 61, 62, 63, 64, 65, 66], using as inputs NOON states defined as [18]

(31) |

Here, the state describes a system with particles in the first bosonic mode and particles in the second bosonic mode. For example, the two modes can be two optical modes, or, two spatial modes in a double-well potential. Thus, similarly to GHZ states, the state is a superposition of two states: all particles in the first state and all particles in the second state. However, in this scheme we do not have a local access to the particles. Hence, we cannot easily measure the operator (28), which is a multi-particle correlation operator, and instead the following operator has to be measured

(32) |

The basic idea of the -fold gain in precision is similar to the idea used for the method based on the GHZ state. The expectation value of and the variance of as a function of is the same as before, given in (29). With that, the Heisenberg scaling can be reached. Metrological experiments with NOON states have been carried out in optical systems that surpassed the shot-noise limit [2, 3, 4, 5].

### 2.3 Metrology with a symmetric Dicke state

As a third example, we will consider metrology with -qubit symmetric Dicke states

(33) |

where the summation is over all the different permutations of ’s and ’s. One of such states is the -state for which which has been prepared with photons and ions [67, 68].

From the point of view of metrology, we are interested mostly in the symmetric Dicke state for even and This state is known to be highly entangled [69]. In the following, we will omit the superscript giving the number of ’s and use the notation

(34) |

Symmetric Dicke states of the type (34) have been created in photonic systems [33, 70, 71, 72, 73] and in cold gases [10, 34, 74]. In references [10, 72], their metrological properties have also been verified.

The state (34) has for all For the second moments we obtain

(35) |

It can be seen that is minimal, and are close to the largest possible value,

The state has a rotational symmetry around the axis. Thus, the state is not changed by dynamics of the type Based on these considerations, we will use dynamics of the type (9). Moreover, since the total spin length is zero, a rotation around any axis remains undetected if we measure only the expectation values of the collective angular momentum components. Hence, our setup will measure the expectation value of

(36) |

Note that this is also a collective measurement. In practice, to measure we have to measure many times and compute the average of the squared values.

For the dynamics of the expectation value we obtain

(37) |

The expectation value starts from zero, and oscillates with a frequency twice as large as the frequency of the oscillation was for the analogous case for in (13). This is due to fact that after a rotation with an angle we obtain again the original Dicke state. Following the calculations given in reference [10], we arrive at

(38) |

which again means that we reached the Heisenberg scaling (2). The quantum dynamics of the Dicke state used for metrology is depicted in figure 4.

### 2.4 Singlet states

Finally, we show another example for states that can be used for metrology in large particle ensembles. Pure singlet states are simultaneous eigenstates of for with an eiganvalue zero, that is,

(39) |

Mixed singlet states are mixtures of pure singlet states, and hence for any direction and any power Such states can be created in cold atomic ensembles by squeezing the uncertainties of all the three collective spin components [35, 75, 76]. Since in large systems practically all initial states and all the possible dynamics are permutationally invariant, they are expected to be also permutationally invariant. For spin- particles, there is a unique permutationally invariant singlet state [36]

(40) |

where the summation is over all permutation operators and

(41) |

A realisation of the singlet state (40) with an ensemble of particles is shown in figure 5.

A singlet state is invariant under for any Thus, it is completely insensitive to rotations around any axis. How can it be useful for magnetometry? Let us now assume that we would like to analyse a magnetic field pointing in the -direction using spins placed in an equidistant chain. While the singlet (40) is insensitive to the homogenous component of the magnetic field, it is very sensitive to the dynamics

(42) |

where is proportional to the field gradient. This makes the state useful for differential magnetometry, since singlets are insensitive to external homogeneous magnetic fields, while sensitive to the gradient of the magnetic field [36]. Similar ideas work even if the atoms are in a cloud rather than in a chain. The quantity to measure in order to estimate is again as was the case in section 2.3. This idea is also interesting even for a bipartite singlet of two large spins [77].

## 3 Spin squeezing and entanglement

As we have seen in section 2.1, spin squeezed states have been more useful for metrology than fully polarised product states. Moreover, states very different from product states, such as GHZ states and Dicke sates could reach the Heisenberg limit in parameter estimation. Thus, large quantum correlations, or entanglement, can help in metrological tasks. In this section, we will discuss some relations between entanglement and spin squeezing, showing why entanglement is necessary to surpass the shot-noise limit. We also discuss that not only entanglement, but true multipartite entanglement is needed to reach the maximal precision in the metrology with spin squeezed states.

