Squeezing metrology: a unified framework

TL;DR

This paper develops a unified theoretical framework for quantum metrology using squeezing, demonstrating that squeezing can achieve quadratic resolution gains similar to entanglement-based methods across various quantum systems.

Contribution

It introduces a general theory quantifying the Heisenberg squeezing bound for arbitrary estimation strategies employing squeezing, unifying existing results across different quantum systems.

Findings

Quadratic resolution gain achievable with squeezing comparable to entanglement-based methods.

The framework applies broadly to quantum optics, spin squeezing, and other quantum systems.

Recovery of known results within a unified theoretical approach.

Abstract

Quantum metrology theory has up to now focused on the resolution gains obtainable thanks to the entanglement among N probes. Typically, a quadratic gain in resolution is achievable, going from the 1/sqrt(N) of the central limit theorem to the 1/N of the Heisenberg bound. Here we focus instead on quantum squeezing and provide a unified framework for metrology with squeezing, showing that, similarly, one can generally attain a quadratic gain when comparing the resolution achievable by a squeezed probe to the best N-probe classical strategy achievable with the same energy. Namely, here we give a quantification of the Heisenberg squeezing bound for arbitrary estimation strategies that employ squeezing. Our theory recovers known results (e.g.~in quantum optics and spin squeezing), but it uses the general theory of squeezing and holds for arbitrary quantum systems.

Equations

$$\Delta A^2\Delta H^2⩾$$

$$+|\frac{1}{2}⟨\{A,H\}⟩-⟨A⟩⟨H⟩{|}^2,$$

$$\Delta A^2=\frac{|⟨C⟩|}{2Re\lambda },\Delta H^2=|\lambda {|}^2\frac{|⟨C⟩|}{2Re\lambda },$$

$$\Delta AH=-\frac{|⟨C⟩|Im\lambda }{2Re\lambda },$$

$$\frac{\partial }{\partial \phi }⟨A⟩=\frac{\partial }{\partial \phi }⟨0|e^{-iH\phi }A{e}^{iH\phi }|0⟩=-i⟨[H,A]⟩,$$

$$\Delta \phi =\Delta A/|⟨C⟩|=1/\sqrt{2\lambda |⟨C⟩|},$$

$$N=(⟨H{⟩}_{sq}-E_0)/(⟨H{⟩}_{cl}-E_0),$$

$$\displaystyle\Delta\varphi\geqslant\max\Big{[}\frac{\kappa}{\nu(\langle H\rangle-E_{0})},\frac{\gamma}{\sqrt{\nu}\Delta H}\Big{]}\;,$$

$$N\simeq \frac{\Delta H_{sq}}{\Delta H_{cl}}=\sqrt{\lambda \left|⟨C{⟩}_{sq}/⟨C{⟩}_{cl}\right|},$$

$$\frac{\Delta {\phi }_{sq}}{\Delta {\phi }_{cl}}=\sqrt{\left|⟨C{⟩}_{cl}/\lambda ⟨C{⟩}_{sq}\right|}=\frac{1}{N},$$

$$\frac{\Delta {\phi }_{sq}}{\Delta {\phi }_{cl}}=\frac{⟨H{⟩}_{cl}-E_0}{\kappa \sqrt{2\lambda |⟨C{⟩}_{sq}|}}=\frac{1}{2\kappa }\frac{1}{N},$$

$$\Delta \phi =\Delta A/|\frac{\partial }{\partial \phi }⟨A⟩|=\Delta A/\Vert ⟨C⟩|=1/(2\Delta H).$$

$$N=\frac{⟨P{⟩}_{sq}}{⟨P{⟩}_{cl}}=\frac{\Delta P_{sq}}{\Delta P_{cl}}\simeq \sqrt{\lambda }\Rightarrow \frac{\Delta X_{sq}}{\Delta X_{cl}}=\frac{1}{\sqrt{\lambda }}\simeq \frac{1}{N},$$