### 3.1 Entanglement and multi-particle entanglement

Next, we need the following definition. A quantum state is (fully) separable if it can be written as [78]

(43) |

where are single-particle pure states. Separable states are essentially states that can be created without an inter-particle interaction, just by mixing product states. States that are not separable are called entangled. Entangled states are more useful than separable ones for several quantum information processing tasks, such as quantum teleportation, quantum cryptography, and, as we will show later, for quantum metrology [23, 24].

In the many-particle case, it is not sufficient to distinguish only two qualitatively different cases of separable and entangled states. For example, an -particle state is entangled, even if only two of the particles are entangled with each other, while the rest of the particles are, say, in the state Usually, such a state we would not call multipartite entangled. This type of entanglement is very different from the entanglement of a GHZ state (25).

Hence, the notion of genuine multipartite entanglement [57, 79] has been introduced to distinguish partial entanglement from the case when all the particles are entangled with each other. It is defined as follows. A pure state is biseparable, if it can be written as a tensor product of two multi-partite states

(44) |

A mixed state is biseparable if it can be written as a mixture of biseparable pure states. A state that is not biseparable, is genuine multipartite entangled. In many quantum physics experiments the goal was to create genuine multipartite entanglement, as this could be used to demonstrate that something qualitatively new has been created compared to experiments with fewer particles [52, 53, 54, 55, 56, 57, 33, 70, 71, 73, 58].

In the many-particle scenario, further levels of multi-partite entanglement must be introduced as verifying full -particle entanglement for or particles is not realistic. In order to characterise the different levels of multipartite entanglement, we start first with pure states. We call a state -producible, if it can be written as a tensor product of the form

(45) |

where are multiparticle states with at most particles. A -producible state can be created in such a way that only particles within groups containing not more than particles were interacting with each other. This notion can be extended to mixed states by calling a mixed state -producible if it can be written as a mixture of pure -producible states. A state that is not -producible contains at least -particle entanglement [80, 81]. Using another terminology, we can also say that the entanglement depth of the quantum state is larger than [43].

It is instructive to depict states with various forms of multipartite entanglement in set diagrams as shown in figure 6. Separable states are a convex set since if we mix two separable states, we can obtain only a separable state. Similarly, -producible states also form a convex set. In general, the set of -producible states contain the set of -producible states if

### 3.2 The original spin squeezing criterion

Let us see now how entanglement and multiparticle entanglement is related to spin squeezing. It turns out that spin squeezing, discussed in section 2.1, is strongly related to entanglement. A ubiquitous entanglement criterion in this context is the spin squeezing inequality [82]

(46) |

If a state violates (46), then it is entangled (i.e., not fully separable). In order to violate (46), its denominator must be large while its numerator must be small, hence, it detects states that have a large spin in some direction, while a small variance of a spin component in an orthogonal direction. That is, (46) detects the entanglement of spin squeezed states depicted in figure 2(b).

For spin squeezed states, it has also been noted that multipartite entanglement, not only simple nonseparability is needed for large spin squeezing [43]. To be more specific, for a given mean spin length, larger and larger spin squeezing is possible only if the state has higher and higher levels of multipartite entanglement. Moreover, larger and larger spin squeezing leads to larger and larger measurement precision. Such strongly squeezed states have been created experimentally in cold gases and a 170-particle entanglement has been detected [83].

At this point note that only the first and second moments of the collective quantities are needed to evaluate the spin squeezing condition (46). It is easy to show that all these can be obtained from the average two-particle density matrix of the quantum state defined as [84]

(47) |

where is the reduced two-particle state of particles and

In summary, entangled states seem to be more useful than separable ones for magnetometry with spin squeezed states discussed in section 2.1. Moreover, states with -particle entanglement can be more useful than states with -particle entanglement for the same metrological task. This finding will be extended to general metrological tasks in section 5.

### 3.3 Generalised spin squeezing criteria

The original spin squeezing entanglement criterion (46) can be used to detect the entanglement of almost completely polarised spin squeezed states. However, there are other highly entangled states, such as Dicke states (34) and singlet states defined in (39). For these states, the denominator of the fraction in (46) is zero, thus they are not detected by the original squeezing entanglement criterion.