$$\Delta \hat{N}\Delta \hat{C}⩾\frac{1}{2}⟨\hat{S}⟩,\Delta \hat{N}\Delta \hat{S}⩾\frac{1}{2}⟨\hat{C}⟩,$$

$$\frac{\Delta {\varphi }_{sq}}{\Delta {\varphi }_{cl}}=\frac{\Delta P_{sq}}{\Delta P_{cl}}\frac{|⟨X{⟩}_{cl}|}{|⟨X{⟩}_{sq}|}=\frac{e^{-\xi }|{\alpha }_{cl}|}{|{\alpha }_{sq}|},$$

$$N=\frac{⟨H{⟩}_{sq}}{⟨H{⟩}_{cl}}=\frac{{\sinh }^2|\xi |+{\alpha }_{sq}^2}{{\alpha }_{cl}^2}\simeq \frac{1}{2}{e}^{2|\xi |}+{\alpha }_{sq}^2,$$

$$\frac{\Delta {\varphi }_{sq}}{\Delta {\varphi }_{cl}}=\frac{1}{e^{\xi }|{\alpha }_{sq}|}⩾\frac{2}{e^{2|\xi |}/2+{\alpha }_{sq}^2}=\frac{2}{N},$$

$$N=(-⟨J_y{⟩}_{sq}+j)/{\Delta J_y}_{cl}=j/\sqrt{j/2}=\sqrt{2j}$$

$$\Rightarrow \Delta {\phi }_{sq}/\Delta {\phi }_{cl}=\sqrt{2j}/j=2/N,$$

Full text

1 Introduction

In quantum metrology one studies the resolution gains that can be attained when using quantum effects in the estimation strategy. The usual setting considers an estimation strategy where a parameter $\varphi$ is encoded onto a probe state through a unitary transformation $U_{\varphi}=e^{iH\varphi}$, where $H$ is the probe Hamiltonian. This is a very general setting that encompasses most estimations. While squeezing has been used in quantum metrology for specific systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], there is no general theory of squeezing-based metrology that holds for arbitrary measurements and systems. In the general case, quantum metrology theory [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] focuses on entanglement: it analyzes the situation in which the estimation is repeated $N$ times and shows that there is a quadratic improvement in resolution whenever the probes are entangled. Without entanglement, one can only achieve the standard-quantum-limit resolution $\Delta\varphi\propto 1/\sqrt{N}$ of the central limit theorem, and the error decreases to $\Delta\varphi\propto 1/N$ of the Heisenberg bound using entangled probes [15, 19, 20, 22], Fig. 1a.

In this paper we focus on squeezing and consider the case in which a single probe is squeezed with respect to the observable $A$ that is measured to estimate the parameter $\varphi$. We are not claiming to have discovered that squeezing is useful for quantum metrology (there is plenty of evidence for that in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]). The main result of this paper is a unified framework that describes all previous (and presumably future) metrology protocols that use squeezing in any quantum system and any observable, as it is based on the elegant general theory of squeezing for arbitrary systems [27]. As shown through various examples, the previously known results [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] can be recovered as specific instances of our theory. Squeezing the probe requires an amount of energy $E=\langle s|H|s\rangle$, where $|s\rangle$ is the squeezed state of the probe. As in entanglement-based quantum metrology, a quadratic resolution gain is obtained also in this case, if one compares the resolution attainable with a squeezed probe to the resolution obtainable with $N$ classical probes (i.e. prepared in a coherent state) of total energy $E$, Fig. 1b. By classical probe we intend a probe prepared in a minimum uncertainty coherent state, as is customary. We employ $N$ classical probes with the same energy as the squeezed probe because we need a dimensionless parameter ($N$) to measure the precision enhancement. We compare the quantum and classical strategies with same average energy because, without any energy restriction, one could achieve arbitrary precision by using arbitrarily high squeezing. In essence, our result can be summarized as follows: whereas one needs $N$ classical probes to decrease the error from $\Delta\varphi$ to $\Delta\varphi/\sqrt{N}$, a single squeezed probe that uses the same energy can decrease the error to $\Delta\varphi/N$. We show that this resolution increase is the optimal one (no further enhancement is possible), so we can call it the Heisenberg squeezing bound. We will be neglecting multiplicative constants of order one and consider only the scaling in $N$: it is impossible to give a general theory of the multiplicative constants because these depend on the specific implementations (as is clear from the examples below).