A complete set of entanglement conditions similar to the condition (46) has been determined, called the optimal spin squeezing inequalities. They are called optimal since, in the large particle number limit, they detect all entangled states that can be detected based on the first and second moments of collective angular momentum components. For separable states of the form (43), the following inequalities are satisfied [84] {subequations}

(48) | |||||

(49) | |||||

(50) | |||||

(51) |

where take all the possible permutations of The inequality (48), identical to (18), is valid for all quantum states. On the other hand, violation of any of the inequalities (3.3b-d) implies entanglement.

Based on the entanglement conditions (3.3), new spin squeezing parameters have been defined. For example, (50) is equivalent to [85]

(52) |

provided that the denominator of (52) is positive. The criterion (52) can be used to detect entanglement close to Dicke states, discussed in section 2.3. One can see that for the Dicke state (34), the numerator of the fraction in (52) is zero, while the denominator is maximal [see (35)]. Apart from entanglement, it is also possible to detect multiparticle entanglement close to Dicke states. A condition linear in expectation values and variances of collective observables has been presented in reference [86] for detecting multipartite entanglement. A nonlinear criterion is given in reference [74], which detects all states as multipartite entangled that can be detected based on the measured quantities. The criterion has been used even experimentally [74]. An entanglement depth of particles has been detected in an ensemble of around cold atoms.

The inequality (49) is equivalent to [35, 76]

(53) |

The parameter can be used to detect entanglement close to singlet states discussed in section 2.4. It can be shown that the number of non-entangled spins in the ensemble is bounded from above by

Finally, it is interesting to ask, what the relation of the new spin squeezing parameters is to the original one. It can be proved that the parameters and detect all entangled states that are detected by They detect even states not detected by such as entangled states with a zero mean spin, like Dicke states and singlet states. Moreover, it can be shown that for large particle numbers, in itself is also strictly stronger than [85].

## 4 Quantum Fisher information

In this section, we review the theoretical background of quantum metrology, such as the Fisher information, the Cramér-Rao bound and the quantum Fisher information.

### 4.1 Classical Fisher information

Let us consider the problem of estimating a parameter based on measuring a quantity Let us assume that the relationship between the two is given by a probability density function This function, for every value of the parameter gives a probability distribution for the values of

Let us now construct an estimator which would give for every value of an estimate for In general it is not possible to obtain the correct value for exactly. We can still require that the estimator be unbiased, that is, the expectation value of should be equal to This can be expressed as

(54) |

How well the estimator can estimate ? The Cramér-Rao bound provides a lower bound on the variance of the unbiased estimator as

(55) |

where the Fisher information is defined with the probability distribution function as

(56) |

The inequality (55) is a fundamental tool in metrology that appears very often in physics and engineering, and can even be generalised to the case of quantum measurement. Finally, note that the inequality (55) is giving a lower bound for parameter estimation in the vicinity of a given This is the local approach discussed in section 2.

### 4.2 Quantum Fisher information

In quantum metrology, as can be seen in figure 7, one of the basic tasks is phase estimation connected to the unitary dynamics of a linear interferometer

(57) |

where is the input state, is the output state, and is a Hermitian operator. The operator can be, for example, a component of the collective angular momentum The important question is, how well we can estimate the small angle by measuring

Let us use the notion of Fisher information to quantum measurements assuming that the estimation of is done based on measuring the operator Let us denote the projector corresponding to a given measured value by Then, we can write

(58) |

which can be used to define an unbiased estimator based on (54). Then, the Fisher information can be obtained using (56). Finally, using (58), the Cramér-Rao bound (55) gives a lower bound on the precision of the estimation. A similar formalism works even if the measurements are not projectors, but in the more general case, positive operator valued measures (POVM).

We could calculate a bound for the precision of the estimation for given
dynamics and a given operator to be measured using this formalism.
However, it might be difficult to find the operator that leads to the best estimation precision
just by trying several operators.
Fortunately, it is possible
to find an upper bound on the precision of the parameter estimation that is valid
for any choice of the operator.
The phase estimation sensitivity, assuming
*any* type of measurement, is limited by the quantum Cramér-Rao
bound as [87, 88]

(59) |

where is the quantum Fisher information. As a consequence, based on (7), for any operator we have

(60) |

The quantum Fisher information can be computed easily with a closed formula. Let us assume that a density matrix is given in its eigenbasis as

(61) |

Then, the quantum Fisher information is given as [87, 88, 89, 90]

(62) |

Next, we will review some fundamental properties of the quantum Fisher information which relate it to the variance.