The paper outline: we prove the general theory of squeezing metrology, giving the yields of the possible strategies that employ squeezing; we then illustrate it with some prototypical cases: quadrature squeezing, interferometric phase estimation, and spin squeezing. We conclude with a step-by-step procedure to obtain new squeezing based metrology protocols.

2 Results

Two observables are relevant in an estimation problem: the observable $A$ that is measured and whose outcomes are used to estimate the parameter $\varphi$ and the Hamiltonian $H$ that encodes the parameter onto the probes, namely the generator of translations of $\varphi$ through $U_{\varphi}=e^{iH\varphi}$. It is then natural to consider as “classical” the strategy that uses the coherent states for these two observables [27, 3]. Then the “quantum” strategies are the ones where we squeeze $A$ and anti-squeeze $H$. Squeezed states for two observables $A$ and $H$ are the eigenstates of the operator $L(\lambda)\equiv\lambda A+iH$ [27], where $\lambda\in{\mathbb{C}}$ is a squeezing parameter: the states are squeezed in $A$ and antisqueezed in $H$ for $|\lambda|>1$ and squeezed in $H$ and antisqueezed in $A$ for $|\lambda|<1$ 111In the case in which $i[A,B]$ is not strictly positive or negative, as for spin systems, additional squeezed states may exist [27].. All states with $|\lambda|=1$ are coherent states. [The converse is not true in Perelomov’s definition of coherent states [3], where $|\lambda|\neq 1$ for some. Here we use Trifonov’s notation [27]: all have $|\lambda|=1$.] In the following, among all eigenstates with $|\lambda|=1$, we will consider the minimum uncertainty ones: these are the ones that are customarily considered the “classical” states. These definitions of squeezed and coherent arise from the Schrödinger uncertainty relation

[TABLE]

where $\Delta X^{2}$ is the variance of $X$, $\langle X\rangle$ its expectation value, and $[\,,]$ and $\{\,,\}$ are the commutator and anticommutator. For the eigenstates of $L(\lambda)$, [27]

[TABLE]

where $C\equiv[A,H]$ and $\Delta AH\equiv\frac{1}{2}\langle\{A,H\}\rangle-\langle A\rangle\langle H\rangle$ (and additional dimensionful constant may be present if $A$ and $H$ have different units). By considering a real $\lambda$, we can restrict to the Heisenberg-Robertson inequality [28] $\Delta A\Delta H\geqslant\frac{1}{2}|\langle C\rangle|$ and squeezed states: the best metrological advantage is obtained in this case (this can be easily shown by repeating the derivation below with complex $\lambda$), so we will consider only real positive $\lambda$.

When using the outcomes of $A$ to estimate the parameter $\varphi$, error propagation implies $\Delta\varphi=\Delta A/|\frac{\partial}{\partial\varphi}\langle A\rangle|$. We can evaluate the derivative recalling that the state of the probe is evolved by the unitary $U_{\varphi}=e^{iH\varphi}$, so that

[TABLE]

where $|0\rangle$ is the initial state of the probe. So

[TABLE]

where we used (3). This equation is also valid for coherent states where $\lambda=1$. When repeating the estimation procedure $N$ times, the central limit theorem implies that the rmse reduces by $\sqrt{N}$ to $\Delta\varphi(N)\equiv\Delta\varphi/\sqrt{N}$.