(i) For pure states, from (62) follows

(63) |

(ii) For all quantum states, it can be proven that

(64) |

This provides an easily computable upper bound on the quantum Fisher information.

For quantum states with we obtain that Such a state does not change under unitary dynamics of the type It is instructive to consider the example when for Then, also implies that the state does not change under the dynamics as we could see in the case of the singlet states in section 2.4.

(iii) More generally, the quantum Fisher information is convex in the state, that is

(65) |

(iv) Recently, it has turned out that the quantum Fisher information is the largest convex function that fulfils (i) [91, 92]. This can be stated in a concise form as follows. Let us consider a very general decomposition of the density matrix

(66) |

where and With that, the quantum Fisher information can be given as the convex roof of the variance,

(67) |

where the optimisation is over all the possible decompositions (66).

At this point we have to note that if in the decomposition (66) were pairwise orthogonal to each other, then the decomposition (66) would be an eigendecomposition. For density matrices with a non-degenerate spectrum, it would even be unique and easy to obtain by any computer program that diagonalises matrices. However, the pure states are not required to be pairwise orthogonal, which leads to an infinite number of possible decompositions. Convex roofs over all such decompositions appear often in quantum information science [23, 24] in the definitions of entanglement measures, for example, the entanglement of formation [93, 94]. These measures can typically be computed only for small systems. Here, surprisingly, we have the quantum Fisher information given by a convex roof that can also be obtained as a closed formula (62) for any system sizes.

There are generalised quantum Fisher informations different from the original one (62). They are convex and have the same value for pure states as the quantum Fisher information does [91]. However, they cannot be larger than the quantum Fisher information. This is counterintuitive: the quantum Fisher information is defined with an infimum, still it is easy to show that it is the largest, rather than the smallest, among the generalised quantum Fisher informations. As an example, we mention one of the generalised quantum Fisher informations, defined as four times the Wigner-Yanase skew information given as [95]

(68) |

For pure states, equals the variance and it is convex. There are even other types of generalised quantum Fisher informations. References [96, 97] introduce an entire family of generalised quantum Fisher informations, together with a family of generalised variances.

Analogously to (67), it can also be proven that the concave roof of the variance is itself [91]

(69) |

Hence, the main statements can be summarised as follows. For any decomposition of the density matrix we have

(70) |

where the upper and the lower bounds are both tight in the sense that there are
decompositions that saturate the first inequality, and there are others that saturate the second one ^{1}

Let us now discuss an alternative way to interpret the inequalities of (70), relating them to the theory of quantum purifications, which play a fundamental role in quantum information science. A mixed state with a decompostion (66) can be represented as a reduced state of a pure state, called the purification of defined as

(71) |

Here is an orthogonal basis for the ancillary system, and denotes tracing out the ancilla. Note that all purification can be obtained from each other using a unitary acting on the ancilla. This way one can obtain purifications corresponding to all the various decompositions.

Let us now assume that a friend controls the ancillary system and can assist us to achieve a high precision with the quantum state, or can even hinder our efforts. Our friend can choose between the purifications with unitaries acting on the ancilla. Then, our friend makes a measurement on the ancilla in the basis and sends us the result This way we receive the states together with the label corresponding to some decomposition of the type (66). The average quantum Fisher information for the states is bounded from below and from above as given in (70). The worst case bound is given by the quantum Fisher information. We can always achieve this bound even if our friend acts against us.

On the other hand, if the friend acting on the ancilla helps us, a much larger average quantum Fisher information can be achieved, equal to four times the variance. At this point, there is a further connection to quantum information science. Besides entanglement measures defined with convex roofs, there are measures defined with concave roofs [100, 101, 102]. For example, the entanglement of assistance is defined as the maximum average entanglement that can be obtained if the party acting on the ancilla helps us. Thus, in quantum information language, the variance can be called the quantum Fisher information of assistance over four. Later, we will see another connection between purifications and the quantum Fisher information in section 6.