We now show that a squeezed state with the same energy as $N$ minimum-uncertainty coherent states can allow a quadratic gain when comparing $\Delta\varphi_{sq}$ to $\Delta\varphi_{cl}(N)=\Delta\varphi_{cl}/\sqrt{N}$ (i.e. what can be achieved classically by repeating $N$ times), where the suffix $sq$ and $cl$ indicates that the respective quantities are calculated on squeezed or minimum-uncertainty coherent states. Namely, $\Delta\varphi_{sq}/\Delta\varphi_{cl}\sim 1/N$ which is a quadratic gain over $\Delta\varphi_{cl}(N)/\Delta\varphi_{cl}=1/\sqrt{N}$. Here $N$ is the number of coherent probes that can be produced with the squeezed state’s energy, namely

[TABLE]

where $E_{0}$ is the ground state energy. Clearly we are interested in comparing the best estimation strategies, since one can always get a worse strategy by simply wasting energy resources.

For arbitrary settings, not just the setup here, the best estimation strategies are the ones that satisfy the quantum measurement bound of [29] with equality. This bound implies that

[TABLE]

where $\kappa$ and $\gamma$ are numerical constants of order 1 (see [29]) and $\nu$ is the number of times the estimation is repeated (here we will consider $\nu=1$). The first term in the max expresses a quantum speed limit [30] and the second arises from the time-energy uncertainty [31] (our choice $\nu=1$ follows from these results [29]). Eq. (7) implies that any energy beyond the standard deviation will be wasted for the estimation: if $\langle H\rangle-E_{0}>\zeta\Delta H$ (with $\zeta\equiv\kappa/\gamma$), then the error $\Delta\varphi$ is dominated by $\Delta H$: “too much energy” strategies. To avoid wastes, we should choose $\langle H\rangle-E_{0}\simeq\zeta\Delta H$ (“good strategies”), where the “$\simeq$” sign emphasizes that only the order of magnitude of the two terms is important since Eq. (7) is not a tight bound [29]. Eq. (7) also implies that estimation strategies that have $\langle H\rangle-E_{0}<\zeta\Delta H$ (“too little energy” strategy) have error $\Delta\varphi$ dominated by the energy: they cannot achieve the error of Eq. (5), but are limited to $\Delta\varphi\sim\kappa/(\langle H\rangle-E_{o})$.

The “good strategy” energy requirement $\langle H\rangle-E_{0}\simeq\zeta\Delta H$ can always be enforced when squeezing is used, since the ratio between energy and its standard deviation can be tuned through the squeezing parameter $\lambda$: one can assign an arbitrary fraction of the energy budget to squeezing the probe. Instead, for classical strategies, the requirement of having a coherent state may be inconsistent with good strategies: for example spin coherent states have “too much energy”. Otherwise typically the “good strategy” requirement can be satisfied only for a specific value of the energy, as for the harmonic oscillator. Indeed, Eq. (3) fixes $\Delta H^{2}$ to ${|\langle C\rangle|/2}$, and fixing it also to $\langle H-E_{0}\rangle^{2}$ may be possible for coherent states only for a given energy. For classical strategies with too little energy, using (6) will underestimate $N$ since some of the coherent state energy is not used for estimation, see Eq. (7). For these strategies, one should use $N=(\langle H\rangle_{sq}-E_{0})/\Delta H_{cl}$ in place of (6) to count only for the energy that is actually employed for the estimation in classical strategies. The focus on “good strategies” is not a limitation of our paper, because these are the strategies that are optimal: the ones that do not waste energy.

Consider the “good strategies” first. In this case, replacing $\langle H\rangle-E_{0}=\zeta\Delta H$ into (6) we find

[TABLE]

where we used (3) in the last equality. Using (5), we find

[TABLE]

a quadratic gain over the classical strategy of repeating $N$ times the classical estimation: $\Delta\varphi_{cl}(N)/\Delta\varphi_{cl}=1/\sqrt{N}$. In other words, when comparing the squeezed and classical strategy of equal energy, we find $\Delta\varphi_{sq}/\Delta\varphi_{cl}(N)=1/\sqrt{N}$, the main result of our paper. Since we obtained the quadratic gain when comparing the best estimation strategies, this is optimal: one cannot achieve a larger gain (unless one employs suboptimal classical strategies). This justifies our claim that (9) represents the Heisenberg squeezing bound.

We emphasize that in this derivation we have neglected multiplicative and additive constant factors of order one: as a general theory containing them is impossible (they depend on the specific implementations). In principle, in the abstract scenario we analyzed here, one may consider a single coherent state that is not minimum-uncertainty (namely a state with equal spreads $\Delta A=\Delta H$ which are larger than their minimum value) and devote all the energy to it. In this case, that state would already achieve the optimal precision without any need for squeezing. However, this strategy does not exist in all the examples we analyzed because those coherent states have always a bounded variance and the “good strategy” condition cannot be met. Moreover, these large-uncertainty coherent states would not be recognized as a “classical” strategy.

Consider now bad estimation strategies. For classical strategies with too much energy, the above results still hold if one uses the appropriate $N$ discussed above. For strategies with too little energy, Eq. (5) does not apply and we need to amend the above derivation, but we still obtain the same result: consider first the case of a suboptimal classical estimation strategy, where $\Delta\varphi_{cl}\sim\kappa/(\langle H\rangle_{cl}-E_{o})$, so

[TABLE]

where we used the fact that $N=\Delta H_{sq}/(\langle H\rangle_{cl}-E_{0})$ in this case. Again we find a quadratic gain (apart from an inconsequential numerical constant of order one). We obtain analogous results when comparing two bad estimation strategies or a bad squeezed strategy with a good classical strategy.

3 Discussion

(i) The above proof elucidates why squeezing is beneficial: only a squeezed estimation strategy can attain the equality $\langle H\rangle-E_{0}=\zeta\Delta H$ for all energies. A coherent state typically attains this only for a specific energy, so it is suboptimal when the procedure is repeated $N$ times: the $\langle H\rangle$ term in (7) scales linearly with $\nu=N$, whereas the $\Delta H$ term scales as $\sqrt{\nu}=\sqrt{N}$. Hence, given an energy budget, repeating the measurement is never advantageous and a single-shot classical strategy that optimally uses all resources may exist only for a specific energy. Then, only for this value of the energy, the single-shot classical strategy performs as well as the squeezed one, otherwise squeezing the probe is always better. (ii) The squeezing parameter $\lambda$ is not present in the final result of (9). Nonetheless, the results obtained hold only for $\lambda>1$, corresponding to a reduction in the fluctuations of the observable $A$ that is measured, as expected. Indeed, for $\lambda<1$, corresponding to an increase in fluctuations of $A$, Eq. (8) typically implies that $N<1$ since the optimal squeezed strategy’s energy is less than the classical one, and the quadratic gain would favor the classical strategies in that case. Intuitively, this can be seen by considering the minimum uncertainty states for which $\Delta A\Delta H=|\langle C\rangle|/2$. Then

[TABLE]

Then, for $\lambda>1$ we get better accuracy since $\Delta H$ is increased, whereas the opposite happens for $\lambda<1$. (iii) The theory above refers to the estimation of $\varphi$ from the measurement of an observable $A$, but it applies also to POVM measurements. Indeed a POVM can be extended to a projective POVM through the Naimark dilation theorem [32] without changing the measurement outcome statistics. One can then assign arbitrary “eigenvalues” to each element of this POVM to obtain an observable $A$ to which the above theory applies. (iv) As for entanglement-based quantum metrology [33, 34, 22], the presence of noise complicates the situation enormously and will be analyzed elsewhere.

4 Examples

We now show some simple examples that refer to different (finite and infinite-dimensional) Hilbert spaces and to different observables.