After the discussion relating the quantum Fisher information to the variance, and examining its convexity properties, we list some further useful relations for the quantum Fisher information. From (62), we can obtain directly the following identities.

(i) The formula (62) does not depend on the diagonal elements Hence,

(72) |

where is a matrix that is diagonal in the basis of the eigenvectors of i.e.,

(ii) The following identity holds for all unitary dynamics

(73) |

The left- and right-hand sides of (73) are similar to the Schrödinger picture and the Heisenberg picture, respectively, in quantum mechanics. Hence, in particular, the quantum Fisher information does not change under unitary dynamics governed by as a Hamiltonian

(74) |

(iii) The quantum Fisher information is additive under tensoring

(75) |

For -fold tensor product of the system, we obtain an -fold increase in the quantum Fisher information

(76) |

where denotes the operator acting on the subsystem.

(iv) The quantum Fisher information is additive under a direct sum [103]

(77) |

where are density matrices with a unit trace and Equation (77) is relevant, for example, for experiments where the particle number variance is not zero, and the correspond to density matrices with a fixed particle number [104, 105].

(v) If a pure quantum state of -dimensional particles is mixed with white noise as [106, 107]

(78) |

then

(79) |

Thus, an additive global noise decreases the quantum Fisher information by a constant factor. If does not depend on then it does not influence the scaling of the quantum Fisher information with the number of particles. Note that this is not the case for a local uncorrelated noise. A constant uncorrelated local noise contribution can destroy the scaling of the quantum Fisher information and lead back to the shot-noise scaling for large as will be discussed in section 6.

(vi) If we have a bipartite density matrix and we trace out the second system, the quantum Fisher information cannot increase (see, e.g. reference [108])

(80) |

In fact, in many cases it decreases even if the operator was acting on the first subsystem, and thus the unitary dynamics changed only the first subsystem. This is due to the fact that measurements on the entire system can lead to a better parameter estimation than measurements on the first system. Let us see a simple example with the following characteristics

(81) |

where is defined in (41). Since is the completely mixed state, the right-hand side of the inequality (80) is zero, while the left-hand side is positive. On the other hand, (80) is always saturated if is a product state of the form

(vii) It is instructive to write the quantum Fisher information in an alternative form as [109]

(82) | |||||

(ix) Following a similar idea, equation (67) can also be rewritten as

(83) |

By removing the second moments of the operator from the infimum, we make the optimisation simpler. Similarly, we can also rewrite the formula (69) as

(84) |

(x) Finally, based on (83) and (84), the difference between the variance and the quantum Fisher information over four is obtained as

(85) |

Clearly, (85) is zero for all pure states. It can also be zero for some mixed states. For example, based on (64), we see that for all states for which we have we also have Thus, the difference (85) is also zero for such quantum states.

### 4.3 Optimal measurement

The Cramér-Rao bound (59) defines the achievable largest precision of parameter estimation, however, it is not clear what has to be measured to reach this precision bound. An optimal measurement can be carried out if we measure in the eigenbasis of the symmetric logarithmic derivative [89, 90]. This operator is defined such that it can be used to describe the quantum dynamics of the system with the equation

(86) |

Unitary dynamics are generally given by the von Neumann equation with the Hamiltonian

(87) |

The operator can be found based on knowing that the right-hand side of (86) must be equal to the right hand-side of (87). Hence, the symmetric logarithmic derivative can be expressed with a simple formula as

(88) |

where and are the eigenvalues and eigenvectors, respectively, of the density matrix Based on (62) and (88), the symmetric logarithmic derivative can be used to obtain the quantum Fisher information as

(89) |

For a pure state the formula (88) can be simplified and the symmetric logarithmic derivative can be obtained as

(90) |

It is instructive to consider a concrete example. Let us find for the setup based on metrology with the fully polarised ensemble discussed in section 2.1. In this case, and the quantum state evolves according to the equation

(91) |

where the initial state is

(92) |

For short times, the dynamics can be written as

(93) |

Using the identity with matrices

(94) |

the short-time dynamics can be rewritten as

(95) |

Hence, for this case the symmetric logarithmic derivative is

(96) |

Indeed, in the example of section 2.1 we measured which now turned out to be the optimal operator to be measured.