4.1 Position measurements

Consider the situation where we want to estimate the position $X$. The generator of translation of position is the momentum $P$, so we choose $H=P$. This Hamiltonian is not lower bounded so the average energy is infinite for any state: we need to introduce an energy cutoff, e.g. by supposing that states have negligible negative momenta components. [Alternatively, one could also consider $H=|P|$ as the Hamiltonian.] We consider a position-momentum squeezed state [1], whose wavefunction is a Gaussian [2]. Introducing (arbitrary) constants $m$ (mass) and $\omega$ (angular frequency), we can write $X$ and $P$ in terms of creation and annihilation operators through $a=X\sqrt{m\omega/2\hbar}+iP/\sqrt{2\hbar m\omega}$. The squeezed states are eigenstates of $\mu a+\nu a^{\dagger}$ with $\mu=(\lambda+1)/\sqrt{4\lambda}$ and $\nu=(\lambda-1)/\sqrt{4\lambda}$. As before, the coherent state is obtained for $\lambda=1$, yielding the “standard” coherent states. The rmse are [1] $\Delta X=\sqrt{\hbar/(2\lambda m\omega)}$ and $\Delta P=\sqrt{m\omega\hbar\lambda/2}$. Then, assuming zero-energy ground state, and using the least energetic squeezed and coherent states $\langle P\rangle\simeq\Delta P$ (the ones with smallest energy that still have negligible negative energy components), we have

[TABLE]

a quadratic improvement over the classical strategy of repeating $N$ times the optimal coherent state measurement, which gives $\Delta X_{cl}(N)/\Delta X_{cl}=1/\sqrt{N}$.

4.2 Optical interferometry

As a second example, consider interferometric phase $\phi$ measurements [35]. In this case $H=a^{\dagger}a\equiv\hat{N}$ and $A$ is whatever observable (or POVM) is measured. Since the phase is periodic, the rmse is not an appropriate uncertainty measure [36] unless the uncertainty is small compared to $2\pi$. One must resort to Susskind-Glogower uncertainty relations (SGUR) [37]

[TABLE]

where $\hat{C}=(E_{+}+E_{-})/2$ and $\hat{S}=i(E_{+}-E_{-})/2$ are the “cosine” and “sine” operators with $E_{\pm}=\sum_{n}|n\rangle\langle n\mp 1|$ (in the Fock basis). Both $\hat{S}$ and $\hat{C}$ can be used as the observable $A$ of the theory given above. Indeed, the SGUR can be used because $\hat{S}=\frac{\partial}{\partial\phi}\hat{C}=i[H,\hat{C}]$ and $\hat{C}=\frac{\partial}{\partial\phi}\hat{S}=i[H,\hat{S}]$. [37]. Hence Eq. (5) can be replaced by $\Delta\phi=\Delta C/|\langle S\rangle|=\Delta S/|\langle C\rangle|$ and we can write SGUR as $\Delta\hat{N}\Delta\phi\geqslant\frac{1}{2}$ (meaningful if $\Delta\phi\ll 2\pi$, with an appropriate choice of boundaries far from the average $\phi$). This implies that the minimum uncertainty squeezed and coherent states for SGUR can be employed in the theory presented here to get a quadratic improvement for phase sensing. (The proof is basically identical to the one for $X$ and $P$ presented below, since $\phi$ and $\hat{N}$ can be considered as conjugate variables in the above regime, where $\langle\phi\rangle$ is basically null at the boundaries.) These states were determined in [38], but unfortunately they seem to have no physical relevance, although they can be approximated in particular regimes, e.g. [39, 40]. The usual coherent states $|\alpha\rangle$, eigenstates of the annihilation $a$, are not minimum uncertainty states for SGUR [38], although they approximate them for large average photon number [37].