Let us now see what can be obtained from the explicit formula (90) for the symmetric logarithmic derivative. Together with (94), it leads to

(97) |

As the example shows, (96) and (97) are different, hence is not unique. Nevertheless, the right-hand side of (86) is the same for (96) and (97). This is because the symmetric logarithmic derivative is defined unambiguously within the support of while in the orthogonal space it can take any form as long as (86) is satisfied.

### 4.4 Multi-parameter metrology

The formalism of section 4.2 can be generalized to the case of estimating several parameters. The Cramér-Rao bound for this case is

(98) |

where the inequality in (98) means that the left-hand side is a positive semidefinite matrix, is now the covariance matrix with elements

(99) |

and is the Fisher matrix. It is defined as for the case of a unitary evolution

(100) |

where and are the eigenvalues and eigenvectors of the density matrix respectively [see (61)].

The bound of (98) cannot always be saturated, as it can happen that the optimal measurement operators for the various parameters do not commute with each other. Examples of multiparameter estimation include estimating parameters of unitary evolution as well as parameters of dissipative processes, such as for example phase estimation in the presence of loss such that the loss is given [38], the estimation of both the phase and the loss [110], estimation of phase and diffusion in spin systems [111], joint estimation of a phase shift and the amplitude of phase diffusion at the quantum limit [112], the joint estimation of the two defining parameters of a displacement operation (i.e., and ) in phase space [113], optimal estimation of the damping constant and the reservoir temperature [114], estimation of the temperature and the chemical potential characterising quantum gases [115], estimation of two-parameter rotations in spin systems [116], and the simultaneous estimation of multiple phases [117]. Multiparameter estimation is considered in a very general framework in reference [118]. Note that not all from the examples discussed above carry out a multi-parameter estimation in the sense it was explained in this section.

## 5 Quantum Fisher information and entanglement

In this section, we review some important facts concerning the relation between the phase estimation sensitivity in linear interferometers and entanglement. We will show that entanglement is needed to overcome the shot-noise sensitivity in very general metrological tasks. Moreover, not only entanglement but multipartite entanglement is necessary for a maximal sensitivity. All these statements will be derived in a very general framework, based on the quantum Fisher information. We will also discuss related issues, namely, meaningful definitions of macroscopic entanglement, the speed of the quantum evolution, and the quantum Zeno effect. We will also briefly discuss the question whether inter-particle entanglement is an appropriate notion for our systems.

### 5.1 Entanglement criteria with the quantum Fisher information

Let us first examine the upper bounds on the quantum Fisher information for general quantum states and for separable states. These are also bounds for the sensitivity of the phase estimation, since due to the Cramér-Rao bound (59) we have

(101) |

Entanglement has been recognised as an advantage for several metrological tasks (see, e.g., references [82, 119]). For a general relationship for linear interferometers, we can take advantage of the properties of the quantum Fisher information discussed in section 4.2. Since for pure states the quantum Fisher information equals four times the variance, for pure product states we can write

(102) |

for For the second equality in (102), we used the fact that for a product state the variance of a collective observable is the sum of the single-particle variances. Due to the convexity of the quantum Fisher information, this upper bound is also valid for separable states of the form (43) and we obtain [120]

(103) |

All states violating (103) are entangled. Such states make it possible to surpass the shot-noise limit and are more useful than separable states for some metrological tasks.

The maximum for general states, including entangled states, can be obtained similarly. For pure states, we have

(104) |

which is a valid bound again for mixed states. Thus, we obtained in (103) the shot-noise scaling (1), while in (104) the Heisenberg scaling (2) for the quantum Fisher information Note that our derivation is very simple, and does not require any information about what we measure to estimate Equation (103) has already been used to detect entanglement based on the metrological performance of the quantum states in references [72, 10].

At this point one might ask whether all entangled states can provide a sensitivity larger than the shot-noise sensitivity. This would show that entanglement is equivalent to metrological usefulness. Concerning linear interferometers, it has been proven that not all entangled states violate (103), even allowing local unitary transformations. Thus, not all quantum states are useful for phase estimation [121]. It has been shown that there are even highly entangled pure states that are not useful. Hence, the presence of entanglement seems to be rather a necessary condition.

The quantum Fisher information can be used to define the entanglement parameter [120]