The prototypical squeezed-light interferometric measurement is the one proposed by Caves [41]. While the original proposal is not optimal [22], one with a modified detection strategy is: it has a phase uncertainty $\Delta\phi_{sq}\simeq 1/\langle a^{\dagger}a\rangle$ at the optimal working point [42]. In contrast, coherent states $|\alpha\rangle$ can only achieve the standard quantum limit $\Delta\phi_{cl}\simeq 1/\sqrt{\langle a^{\dagger}a\rangle}$, so the classical strategy that optimally employs the energy resources, i.e. the one for which $\langle a^{\dagger}a\rangle\simeq\Delta H$, is the one that employs coherent states with $\langle a^{\dagger}a\rangle=1$ since the coherent state’s Poissonian statistic implies that $\langle a^{\dagger}a\rangle=\Delta H^{2}$. Then a quadratic gain over this strategy (when repeated $N$ times) follows: the energy of the repeated strategy is $\langle a^{\dagger}a\rangle_{N}=N$, so that $\Delta\phi_{sq}/\Delta\phi_{cl}(N)=\sqrt{N}/\langle a^{\dagger}a\rangle=1/\sqrt{N}$ or, when comparing with the single-shot optimal classical strategy, $\Delta\phi_{sq}/\Delta\phi_{cl}=1/\langle a^{\dagger}a\rangle=1/N$.

As a further example of phase estimation that is optimal only in certain regimes, consider quadrature squeezing for phase estimation (quadrature squeezing is different from SGUR squeezing). Namely, estimate phase shifts $\phi$ generated by $H=a^{\dagger}a$, by measuring the quadrature $P=i(a^{\dagger}-a)/\sqrt{2}$. Then $\Delta\phi=\Delta P/|\langle X\rangle|$, with $X=(a+a^{\dagger})/\sqrt{2}$ since $\frac{\partial}{\partial\phi}P=i[a^{\dagger}a,P]=-X$. For a quadrature squeezed displaced state we find $\langle X\rangle=\sqrt{2}Re(\alpha)$, and $\Delta P=e^{-\xi}/\sqrt{2}$, where $\alpha$ and $\xi$ are the displacement and squeezing parameters and where we choose real $\xi$ (the only interesting case here). Since $\langle H\rangle=|\alpha|^{2}+\sinh^{2}|\xi|$ [44], it is clear that the good strategies (the ones which do not waste energy) are the ones where $\alpha_{sq}$ and $\alpha_{cl}$ are real. For these,

[TABLE]

where the last equality holds for large squeezing $|\xi|\gg 1$ and for the optimal classical strategy that, again, is for $\langle a^{\dagger}a\rangle=\Delta H=\Delta H^{2}={\alpha_{cl}^{2}}=1$. Since $x^{2}+y^{2}\geqslant 2xy$ for any real $x,y$, we find

[TABLE]

if we choose $\xi>0$ (i.e. $P$-direction squeezing). The inequality (16) becomes an equality for $\alpha_{sq}^{2}=\frac{1}{2}e^{2|\xi|}$, namely if we devote half of the energy to squeezing and half to displacing, which is known to be the best way to allocate the energy [43]. So, in the limit of large $P$-direction squeezing, we have optimality also in this case. Many other optical phase estimation strategies based on squeezing are known, e.g. [45, 46, 10, 13].

4.3 Spin squeezing

As a final example consider spin squeezing, with $A=J_{x}$ and $H=-J_{y}$. In this case, not all squeezed states are eigenstates of $L(\lambda)$ since $C$ is not strictly positive or negative [27]. Thus, instead of limiting ourselves to the eigenstates of $L(\lambda=1)$, we will employ the $su(2)$ coherent states [3] which are a larger class [27]. These are defined as $|\beta\rangle\equiv\exp(\beta J_{+}-\beta^{*}J_{-})|j;-j\rangle$, with $J_{\pm}\equiv J_{x}\pm iJ_{y}$, $\beta\in{\mathbb{C}}$, and $|j;-j\rangle$ the lowest-weight eigenvector of $J_{z}$. The latter state is the only eigenstate of $L(\lambda=1)=J_{x}-iJ_{y}\equiv J_{-}$ (this is the reason for the minus sign in the definition of $H$, without which we would obtain the equivalent coherent state class originating from $|j;+j\rangle$). The states $|\beta\rangle$ have $\Delta J_{x}^{2}=j(1-\sin^{2}\theta\cos^{2}\phi)/2$, $\Delta H^{2}=\Delta J_{y}^{2}=j(1-\sin^{2}\theta\sin^{2}\phi)/2$, $\langle-J_{y}\rangle-E_{0}=j(1+\sin\theta\sin\phi)$, and $|\langle[J_{x},J_{y}]\rangle|=|\langle J_{z}\rangle|=j|\cos\theta|$, where $\theta$ and $\phi$ are defined as $\beta=-e^{i\phi}\tan(\theta/2)$ [3]. All these coherent states give rise to a “too much energy” strategy, since $\langle H\rangle_{cl}-E_{0}>\Delta H$ (except for the case of $|\beta\rangle$ eigenstate of $J_{y}$ which is obviously useless for estimation). Thus, we need to define $N=(\langle H\rangle_{sq}-E_{0})/\Delta H_{cl}$ in order to count only the energy that is actually employed in the estimation in the classical strategy. The precision achievable in estimating a rotation by an angle $\varphi$ around the $y$ axis is $\Delta\varphi=\Delta J_{x}/|\langle J_{z}\rangle|=\sqrt{\frac{1-\sin^{2}\theta\cos^{2}\phi}{{2j}\cos^{2}\theta}}\geqslant\frac{1}{\sqrt{2j}}$, where the last inequality becomes an equality on the eigenstates of $J_{z}$ (i.e. for $\theta=0,\pi$) as expected. The squeezed strategy should employ squeezed spin states [6] with reduced fluctuations in $J_{x}$, namely $\Delta J_{x}=1/\sqrt{2}$, $\Delta J_{y}=j/\sqrt{2}$ for which $\langle-J_{y}\rangle+j=j$. This is a “good strategy”, since $\langle H\rangle_{sq}-E_{0}=j\simeq\Delta H_{sq}=j/\sqrt{2}$. It achieves the Heisenberg squeezing bound precision $\Delta\varphi\simeq 1/j$ [6]. The comparison between the classical and the squeezed strategies follows:

[TABLE]

again a quadratic improvement (apart from a constant of order one) over the ${\Delta\varphi_{cl}(N)}/{\Delta\varphi_{cl}}=1/\sqrt{N}$ precision obtained by repeating the classical strategy $N$ times. Note that in all spin-squeezing literature $N$ is defined differently, as the number of elementary spin-$1/2$ particles equivalent to a $j$-spin system [6]. Thus elsewhere spin squeezing is analyzed in terms of the entanglement among these particles, using the entanglement-based theory of quantum metrology [15].

5 New protocols

We now detail the step-by-step procedure that one can employ to devise a new metrology protocol based on squeezing, employing the results of our paper:

Identify the Hamiltonian $H$ that generates the translations of the parameter of observable that we want to estimate. Identify the observable $A$ that we can employ in the estimation. Note that $A$ is in general not unique (as shown in the above examples). 2. 2.

Solve the eigenvalue equation for the non-Hermitian operator $\lambda A+iH$ to find the squeezed states of the system (the intelligent states) [27]. 3. 3.

Use all the energy available to the estimation procedure to prepare a single probe in such squeezed state. 4. 4.

Let the probe evolve with the Hamiltonian $H$ and then measure the observable $A$. 5. 5.

The results of our paper guarantee that the outcome will have an error on the parameter or observable to be estimated that scales as $1/N$, where $N$ is the number of classical (coherent) probes that could be prepared using the same energy. These classical probes would give an error that scales as the standard quantum limit $1/\sqrt{N}$ because of the central limit theorem. Namely, the squeezed strategy described here has a quadratic gain.

6 Conclusions

In conclusion, we have proposed the general theory of quantum metrology that is focused on squeezing, instead of on entanglement. Our framework applies to arbitrary quantum systems, since it uses the general theory of squeezing. We have also provided various examples of applications of such theory. We believe that this theory will be of use to theoreticians to develop new protocols, rather than to experimentalists.

We acknowledge funding from Unipv, “Blue sky” project - grant n. BSR1718573 and the ATTRACT project funded by European Union’s Horizon 2020 research and innovation programme under grant agreement No 777222.